THE LIBRARY OF THE UNIVERSITY OF CALIFORNIA Education IN MEMORY OF Professor George D. Louderback 1874-1957 APPLETONS' STANDARD ARITHMETICS NUMBERS APPLIED A COMPLETE ARITHMETIC BY Andrew J. Rickoff, A.M., LL.D. NEW YORK, BOSTON, AND CHICAGO D. APPLETON AND COMPANY 1887 Copyright, 1886, By D. APPLETON AND COMPANY. £duo. Lib. GIFT PEEFACE. Tms work is not the result of any ambition on the part of the publishers to add another title to their already long list of text-books, but of a desire to meet a wide-spread and growing demand for a treatise on arithmetic adapted to the objective methods of instruction now so common in all edu- cational institutions which have been reached directly or indirectly by the influence of normal schools, teachers' institutes, etc. In its preparation the author has kept steadily in view these two thoughts : (1) That words are useless in the ratio that they fail to call up in the mind vivid images of the things signified. Hence the aim to vitalize the relation of words and things by the aid of the best practicable illustra- tions at every point ; and (2) That, to the learner, the operations of arith- metic are apt to be but manipulations of figures after prescribed models, unless he realizes the fact that they are representative of processes that may be applied to material objects. The book is intended to be put into the hands of the learner as soon as he has completed a course in primary arithmetic ; but it would be well for him to begin the study of it with the first chapter, that he may get a better technical knowledge of the fundamental rules and their relations to each other, and that he may become rapid and reliable in computations involving integers before he takes up the more complicated subject of fractions. Great care has been taken to adapt the work as far as possible to the needs of the great number of children who are withdrawn from school before a full course in arithmetic can be completed. With this object in view, the more useful business applications of elementary principles are made as soon as they are learned. Thus, familiar measures are introduced before reduction is mentioned ; federal money before decimals ; many prac- tical measurements before mensuration ; and questions even in percentage and interest are to be met with before those subjects are reached in due course. The conditions of these problems are so presented as to be within 965 iv PREFACE. the easy understanding of the pupil, while their solution requires only such arithmetical operations as he has already learned. Attention is respectfully called 1. To the simple treatment of the decimal system of notation, and the great number of exercises intended to familiarize the pupil with the facili- ties for calculation which it affords. 2. To the multiplicity of short exercises that can be performed without the aid of the pencil, or that require but few figures in their solution. Longer ones are not wanting to test the perseverance of the pupil. 3. To the directions to the pupil, having in view the formation of right habits of computation. The ''making up " method of subtraction, and the so-called "continental" method in division, though not obtrusively pre- sented, are worthy of the attention of teachers. The latter furnishes an excellent mental exercise. 4. To the suggestions for original problems, now so commonly resorted to by the best teachers to stimulate the interest of their pupils, and to give them a better understanding of the subjects to which they relate. Their usefulness as short practical exercises in penmanship, spelling, and com- position, will be appreciated by all. 5. To the simple and direct methods of treating the fundamental rules, common and decimal fractions, percentage, interest, proportion, square and cube roots, the problems of mensuration, etc. 6. To the rigorous adherence throughout the work to the inductive methods of instruction. The number and variety of exercises and problems in this work are so great as to supersede any necessity for a supplementary book of exercises. It is earnestly recommended that the pictured illustrations may be regarded as merely suggestive of the objective demonstrations which the student should be encouraged to get up for himself. As far as possible, let the learner furnish all the apparatus needed. While he is engaged in preparing it, the principles to be illustrated will present themselves to his mind more forcibly than in the repetition of definitions and rules in which he can take but slight interest till he appreciates their significance. If this course be taken, the pupil will, in most cases, be able to make out his own analyses. These may be crude at first, but they will be the better for being his own. Observation and experience will guide to better forms. Such a method will give him a mastery of the subject, develop mental power, and cultivate a taste for independent investigation. New Yokk City, May 15, 1886. CONTENTS PAGE Notation and Numeration ... 3 Roman Notation 18 Addition .... * 19 Blackboard Exercises .... 26 Original Problems 34 Subtraction 35 Miscellaneous Examples ... 4*7 Original Problems 50 Multiplication . . . . . . . 51 Familiar Measures 65 Original Problems 70 Division VI Original Problems 93 Self-Testing Exercises .... 96 Miscellaneous Examples . . . . 97 United States Money 107 Making Cbange 121 Factors and Divisors 123 Factors and Multiples . . . .131 Common Fractions 135 Introductory Exercises . . . .135 Reduction . . , 140 Addition 143 Subtraction 145 Multiplication 149 Division 155 Aliquot Parts 164 Miscellaneous Examples . . .166 Decimal Fractions 173 Introductory Exercises . , . .173 Addition . 180 Decimal Fractions {continued). Subtraction . . . . . . .181 Multiplication ....... 182 Division 184 Reduction 186 Miscellaneous Examples . . .191 Bills and Accounts 194 Original Problems ..... 200 Measures 201 Of Extension 201 Of Capacity 205 Of Weight . . . . • . . . .207 Of Value ........ 210 Of Arcs and Angles 212 Of Time 213 Miscellaneous Problems . . . 214 Compound Denominate Numbera .215 Reduction .218 Addition 224 Subtraction 226 Multiplication 227 Division 228 Longitude and Time 236 Application and Review . . . 237 Miscellaneous Problems . . . 242 The Metric System of Weights and Measures 249 Reduction 254 Square Measure 256 Cube Measure 258 Wood Measure ...... 259 VI CONTENTS. PAGE Practical Measurements .... 262 Lumber 262 Masonry and Brickwork . . . 263 Flooring 264 Plastering 265 Painting and Kalsomining . . . 265 Paper Hanging 266 Carpeting 266 Paving 267 Bins, Tanks, and Cisterns . . . 268 Estimating the Weight of Hay in a Mow 268 Miscellaneous Problems . . . 269 Original Problems 270 Percentage 271 Loss and Gain 277 Trade Discount 285 Insurance 286 Commission and Brokerage . . 288 Stocks 290 Taxes 292 Miscellaneous Problems . . . 295 Original Problems 800 Interest . . 301 Present Worth 316 Exact Interest 317 Common Business Method . . . 318 Bank Discount 319 Promissory Notes 321 Partial Payments 323 United States Rule for Partial Payments 324 Mercantile Rule for Partial Pay- ments 326 Annual Interest 327 Miscellaneous Problems . . . 328 Compound Interest 331 Equation of Payments . . . .333 Debit and Credit Accounts . . 340 Original Problems ..... 342 PAGE Proportion 343 Inverse Proportion 345 Ratio and Proportion .... 347 Compound Proportion .... 352 Miscellaneous Problems . . . 355 Squares and Cubes 357 Square Root 363 Cube Root £68 Extraction of Roots by Factoring. 371 Geometric Solution of the Prob- lem of Square^ Root .... 372 Geometric Solution of the Prob- lem of Cube Root 373 Applications of Square and Cube Root 374 Right-angled Triangles .... 375 Mensuration 377 Of Plane Surfaces 377 Of surfaces of Prisms, etc. . . 386 Of Volume or Contents of Solids. 390 Duodecimals 392 Original Problems 394 Exchange 395 Domestic Exchange 395 Foreign Exchange 400 Duties or Customs 403 Bonds 404 APPENDIX. Testing the Accuracy of Addition, Subtraction, Multiplication, and Division 407 Casting out Nines 407 Greatest Common Divisor . . . 409 Circulating Decimals 41 Progression 411 Arithmetical Progression . . .411 Geometrical Progression . . .413 Values of Foreign Coins . . . .415 Legal Rates of Interest . . . .416 ^PPLETO^S' Standard ^ritlmietic. CHAPTER I. NOTATION AND NUMERATION. The Writing and Reading of Numbers. NUMBER. OBJECTS. NUMBER. • One 1 • • • • • • Six 6 • • Two 2 :•: • • Seven 7 • • • Three 3 • # • • • • e • Eight 8 • • • • Four 4 v •• • • • • Nine 9 :•: Five 5 V V • • • • Ten •• (. The signs 1, 2, 3, 4, 5, 6, 7, 8, 9, are called the nine digits; because first used to represent a number of fingers. The word digit is sometimes used for the word finger. 4 STANDARD ARITHMETIC. The following are the written forms of the digits : /. z. <3. a. & 6, y. fi Suggestion. — Let the pupils make original notation exercises similar to the above, arranging the objects in groups, and noting the number both in words and figures. 8. Hundreds. — If we count one more than ninety-nine we shall have nine tens and ten ones, or ten tens. Ten tens make one hundred. To express one hundred in figures we write 1 in the third place, thus, 100, filling the places of tens and units with ciphers. The 1 now stands for one hundred. A digit in the third place from the right stands for hundreds, and hence we write : 100 (one hundred), 400 (four hundred), 700 (seven hundred), 200 (two hundred), 500 (five hundred), 800 (eight hundred), 300 (three hundred), 600 (six hundred), 900 (nine* hundred). If with the hundreds we have to write any number of tens, as three hundreds and seven tens, we place the digit representing 8 STANDARD ARITHMETIC. the tens in the tens' place ; thus, 370, read (3 hundreds 7 tens), three hundred seventy. Again, if with the hundreds we have to write any number of ones, as three hun- dreds and five ones, we place the figure re- presenting the ones in the ones' place ; thus, 305. Three hundreds, seven tens and five ones here represented are written thus : 375, and read three hun- dred seventy-five. We have learned, 1st, that ten ones make one ten, and ten tens make one hundred ; 2d, that in writing numbers the place on the right is the ones' place ; the next, the tens' place; and the next, the hundreds' place. EXERCISES IN WRITING AND READING NUMBERS. 1. Express in figures : One hundred, six hundred, nine hun- dred, seven hundred, four hundred, two hundred, etc. 2. Also, one hundred thirty, six hundred twenty, five hun- dred eighty, three hundred fifty, two hundred seventy, etc. 3. Also, one hundred sixty-five, three hundred eighty-four, nine hundred seventy-one, four hundred thirty-three, etc. 4. How many hundreds in 481 ? How many tens ? How many ones ? How many of each in 385, 610, 974, 572, 137, 448 ? 5. Write the following numbers in a column, and opposite each the same number in words : 218, 117, 916, 675, 854, 370, 523, 388, 446, 770, 978, 101, 340, 620, 304. 6. Eead, taking in at one glance as many numbers as possible : 100 201 310 404 500 691 700 800 909 102 204 320 .440 572 673 719 808 910 Note. — The foregoing numbers may be read in lines or columns, forward or backward, as the teacher may direct. NOTATION AND NUMERATION. 9 7. Write in figures the number of jack-straws represented in each of these groups. %V».v 8. Read 792. What does the 2 stand for ? The 9 ? . The 7 ?— Show what each figure stands for in the following numbers : 439, 562, 101, 760, 875, 460, 140, 104, 583, 61. 9. In a number of three places, which figure, is read first ? Which represents the highest order ? How would you write three hundred nine, having no tens ? How would you write 7 hundred twenty, having no ones ? Will it do to leave the place of the ones or tens vacant ? Why ? 10. What is the largest number that can be represented by three figures ? What is the smallest whole number ? 11. Write in figures : Three hundred fifty, six hundred eighty, two hundred seventy, eight hundred fifteen, four hundred twenty- eight, nine hundred nine, one hundred ninety-six. 12. Copy the following, glancing at each number but once : (Think of the numbers represented, not merely of the figures to be written.) 107 400 212 560 309 653 356 365 635 536 801 118 180 870 357 429 560 608 742 897 215 419 711 999 233 100 677 822 301 405 103 205 340 409 583 655 728 846 979 893 13. Note.- —The teacher ma y also dictate two or more of the i f oregoii lg num- bers at once, thus quickening the attention of the pupils. 14. Write in regular order the numbers from 150 to 199 ; from 260 to 299 ; from 307 to 328 ; from 480 to 499 ; from 585 to 602 ; from 687 to 706 ; from 791 to 809. 15. Write in words the numbers from 337 to 345 : also from 10 STANDARD ARITHMETIC. 883 to 890 ; from 555 to 563 ; from 98 to 104 ; from 872 to 883 ; from 190 to 205. 9. Thousands. — The greatest number we have written thus far is 999, or 9 hun- dreds, 9 tens, 9 ones. If we count one more, the nine ones at the right hand will become one ten ; and putting this with the nine tens, we have ten tens, or one hundred. Putting this one hundred with the nine hundreds, we have ten hundreds; and, as we made one ten out of ten ones, and one hundred out of ten tens, so we make one thousand out of ten hundreds. Thus, after adding one stick to the nine hundred and ninety-nine shown on the table above, the result would be as represented in this picture. To express one thousand in figures, we write 1 in the fourth place ; thus, 1000, filling the places of hundreds, tens, and units with 0's. The 1 now stands for one thousand. A digit in the fourth place stands for thousands, hence we have 2000 (two thousand), etc. Hundreds, tens, and units, if any, fill their proper places. EXERCISES IN READING AND WRITING NUMBERS. 1. Read 77, 15, 93, 106, 601, 810, 7080, 9107, 5006, 561, 3091. 2. Write at dictation and read : 5783 2100 9009 1706 5430 8071 2360 3902 1003 5701 6702 4000 3201 4300 5701 7010 8090 9100 1901 7707 3. Write ten such numbers as you please, and read them. NOTATION AND NUMERATION. 11 4. In the following numbers, how many units, tens, and hun- dreds are expressed by the figures in those orders ? 2375 1318 2380 3689 7401 8120 9000 7895 3624 5074 6138 8376 2380 8016 7980 1234 5. What is the smallest whole number that can be expressed by four figures ? What is the greatest ? 6. Read; 6789 9867 5432 4756 8912 3129 7891 4321 6000 6600 6660 6666 7654 6035 8765 8003 1098 2076 3054 4032 5010 7123 7009 5273 9002 9387 8793 1100 , 4002 7628 9347 6102 7. Read the foregoing columns downward and upward, and the lines from right to left and left to right. They may also be written at dictation and read. 8. Read the numbers of Exercise 4, reading the thousands and hundreds together as hundreds. Thus, 23 75= twenty- three hundred seventy-five. 9. Copy the numbers of Exercise 6. No copying of single figures should be allowed ; the number should be recognized and written as a whole. 10. Which figure in a number of four places is read first ? Which represents the highest order ? Which the lowest ? 11. How many ciphers are needed in 4 thousand 17 ? Why ? 12 STANDARD ARITHMETIC. What difference is there between the written forms 468 and 4608 ? Between 375 and 3705 ? 12. Is there any difference between the numbers indicated by 46 in 468 and in 4608 ? Is the value of 8 in one number different from its value in the other ? Why ? 13. What number is expressed by the figure 9 in 7009 ? In 7900 ? In 7090 ? Review Exercises. 1. Write in columns the figures which express the following numbers : four hundred one hundred five hundred two hundred nine hundred six hundred three hundred eight hundred four one five two nine six three eight four thousand one thousand five thousand two thousand nine thousand six thousand three thousand eight thousand forty ten fifty twenty ninety sixty thirty eighty 2. How can you make the digits in your first column express tens ? (Answer : By annexing a cipher.) How hundreds ? How can you make the digits in the second column express units ? How hundreds ? How can you make the digits in the third column express tens ? How ones ? (Answer : By erasing two ciphers.) Make the digits of the fourth column express hun- dreds ; also tens. Will it change the value of the digits to place a cipher at their left ? 3. Express in figures: sixteen one hun. seven twenty-nine fifty-two thirty-six seventy-eight forty-five ninety-four eighty-three three hun. eighteen five hun. twenty-six seven hun. sixty-four eight hun. fifty-six nine hun. thirty-eight two hun. eighty-nine four hun. forty-five three thou, seven hun. eight six thou, one hun. twelve one thou, six hun. thirteen eight thou, two hun. twenty three thou, four hun. fourteen four thou, nine hun. ninety two thou, six hun. ten five thou, three hun. thirty NOTATION AND NUMERATION. 13 4. Make 60, 40, 80, 10, 30, 50, 90, 70, 20 larger by 100 ; by 300 ; by 500 ; by 400. 5. Make 61, 42, 53, 74, 85, 26, 37, 18 larger by 400 ; by 200 ; by 700. 6. Would it alter the value of 8 in 81 if you were to place a cipher on the right of the 1 ? Answer similar questions in regard to the first figures in 29, 36, 45, 19, 58, 67, 71. 7. How many thousands, hundreds, tens, and units are expressed in each of the following numbers : 3624, 5781, 9010, 8107 ? 8. Express by figures the number of sticks represented in the 1st or units' group ; also in the 3d the 4th and 3d the 2d and 1st the 3d, 2d, and 1st the 2d the 3d and 1st the 4th and 1st the 4th, 3d, and 1st the 4th the 3d and 2d the 4th and 2d the 4th, 2d, and 1st What number is represented in all the groups together ? Note. — Pupils should prepare suitable objects for such illustrations. Bundles of tens, hundreds, etc., with single objects, should often be arranged promiscu- ously, and the learner be required to write the number in figures. Let him observe that we may estimate the value of the bundle by its size, but whether a digit repre- sents tens or thousands depends on the place it occupies. 9. Write in columns, of ten each, all the numbers from one hundred to one hundred fifty-nine. Also, from two hundred fifty-one to three hundred. Also, from seven hundred eighty- three to eight hundred thirty-two. Also, from six hundred twenty-two to seven hundred one. 10. Write all the numbers from one thousand six hundred seventy-three to one thousand seven hundred two. Write in col- umns of ten numbers each. 14 STANDARD ARITHMETIC. 10. After Thousands.— 1. We have thus learned, First, the names of the orders to thousands ; Second, that ten of any order make one of the next higher ; and, Third, that the order of a figure — that is, whether it represents units, teus, hundreds, or thousands — is known by the place which it occupies, num- bered from the right. 2. When we reach a thousand we begin to count the thousands as we did the units or ones : that is, we count 1 thousand, 2 thou- sand, up to 999 thousand, and when we have a thousand thousand we call the number one million. Millions we count in the same way : that is, 1 million, 2 millions, etc., up to 999 millions. When we reach a thousand millions, we call the number a billion. Billions we count in the same way, so trillions, quadrillions, etc. 3. In writing these numbers we might write the number of thousands as we do the numbers up to a thousand, and attach the word thousand to each number ; thus, 8 thousand, 76 thousand, 999 thousand, etc., etc. But just as we avoid writing the words units, tens and hundreds, by giving to each order its place, so we avoid writing the word thousand by giving thousands the three places to the left of hundreds. In the same manner we give millions the three places to the left of thousands. 4. In this way it comes that, when more than three figures are employed to express any whole number, they are divided into groups, the first of which, numbering from the right, is used to denote any number from 1 to 999 units ; the second, from 1 to 999 thousand ; the third, from 1 to 999 million, etc. These groups are called Periods, and, for convenience in reading, are sometimes separated from each other by commas. 5. Thus, beginning at the right, we have the first ^period, con- sisting of ones, tens and hundreds of units ; the second period, ones, tens and hundreds of thousands ; the third period, ones, tens and hundreds of millions. The fourth period is that of billions ; the fifth, trillions ; the sixth, quadrillions ; each period contain- ing ones, tens, and hundreds of that period. NOTATION AND NUMERATION 15 li. Numeration Table. S3 °3S 2 a » So* So* So B ~ 3 S r 3 S n 3 £ a Said 3 SI WHO WHO WHO WHO WHO WHO Quadrillions Trillions Billions Millions Thousands Ones Sixth Fifth Fourth Third Skcond First 741 852 963 074 197 581 7 299 977 802 814 356 865 243 576 006 050 182 653 578 00 769 378 31 840 999 000 642 10 000 000 996 342 876 875 763 954 123 508 627 EXERCISES IN READING AND WRITING NUMBERS. Read the foregoing numbers, consulting the headings, till you get accustomed to the names of the periods. Read also as follows, or as may be directed by the teacher. a. Read the tens' and ones' columns in the first period on the right. b. Read all the numbers of the ones' period. c. Read the right-hand column in the thousands' period. d. Read the tens' and ones' columns in the thousands' period. e. Read the two right-hand periods. /. Read the right-hand column of the millions' period with the left of thousands, as hundred thousands. g. Read the millions' and ones' periods, omitting the thousands' period as if filled with ciphers. Note. — These exercises may be varied to almost any extent. 1. Read these numbers : 10,000 83,000 60,000 75,000 150,000 756,000 23,600 14,900 46,300 65,100 294,000 632,480 47,225 83,720 85,493 62,340 392,500 290,405 80,027 90,008 84,003 60,050 576,168 161,002 790,000 206,960 410,000 359,200 941,000 562,387 684,210 678,800 143,576 500,007 246,890 635,794 16 STANDARD ARITHMETIC, 2= Write and read : 800,000 450,000 743,000 200,000 375,670 237,090 135,791 987,654 3. Write and read 2,000,000 ; 5,000,000 ; 7,000,000 ; 4,000,000. Fill the places occupied by ciphers with any digits yon choose, and then read the numbers thus formed. Do this in various waySo 4. Copy the following numbers, and read them ; then erase the digit at the right hand, and arrange the periods anew, by placing the commas where they should be, and read : 1,635,987 416,429,863 134,764,211 7,763,664 29,876,354 Continue this exercise by erasing the digits one by one, and j)oint- ing off the periods correctly. 5. How many tens, hundreds, thousands, ten- thousands, hun- dred-thousands and millions are expressed in those places respec- tively, in the numbers of Exercise 4 ? Review Exercises. Hundreds.— l. Write 300 and 20 and 7 as one number, ex- pressed by three figures. In the same way write : 400 and 50 and 3 ; 100 and 70 and 6 ; 200 and 80 and 2 ; 300 and 60 and 6; 600 and 90 and 4; 900 and 10 and 1. 2. Write : 1 hundred 4 tens 6 ones; 2 hundreds 6 tens 3 ones; 1 »« 8 " 7 " 4 " 7 " 5 " 1 " 9 " 2 " 8 " " " 3. Read; 527 723 168 365 134 524 340 729 792 843 290 209 902 299 313 901 910 109 953 646 728 Thousands. — 4. Make the following numbers larger by one thousand : 328, 456, 508. By three thousand. By five thousand. 5. What number is next greater than 1599, 3019, 4091, 8400, 6379, 4599, 9999, 8765, 9109, 3099, 4098 ? NOTATION AND NUMERATION 17 6. What number comes next before 2000, 7000, 4600, 5060, 3010, 2790, 8970, 1000, 1010, 7801 ? 7. Count and write from 996 to 1006; from 3189 to 3200; from 7990 to 8012 ; from 3001 back to 2989 ; etc. 8. How many hundreds and tens in 43 tens ? In 68, 37, 56, 27, 49, 168, 434 tens ? How many in 386 units ? In 468, 125, 632 units ? In 354, 538, 624 tens ? 9. How many thousands and hundreds are in 25 hundreds ? In 61, 52, 47, 56 hundreds ? How many in 250 tens ? In 310, 161, 289, 364, 543 tens ? How many in 6987 units ? Tens and Hundreds of Thousands. — 10. Prefix first twenty, then sixty, then forty thousand to 438, 132, 596, 100. 11. What number next greater than 25,999? 130,109? 199,999? 888,889? 986,290? 18,400? 689,999? 12. What number next less than 300,001 ? 700,000 ? 147,000 ? 354,989? 500,790? 100,000? 600,999? 489,123? 500,000? Definitions. 12 . A unit is one of any order or kind. 13 . A number is a unit or collection of units. 14. Notation is the expression of number by figures or letters, 15. Numeration is the reading of numbers written in letters or figures. 16. All the digits have a Simple Value and a Local Value. A simple value, when they represent units or ones ; a local value, when used to express tens, hundreds, etc. This value is called local because it depends on the place which the digit occupies (its locality). 17. The nine digits are signs of number, hence they are called Significant Figures. In this sense, the cipher "0" is not a significant figure. 18 STANDARD ARITHMETIC. Roman Notation. 18. The following table gives a complete view of a method of representing numbers by letters. This is called the Roman method because first used by the Eoman people. Table. Uuits. IV V VI VII VIM IX Note 1. — It will be noticed that in writing four and nine of each order, a letter of less value is placed before one of greater value. In this case the less value is deducted from the greater. Thus, XL (ten less than fifty) is written for XXXX. CD (one hundred less than five hundred) is written for CCCC, etc. This mode of abbreviation is common, not universal. Note 2. — A bar over a letter, or combination of letters, increases its value a thousand times. For writing numbers in Roman numerals, we have the following 19, Mule. — Write the several terms in order as given in the table. EXERCISES IN THE ROMAN NOTATION. 1. Eead XXIV, IX, XIX, XV, XIV, LX, XLIV, LXXXIX, XO, XCIX, CCI, CCCXOIX, CD, CDLVIII, CDLIX. 2. DV, CDXCIX, DXLVI, DCCCIX, CMXCIX, MD, MIX, LIX, DIX, MDCCCLXXXIV, MCDLXX. Thousands. Hundreds. Tens M c X MM cc XX MMM ccc XXX IV CCCC or CD XL V D L VI DC LX VII DCC LXX VIII DCCC LXXX IX CM xc X- -10000 3. Write in Roman numerals, 54, 72, 83, 59, 119, 72, 38, 49, 63, 98, 75, 69, 43, 91, 108, 319, 444, 333, 991, 3847, 2563, 3482. CHAPTER II. ADDITION. Examples. — l. A hunter shot 6 rabbits on Monday, 7 on Tues- day, 8 on Wednesday, but only one on Thursday. How many rabbits did he shoot ? 2. Charles is 9 years old. How old will he be in 6 years ? In 5 years ? In 8 years ? 3. Fred had 8 dollars in his bank ; he received 7 more on his birthday, and 4 at Christmas. How much had he then ? 4. Grandfather was 53 years old when his grandchild was born. How old is he now that his grandchild is 9 years old ? 5. The sun rose at 6 o'clock this morning ; that was 3 hours ago. What o'clock is it now ? What o'clock 5 hours after sun- rise ? 4 hours ? 6 hours ? . 6. William read 7 pages in the morning, 3 in the afternoon, and 4 in the evening. How many pages did he read that day ? 7. Sarah goes up and down stairs 8 times in the morning, and 5 times in the afternoon. How many times in the day ? 8. Count to one hundred. — Count by twos to 100. — Count by threes to 99. Count by fours to 100. — Count by fives to 100. — Count by sixes to 96. — Count by sevens to 98. — Count by eights to 96. — Count by nines to 99. The pupil may first write the result of each successive addition, and afterward go through the exercise orally. 9. How many units in 10 twos ? (Count by 2's till you find out.) How many in 10 threes ? In 10 fours? In 10 fives ? etc. 20 STANDARD ARITHMETIC. 10. Draw lines upon your slate, so as to divide it like a checker-board, but make ten squares instead of eight in each row, and as you count by l's, write the results in the first line of squares from left to right ; as you count by 2's, write the re- sults in the second line of squares ; as you count by 3's, write the results in the third line, and so on. Definitions. 20. Addition in arithmetic is a process of finding the sum of two or more numbers. 21. Signs.— 1. The sign + is read plus, and indicates addi- tion ; thus, 5 + 3 means 5 and 3 more. 2. The sign = is read equals, or is equal to ; thus, 5 + 3 = 8 is read, 5 plus 3 equals or is equal to 8. ORAL EXERCISES. Write on slate or paper these two lines of figures. 4, 7, 2, 8, 8, 5, 6, 3, 9, 2, 6, 6, 8, 7, 9, 5, 3, 3, 4, 2, 5, 5, 3, 7, 7, 5, 5, 4, 6, 9, 9, 8, 4, 4, 2, 3, 8, 5, 4, 9, 6, 7. 11. To each number represented add 2, add 4, add 6, add 8. Caution. — Do not say 4 and 2 are 6, but speak only the results, as 6, 9, etc. In 13 (below) give results directly, as 11, 9, 10, 16, etc. 12. Add 3, add 5, add 7, add 9, to each one. 13. Add the first to the second ; add the second to the third, etc., beginning at the left — beginning at the right. 14. Add each number in the lower line to the one above it, proceeding first from left to right, and then from right to left. 15. 4 + 5 + 6= 20. 2 + 3 + 4 + 5= 25. 4 + 4 + 4 + 4= 16. 5 + 6 + 7= 21. 3 + 1+4 + 8= 26. 6 + 3 + 6 + 3 = 17. 6 + 2 + 8= 22. 4 + 7 + 6 + 3= 27. 5 + 5 + 5 + 5 = 18. 4 + 9-l-3= 23. 9 + 3 + 2 + 3= 28. 7 + 2 + 7+2= 19. 7 + 4 + 9= 24. 7 + 4 + 6 + 3= 29. 8 + 4 + 5 + 4= ADDITION. 21 30. 6 + 5 + 3: 8 + 4 + 2= 9 + 7 + 4: 5 + 8 + 3: 7 + 4 + 7: 33. 10 + 6: 30 + 5: 40 + 4: 50 + 3 = 60 + 2: 80 + 7: 31. 5 + 4 + 8 + 2 = 2+4+6+8= 3+5+7+4= 3+6+3+7= 8 + l + J* + 2 = 34. 20 + 9= 50 + 7= 70 + 5= 90 + 3= 30 + 2 = 60 + 8= 35. 90 + 2: 80 + 4: 70 + 6: 60 + 3: 50 + 5: 40 + 9: 32. 5 + 3 + 4 + 2: 9 + 7+1+2: 6 + 4+4+3: 9 + 4+4+3= 4+9.+ 3 + 4= 36. 20 + 2+4= 40 + 3 + 2 = 60 + 4 + 5= 80 + 5 + 2= 30 + 1 + 7= 70 + 2 + 6= 37-117. Add 1, 2, 3, etc., up to 9, separately to each number in each line. Observe the units of the results. 1 11 21 31 41 51 61 71 81 2 12 22 32 42 52 62 72 82 This may be done orally, or on the slate, thus : 13 23 33 43 53 63 73 83 (37.) 1 + 1 = 14 24 34 44 54 64 74 84 15 25 35 45 55 65 75 85 16 26 36 46 56 66 76 86 17 27 37 47 57 -67 77 87 8 18 28 38 48 58 68 78 88 9 19 29 39 49 59 69 79 89 11 + 1 = 21+1 = 31 + 1 = 41+1 = 51 + 1 = etc. 118. 32 + 9= 43 + 8= 54 + 7= 65 + 6 = 76 + 5= 87+4= 91 + 8= 119. 33 + 9= 44 + 8= 55 + 7= 66 + 6= 77+5 = 88 + 4= 22 + 7= 120. 87 + 6 = 78 + 5= 45 + 8= 54+7= 65 + 9= 56+4= 37 + 8= (117.) 9 + 9= 19 + 9= 29 + 9 = 39 + 9= 49 + 9= 59 + 9= etc. 121. 56 + 8= 65 + 6= 29 + 9= 92 + 5 = 87 + 7= 78 + 6= 47 + 6= 122-127. Add 6 to 21, 18, 36, 48, 54, 63, 17, 82, 88. Add also 8 ; 4 ; 5 ; 7 ; 9. 123-131. Increase the numbers 14, 19, 23, 25, 48, 84, 56, 37, 64, 83, 52, 38, 90, 87, 75, 61, 47, 79, 39, 59, 27, 69, 89, by 4 ; by 6 ; by 8 ; by 10. 22 STANDARD ARITHMETIC. 133-135. Increase each of the numbers 23, 35, 48, 64, 56, 37, 41, 90, 82, 52, 61, 73, 84, 59, 47, 36, by 3 ; by 5 ; by 7 ; by 9. Direction. — The foregoing exercises should be so thoroughly practiced, both orally and in writing, that the pupil can announce the sum of any two numbers expressed by single digits as readily as he can read them. — If he has to count his fingers in addition, he can proceed but slowly. He might as well spell every word as he reads, or crawl on his hands and knees instead of walking. Again, if he says "9 and 7 "are 16," he uses five words where one, "sixteen," would be better. Applications. — 136. An hour has 60 minutes, and a half-hour has 30. How many minutes are there in one hour and a half ? 137. There were hanging on a Christmas-tree 10 oranges, 20 apples, 30 nuts, 20 sugar-plums. How many gifts in all ? 138. There are at work in a factory 40 men on the ground- floor, 30 on the second floor, 20 on the third floor, and 7 in the office. How many men are at work in the factory ? 139. Grandmamma is 60 years old, mamma 30, and I am 7 years old. What is the sum of our ages ? 140. An overcoat costs 30 dollars, a coat 20 dollars, a vest 4, and a pair of trousers 8 dollars. How much does the whole suit cost? 141. A fisherman caught in his net 36 pike, 30 bass, and 10 trout. Can you tell how many fish he caught ? 142. A butcher bought two calves ; one weighed 53 pounds, the other 47. How much did they weigh together ? ORAL EXERCISES. Direction. — In adding, do not say (see 1st example) 4 and 5 are 9 and 6 are 15, etc., but give results at once ; thus, 4, 9, 15, 23, etc. 143-168. Add by columns and lines. 4+5+6+8+9+4+3+7= 9 + 6 + 8 + 2+7 + 5 + 9 + 3 = 3+2+1+9+7+6+2+8= 6+7+4+9+3+1+7+5= 8+4+3+9+5+7+8+6= 8+5+3+6+2+8+4+7=" 6 + 6 + 8 + 3 + 7+4 + 4 + 5= 34.5 + 4 + 9 + 7 + 2 + 8 + 2= 7+8+2+7+6+9+6+5= 3+4+2+5+3+1+6+2= ADDITION. 169. 170. 171. 172. 10 + 40= 90 + 10= 20 + 70= 40 + 15: 20 + 50= 80+10= 30 + 60= 30 + 26= 30+40= 60 + 30= 40 + 50= 60 + 38; 40 + 50 = 70 + 20= 50 + 30= 50 + 47 50 + 30= 50+40= 60 + 15 = 70 + 28: 60+40= 40+40= 70 + 26 = 40+48 70 + 20= 80 + 20= 80 + 17= 60 + 37 90 + 30= 20 + 60= 20+45= 70+46: 50 + 50= 30 + 30= 40 + 4:1 = 80 + 11 23 174. 30 + 20 + 10 + 30= 175. 20 + 10 + 30 + 20= 176. 10 + 30 + 20 + 30= 177. 40 + 20 + 30 + 10= 173. 56 + 30= 44 + 20= 38 + 60: 29 + 50: 14+80= 63 + 30: 74 + 20: 88+40= 30 + 53: 178. 30+15 + 20 + 15 + 10 + 3+ 7= 179. 10 + 25 + 30 + 10 + 20 + 2+ 2 = 180. 20 + 25 + 10 + 15 + 10 + 3 + 14= 181. 15 + 25 + 30 + 10 + 12 + 4+ 4= Suggestions. — If, from this point to the rule on page 28, the examples seem too difficult, they may be omitted, to be taken up under the rule, but let the oral work be carried as far as possible. The learner who is left to himself to work out all his exercises on the slate is apt to form habits fatal to accuracy. Applications. — 182. There are 25 girls and 23 boys in a school- room. How many pupils in all ? 183. On one side of Blair Street there are 34 houses, on the other side, 59. How many on both sides ? 184. Mr. H. bought a horse for 73 dollars ; he sold it and gained 19 dollars. For how much did he sell it ? 185. Our house has 17 windows, that of one neighbor has 26, and that of another neighbor has 13. How many windows are there in the three houses ? Exercises. — 186-206. Add rapidly, by columns and lines, giving results only : 4+6+4+8+7+4+9+4+3+5+2+3+1+8+9= 8+5+3+9+6+8+5+3+6+7+9+7+6+9+1= 4+9+5+6+9+7+6+2+1+8+3+4+2+3+9= 7+6+5+4+3+2+1+9+7+5+6+4+9+4+8= 4+3+2+1+2+3+4+5+6+7+8+9+7+6+7= 8+4+6+9+1+7+3+5+7+9+6+8+7+5+6= 24 STANDARD ARITHMETIC. Note. — Let the pupil illustrate examples 207 to 226 by the use of buttons, acorns, or other objects which he can tie into bundles or string together in collections of ten; or let him make marks upon the slate such as these at the right, designed to illustrate example 207. Thus, understanding well the nature of the thing to be done, he will need no rule for the sim- ple operations here required. Let him first add the tens, and to the sum let him add the numbers in units' place. 207. 17 + 15+10 + 18 + 19= 217. 208. 14 + 16 + 18 + 20 + 13= 218. 209. 12 + 13 + 20 + 14+16= 219. 210. 20 + 19 + 11 + 18 + 13= 220. 211. 18 + 15 + 14 + 17 + 16= 221. 212. 12 + 14 + 16 + 13+15= 222. 213. 15 + 17 + 19 + 16 + 14= 223. 214. 19 + 12 + 13 + 17 + 16= 224. 215. 17 + 18 + 14 + 17 + 18= 225. 216. 20 + 19 + 19 + 12 + 13= 226. WWW// /r MMM /s MM /o MM M// /s MMMM/& 12 + 15 + 18 + 10 + 19 + 17 + 13 + 20 + 23 + 17 + 8 + 12 + 11 + 17 + 27 + 19 + 12 + 24 + 20 + 13 + 9 + 10 + 13 = 17+ 9 + 18= 15 + 13 + 11 = 7 + 16 + 12 = 18+ 2 + 15 = 13 + 19 + 21 = 28+ 6 + 18= 8 + 20 + 16: 18+ 6 + 17: 16 + 10 + 25: Suggestion. — Exercises in numeration should precede the following examples. Applications. — 227. There were at a party 50 gentlemen, 60 ladies, and 70 children. How many people were there ? 228. A farmer raised 80 bushels of wheat; 39 bushels of oats, and 10 bushels of barley. How many bushels in all ? 229. There are in an orchard 63 plum-trees, 75 apple-trees, and 11 peach-trees. How many trees in all ? 230. A book-case has on the first shelf 48 books, on the second 57, and on the third 75. How many on the 3 shelves ? 231. There are 68 boys in one room of a school-house, 73 girls in another, and 87 girls and boys in a third. How many pupils are there in the school ? 232. The first book of Moses has 50 chapters, the second 40, the third 27, the fourth 36, and the fifth 34. How many in the 5 books ? ADDITION. 25 Add without the use of the slate : 233. 50 + 60= 234. 40 + 60 + 30= 235. 60 + 80= 70 + 40= 80 + 30= 90 + 50= 40 + 90= 50 + 30 + 50= 60 + 90 + 70= 70 + 40 + 90= 80 + 70 + 40= 90 + 50 + 60= 80 + 38= 70 + 59= 90 + 67= 50 + 74= 60 + 83 = 70 + 92= 236. 64 + 50= 76 + 60= 85 + 90= 59 + 70= 68 + 80= 75 + 40= Note. — The examples in 235 and 236 require only one oral step, that is, the direct announcement of the result ; as, for instance, in adding 80 and 38, think 80 and 30 (=110) and 8, but say at once 118. In examples 237 to 241, two steps are enough ; thus, in adding 59 and 32, first think 59 and 30, and say 89, then 89 and 2, and say 91. In examples 242 to 244, four steps may be necessary for the learner ; thus, in adding 25, 38, and 49, say 55, 63, 103, 112. 237. 238. 239. 240. 241. 59 + 32= 36 + 42= 25 + 22 = 57 + 28= 26 + 34= 44 + 27= ' 47 + 33= 24 + 35= 28 + 39= 24 + 16= 36 + 24= 38 + 24= 33 + 19= 45 + 37= 74 + 24= 47 + 25= 58 + 25 = 42 + 35 = 36 + 38= 25 + 33= 88 + 26= 64 + 26= 53 + 43= 25 + 47= 47 + 29= 45 + 37= 47 + 27= 65 + 26= 34 + 39= 58 + 33= 62 + 25= 29 + 34= 35 + 36= 27 + 48= 244. 34 + 28= 242. 243. 65 + 36 + 3 = , 85 + 47 + 5 = 25 + 38 + 4S 1= 85 + 74 + 5= 83 + 68 + 6= 97 + 23 + 62 67 + 63 + 7= 94 + 42 + 9= 56 + 44 + 68 58 + 54 + 6= 86 + 29 + 3= 83 + 28 + 25 73 + 72 + 9= 92 + 56+5 = 62 + 37 + 27 68 + 45 + 4= 75 + 65 + 6 = 56 + 48 + 39 i — 245-259. Add by columns and lines : 40 + 60 + 30 + 70 + 80 + 90 + 60 + 80 + 80 + 30 50 + 80 + 90 + 80 + 40 + 50 + 90 + 70 + 20 + 50 60 + 40 + 70 + 90 + 20 + 70 + 70 + 60 + 70 + 30 70 + 90 + 20 + 60 + 30 + 30 + 40 + 50 + 90 + 70 80 + 70 + 60 + 50 + 40 + 30 + 20 + 40 + 30 + 50 STANDARD ARITHMETIC. Suggestions for Blackboard Exercises. 22. In drill exercises, the double star affords some ad- vantages over the circle, and at the same time facilitates the learning of the several series arising from successive addi- tions of %% 3's, 4's, etc. Direction. — Beginning at the unit figure of any given number, the unit figures of the successive sums will be found as follows : 1. In adding 3's, at the next point to the right, and so on ; in adding Ts, at the next to the left. 2. In adding 6's, at the second point to the right, and thus on, from point to point, of the same star. In adding 4's, at the second point to the left, and so on. 3. In adding 9 ? s, at the third point to the right. 4. In adding 2's, at the fourth point to the right, and thus on (following the line at the right of the last unit figure). In adding 8's, at the fourth point to the left (following the line at the left of the last unit figure). 5. In adding 5's, at the point directly opposite the unit figure of the given number, and thus to and fro. 23. Other Uses of the Figure. — A suitable number being writ- ten at the center, the numbers at the points can be combined with it, in addition, subtraction, multiplication, or division, as may be desired. The number at the center being changed from time to time, there is no end to the variety of exercises that may thus be had at little expense of time or labor on the part of the teacher. Exercises in common and decimal fractions may be given in the same way. ADDITION. 27 Addition of Higher Orders. 24"« Note. — The illustrations of this book are not intended to be merely ob- served and read about, but they are designed to picture to the eye, as far as possible, the actual work which it is intended shall be done by the pupils with objects. These objects should be supplied by the school authorities, or, with slight suggestions by the teacher as to what is best or most available, according to the circumstances of the school, they may be brought in by the pupils. They should be as large as pos- sible, and yet not inconvenient to handle in great numbers. SLATE WORK. Example.— Find the sum of 738, 236 and 573. One who knows nothing more of arithmetic than how to count to ten might find the sum of these numbers by some such means as the following : Suppose that he has a large num- ber of sticks, some of them tied up in bundles of ten, and some in bun- dles of a hundred each, and that he has, besides, some single sticks. If these were placed in rows or shelves, as in the picture at the right, he might count first the single sticks, taking them in his hand as he does so, and when he has reached ten, tie them in a small bundle, leaving the remaining single sticks at the right on the shelf below. He could then count the bundle which he had just made, with the tens' bundles on the shelves, and tie each ten of these bundles into larger bun- dles of a hundred each, and leaving the odd bun- dles of tens on the shelf below where they formerly were, he could ii B - STANDARD ARITHMETIC. count all the bundles of hundreds together, ten of these larger bun- dles into one, he would have the sticks arranged as here represented ; that is, one bundle con- taining a thousand, five of a hundred each, four of ten each, and seven single sticks. 25. The foregoing method is the same as that which is indicated in the following arithmetical process : <&u?ztd (7Atm& ofcnJ. <2&U&. ct/itntJ. (?/&*zd. C&**4. fl&Utt r 3 *u 7 Z 3 els / //+/ j+v 7 Having seen how it is, that this process really produces a number equal to the sum of the numbers added, the pupil is prepared for the rule for addition. 26. Bule.—l. Arrange the numbers to be added so that the figures of the same order shall stand in the same column, units under units, tens under tens, and so on. 2. Begin at the lowest order, and add each column separately. If the sum of any column is less than 10, write it underneath. If it is equal to or greater than 10, place the right-hand figure of the sum under the column added, and unite the left-hand term or terms with the next column. Vroof. — In order to be quite sure that the addition is correct, add each column both upward and downward. If the two results are the same, there is little danger of error. ADDITION. 29 SLATE EXERCISES, Examples l-ll. Find the sum of 27 37 29 68 68- 92 53 27 57 91 84 58 28 36 43 79 72 67 72 46 24 37 90 64 39 29 25 68 85 28 90 56- 62 56 47 76 46 97 73 39 57 51 45 78 12-22. Find the sum of 48 83 69 21 43 96 81 49 36 85 20 47 59 56 53 28 45 47 57 17 24 38 56 62 67 38 57 51 63 42 58 66 47 29 48 58 73 46 23 86 56 32 77 93 23 -77. Add 345 to each, 639 542 894 457 837 910 735 628 246 802 135 429 900 312 163 524 254 736 547 749 630 757 372 713 457 298 337 698 507 192 293 394 495 345 298 110 206 387 471 289 509 135 398 211 213 361 425 357 862 135 779 387 600 731 877 78-88. Add together the numbers in each column. 89-93. Arrange each line of numbers in column and add. 94. 95. 96. 97. 93. 99. 100. Apples. Nuts. Oranges. Peaches. Lemons. Plums. Books 136 268 204 268 301 718 593 241 194 237 473 275 629 868 217 187 168 118 478 446 687 153 253 352 323 262 537 774 302 145 249 248 164 855 956 Add the following numbers, first arranging them in columns 101. 125, 126, 138, 139, 140. 103. 83, 194, 56, 168, 473. 102. 87, 9, 55, 394, 225, 194. 104. 336, 195, 987, 9, 11. 105-114. Add by columns. Also by lines, 2123 + 2364 + 7025+ 428+ 20 + 2103=. 6354 + 2559+ 843 + 1125+ 359+ 23= 698 + 1994 + 1427 + 2496 + 2478+ 437= 1927+ 49 + 7917 + 6579 + 1000 + 3706= 30 STANDARD ARITHMETIC. 115-124. 3254 + 4015 + 7348 + 1570+ 439 + 7986= 968+ 916 + 3407 + 4630 + 1690+ 375= 725 + 1207+ 197 + 1820+ 420+ 9= 4302+ 885 + 8329+ 7 + 7756 + 8975 = The following may be solved first without the use of the slate. Only results should be pronounced. (In the last line, Ex. 127, for instance, say 892, 952, 956.) 125. 300 + 600= 126. 600 + 70 + 38= 400 + 900= 700 + 50 + 49= 500 + 100= 800 + 80 + 76= 600 + 400= 900 + 90 + 47= 700 + 700= 300 + 60 + 83= 800 + 200= 200 + 80 + 72 = 127. 300 + 56 + 84= 400 + 72 + 65= 500 + 83 + 49= 600 + 64 + 57= 700 + 53 + 88= 800 + 92 + 64 = 128-148. A dd by columns and by lines. 568 + 487+2000 + 5872 = 435 + 675 + 6060 + 9321 = 357 + 894 + 7009 + 7234= 479 + 383 + 5800 + 5321 = 692 + 596 + 3750 + 6947= 824 + 776 + 4680 + 3579 *= 546 + 484 + 8541+2468= 149. 345,271 150. 4,391,002 151. 4769 + 634 + 2465 = 1250+ 4 + 1975= 3456+ 27+ 888= 5861+ 5+ 935= 9642+ 95+ 576= 8347+ 7 + 6491 = 4936 + 1S3+ 587= Find the i 3i//77 Of . 345,271 150. 4,391,002 65,382 686,975 7,491 68 83,257 4,937 496,350 53,286 1,849 487,659 65,472 7,321,445 4,622,715 9,874,963 8,472,465 8,010,706 4,506,080 432,741 98,653 152. 5,324,681 1,964,735 28,497 68 5,834 4,793 56,689 153. Add 768, 5,643, 12,354, 678,901, 5,847, 2,146,353, 975,321, 64,387,510. 154. Add nineteen, ninety, seventy thousand four hundred eight, 87, 1,625,847, 269,751, 3,894, twelve hundred sixty-one, 5,050,050, six hundred thousand six. ADDITION. 31 155. Add 198,725, 918,273, 1,928,370, 4,354,651, 34,234,534, 6,712,893, 647, 19, 1,345, 67,351. 156. Add 283,857, two thousand twenty, 998,722, five mill- ions fifty thousand fifty, eight hundred thousand seven hundred twelve, 27, 192,875, 909,090, six hundred eight thousand four hundred ten, 34,827, fourteen hundred fourteen. 157. 1,783 158. 4,328 159. 42,235 160. 9,999,999 19,456 369,300 10,305,236 8,000,000 5,788 53,528 84,165,352 4,730,876 94,374 51,279 1,236,536 12,359,776 100,855 13,975 163,021 76,305,864 456,788 34,975 29,363,987 32,467,209 872,543 124,900 37,903,210 57,264,902 321,354 1,243,651 16,988,710 98,537,873 Direction. — In adding these columns, do not say (see Ex. 157) 4 and 3 are *7, and 8 are 15, and 5 are 20, and 4 are 24, etc., but simply speak results ; thus: 4, 7, 15, 20, 24, 32, 38, 41. The repetition of the numbers to be added in- creases liability to error. Some can learn to add mentally numbers of even three places. Treating tbe tens and hundreds as units and tens (see note, p. 25), they would say, in Ex. 161, 27, 57, 65 tens = 650, 655, 661, and set down the answer at once. 27. Adding two or more columns of figures at once is valu- able practice in "mental arithmetic." It should be carried as far as time and the ability of the pupil will permit. 161-169. 275 369 .876 629 483 987 797 519 357 386 625 529 875 759 654 686 982 678 Examples for Practice and Review. Applications. — l. I gave 83 marbles to Lewis, 34 to William, and 97 to Charles. How many did I give away ? 2. In one book there are 89 pages, in another 246, and in a third 387. How many pages in all ? 3. A certain tract of land was divided into four farms, one containing 113 acres, another 237, a third 180, and the fourth 320 acres. How many acres did the original tract contain ? 32 STANDARD ARITHMETIC. 4. There was a large number of cents in a bag. I took out of it first 289 cents, then 397, then 478, then 693, and then I found 134 cents left in the bag. How many cents did it contain at first ? 5. Our school-house contains 6 rooms. In room No. 1 there are 25 boys and 31 girls ; in No. 2 there are 18 boys and 29 girls ; in No. 3, 21 boys and 37 girls ; m No. 4, 29 boys and 19 girls ; in No. 5, 45 boys ; in No. 6, 53 girls. How many boys in our school ? How many girls ? How many children in all ? 6. In a certain township there are six farmers. The first has 5 horses, 12 cows, 35 sheep, and 20 hogs. The 2d has 5 horses, 10 cows, 18 sheep, and 12 hogs. The 3d has 3 horses, 6 cows, 27 sheep, and 9 hogs. The 4th has 4 horses, 8 cows, 25 sheep, and 14 hogs. The 5th has 1 horse, 2 cows, and 6 hogs. The 6th has 8 horses, 17 cows, 45 sheep, and 27 hogs. (1) How many horses have the 6 farmers ? (2) How many cows ? (3) How many sheep ? (4) How many hogs ? (5) How many head of live stock has the first, the second, the third, the fourth, the fifth, the sixth ? (6) How many head of live stock on the 6 farms ? For the solution of this and similar examples, follow the arrangement given here. It is called " Tabulating," or arranging in tables. FARMERS HORSES COWS SHEEP HOGS NO. HEAD The First 5 12 35 « 20 The Second 5 10 18 12 The Third 3 6 27 9 The Fourth 4 8 25 14 The Fifth I 2 — 6 The Sixth 8 17 45 27 Total Note. — If the sum of the footings in the last line is not equal to the sum of the extensions in the last column, the work is incorrect. Why ? ADDITION. 33 7. Three farmers have fruit-trees as follows : The first, 72 apple-trees, 108 peach-trees, 18 quince-trees, 16 plum-trees, 19 cherry-trees. The second, 38 apple-trees, 219 peach-trees, 9 quince-trees, 27 pear-trees, 38 plum-trees, 3 cherry-trees. The third, 19 apple-trees, and 43 peach-trees. (1) How many trees of each kind ? (2) How many fruit-trees has each farmer ? (3) How many trees have all ? 8. In the year 1880, Cincinnati had 255,139 inhabitants ; Cleveland 160,146; Toledo 50,137; Columbus 51,647; Dayton 38,678; Sandusky 15,838 ; Springfield 20,730; Hamilton 12,122 ; Portsmouth 11,321. How many inhabitants in these 9 cities of Ohio ? 9. Three farmers, last year, sold, fruit as follows : The first, 79 bushels of apples, 391 bu. of peaches, 8 bu. of quinces, 39 bu. of pears, and 8 bu. of cherries. The second, 43 bu. of apples, 539 bu. of peaches, 19 bu. of quinces, 37 bu. of pears, 47 bu. of plums, and 6 bu. of cherries. The third, 27 bu. of apples, and 87 bu. of peaches. (1) How many bushels of each kind of fruit were sold ? (2) How many bushels of fruit did each farmer sell ? (3) How many bushels did all of them sell ? 10. North America is inhabited by 54,566,936 people ; the West Indies by 4,316,718 ; South America by 26,913,531 ; Europe by 311,694,029; Asia by 791,031,473; Africa by 199,921,600; Oceanica by 38,318,771. Find how many in all. (Statistics 1885.) 11. Three farmers divided their land as follows : The first had 6 acres in rye, 20 acres in wheat, 69 acres in corn, 2 acres in pota- toes, 26 acres in meadow land, 19 acres were lying fallow, and 3 acres were in grapes. The second had 12, 39, 136, 5, 75, 26, 2 acres (take them in the same order). The third had 4, 19, 53, 19, 5, 4, 9. (1) Find., how many acres of each kind ; (2) how many acres to each farmer ; (3) How many acres in all. 12. Lake Superior has an area of 32,000 square miles ; Lake Michigan 24,000, Lake Huron 20,400, Lake Erie 9,600, Lake Ontario 6,300. What is the total area of the five great lakes ? 34: STANDARD ARITHMETIC. Original Problems. 28 a Note. — Problems such as the first may be made up under the direction of the teacher with the aid of all the pupils of the class, due notice having been given of the kind of contributions desired. One problem per day, if possible, should be required of each pupil. 1. Find the number of pages read by all the pupils in books not used in school. (Each reports for himself, all set down the items and find the sum.) 2. Find how many examples all have solved within any given time ; how many lines all have read ; how many words all have spelled or missed ; how many chestnuts, walnuts, hickory nuts, acorns, etc., all have gathered. 3. In a rural district an enumeration may be made of the num- ber of horses, cows, sheep, hogs, chickens, etc. ; of the apple, peach, cherry trees, etc. ; of the eggs gathered, etc., etc., in the district, or on all the farms from which the children come. Let them report in writing, so that no sensitive child or parent be offended. Note. — Such problems as the following should be written out by the pupils separately, and without the aid of the teacher. Paper, cut to uniform size, should be used for the purpose, and the exercises corrected, as in other language lessons. 4. Each pupil may imagine himself to be a store-keeper in any line of trade he pleases, and make problems in regard to it. He should write out each problem with the utmost care, and be sure that he knows the answer before he gives it to the class. 5. Questions may be prepared with the aid of a text-book in geography ; as, for instance : What is the population of the largest cities of ? (the pupil's own state, the United States, any foreign country, or the world). 6. The teacher can at sight judge of the correctness of answers to such questions as the following : If there are ten wagons, and each one contains 23 bushels of apples and 27 bushels of potatoes, how many bushels of potatoes and how many bushels of apples in them all ? CHAPTER III. SI) BTRACTION. Units and Tens. Examples. — l. Fred had 13 cents, and bought a top costing 80. How many cents had he left ? How many would he have had left if the top had cost 7^ ? 6^ ? 4^ ? 5^ ? 9^ ? 3^ ? (!i H> When the above work is neatly done, copy the following para- graphs, carefully filling all the blanks : 1. In each full line there are dots, which make groups (of 7), and — dots over. 2. In the ten lines there are — times as many dots, or dots ; and — times as many groups, or groups ; and there are also — times as many over. 3. The ten 6's over and the 3 in the short line beneath make together — dots. Out of 63 dots we can make groups of 7 dots each. Hence there are times 7 dots in 273, and times 7 cents in 273^. Therefore, at balls of twine for 273^. a ball, we can get DIVISION. 75 36. How many pine-apples can be bought for $27.78, or 2778^, at 6^ apiece ? Make 27 dots, and separate them into groups of 6 each ; thus, £»».*«* ) ( ) ( ) ( ») • • • Note. — With this single line of counters before him, the pupil should now be able to answer the following questions : 1. How many such lines would represent the number of cents expressed by the first two figures of 2778^ ? 2. How many dots would there be in 10 such lines ? In 100 ? How many groups of 6 in 10 lines ? In 100 ? How many over in 10 lines? In 100? The 300 dots over and 78 dots would make how many dots, etc. ? (From this point the process is exactly the same as in the preceding Ex. The pupil may complete the work, beginning with the making of 37 dots, if necessary, etc.) 66. The Arithmetical Solution. — 6 is not contained in 2778 any thousands of times, for even one thousand times 6 would be 6000, but it is contained some hundreds of times, for 100 times 6 is only 600. The question is, how many hundreds of times ? In a line of 27 dots we made 4 groups of 6 dots each, in ten lines (270 dots) there would be 40 groups, and in 100 lines, 400 groups. Hence, in the arithmetical 6)2778 process we write 4 in hundreds' place. It shows 4.. how many hundred groups of 6 can be made of 27 hundred dots. There are 3 hundred remainiug. These, with the 78, make 378. How many groups can be made of 378 dots ? In a line of 37 dots we could make 6 groups, with one dot over, and in ten lines we could make 60 groups, with 10 over. We write the 6 in tens' place. It 6 )2778 shows how many times 10 groups of 6 each can 46. be made of 37 tens. The ten over and the 8 make 18, in which 6 is contained 3 times. Hence, 6 is contained 463 6)2778 times in 2778, ""463 76 STANDARD ARITHMETIC. SLATE EXERCISES Find 37. 4 in 296 348 264 192 356 How 4 in 42. 3124 43. 4796 44. 5340 45. 9516 46. 6779 47. 8040 48. 1596 49. 7120 how many times 38. 6 in 258 462 354 408 516 many times 6 in 50. 3510 51. 2578 52. 1602 53. 5958 54. 1086 55. 2802 56. 4556 57. 3062 39. 7 in 656 483 598 651 619 7 in 58. 7056 59. 4606 60. 6110 61. 2072 62. 5408 63. 4260 64. 1645 65. 1113 40. 8 in 784 608 527 770 664' 8 in 66. 1707 67. 9608 68. 7527 69. 5768 70. 3664 71. 6856 72. 2392 73. 2216 41. 9 in 399 432 873 . 778 617 9 in 74. 5373 75. 2961 76. 4212 77. 8136 78. 7008 79. 4905 80. 2220 81. 2052 Definitions. 67. Division is a process of finding how many times one number is contained in another, or of finding one factor of a number when the other factor is known. Division has two uses : 1. To find how many equal parts there are in a number when we know what one part is. 2. To find one of the equal parts of a number when we know how many parts there are. 68. Terms used. — The Dividend is the number to be divided. 69. The Divisor is the number by which we divide. 70. The Quotient is the number of times the divisor is con- tained in the dividend. 71. A part of the dividend remaining undivided is called the Remainder. ^v DIVISION. 77 72. Signs. — There are three signs used to indicate division : 1. The divisor is placed on the right of the dividend with the sign -f- between them ; 84 -r- 7 is read 84 divided by 7 ; or, 2. The divisor is placed on the left of the dividend, with a curved line between them ; thus, 7)84 is read 84 divided by 7 ; or, 3. The divisor is written beneath the dividend with a hori- zontal line between them ; thus, -^ is read 84 divided by 7. Note 1. — Multiplication is taking one number as many times as there are units in another. Division is finding how many times one number is contained in another. Hence division is the reverse of multiplication. Note 2. — One number can be taken from another as many times as it is con- tained in it, hence by division we find how many times one number can be sub- tracted from another. Division of Numbers into Parts, 73. If a single thing or a number of things is divided into two equal parts or lots, the parts are called halves. Thus, if a boy shares an apple equally with a school-mate, he cuts it into two equal parts : each part is a half of the apple ; or, if he shares a number of chestnuts equally with another, he divides the chest- nuts into two equal lots : each lot is a half of the whole number. If a single thing or number of things is divided into three equal parts, the parts are thirds; if into four equal parts, the parts are fourths, etc. Such equal parts are called fractions. 74-. One half is written in figures thus : % ; the figure above the line represents one, the figure below represents the word half. One third is expressed by 1 above the line and 3 below it, thus, y 3 ; two thirds is written % ; the 1 and the 2 represent the number of thirds, the 3 stands for the word thirds. ORAL AND ILLUSTRATIVE EXERCISES. l. A mother has a basket of pears which she wishes to divide equally among three children : what part of the whole number will she give to each one ? What part would be given to each if 78 STANDARD ARITHMETIC. they were equally divided between 2 children ? Among 5 children? 4 children ? 2. If a father divides 8 one-cent pieces among 4 children, what part of the number does each one receive ? How many is that ? 3. If he divides 18^ among 6 children, how many does each one receive ? What part of the whole number is that ? 4. Illustrate with strips of paper, apples, or other objects, what is meant by %, y 3 , %, y 4 , %, 3 / 4 , of anything or number. 5. Show that 2 / 2 , 3 / 3 , 4 / 4 , of anything are equal to the whole of it. 6. Illustrate with objects what is meant by 3 / 4 of 24 ; also, four sixths, three sixths, one sixth, two sixths, six sixths, three tenths, five sixths, Vs of 32, V. of 27, V, of 54, M>V/£e in figures. one third, % of 28, % of 56, y 4 of 36, three thirds, two tenths, six sevenths, two sevenths, four tenths, six fourths, three sevenths, five sevenths, one seventh, seven sevenths, six sevenths, four eighths, seven tenths, % of 24, % of 72, % of 50. five eighths, eight ninths, seven eighths, five ninths, nine tenths, three ninths, seven fifths, The Two Uses of Division Compared. 75. The two uses of division which are represented in the following problems are often confounded. The figures employed in the arithmetical solutions, and the digits in the answers, are exactly the same for both, yet the answers are really different, and the explanations of the process by which they are obtained should vary accordingly. l. How many times is 5 2. One fifth of 45 is how contained in 45 ? many ? The arithmetical solution to both is the same, thus 45 -i- 5 = 9. DIVISION. 79 The Solution with Counters. — That the pupil may clearly understand the essen- tial differences between the two problems, and yet why their modes of solution should be the same, let him solve them with counters, thus : 1. To find how many times 5 are 2. To find J / 5 of 45 he does pre- contained in 45, he puts down 45 cisely the same as before, and then counters, arranging them in groups takes one counter from each group, of 5 as he does so. thus getting l /j of all the groups, or V* of 45. The answer to this is 9 groups • • • of 5, or 9 Jives. The answer to this is 9 units. In this way it is shown that there are as many units in 1 / 5 of 45 as 5 is contained times in J+5. Show with counters how to find 1. y 3 of 18 4. % of 36 7. % of 18 10. % of 54 2. y 4 of 28 5. % of 42 8. Vt of 49 11. % of 72 3. % of 30 6. % of 55 9. % of 40 12. % of 54 OBJECTIVE EXERCISES. Work out the following problems with the aid of counters, without using your knowledge of the multiplication table, and in each case state whether it is your object to find how many there are in a lot ; or, how many lots there are. 1. Six gentlemen on a fishing excursion catch 48 fish, and divide them equally among themselves, how many does each one receive ? How many poor families would they supply if 6 fish were sent to a family ? 2. A dairyman has 96 pounds of butter to be sent to market, how many jars will he need if he puts 8 lbs. in a jar ? How many pounds must he pack in a jar if he has but 8 jars ? 3. A lady pays $42 for 14 yards of silk, how much does she pay a yard ? If instead of the silk she buy velvet at $6 a yard, how many yards would she get for the same money ? 80 STANDARD ARITHMETIC. 4. A teamster hauls 9 barrels of coal oil at a load; how many loads does he make of 126 barrels ? He puts them into 3 freight cars; how many in a car ? ORAL AND SLATE EXERCISES, l. Five ninths of 54 are how many ? Analysis. — x ( 9 of 54 is 6 ; and 5 / 9 of 54 are 5 times 6 s= 30. How many 2. 3 /s of 72 4 / 9 of 36 Vs of 48 are 3. % of 54 4. % of 32 % of 42 % of 27 % of 45 % of 32 5. Vio Of" 90 % of 28 Vo of 81 0. % of 72 5 / 9 of 63 V 8 of 64 7. */t of 63 8. % of 35 %' of 56 % of 48 3 / 4 of 32 Vs of 20 9. % of 49 % of 72 Vis of 84 How many 10. 2 / 5 of 725 are 15. % of 8008 20. V 5 of 18365 11. 2 /s of 891 16. % of 4392 21. Vs of 93208 12. % of 582 17. "As of 2196 22. 5 / 7 of 98098 13. *// of 861 18. 7 T of 4011 23. % of 35172 14. 3 /s of 520 19. 9 / 10 of 7680 24. % of 31738 Remainders, and how to Treat them. 76. l. To divide seven apples equally between two persons, we would divide 6, the greatest number of them that can be so divided without cutting any, and then we would cut the remain- ing apple into two equal parts, and give one part to each person. In like manner a half of 7 is obtained arith- metically by first finding how many there are in 2)7 a half of 6, and adding thereto one half of the re- 3 7s mainder. 2. If 11 apples were to be divided equally among three chil- dren, we would divide 9, the greatest number of them that can be so divided without cutting, into 3 equal lots ; then cutting DIVISION, 81 each of the two remaining apples into thirds, we would put two of the parts with each lot. In like manner the */ 3 of 11 is found arithmetically 3)11 by finding first the % of the greatest number in 11 3% that can be divided by 3 without a remainder, and then adding thereto */ 3 of the remainder. ILLUSTRATIVE EXERCISES. \ Show by the division of two strips of paper of the same length and breadth, or by two lines drawn side by side and of the same length, that 2 / 3 of 1 are equal to the y 3 of 2. 1. Divide 7 sticks of candy equally among 5 children. Illus- trate the actual division by lines upon the slate ; also perform the example arithmetically. 2. Divide 9 pencils among 4 boys ; 7 yards of ribbon among 5 girls. (Illustrate as above.) 3. Divide 11 dollars among 4 persons ; 13 dollars among 3 persons ; 17 dollars among 5 persons. (Illustrate with counters.) 4. Divide 7 feet into 3, 9 inches into 4, 13 yards into 6, 19 feet into 7 equal parts. 5. Divide 16 min. into 5, 21 h. into 6, 39 d. into 9 equal parts. SLATE EXERCISES. These examples may be read thus : find y 3 of 235 ; */ 4 of 457, etc., etc., or how many times 3 in 235. 6. 235-1-3= 14. 4934-3= 22. 5674-8= 30. 4314-8= 7. 4574-4= 15. 7434-6= 23. 3854-6 = 31. 5074-8= 8. 3684-5= 16. 6214-7= 24. 7454-6= 32. 620-4-7= 9. 2794-6 = 17. 349-4-3= 25. 946-4-7= 33. 4704-7= 10. 4634-7= 18. 5734-4= 26. 8534-7= 34. 5834-6= 11. 7494-8= 19. 8314-4= 27. 3454-8= 35. 4974-6 = 12. 8354-9= 20. 7564-5= 28. 7004-9= 36. 3494-5= 13. 4534-8= 21. 4924-5= 29. 6004-9= 37. 4634-4= 82 STANDARD ARITHMETIC. 38. 4321-5-3 = 48. 4567-5-5 = 58. 76543-5-4= 68. 44312-4-7: 39. 5678-4-4= 49. 8765-5-4= 59. 49021-4-5= 69. 57368-5-8: 40. 4566-5-5 = 50. 9463-4-5= 60. 74935-5-6 = 70. 49564-5-9: 41. 8321-5-6= 51. 4407-4-6= 61. 68427-5-7= 71. 87310-5-8= 42. 4720-5-7= 52. 8371-4-7= 62. 54379-5-8= 72. 40302-4-7 = 43. 5008-4-8= 53. 9462-5-9 = 63. 48628-5-9= 73. 50000-4-4: 44. 6000-5-9= 54. 4587-5-8= 64. 34567-5-8= 74. 46738-5-6 = 45. 4725-5-8= 55. 5349-5-9= 65. 50021-4-7= 75. 27493-5-3 = 46. 9613-5-7= 56. 4623-5-8= 66. 38745-5-6= 76. 56843-5-5 = 47. 7895-4-6= 57. 7000-5-7= 67. 74960-5-5= 77. 74219-4-8= Applications. — l. How many pounds of beef can be bought for 1854^, at 9^ a pound ? 2. There are 12 boys on 6 sleds; what part of the boys to each sled ? A regiment of 531 men is transported in 9 cars ; how many men to each car ? What part of the regiment ? 3. A man divides $58424 among his 8 children ; how much does each one get ? What part of the whole is that ? 4. A carpenter cuts a strip of molding 192 inches long into 8 equal pieces. How long is each piece ? Suppose he cuts it into 4 equal pieces; how long will each be ? 5. There are 5280 feet in a mile, how many in y 4 of a mile ? 6. If a boy's hoop measures just 8 feet in circumference (around it), how many times will it revolve (turn round) in a half mile ? 7. If I pay 150 for 3 lead-pencils, what part of the money do I pay for one ? If I pay $15759 for 9 lots, how much do they cost me apiece ? What part of the whole sum does each lot cost ? 8. If a ship sails 1918 miles in two weeks at a uniform rate of speed, what part of the distance does she sail in one week ? How many miles ? What part of the whole distance does she sail in one day ? How many miles ? What part of the whole distance in two days ? How many miles ? DIVISION. 83 Long Division. 77. l. How many times is 37 contained in 9386 ? Here we have a divisor which does not occur as a factor in the multiplication table, hence we construct a table specially for it. Having done this, we proceed with the division exactly as we do with divisors less than 10, except, 1st, that we write down the products and remainders because too large to carry in the mind ; and 2d, that we place the quotient over the dividend that it may be out of the way of the written work which is to follow. We first find in this table Table of Multiples of 37. that 2 times 37 = 74 J hence Process of Division. 1X37= 37 we know that 37 is contained 253 25 / 37 2 X 37= 74 two times in 93, and therefore 37)9386 3X37=111 2 hundred times in 93 hurt- Uoo 4x37=148 dred. We place the 2 in the 1986 5x37=185 order of hundreds (over hun- 1850 6X37=222 dreds' place in the dividend), 136 7X37=259 and subtract the 74 hundred 111 8x37=296 from the 93 hundred, and ob- 25 Rem. 9x37=333 tain the remainder, 19 hun- dred. Now, instead of carrying this to the 8 mentally, we annex the 8 to the 19, and thus obtain 198 tens for the second partial dividend. Again, by referring to our table, we find that 37 is contained in 198 (tens) 5 (tens) times, this we write in the order of tens, and, subtracting 185 from 198, we get 13 (tens) for a remainder. To this we bring down the 6 units and proceed as before. Thus we find that in 9386, 37 is contained 253 times, with a remainder of 25. This can be proved, for 253 times 37 are 9361, and the remainder, 25, being added to 9361, the sum is 9386. Note. — The terms of the several partial dividends that are at the right of the first figure brought down, and the ciphers annexed to the several products, may be omitted in the work, since they have no effect in the result. In the process above, the figures that may be omitted are printed in italics. EXERCISES. 8. 2035 11. 5817 9. 8980 12. 3542 10. 6050 13. 9273 84 STANDARD ARITHMETIC. SLATE Find how many times 37 in 2. 9860 5. 8376 3. 7935 6. 9098 4. 5047 7. 5672 Note 1. — In the following exercises, 14-40, construct a table of multiples for each divisor. These exercises can be carried to any desirable extent. The divisors remaining the same, the same table of multiples will suffice for thousands of ex- amples. It will be well to practice the pupils in this way till they are thoroughly familiar with the process of long division. They will then find little difficulty in obtaining quotient figures without the aid of tables. Note 2. — The table of multiples may be formed by adding the divisor to itself for the second multiple, next by adding the divisor to this sum, and so on, till the tenth multiple is reached. If this be the same as the divisor, with a cipher annexed, the result is a good test of the accuracy of the table. Find how many times 54 in 49 in 64 in 83 in 98 in 14. 872 19. 658 24. 7856 29. 87506 34. 296725 15. 953 20. 9030 25. 4785 30. 84378 35. 875682 16. 428 21. 720 26. 9378 31. 59643 36. 937865 17. 397 22. 692 27. 2704 32. 23232 37. 468728 18. 685 23. 5377 28. 1921 83. 12345 38. 321485 39-40. How many 1 times 98 in 576432? 1694761? In 31 18674930006? Definitions. 78. When the steps of the solution are all written, as in the preceding examples, the process is called Long Division. 79. Any part of a dividend used to obtain a quotient figure is called a Partial Dividend. (It is only a part of the entire dividend. See also partial product.) 80. The use of the multiple tables is convenient when we have to employ the same divisor many times successively, as in the foregoing exercises, but it would involve a great deal of un- necessary labor to construct one for every divisor we may happen DIVISION. 85 to have. Hence it is desirable to learn how to obtain a quotient figure without the aid of such a table. In doing this the learner should ask himself, "What must I multiply the divisor by, so as to obtain a product not greater than the partial dividend, and not so small that the remainder will be equal to or greater than the divisor ?" Note. — If the product is greater than the partial dividend, the term proposed for the quotient is too great, and if the remainder is equal to or greater than the divisor, the proposed term is too small. SLATE EXERCISES. 1-5. 6-10. 11-15. 16-20. 234-^-11 = 5364-31 = 743-4-62 = 3456+51; 548+11= 6354-41 = 6344-72= 2345+51: 754+21 = 8744-41 = 5494-82 = 3856+51: 638-4-21 = 504-j-52= 6384-53= 7432^61: 4974-31 = 970-4-52= 5434-95 = 1579+61: 21-25. 26-30. 31-35. 36-40. 3842-4-71 = 34614-82 = 234-r-19= 6844-69: 65484-71 = 71114-73 = 7654-24= 943+13: 74324-81 = 90004-64= 8014-35 = 976+25 = 9465+81 = 40504-55 = 7434-46= 564+36: 4567-^91 = 60314-46= 2574-58= 310+47= 41-45. 46-50. 51-55. 56-60. 2404-58= 36544-57= 35794-54= 64924-88: 5894-69= 78904-65= 13574-29= 7483+73 = 4324-88= 23454-78= 46824-37= 6294+97= 3454-77= 79374-47= 70384-76 = 73854-68= 6784-59= 24684-38= 49254-89= 4291+51 = 61-65. 66-70. 71-75. 76-80. 4064-23 = 5004-74= 4000-4-32= 4000^87= 7094-34= 400-r-83 = 2000+43= 3000+96= 3054-54= 3004-92= 6000+54= 40004-65= 407+56= 5004-94= 3000+65= 7000-j_44= 8084-65= 800-4-52= 6000+76= 9000+33= 80 STANDARD ARITHMETIC. Find how many times 81. 326 in 1630 82. 251 in 2362 83. 347 in 1829 84. 628 in 2654 85. 592 in 1867 96. 428 in 12415 97. 326 in 24081 98. 234 in .13462 86. 489 in 4375 87. 981 in 982 88. 873 in ^756 89. 784 in 7830 90. 892 in 8000 99. 435 in 15781 100. 1723 in 344680 101. 2938 in 357264 91. 384 in 684 92. 721 in 1223 93. 876 in 1676 94. 988 in 9875 95. 876 in 8759 102. 9321 in 993280 103. 8746 in 785463 104. 5932 in 593175 General Rule for Division. 81. Utile.— 1. Write the divisor at the left of tho dividend with a right curve between them. 2. For the first partial dividend take only as many figures at the left of the given dividend as would, if considered apart from the rest, express a number great enough to contain the divisor. 3. Find the greatest number by which you can multiply the divisor to make a product not greater than this partial dividend, and place it in the quotient. 4. Multiply the divisor by this number, subtract the product from the partial dividend, and to the remainder annex the next figure of the dividend. If the result is equal to or greater than the divisor, it is the second partial dividend, but if less, continue to annex figures from the dividend in their order, placing a cipher in the quotient for each figure brought down, till a partial divi- dend is formed; or, till all the figures of the dividend have been brought down. 5. Proceed with the second partial dividend as with the first, and so on, till all the terms of the dividend have been used. The result will be the quotient sought. 6. If, after the last division, there be a remainder, place it with the divisor underneath, at the right hand of the quotient. Froof.— Multiply the divisor by the quotient, and to the pro- duct add the remainder, if any. The result will be equal to the dividend if the work is correct. Note. — The learner should write each term of the quotient over the last figure of the dividend from which it was obtained. It will save him from some mistakes to which he is liable. DIVISION. 87 Applications.— l. Distribute $13425 equally among 27 sailors. How much will each one receive ? 2. If a locomotive can run 513 miles in 19 hours, how far can it run in one hour ? In two hours ? In 10 hours ? 3. The steamer Suevia makes the trip from New York to Hamburg in 12 days. The distance is 3408 miles. How many miles per day does she make ? How many in 6 days ? 4. How many pounds are there in 352 ounces ? 5. How many days in 3567 hours ? In as many minutes ? 6. How many hours in 4628 minutes ? How many days ? 7. If 20 horses eat 1940 bushels of Oats in a year, how many will one horse eat in the same -time ? 8. If one boy picks a barrel of apples in an hour, what part of the time ought it to take two boys to do the same work ? Three boys ? If one man can dig a ditch in 54 hours, how long will it take 9 men to dig it ? 9. What number multiplied by 23 will give 36087 ? 10. How many times can 27 be subtracted from 62397 ? 11. If the product of two factors is 21015, and one factor is 45, what is the other factor ? 12. When potatoes are 75^ a bushel, how many bushels can I purchase for 675^ ? 13. A quire of paper has 24 sheets. How many quires are there in 5631 sheets ? In 1436 sheets ? 14. What is the price of a barrel of apples, if 36 barrels cost $90 ? 15. At a post-office there are 812 boxes in 14 rows, how many are there in a row ? 16. If you weigh 1476 ounces, how many pounds do you weigh ? How many pounds does your sister weigh, her weight being 133 ounces less than yours ? 17. Bought 897 acres of land for $77142. How much did I pay per acre ? 88 STANDARD ARITHMETIC. When the Divisor has One or more O's at the Right. 82. A boy employed at a toy-shop had to put a large number of marbles into little canvas bags, which were to be sold with the marbles. He put ten marbles into a bag, and when he had thus filled ten bags, he put them into boxes, and ten of these boxes he put into a basket to be taken to the store-room. When the work was done there were gy §s»- li». <&# Sfc*- •Vir Express the number of marbles in figures. ILLUSTRATIVE AND SLATE EXERCISES. 1. a. How many baskets full in the lot of marbles above repre- sented, and what would be left if they were taken away ? How many marbles would remain ? b. How many thousands in 4765, and how many over ? Arithmetical Process. 11000 )4J765 4-765 Rem. c. What do you notice on comparing the figures of the quo- tient and remainder with the figures in the dividend ? 2. a. How many times 2 baskets full in the lot, and how many would remain if two times 2 baskets full were taken away ? b. How many times 2000 in 4765, and how many over ? Arithmetical Process. 2000)4765(2 Or, 21000 )41765 4000 2-765 765 Note. — The quotient figure is the same as if 4 were divided by 2. The figures of the remainder are the same as the 3 right-hand figures of the dividend, which stand for the boxes, bags, and single marbles that would be left if 2 times 2 baskets were taken away. DIVISION. 89 3. a. How many times 3 baskets full in the lot, and what would be left if they were taken away ? b. How many times 3000 in 4765, and how many over ? Arithmetical Process. 3000)4765(1 Or, 3|000)4|765_ 3000 1-1765 Rem. 1765 Eem. Note. — Here it will be noticed that the result is the same as if 4 were divided by 3 and the remainder prefixed to the figures cut off from the dividend. 4. a. How many boxes in the lot, including those in the baskets, and the single boxes represented ? b. How many times 23 boxes in the lot, and what would be left if 2 times 23 boxes were taken away ? c. How many times 2300 in 4765, and how many remaining ? Arithmetical Process. 2300)4765(2 Or, 23i00)47|65(2 4600 46 165 Rem. 165 Rem. Note. — The result is the same as if 47 were divided by 23, and the remainder prefixed to the figures cut off from the dividend. Hence, to shorten the work of division when the lower orders of the divisor are filled with ciphers, we have the following 83. Rule. — Cut off the O's of the divisor, and as many figures at the right of the dividend. Divide as if the parts left were the entire divisor and dividend. The remainder, if any, and the figures cut off from the dividend, form the true remainder. SLATE EXERCISES. 1. 5674-40= 7. 4478^-80= 13. 67834-80= 19. 34541-4-80= 2. 8764-50= 8. 2345-4-60= 14. 45714-70= 20. 26483-4-90= 3. 3934-60= 9. 67894-70= 15. 783514-20= 21. 987654-80= 4. 5844-70= 10. 34564-80= 16. 462284-30= 22. 1234564-70= 6. 7484-80= 11. 74824-90= 17. 571354-40= 23. 7001234-60= 6. 5094-90= 12. 39254-90= 18. 462874-70= 24. 8456794-50= 93 STANDARD ARITHMETIC. Shorter Method of Computing in Long Division. 84. Many rapid accountants dispense with written products in long division. They form the remainder by writing down at once what the several terms of the product lack to make up the partial dividend. Example. — l. How many times 956 in 3681 ? Explanation. — The work at the left exhibits the steps of the 3 operation as already learned. How the written 956)3681 product may be dispensed with is shown in the work 3 2868 at tne "S nt > f° r w ^ich the following wording is a 956)3681 sufficient explanation. 01 o 813 \ 813 Wording. — 18 and 3 are 21 (carry 2); 15, 17, and 1 are 18 (carry 1) ; 27, 28, and 8 are 36. Don't say carry 2, etc., but do it. The numbers in heavy italics, occurring after the word "and," are written while they are being pronounced. 2. How many times is 217 contained in 7507083 ? Explanation.— 217 being contained 3 times in 750, Written Work. we mu itiply, and write down what the product lacks to 34594 185 / 217 make up 750. 217)7507083 Wording.— 21 and 9 are 30, carry 3 ; 3, 6, and 9 are 15, carry 1 ; 6, 7. 997 Annexing the next figure of the dividend, we have 2058 997 for the second partial dividend ; 217 being contained 1053 in 997 four times, we set 4 in the quotient, and proceed ~Tax as before. Wording. — 28 and 9 are 37, carry 3 ; 4, 7, and 2 are 9 ; 8 and 1 are 9. For Practice in the Shorter Method. How many times 3. 72 in 856 6. 56 in 934 9. 333 in 6931 4. 83 in 984 7. Ill in 5935 10. 235 in 9871 5. 87 in 899 8. 222 in 7356 11. 354 in 98763 DIVISION, 91 1. 45600-5-100= 2. 72400-5-300= 3. 456004-500 = 4. 836004-700= 5. 73500-f-900= 6. 47400-5-400= 19. 42764-201 = 20. 5318-5-102= 21. 37254-305 = 22. 4943-5-406 = 23. 37564-507= 34. 87654-5-743 = 35. 94615-5-685 = 36. 34641-5-567= 37. 64925-5-784= 46. 3654701-5-4372= 47. 2043217-7-6482 = 48. 4700031-5-6395= 49. 6127421-5-9362 = SLATE EXERCISES 7. 43000-5-1000= 8. 234000-j-4000= 9. 6450004-6000= 10. 840000-5-8000 = 11. 375000-4-3000= 12. 4687000-5-5000= 24. 35312-^-342 = 25. 44325-4-429 = 26. 73812-5-368 = 27. 44831-^-493 = 28. 34052-4-504= 38. 362874-1926= 39. 400324-1835= 40. 506074-1749 = 41. 483254-1683 = 50. 5432101-4-7408= 51. 4382146-5-8432 = 52. 7040047-5-9069= 53. 2468301-5-7456= 13. 991204- 590= 14. 858004- 780= 15. 2079004- 630= 16. 108004- 270= 17. 2090000-4-3800= 18. 1617000-5-4900= 29. 365212-4-7040= 30. 456721-5-8050= 31. 8439564-9002= 32. 4334214-6302= 33. 3465494-5900= 42. 346819-5-4297= 43. 5437264-7453 = 44. 4925704-6853= 45. 7492564-9469= 54. 7654321-5-6435= 55. 5043062-5-4372= 56. 3489719-5-9348= 57. 71543274-8745 = Self-Testing Problems. Note. — Divide each dividend by all the divisors. The remainder, if any, will, in each example, be divisible by 9. 1. 53146827 4-549 2. 61327548 4-558 3. 1287613534-567 4. 123456789-5-576 5. 9876543214-585 6. 963187452-5-594 7. 7123456894-711 8. 7239186454-729 9. 7913524684-738 10. 356912478-5-747 11. 981762345-5-756 12. 7654321894-765 13. 781965423-5-774 14. 7839934564-783 15. 7923456814-792 16. 8297134564-828 17. 8461235794- 846 18. 8641235974- 864 19. 7090054744- 882 20. 4700495704- 918 21. 3571146364- 936 22. 9876543214- 954 23. 9765483214- 972 24. 981234567-5- 981 92 STANDARD ARITHMETIC. Applications. — l. If a clock ticks 29,347,095 times in a year of 365 days, how many times does it tick in a day ? 2. Divide nine million nine hundred ninety-eight thousand five hundred fifty-seven by seven thousand eight hundred forty-two, and write out the answer in words. 3. The Valley Railroad is 271 miles long, and cost $5,272,305. What was the cost per mile ? 4. If a farmer had 138 acres in wheat, from which he har- vested 3692 bushels, how many bushels did he raise per acre ? 5. A milliner cuts 7 pieces of ribbon, each 10 yards long, into pieces each 27 in. long. How many such pieces has she, and how many and how long are the waste pieces ? 6. In a week a boy gathers 192 bushels of apples ; how many bushels does he average per day ? 7. A farmer raises 1875 bushels of wheat, which he exchangee for flour, at the rate of 5 bushels of wheat for one barrel of flour. How much flour does he receive ? 8. Find how many gallons of milk in 8 cans that hold respect- ively 92, 102, 170, 89, 97, 125, 106, and 56 pints ? 9. Suppose that two cans of equal size together hold 376 pints; how many gallons are there in each ? 10. How many poor families may be supplied from 37 barrels of flour, allowing 28 pounds to each, a barrel of flour weighing 196 pounds ? 11. If you take 86 steps in a minute, how many hours and minutes will it take you to walk 38,270 steps ? 12. A train of 28 cars carries 493,920 pounds of flour in bar- rels, each barrel containing 196 pounds. How many barrels to each car ? 13. In 3 weeks, a certain oil-well is said to have produced 35,000 barrels of oil. How much was that per day ? 14. How many thousand make one million ? DIVISION. 93 Original Problems. Note to the Teacher. — Let the yard-stick and foot-rule be as freely used as the circumstances of the school will allow. If the foot or yard measure does not " come out even," let dimensions be given in inches, but let no account be taken of the frac- tions of an inch at present. No pupil should be allowed to give a problem which he has not solved himself, and the answer to which he docs not know. 1. Suppose yourself to be building a pile of cubic blocks, each measuring one inch, foot, or yard on each side, the pile to be as many inches, feet, or yards, on each side, as you please, and ask to know how high you can build it with a given number of blocks. 2. If you see an oil-cloth or carpet with square figures cover- ing a floor, measure one square, count the number on the side and end of the room, give the facts to the class, and ask them to find how long and wide the floor is. 3. Ask questions similar to these : How many states of the size of Delaware might be made out of the state of Georgia ? How many cities of the population of Albany (N. Y.) might be made of the city of New York ? The members of the class should hunt up the facts for themselves. 4. Give the height of one step of a stair-way, and the distance from the first to the second floor, second to third, etc., in some house just building, and ask how many steps will be needed. 5. State the cost of any number of things, as yards of cloth, horses, etc., etc., and the price of one to find the number. State the cost and the number, and ask the price of each. 6. Construct questions about changing hours to weeks, equal parts to wholes, etc. 7. How many days sail from to for a vessel which runs — miles per day ? How many hours run for a railway train from to , at — miles per hour ? (Find distances from your text-book in geography, and rates of sailing from your friends.) 8. A railway train goes from to in — hours. How many miles an hour ? Note. — The newspapers often suggest interesting problems. 5 94 STANDARD ARITHMETIC. Principles of Division. 85. A convenient number of counters being equally distrib- uted among 6 or 8 members of a class, let the following ques- tions be proposed : 1. If there were twice as many counters equally distributed among the same pupils, how would each one's share be changed? If there were only half as many counters ? 2. If the same number of coun- ters had been distributed among twice as many members of the class, how would the share of each be changed ? If among half as many ? 3. If twice the number of coun- ters had been given to twice as many members of the class, how would the share of each be changed ? If one- half as many had been given to half as many persons? 1. How does it affect the value of a quotient to multiply the divi- dend by 2, by 3, by any number, while the divisor remains unchanged? To divide the dividend by 2, by 3, etc. 2. How does it affect the value of a quotient to multiply the divisor by 2, by 3, by any number, while the dividend remains unchanged ? To divide the divisor by 2, by 3, etc. 3. How does it affect the value of a quotient to multiply divisor and dividend by the same number ? To divide both divisor and dividend by the same number? ORAL EXERCISES 1. How many 13's in 78 ? In 5x78? In % of 78 ? In 9x78? In Vs of 78 ? 2. How many times is 18 contained in 360 ? % of 18 in 360 ? % of 18 in 360 ? 4x18 in 360 ? % of 18 in 360 ? 3. How many times 24 in 96 ? 5x24 in 5x96 ? % of 24 in % of 96? 24x24 in 24x96? 4. Divide 224 by 28 ; 224 by 2x28 ; 224 by % of 28 ; % of 224 by 28 ; 3X224 by 3x28 ; % of 224 by % of 28 ; 13x224 by 13 X28. SLATE EXERCISES How many times 3119 in 1197696 4316 in 1031524 7. 4316 in 1031524^-52 9. 4316 x 718 in 1031524 x 718 2. 24x3119 in 1197696 5. 4316-=-52 in 1031524 8 10 3. 3119 in 1197696-1-24 6. 4316 in 1031524x52 4316x52 in 1031524 4316-J-52 in 1031524-4-52 DIVISION. 95 Division by Composite Numbers. 86. Let the pupil show, by various examples, that division by a composite number (product of two or more factors) may be per- formed by dividing successively by its factors. Thus, that 7,756 63)756(12 9|108 is equivalent to 12 126 126 Divide in both ways 1. 18576 by 48 3. 30375 by 81 5. 391272 by 56 2. 37656 by 72 4. 24678 by 54 6. 629937 by 63 Why the results of the two methods should be the same, and how to deal with remainders when they occur in the division by factors, is shown in the solution of the following example. 7. Divide 59 by 42. Solution with Counters. — In 59 counters there are 29 twos and 1 counter re- maining ; in 29 twos there are 9 sets of 3 twos and 2 twos over ; in 9 sets of 3 twos each there are 1 group of 7 sets and 2 sets of 3 twos remaining; all of which is shown as follows. /?/7/7/?/7///7/?A//7/?/7r//?/^/?/?/7/?/7/7r//7/(/?/?/7/?/7/ Arithmetical Process. Explanation. — As may be seen in the illus 2 3 59 tration, the first divisor is 2 units and the re- «q -t -j mainder is a unit; the second remainder is 2 — groups of 2 each, and the third is 2 larger groups 9 — 2 X < = 4 . . 4 of 3 twos each. The sum of these remainders is 1 — 2 X 3 X 2 = 12 17, the same as that obtained by dividing 59 at P - j Tiy once by 42. Thus we obtain the rule for dividing any number by the factors of composite divisors. 87. Rule, — 1. Divide the dividend by any factor of the divisor, divide the quotient by another factor, and so on, till an entire set of factors has been employed. 2. If remainders occur, multiply each by all the divisors preced- ing the one that produced it. The sum of the products, added to the remainder, if any, resulting from the first division, will be the true remainder. 96 STANDARD ARITHMETIC. Self-Testing Exercises. To be Constructed by the Teacher for Dictation. Addition and Subtraction.— 1. Write any set of numbers, each of which shall be greater than the preceding, as, for example, 83, 237, 250, 592, 728, 851, 9872, 18589. Subtract the first from the second, the second from the third, etc. To the sum of the remainders add the first number. If the work is correct, the sum will equal the last number. Multiplication. — 2. Take any set of numbers the sum of which is 10, 100, 1000, etc., multiply each by any given number, and add together the products. (See examples 7-18, p. 64.) Division. — 3. Take 17 and 19. Annex Jf ciphers to each. Di- vide each number thus formed by the sum of 17 and 19. If the sum of the quotients, including fractions, be expressed by 1, with Jf ciphers annexed, the work is correct. 4. Separate any number expressed by 1 with 1/. ciphers annexed into any two parts, each represented by If figures. Take any two convenient smaller numbers, as 29 and 88. Prefix the 29 to either of the former numbers, and the 88 to the other ; thus, 003276 and 886724, or 88327Q and 006724. Divide both numbers of either pair by 68, that is, the sum of 29 and 88 increased by 1. The test of accuracy is the same as in 3. The last method being somewhat complicated, the following additional example is given. We divide b}' the shorter method for the sake of space. Explanation.- 40168 2 / 10 |, 59831 1OT / 10a ll"Z £ 109 )43,78814 109)65^1686 parts of 100000, and 183_ 1071 prefix 43 to one and 741 906 by the sum of 43+ 59831 10 V 109 2 216 65 + 1- 100000 (The teacher adds the quotients.) 107 Note. — In 3 and 4, other numbers may be substituted for those in italics. CHAPTER VI. MISCELLANEOUS EXAMPLES. Addition, Subtraction, Multiplication, and Division. 1. George Washington was born in 1732, and was 67 years old when he died ; what was the year of his death ? Abraham Lincoln was born 77 years later than Washington ; when was he born ? President Lincoln lived 56 years ; in what year was he killed ? 2. In what number is 244 contained 28 times ? 3. How many strokes does the hammer of a clock make from 1 till 12 o'clock, if it strikes only the hours ? How many in a day ? 4. A man died leaving $5200 to his wife and three children. The widow received $2500, and the children shared the rest equally. How much did each one receive ? 5. A dealer proposes to ship 100000 eggs in boxes containing 40 dozen each. How many boxes will he require ? 6. One hundred and thirty-eight boxes of equal capacity con- tain 76176 eggs. How many dozen eggs in each box ? 7. If I pay 45^ for lead-pencils, at 30 each, how many pencils do I buy ? How many if I pay 50 each ? 8. A drover has $150. How many cows can he buy at $50 each ? $25 each ? How many could he buy at $45 each, and how much would he have left ? 9. A son is born when his father is 33 years old. When the father is 36 years old, how much older is he than the son ? How many times as old ? 98 STANDARD ARITHMETIC. 10. Twenty-four sheets of paper make a quire. How many- quires in 1824 sheets ? In 1 / 2 as many sheets ? In y 4 as many ? In 3 times as many ? In 5 times as many ? 11. How many hours are there in 9480 minutes ? In twice as many minutes ? (Always find your answer in the shortest and most con- venient way.) 12. How many days are there in 14088 hours ? In ten times 14088 hours ? 13. Out of 796 logs 3980 planks were sawed. How many planks were cut from each log, supposing them to have been of equal size ? 14. Four boys agreed to sweep a school-house two weeks for $24, but at the end of the first week, three of them gave it up, and left the remaining boy to complete the work. How much should the last boy receive ? How much each of the others ? 15. The managers of an orphan asylum spend $239 per year for each child. The expenses one year were $7170. How many orphans in the asylum that year ? 16. The manager of a concert sold 534 tickets at $1 apiece, 936 tickets at $2 apiece, and 257 at $3 apiece. He gave out 34 free tickets. The hall cost him $120 rent, and for gas and fuel he paid $19 extra. How much was left after all expenses were paid, including $2100 for the performers ? 17. A congregation intends to build a church, which is to cost $12000. The collections already made are $524, $726, $837, $632, $439. How much is lacking ? 18. Mr. Brown earns $28 while Mr. Black earns $15. How much will Mr. Brown earn while Mr. Black earns $105 ? 19. Express in words the product of the sum and difference of 8765 and 5678. 20. A train of 9 cars has in each car 63 passengers, of whom 4 are children. How many passengers altogether, how many adults, and "how many children ? MISCELLANEOUS EXAMPLES. 99 21. January 4th, paid into savings bank, $14 ; February 1st, paid in $13 ; February 28th, drew out $11 ; March 14, paid in $19 ; March 31st, drew out $25 ; April 24th, paid in $17 ; May 3d, paid in $9 ; May 25th, drew out $15 ; June 1st, paid in $16. How much had I then in bank ? 22. A number of boys in a work-shop earn $7 each per week, and an equal number of younger ones earn $5 each per week. How many boys are there if their wages amount to $132 per week ? Suggestion. — Suppose they work in pairs, an older and a younger boy in a pair, how much would a pair receive ? How many pairs ? 23. How many times can a 5 gal. pail be filled from a cask containing 150 gal. ? How many times from a cistern holding 12 such casks ? 24 casks ? 24. On Tuesday the Opera was attended by 2486 persons ; on Wednesday by 3574 persons. How much more money received on the second day than on the first, at $2 per ticket ? 25. Which is the greater, and by how much, one fifteenth of 645, or one sixteenth of 992 ? 26. If a man takes 92 steps in a minute, how far will he walk in 3 hours if he advances 5 feet in taking 2 steps ? At the same rate, how far would he travel in 2 days of 9 hours each ? 27. A railroad conductor makes two trips every day (except Sunday) from Philadelphia to New York and back. If these cities are 90 miles apart, what distance does he travel in a week ? In a year, if he has two weeks vacation ? 28. How many times will a cart-wheel, 16 feet in circumfer- ence, revolve in going a mile (5280 feet) ? 29. A drover paid $780 for cows and sheep. Of this sum he gave $360 for 9 cows. If a cow cost 8 times as much as a sheep, how many sheep did he buy ? 30. How many feet of telegraph wire are needed to connect two stations with a double line, the stations being 37845 yards apart ? How many poles would be required if set 45 yards apart ? 100 STANDARD ARITHMETIC. 31. A healthy child's pulse beats 78 times a minute. How often does it beat in an hour ? In 4 hours ? In 8 hours ? 32. Find the smallest number that must be subtracted from 9904, to leave a number that can be divided by 173 without re- mainder. 33. There are 35 regiments in an army, averaging 693 men in each; how many men in the army ? 34. If a man earns $3 a day, how many weeks will it take him, working 6 days a week, to earn $567 ? $681 ? 35. A man bought a piece of land for $1564 ; he built a house on it for $642, a barn for $273, and made other improvements costing $148. He then sold it for $3000. How much did he gain ? 36. A railroad 270 miles long has a station every 10 miles; how many stations has it ? (There must be a station at both ends of a road.) 37. What is the length of a railroad that has 18 stations, at an average distance of 17 miles apart ? 38. A family uses 7^ worth of milk a day. What was the cost of the milk used the last 4 months of the year ? (See 27, p. 49.) 39. If a man pays 7^ a year for the use of a dollar, how much does he pay for the use of $5 for one year ? Of $10, $50, $90, $200, $750 ? How much for each for a half year ? 40. If a man pays 6 in his savings bank, Ber- tha $2 3 / 4 , Carl $1%, May $3%, Emma $0.50. How much have they in all ? 74. Mrs. Duncan paid the butcher on Monday $0.60; on Tuesday, $0. 78 ; on Wednesday, %, dol. ; on Thursday, % dol. ; on Friday, % dol.; and on Saturday, $1.10. How much in all ? 75. My new Eeader cost % dol., my Speller % dol., my Arith- metic % dol., and my new slate 15*. How much did I pay for them all ? 76. I bought a pair of shoes for $4%, a hat for $7%, and an umbrella for $1%, and gave the merchant a $20 bill. What change did I get ? 77. In paying an account of $34 4 / 5 , I gave the dealer 4 ten- dollar bills. What change was due me? If I had given him a $50 bill, what change should I have received ? 78. A boy had bought four articles at the grocery worth $2.25. But not having so much money, he handed back one of the arti- cles costing 9 /io of a dollar. How much did he pay for the other three articles ? 79. John buys coffee for 11%, sugar for 70*, cheese for % dollar. He sells the grocer potatoes for $1.50. How much does John have to pay, after deducting the price of the potatoes from his bill ? 114 STANDARD ARITHMETIC. Multiplication of United States Money. Illustrative Example.— l. If the price of rye is $1,355 per bu., what will 65 bushels cost ? Suggestive Questions. — If $1,355 were written Operation, in column 65 times, and an addition of all made, *-, qkk • u where would the separatrix fall ? How many places * lt 6i)b P rlCe P er DU * would there be to the right of the point V What 65 no. of bushels. denominations would they represent ? 6775 0?*, How many mills are there in $1,355 ? In q-. qrj 65 times $1,355 ? Keduce the result to dollars. How many places to the right of the point ? What $88,075 Cost of 65 bu. denominations do they represent ? 94-. Rule.— Multiply as in simple multiplication. The product will be of the same denomination as the lowest order of the mul- tiplicand. If it be in cents or mills reduce to dollars and prefix the dollar mark. SLAT E EXERCISES. 2-7. Multiply $87.74 by 27.— $324,034 by 56.— $1.95 by 18.— $27,341 by 35.— $0,934 by 746.— $0.34 by 61. 8-22. Multiply each of the following sums of money by 8 ; by 37 ; by 368 : $0,398 ; $20.03 ; $47,731 ; $621.70 ; $0,604." 23-37. Find first 13, then 49, then 387 times $34.75; $967.03; $309.08; $7654.60; $190.10; and subtract $99,999 from each product. 38. What is the product of $4.37 by 18x27x15 ? Applications. — 39. How much must the government pay for 327 horses, at $135.50 each ? 40,41. What will 426 sheep cost at $4.87% per head? At $3.62 1 / 2 ? (Express fractions of cents in mills.) 42. What is the amount of a contribution if 157 persons con- tribute each $5 y 4 ? 43-45. What will 476 lb. of tea cost at 65^ a lb. ? At 50^ ? At%dol.? UNITED STA TES MONEY. 115 46. A merchant bought 3 bales of cloth containing respectively 485, 492, and 497 yards, at $1.87 % a yard. Find the cost ? 47. The merchant just spoken of sold the two bales first men- tioned at $2.25, and the last one at $2.75 per yard. What did he receive for the cloth ? 48. A grocer buys 38 doz. loaves of bread on Monday, 35 doz. on Tuesday, and 36 doz. on each remaining day of the week, at 60$ per doz., and sells at 7^ per loaf. How much does he gain ? 49. A grocer bought 9 barrels of cider, each barrel containing 30 gallons, at 12 1 / 2 ( f per gallon. What did the cider cost ? 50. What is the cost of 4 barrels of sugar, weighing 495 lb. each at 7 y 8 # a pound ? 51. What is the cost of 156 acres of land at $20.50 per acre ? 52. Mr. Jones bought of Messrs. Taylor, Kilpatrick and Co., 16 yd. silk, at $2.25 a yd.; 6 yd. gingham, at $0.19 a yd.; 27 yd. linen, at $0.37 a yd.; 39 yd. muslin, at $0.13 a yd. What was the amount -of the bill ? 53. Mr. Taylor bought of Mr. Watts the following implements : a spade for 3 / 4 dol., a rake for % dol., a hoe for % dol., a shovel for 45^', and a lawn-mower for 14% dols. Find the amount. 54. Mr. James went to market with $5. He spent for eggs 45^, for butter 98^, for fruit 25^, and for flour $1.20. How much had he left of the $5 ? 55. A laborer earns $50 and spends $39.75 a month. How much will he have saved at the end of six months ? 56. What are the profits of a concert, if 3427 tickets are sold at $1 y 2 each, and the expenses are $938.40 ? 57. Mr. Mills sold 837 shade-trees for $0.65 each. How much did he receive for the lot ? 58. A drover bought 265 head of cattle at $43.75 a head, and paid $8.35 a head to get them to market, where he sold them at $56.80 a head. How much did he gain by the transaction ? 116 STANDARD ARITHMETIC. Division of United States Money. ILLUSTRATIVE EXAMPLES. 1. If a builder pays $693.68 for lumber at % t ]^f per foot, how many feet does he buy ? 27747 % 5 times. 25 m.)693680 m. 50 193" 175 186 175 118 100 180 175 2. If $693.68 be equally di- vided among 25 men, what will be the exact share of each ? $27.747 5 / 2 5 25)1693.68 50 193 175 186 175 118 100 Note 1. — Every time 2 ! / 2 ^ can be taken from $693.60, a foot of lumber can be bought. Hence, to find the num- ber of feet, we find how many times 2 1 / 2 $ is contained in $693.68. To do this, we change both the 2 x / 2 ^ an( i the $683.68 to mills, and divide the second by the first. A shorter way would be to change both sums to half cents, and divide, but the first way is generally the better. Try both and see which you like best. Note 3. — Thus we find that division of United States money is, first, the process of finding how many times one sum of money is contained in another; and, second, of finding a required part of a given sum. (See also Art. 75, p. 78.) 95. Rule I. — To find how many times one sum is contained in another, change both to the lowest denomination in either, and divide as in simple division. The quotient will be in integers. Rule II — To find a required part of a given sum of money, divide the sum by the number of parts, as in simple division. The place of the separatrix in the quotient will be directly over the separatrix of the dividend. 180 175 5 Note 2.— If $693 be divided into 25 equal parts, there will be $27 in each part, and $18 undivided. $18 = 180 dimes, 180 dimes 4- 6 dimes = 186 dimes. If 186 dimes be divided into 25 equal parts, there will be 7 dimes in each, and 11 dimes will remain undi- vided. To this we add the. 8^, and then proceed as before till we come to a re- mainder of 5 mills. For the present, re- mainders should be disposed of as direct- ed in Art. 76, p. 80. UNITED STATES MONEY, 117 SLATE EXERCISES. 3. Divide $100.50 by 5. Also by 7, by 9, by 65. 4, 5. Four persons are to have equal shares of $4412.88. How- much will each one receive ? How much would each receive if there were 12 persons ? 6-11. Divide $369,009 by 3, by 9.— Also by 8, by 6, by 5, by 15. 12-16. Divide $3759.91 by 19, by 35, by 54, by 67.— Are you here required to find certain parts of $3759.91; or, how many times that sum contains the several divisors ? 17. Which is greater, % or % of $90.50 ? How much ? Find 18. y 2 of $ 97.78. 22. Vic of $8775.36. 26. 13 / 16 of $10391.52. 19. y 4 of $ 363.68. 23. % of $ 436.50. 27. 15 / 27 of $ 8335.71. 20. y 5 of $ 728.15. 24. % of $ 410.41. 28. %^ apiece, how many can be had for $3.60, for $3.78, for $36.00, for $0.90 ? 36. A dozen chairs can be bought for $11.40. How much does 1 chair cost ? 37. A carpenter has 34 men at work ; at the end of the week, 17 of them receive $211.65 wages ; the other 17 receive only $184.45. How much does each one of the two classes of work- men receive per week ? 38. A street commissioner had 347 men at work. At uniform wages the weekly pay roll amounted to $2602.50. What did each man receive per day ? Six workmen received pay for 26 days' work as follows : 39. The carpenter, $55.25. 40. Painter, $52. 41. Bricklayer, $65. 42. Plumber, $68.25. 43. Laborer, $42.25. 44. Plasterer, $63.70. What were the daily wages of each ? 118 STANDARD ARITHMETIC. Miscellaneous Examples. 1. Mr. Jacobs paid me $27.34 ; Mr. Niel, $79.14 ; Mr. French, $34.27; Mr. Myers, $647.79. My expenses on the tour of col- lection were $19.68. When I started out I had $50.75 in my purse. How much money ought I to have had on my return. 2. In 1873 I paid $52.23 taxes ; in 1874, $50.79 ; in 1875, $46.27 ; in 1876, $44.83 ; in 1877, $42.21 ; and in 1878, $40.90. How much in these 6 years ? The repairs on my house in the meantime cost $238. 65. For the first 3 years I received $650 per year rent, for the last 3 years $700 per year. How much did I receive in the 6 years clear of expenses ? 3. A lady had $30. She bought a dress for $9.15, shoes for $3.40, a bonnet for $4.50, and 23 yd. muslin at 25 doz. egg cups, at 900 a doz. ; 3 cups and saucers, at 600 a pair ; 2 doz. fruit jars, at 90 apiece. What is the bill ? 12. If your mother sends you to the grocery with $5 to buy % lb. of tea, at 900 a lb. ; 1 lb. of coffee, at 400 a lb. ; 5 lb. of granulated sugar, at 110 a lb. ; 3 lb. of lump sugar, at 120 a lb. ; 1 small bag of salt, at 90 ; 4 loaves of bread, at 60 a loaf ; 1 peck of apples, at 800 a bu., what change will you receive ? 13. If you are sent with $2 to buy 3 lb. rice, at 100 a lb. ; 20 lb. of flour, at 50 a lb. ; 6 lb. of cheese, at 90 a lb. ; 5 lb. of prunes, at 160 a lb. ; 1 gal. coal-oil, at 80 a quart, will you have money enough ? If not, which article must you omit to keep the sum total within $2 ? 14. Twelve tons of coal cost $75.00, how much is that per ton ? What would 37 tons cost at that rate ? 15. A person sells 5 cows at $55 each, and a yoke of oxen at $125. He agrees to take in payment 80 sheep. How much do the sheep cost him per head ? 16. What is the cost of 39000 feet of planed pine lumber at $40 per thousand feet ? Of 16000 shingles at $2.75 per thou- sand ? 17. Find the cost of 18. Required the cost of 60 pr. overshoes, © $ .65 9 lb. lard, © $.08 15 " boots, © 4.25 15 lb. butter, © .22 17 " gaiters, © 3.85 18 lb. pork, © .06 37 " slippers, © 1.75 20 lb. rice, © .09 230 " mittens, © .34 12 lb. raisins, © .20 2 doz. pr. slippers, © 9.00 4 cans oysters, © .35 2 pr. boots, © 9.50 10 lb. codfish, © .10 120 STANDARD ARITHMETIC. 19. Mr. White bought 6 tubs of butter, containing 58 lb. each, for $80.04. How much did he pay per lb. ? 20. He sold the butter at a profit of \%f a pound. Deduct- ing 7 lb., which he used in his family, how much did he get for it ? 21. When coal is $4% per ton, how many tons can be bought for $238 ? How much would be saved by buying at $4 per ton ? 22. A farmer buys goods amounting to $235.75. He pays in cash $58.25, and agrees to pay the balance in rye, at $1.25 a bushel. How many bushels will be required ? 23. How many pounds of cheese, at 15^ a lb., must be given in exchange for 14 yd. of gingham, at 30^ a yard ? 24. Subtract $37.87 from $237.37 ; from the remainder sub- tract $37.87, and continue subtracting till the remainder is less than the subtrahend. What is the remainder ? Is this the short- est way to find the remainder ? 25. Multiply $4.35 by 2 ; multiply the product by 3 ; multiply the second product by 4 ; the next by 5 ; the next by 6 ; and the next by 7. What is the last product ? A r range each line in column, and add : 26. $13.44, $300, $55.25, $288.39, $19.50, $31.67, $509.07. 27. $67.31, $180.61, $79.03, $152.70, $14,23, $11.12, $50.22. 28. $88.75, $264.16, $44.56, $76.82, $30.50, $72.39, $142.33. 29. $10.13, $7.56, $2.18, $55.44, $11.19, $70.25, $312, $9. 30. $13.33, $72.69, $15,437, $34,805, $125,595, $77,666. 31. Add together all the sums of money given in examples 26 to 30, inclusive. • 32. A store-keeper, who was about to pay some debts, found that he had $37.45 in change and $76 in bank-notes in his money- drawer, $318 in his safe, and $98.36 in his pocket-book. How much had he left after paying 5 bills of $56.10, $38.05, $48.00, $213, and $78.90, respectively ? UNITED STA TES MONEY. 121 33. The treasurer of a street railroad took 400 dimes, 800 quarter-dollars, 23 twenty-cent pieces, 600 half-dollars, 1000 five- cent pieces to be exchanged for $5 bills. How many did he get ? 34. A grocer exchanged bills for small change. How many 50 pieces could he get for $5, $10 ?— How many dimes for $5, $10 ? — How many quarters for $25, $45 ? — How many half-dollars for $37, $54, $96 ? Making Change. 96. l. You buy three pounds of rice at 90 a pound, and hand the grocer in payment a dollar bill ; how does he count the change due you ? Answer. — Giving you the rice he would count that as 27^ ; then placing in your hand successively 3^, 10^, 10^, and 50^, he would count 30, 4.0, 50 $1. This is the most convenient way, and least liable to error. It is similar to the " making up" method in subtraction, which is recommended on page 41. 2. Having only 50 and 100 pieces, how will the change be counted, taking 150 out of $1 ? 3. Having 10, 50, 100 pieces, and $1 bills, how would you make the change for 350 out of $2 ? For 850 out of $5 ? For 75^ out of $10 ? For $1.12 out of $1.50 ? For $6.03 out of four two-dollar bills ? For 670 out of $2 ? For $3.33 out of $5 ? 4. If the merchant has no change except 250, 500, and $1 pieces, how can he make change for $2.75 out of $5 ? 5. A collector presents a bill for $1.90 ; you have only two $1 bills, one 100, and one 50 piece = $2.15. The collector has only large bills and quarter dollars. How can the change be made ? (If you were to give him your $2.15, could he then make the change?) 6. You owe $2.75, but have only three dollars and a quarter. How can change be made, the collector having none less than a half dollar ? 7. If a grocer has only small change, namely, 10, 20, 30, 50, 100, 200, 250, 500 pieces, how can he make change for $2, the goods you have bought costing 160 ? 122 STANDARD ARITHMETIC. 8. If a merchant has only $1, $2, and $5 bills, and l%, *%, % 3 %, s %, % %, •%, 5 % ? 12. If each sixth were divided into two equal parts, what would the new parts be called ? How would you express one or more of them ? If the line is 12 inches long, what is the length of each part f 13. How many of them in % of a line ? In 2 / 3 , % %, % % ? 14. How many twelfths in 2 lines and % ? In 3 3 / 12 ? In 4 5 / 12 ? In7Vi2? Inl0y i2 ? Inll n / 12 ? How many twelfths in 1% 2% 4%, 7 3 / 12 , 9% lines? FRA CTI0N8. 137 Questions upon the Rules in the Margin. 125. Note. — The following questions arc designed to be only suggestive of exercises that may be given. A foot-rule or a yard-stick will afford many others. 1. Into how many parts is the first of these two measures divided by the horizontal line in the middle ? What do you call the parts ? Into how many parts is the measure at the right di- vided by the longest horizontal lines ? What do you call these parts ? Why ? 2. Which is greater, % or y 3 ? How can you tell without seeing or measuring the parts ? 3. What parts of the whole do you get by dividing y 3 into 2 equal parts ? Why ? 4. How many sixths in y 3 , %, 3 / 3 ? What part of y 2 do you get by dividing it into 2 equal parts ? What part of the whole is 1 / 2 of 1 / 2 ? 5. How many parts do you get by dividing each of the fourths into 2 equal parts ? What are these new parts called ? Why ? 6. Which is longer, y 6 or y 8 ? Suppose you could not see, nor measure, would you know which is the greater, 3 / 6 or 3 / 8 ? How ? 7. The sixths in the second measure are di- vided each into 2 equal parts. What is their name ? Why ? 8. Are these twelfths as large as the eighths in the other measure ? 9. What are the smallest parts of the second measure ? How many are there ? 10. Are the smallest parts of the first meas- ure as large as the smallest parts of the second measure ? Can you tell by counting them ? 138 STANDARD ARITHMETIC. ORAL EXERCISES. 1. Explain how it is that y 2 is equal to %. Note. — Divide any whole thing or number into fourths, and show that one half is equal to 2 fourths. 2. Is % equal to %, to 4 / 8 , to 6 / 12 , to % ? State why. 3. Name some other parts equal to y 4 ; also parts equal to y 6 , to y 8 , to y 3 , to y 12 , to % to %, to % to %, to % to %. 4. If you had a line divided into sixths, how could you change the sixths into twelfths ? The twelfths back to sixths ? 5. How many sixths in y 3 , 2 / 3 ? How many eighths in 3 / 4 , */, ? 6. How many twelfths in % % % % ? In % % % % ? 7. How many twenty-fourths in 3 / 8 ? In 5 / 8 ? In 7 / 8 ? How many in % % % % ? How many in % % % ? How much is s. y 2 of % ? 9. V, of y. ? io. % of % ? % of % ? >/, of % ? % of 'A ? V, of'/s? V s ofy ls ? y 3 ofy 4 ? 11. VsOf %? %of y 3 ? %of »/,? Note. — The following representation of a fraction rule will suggest other exer- cises. The figures at the left show how many parts each side is divided into. Tl^T^T! 12. How many wholes and ninths are in 15 / 9 , 2 %, 21 / 9 , 3 %, 47 / 9 ? 13. Which makes the larger parts, dividing an apple into lOths or 12ths ? Which is the greater, y 8 or % of a thing ? % or % ? % or % ? y 4 or % ? 14. Which is the greater, % or % ? % or % ? 73 / 145 or 29 /u 5 ? 5 / 8 or 3 / 8 ? y i8 or 10 / 18 ? Why? 16. Draw two lines of equal length. Divide one into thirds, the other into fourths, and find how many more twelfths there are in y 3 than in y 4 . FRACTIONS. 139 Definitions. 126. A Fraction is one or more of the equal parts of a unit or whole. (27. The unit of the fraction is the unit which is divided. One of the equal parts is a fractional unit. (28. Fractions obtained by the division of the unit into tenths, tenths of tenths or hundredths, etc., are called Decimal Frac- tions. All others are called Common Fractions, to distinguish them from decimals. (29. Common Fractions may be expressed by words, as tivo thirds, or by figures, thus, %, the upper number standing for "two," the number of parts, and the lower one for "thirds," the name of the parts. (See Art. 73, page 77.) (30. The number of parts and the name of the parts are called the terms of the fraction. 131. The term which expresses the number of parts is the numerator (counter or numberer). The term which indicates the name of the parts is the denominator (namcr). Note. — Since the denominator indicates the name of the parts by showing how many parts there are in a unit, it may be treated as a number as well as a name. 132. A simple fraction is one whose terms are both integers, ^ /9> /20> GtC. 133. A proper fraction is one whose numerator is less than the denominator, as 2 / 3 , 3 / 4 , etc. (34. An improper fraction is one whose numerator is equal to or greater than its denominator, as %, %, etc. 135. A mixed number is one which is composed of an integer and a fraction, as 3y 2 , 5 3 / 7 , etc. (36. An integer may be expressed in the form of an improper fraction by writing it as a numerator, with 1 as a denominator. Thus, 5 may be written 5 /j, which is read 5 ones or 5. Slate Work. 67 140 STANDARD ARITHMETIC. Reductions. Changes of Form, not of Value. 137. To reduce integers or mixed numbers to improper fractions, and the contrary. Example.— l. Eeduce 67% to fifths. Analysis. — In 1 there are 6 fifths, hence in 67 there are 5 67 times 5 fifths=335 fifths; 335 fifths* 8 fifths=338 fifths. 335" Note. — The result being the same, we multiply 67 by 5, ^_ as the shortest way of obtaining 67 times 5. 338 Example. — 2. Reduce 19 / 5 to an integer or mixed number. Analysis. — 5 fifths = 1. Hence 19 fifths contain as many units as there are times 5 fifths in 19 fifths = 3 4 / 5 . Slate Wcrk. Suggestion. — For the rules in these cases the pupil may 5)19__ be required to state the processes by which he obtains the 3 4 / 5 results. Note. — In oral exercises the pupil should be required to announce results at once if possible, except when specially directed to give an analysis. Reduce to improper fractions 3. 4. 5. 6. 7. 8. 1% 2% 5% 4% 197 3 31% ?7a 1% 8% • 3% 18% 29% 5% 3% 7% 8% 37% 33% Reduce to integers or mixed numbers 9. 10. 11. 12. 13. 14. Vi "A "/• "A 3 % "A "A % 15 A "A "A 17/ 32/ 53/ 39 % % *% 3 % 6 % 3 % Reduce mixed numbers to improper fractions, and improper fractions to integers or mixed numbers. 15. 28% 6 18. 42 % 21. 100% 24. 1828 / u 27. 487% 16. 31% 3 19. 35 % 22. 375 10 / 23 25. 132 %3 - 28. 723 9 / 10 17. 24 8 / 17 20. 21 % 23. 841 %i 26. 4563 / 15 29. 891 9 / 13 30. How many yards in 35 % of a yard ? FRACTIONS. 141 Reducing to Higher and Lower Terms. Let it be remembered that the value of a fraction is not changed by multiplying or dividing both terms by the same num- ber ; thus — 4x3_12 • 4-s-2_2 6x3~ 18 6-*-2~ 3 For it is clear that 7*\ I I I is equivalent to ^ | 1 1 I 1 1 1 or to % \ i i 1 i i | i m 1 i i 1 , | 1 138. To reduce a fraction to higher terms (greater numerator and denominator). Example.—i. Eeduce % to twelfths. Process. Explanation. — If each fourth of a slip of paper be g x o q divided into three equal parts, the whole slip will contain 4 j v q = T9 times 3 parts, or 12 twelfths, and 3 fourths will contain 3 times 3 parts, or 9 /, 2 . Hence the following Utile. — To reduce a fraction to higher terms, divide the required denominator by the denominator of the given fraction, and multiply both of its terms by the quotient. ORAL EXERCISES. Reduce Reduce . 2- % % % % to 12ths. 7. % % % V, 2 , Vis, to 72da. 3- 1, %, % V* " 8ths. 8. % 7e, % '%, 14 /.8, ". 54ths. *• Yt> V* V„ % " 18ths. 9. % % '/ 9 , % */ n , " 45ths. 5- % %, %, 'A, " 24ths. 10. •/* % %, % »/„, " 48ths. 6- % %, %, Vu. " Wths. 11. % % 'As, "A, "As, " 36ths. Let the first three examples be illustrated by division of lines or folding of paper. 12. Change % % % % % and % to tenths. ' 13. Changed hundredths & % %* 3 / 20 , % %, %, % % »/* U2 STANDARD ARITHMETIC. A 139. To reduce fractions to lower terms (smaller numerator and de- nominator). Note. — In the preceding case (p. 141) we computed the numerical result of dividing any given equal parts of a thing or number into smaller equal parts. In this case, we are to find the result of uniting smaller into larger ones. Example.— l. Reduce % to lower terms. Process * Explanation. — Uniting each 3 twelfths of any object into 1 g|__3/ larger part, we have 4 larger parts (fourths), and in the 9 twelfths Il2 /4 there are 3 of them. Hence 9 /i 2 = 3 /4- Reduce to lower terms 41/ /164 196/ '144 /444 140. Thus we find that fractions are reduced to lower terms by dividing both terms by any common factor. And, that they are reduced to their lowest terms by dividing them, successively, by all the prime factors common to the two ; or, by the continued product of all, the latter being their greatest common factor , and hence their greatest common divisor. 141. When the terms of a fraction are large, or not readily resolved into factors, the following method of finding the greatest common divisor will be convenient. 4. Ileduce 475 / 589 to lowest terms. 2. 12/ /24 64/ In /90 66/ /240 59/ /in 10/ /50 48/ In 15/ fn % 3. 15, /35 6 / 718 12/ /96 72/ /144 480/ /500 15/ fit 31/ /93 34/ /68 % Process. Explanation. — We divide the greater 'number by the less, and the divisor by the 475)589(1 remainder, and so on till there is no re- 475 mainder. The last divisor (19) is theg, c. d. 114)475(4 sought ' Why it is that we can always depend 456 19 on such a process to find the g. c. d. is not 475 25/ 19)114(6 readily understood by the young learner. koq = /3X 214 ^ nc demonstration is therefore reserved for the appendix. No formal rule is necessary. Reduce to lowest terms K 1645/ 17 1363/ 8903/ -,-, 1261/ 10 1989/ B « /1833 '• /1739 »• /13201 »*• /1649 ld ' /2873 ft 1589/ p 8903/ , n 1945/ 10 2813/ i a ™*5 1 «= /2724 o- /10991 10- /3501 12. / 378 3 14. /350I FRACTIONS. 143 Addition of Common Fractions. ORAL EXERCISES. I. Mary takes % of a pie for lunch at school, William takes %, John %, and Henry 2 / 6 . How many sixths do they all take ? Find the sum of 2. % + % = 3. % + % = 4. •/„+ 8 /u= 5 . 4 /s + % +>/ _ *A + % = %0+ VlO= % 2 + Vl2= % + V8 +% = % + Vs = Vi3+ %3= 7n+ Vn= % + % +%= 6. Sarah has 5 / 8 of a yard of ribbon, and Lucy has % ; how many 8ths have they together ? How many yards, and what part of a yard ? Find the sum of 7. 7%+%* 8. 5%+ 3 /,= 9. 6% +3 % = 10. 16 y„+ •/»= 3%+%= 4%+%= 8V„+BVi.= U»/»+%= 5 4 A+ 3 A= 8%+%= 6V„+8»/„= 18"/«+"/i.= II. One piece of cloth contains % of a yard, and another 2 / 3 of a yard. How many yards in the two pieces ? Oral Solution. — 3 / 4 is equal to 9 / 12 , 2 / 3 is equal to 8 / 12 , 9 /i 2 and 8 / 12 together are equal to 17 /i 2 . 17 /i2 = l 5 /i2> hence, 3 / 4 and 2 / 3 of a yard of cloth equal 1 5 / 12 yd. Illustration. — 3 fourths and 2 thirds of a sheet of paper make neither 5 fourths nor 5 thirds, but subdividing both into twelfths we find that they are together equal to 17 / 12 or l 5 /i 2 . 14-2. Fractions to be added together must have a common denominator. Find the sum of i2. v 2 +y 3 = i3. %+«/,= i4. v 3 +7o= i5. 76+7, 2 = V«+7t= 7*+7 8 = 7+7,= ...■ 7s+7*= 7+7 6 = 7s+7«= */*+%= %+7.4= %+%= V 3 +7,= %+ 744? x Vu» 2 7i4> and n {m from 57m* 12. Subtract the sum of 6 / 19 , 3 / 10 , y i0 , "/w, and 15 / 19 , from 7%. 13. Subtract the sum of 1 3 / 100 , 47ioo> 5^/100, 6 n /ioo, and 2y 100 , from 497ioo- 14. Sarah has 76 of a yard of velvet. How much will she have left if she uses 74 of a yard for trimming a dress ? Oral Solution. — 5 / 6 is equal to 10 /i 2 , 3 U 1S equal to 9 / 12 ; 9 /i2 being sub- tracted from 10 /] 2 leaves x / lf . Hence, Sarah would have 1 / l2 of a yard re- maining. 145. That one fraction may be subtracted from another, the two must have a common denominator. 1 4-6, Mule, — 1. Reduce the fractions, if necessary, to fractions having* a common denominator. 2. Subtract the numerator of the subtrahend from the numerator of the minuend, and write the remainder over the common denomi- nator. 3. In the subtraction of mixed numbers, if the fraction in the subtrahend is greater than that in the minuend, take 1 unit from the whole number in the minuend and add it in fractional form to the fraction of the minuend, and then subtract. FRACTIONS. " 147 ORAL AND SLATE EXERCISES. Note. — The oral exercises may be carried as far as the ability of the pupU will permit. 15. 16. 17. 18. id. %-% = .%-% = % - 7s = 1%-% = 3%-%= Va-Vc = 7a-V. 2 = % - %= 27a - % = 2% - 3 A= %-% = i/ _v - /4 /24 — % — Vl2= 4% -'7.8= n -%= /5 — Ao = . Vs— Vl6 = 4/ 5/ — /9 — /18 — 67s - 7.6= 9 /is— % = %-% = /5~~ /25 = % ~~ 716 = < /l2 — /24 = ?%*-%= 73-% = 7,-%.= 9/ 11/ — '10 — /20 — 5 /ll — /22 = 67,.-%= /$— lti = % — %6 = % — %4 = 8%5~ %C = 57,5-%= 20. 21. 22. 23. 478-37 16 = ?v«- -«%= 6% -1% = 5 % -3% = 8%-l%' = 137s- -1%= 24% -2% = 7 /n— 3% 2 = 8V.-1V8 = 25%-2%= 37% 1 -3% = 9% -5% 7 = 24. Subtract $2% from $7%, $6%, $5«%, $9%, $4%. 25. Subtract 1% lb. from 6% 6 , 7%, 9 5 / 16 , ll 7 /^, 5% 2 lb. 26. Subtract 3% qt. from 8%*, 9% 8 , 11%, 7 7 / 30 qt. 27. 2% 6 -l 3 % 8 = 34. 56% -27%= 41. 48 % -7 n / 12 = 28. 3 % -l 1 7i 2 o= 35. 165 : % -39% 2 = 42. 824 % -6 % 2 = 29. 74 % 5 -2 1 % = 36. 283 % -46% 4 = 43. 936 %-±% = 30. 68"/ 20 -9 % = 37. 394 % 5 -53% = 44. 141 % -3 »/ tt s= 31. 15 13 / 18 -9 % = 38. 443 % -31% 6 = 45. 475 8 / S6 -7 % s = 32. 60 15 / 16 -3 % = 39. 527 % 7 -13% 2 = 46. 718 % 2 -8 %,= 33. 25 1 % 6 -2 7 / 12 = 40. 136%- 4% = 47. 248 %-5 % = 48. From 585% 1 +456 1 % 2 +354 1 % take 8%+0%+lYi* 49. Take 7 7 / 8 inches from 16 5 / 6 in., from 18 7 / 23 in., from 21% 7 in., from 23 7 / H in., from 29 7 / 29 in. 50. Take 13 7 / n minutes from 23%, 47 % 9 , 31%, 56 17 / 23 , 35 % minutes. 61. Subtract each number from the next to the right ; add the first number and all the remainders together. (Sec No. l, p. 96.) % *V» .2%, 3%, 3%, 4%, 5% 6 , 6%, 300. 148 STANDARD ARITHMETIC. Applications. — l. A train reached Chicago at 10 o'clock ; it had made the trip from Milwaukee in 3 % hours. At what time did it start from Milwaukee ? 2. A person bought a piece of linen, measuring 60% yards. After the linen was washed it measured only 59% yd. How much had it shrunk ? 3. A grocer had 2 cheeses, one weighing 38% and the other 45 17 / 32 pounds. He sold 7% lb. of each. What was the differ- ence between the weights before and after the sale ? 4. What is the difference in age of two persons, now 73 7 / 12 and 46 % years old respectively ? What will it be 10 years hence ? 5. A thermometer showed at noon 73 %> degrees above zero. At 6 o'clock r. m. it showed 65 % degrees. How much had it fallen ? 6. A grocer received a box of tea weighing 247a pounds. The weight of the box alone was 3 % 6 lb. How much did the tea weigh ? 7. From $120 % the following sums were taken : $6%, $12%, $26 17 / 20 , $20 n / 50 . What was the remainder? 8. Last fall we received 17 % tons of hard coal ; in the spring 1% tons were left. How much had we used during the winter ? 9. A boy said, "If I had $5% more than I have, I should have $21%." How much did he have ? 10. A flag-staff 48 3 / 4 feet high was broken off near the top by a storm, so that it measured only 41% ft. How long was the piece that had been broken off ? 11. A farmer, owning 388% acres of land, bought in addition 251%, and then sold parcels containing, respectively, 84%, 26%, 38 %, 29 n / 12 , 93%, and 84% acres. How much had he left ? 12. A grocer cut 1%, 2%, 2%, 3% 2 , 1%, 5%, 9%, 1%, 2%, %6, kz, 2% pounds from a cheese which weighed 43 7 / 8 lb. How many pounds remained ? 13. One man cuts 3% cords of firewood per day, another 3% cords per day. The first works 7 days, the second 6 days. How much does one cut more than the other ? FRA CTIONS. 149 Multiplication in Common Fractions. Example. — l. Three times 3 / 4 inch are how many inches ? Solution.— 8 /iX% in.= 9 /4 in. =274 in. 2. Three times 2 3 / 4 inches are how many inches ? Analysis.— 3X2% in.= s /iX 11 /4 in.= 8 % in.=8y 4 in. 3. % of 2 are how many ? Analysis.- s / 4 of */ 1 =%=lV*. Note. — After a simple fraction the sign x should be read "of" not "times." 4. 3 / 4 of 3 / 4 are how many ? Analysis—I/, of y_, /w< . % of %= , /jt; «/4 Of %=%,. 5. 3 / 4 of 2% are how many ? . Illustration.— The line of the arrow cuts off 1 / A of the 2 3 / 4 squares represented, leav- ing 3 / 4 of the 2 3 / 4 squares "^ *-> > >;; above it. Analysis.-^ of 2%=% of % ; % of %=% ; % of «%»f»Ai ; % of "U=»/ 1% =2V t .. 6. 3 3 / 4 x2 3 / 4 are how many ? Illustration. — Copy the last diagram four times, omitting the arrow in each line of squares except the fourth. Thus, above the line of the arrow 3 3 / 4 times 2 3 / 4 . squares will be represented, illustrating the following analysis. Analysis.-3 3 /4X2%= 15 / 4 of »/ 4 J V* of %=% J % of %= U U S 15 /4 0f »/4= 166 /l«=10 6 /l«. The parts of the several analyses printed in heavy type are the only parts needed in a written solution. Whence the 14-7. Utile,— Reduce integers and mixed numbers to improper fractions. Multiply the numerators together for the numerator of the product, and the denominators together for the denominator of the product. 150 STANDARD ARITHMETIC. For practice in multiplication of fractions, the pupil may complete the follow- ing tables, and construct similar ones. When the multiplier is a fraction, he should substitute the word "of" for "times" in all oral exercises. 1 2 3 4 5 6 7 1/ /% 1 IX 2 ^ 3 3X A A 1 IX IX 2 1/ 7l A 3/ A 1 1/ /5 /5 3/ /6 1/ v% a 1 X X tT 1 */ 75 % 6/ 7/ /7 /7 2 1 IX IX B< 3 J# 2 ^X 4 2 2% 3 i 5 2% 6 3 7 1 1/ /2 A 7* X X 1/ /7 X 1/ 7± /6 /8 X, 1/ /3 X 1/ /9 A 1/ /8 A &c. 1 A A 1/ /7 X /f 6 /< X X As A* 3/ 735 4/ /35 X As ±7 /35 A 77 A X X A A A 1 ORAL EXERCISES. Note. — Let the oral exercises be carried as far as possible. 1. 2. 3. 4. ?xy 2 = 7^X26 = 6X%= VioX23 5X%= V 6 iX72= 7X%= "/15X6O 4X%= V 25 X36= 9X%= 7 / 12 X29 6xy 4 = V3 8 X47= 3X%= Vs X13 8X%= V 27 X54= 5X 6 A = 9 /nXl5 9xVr= V 56 X63= 4X%= % X25 iox%= V 2 ,X33 = 8X%= % 5 X39 FRACTIONS, 151 Note. — In written work, always cancel factors that are common to both nu- merator and denominator. 5. How much is 6 X % day ? % d. ? % d. ? 6. How much is 9x% lb.? % lb. ? 17 / 16 lb. ? 7. How much is 17X* 1 /* ? $ 7 /io ? * S A ? 8. What is % of 1 hour ? 6 h. ? 9 h. ? 9. What is 7« of 3, 5, 7 feet? 10. 11. 12. 13. V, of %= % X%= "A of %= %iX 17= V, of •/,= /igX /«= % " %= 19x 13 / 25 = V. of %= % xy 5 = 4/ << 3/ /9 /8— . 8 /„X 28= 'A of y 4 = A2X / 6 = 2/ " 5/ — /3 /6 — 51X 14 / 29 = % of •/«= 3 A xVs= Ve " %= /63X 35= % of %= Ve X%= Vi " %= 46 X 33 ^- *A of y,= 7s xv«= 4/ << 5/ _ %X 29= 14. 15. 16. 17. 3%X5= 6 5 / ls X8= 13 / 3 x 1 V 6 = u Ax"A« 4%X6= 7 ? / 8 X9= 19 /5X 19 / 8 = M Ax ,3 / 9 = 5%X7= 9% X8= %x%= 7.x»A= 18. What is Vb of $% ? % of $ V 10 ? % of I % ? Vs of *% ? 19. What is % of % hour ? % of % h. ? % of % h. ? 20. What is y 16 of % lb. ? y i6 of % lb. ? Vie of % lb. ? 21. What is % of % ft. ? % of % ft. ? % of % ft. ? 22. What is % of % week ? % of % wk.? % of % wk. ? 23. What is % of % qt.? % of % qt.? % of 9 / 10 qt. ? 24. Multiply % by 1% ; % by 2% ; % by 3% ; 6 / n by 4 4 / 7 . 26. 3 % X %= 26. % X 7 %= 27. 8 % X %= 28. % X 6 %= 2 2 4xy 5 = %X3%= 7y 4 X%= %X2%= 4 2 / 5 xy 6 = y 3 X5y 2 = 54%X 3 A= 4 A X8%= 3 3 Axy 8 = y 5 X4 2 / 5 = 25 3 / 8 X%= VnX7V 7 = 29. Multiply 3%, 7%, 9%, 6%, 8%, 5%, 4%, each by %, 30. What is % of 3 n / 12 ? of 5%i ? of 6% ? of 10 2 / 3 ? of 9% ? 152 STANDARD ARITHMETIC. (4-8» Since mixed numbers may be reduced to improper fractions, and inte- gers may be expressed in fractional form, the general rule may be applied to all cases of multiplication in which fractions are involved, but when the numbers arc large, the method is awkward, and requires more figures than the following pro- cesses. In business calculations, the rule is seldom followed. Example.— l. Multiply 85 by 17%. 85 1' k Analysis. 3)170 =2x85 5fj% = 2 / 3 x85 595 85 (- x85 1501% = 17 2 / 3 x85 3. Multiply 645% by 328 %. 645% 328% Analysis, 4)1935 3) 656 483% % = 3 Ax 2 / 3 = 3 / 4 x645 r=328x 2 / 3 =328x645 212262 % = 328 3 / 4 x645 2 /. 2. Multiply 58% by 29. 58% jf Analysis. 4)87 =29x3 21% =29x 3 / 4 522 116 1703% =29x58 3 / 4 \- 29x58 Explanation. — Beginning at the right, as in multiplication of integers, we multiply separately the fraction and integer of the multiplicand, Jlrst by the fraction and second by the integer of the multiplier. The work at the left indicates the steps by which we obtain the product of the integers and frac- tions. Multiply 1. 8% by 12 15% by 16 7% by 9 6% by 14 5. 2% by 3 4% by 5 7% by 6 ORAL EXERCISES, 13% by 17% by 18% by 15% by 12 6. 4% by 5 6% by 8 8% 2 by7 3. 6 by 2% 12 by 3% 14 by 6% 16 by 4% 7. 9 by 3% 13 by 6% 18 by 3% 4. 12 by 7% 18 by 4% 21 by 6% 15 by 7% 8. 37 by 2% 16 by 6% 27 by 2% FRACTIONS. 15S SLATE EXERCISES 8y 5 x6%= *%X3%= 5%X2%= 7%X3%= . 5. 6. 7. 8. 6%X4%= 44%X5%= 82%X7%= 27% X 3%= 1 /25X3 %3 — 6^X9%= 9 ,8 /i9X7%= 3. 14%Xll% = i3%x25% = 16%Xl3% = 24%X28% = Find the product of 2%X$30% ; 1%X$25% ; 4 7 / 8 X*19%. Multiply $18% by 15 3 / 7 ; $27 13 / 21 by 38 17 / 20 ; 41% by 20%. 2% 5 X5 3 / 4 X16%=? % of % of % of y 21 of %=? What number is 3% times 35% ? 5% times 6 3 / 8 ? 9. There are 4y 8 lb. in a package. How many pounds in 8y 4 such packages ? 10. Eeduce 2 /3X 3 / 4 X 4 / 5 X 5 /9X 9 /32X 1 %7 to a simple expression. Applications. — Note. — In business it is customary to drop a fraction in the result, if less than l /t$, and to add \f to the integer, if the fraction is equal to or greater than J /20- The pupil should here be required to obtain the exact an- swer, and to write the result in business form under it. 11-38. What is the cost of 37y 2 bushels of potatoes at 75%£ ? 345 yards of cloth at 90 %£ ? 17% feet of oilcloth at 3%^ ? 43 y 2 quarts of cider at 5%jfi ? 387 bushels of oats at 43%$* ? 40% 6 pounds of starch at 18*/*^ ? 345 barrels of apples at $3 2 / 5 ? 47 y 2 sacks of flour at $2 3 / 5 ? 53 y 2 pounds of cheese at 9 3 / 4 ^ ? 17 sacks of rice at $14 3 / 5 ? 64 pecks of beans at 17 3 / 5 ^ ? 27 4 / 5 pecks of potatoes at 23 y 8 # ? 328 3 / 4 pounds of butter at 43 3 /rf ? 17 9 /j6 pounds of bacon at 12 y 2 ^ ? 56y 2 pounds of tea at $ 3 / 4 ? 6 3 / 4 tonsof coal at $3 3 / 4 ? 15% yards of ribbon at 37%^ ? 24% gallons of oil at 85^ ? 3 quarts of nuts at 12 y 8 # 300 bushels of rye at 94y 8 ? 12 y 2 yards of lace at $5% ? 17 3 / 4 pounds of honey at 18%$* ? 4y 2 barrels of herring at $3 y 4 ? 325 pounds of beef at 11 %£ ? 63 bl. of cranberries at $12 y 8 ? 17 3 / 8 bu. of strawberries at $4 5 / 8 ? 37% yards of velvet at $4% ? 48 3 / 4 yards of carpet at $l 3 / 4 ? 154 STANDARD ARITHMETIC. 39. How many square feet in a square, each side of which measures 3 % feet'? 3 % inches? 3% yards? Draw a square, each side measuring 2 3 / 4 inches. Divide each side into fourths of an inch, and draw lines, cutting the square into small ones, each a fourth of an inch square. How many of these small squares ? How many square inches ? 40. Find the number of square inches in a rectangle 5% ft. long and 7% inches wide. 41. What is the cost of 9 tons of coal at $3 %, with cartage at $y 4 per ton? 42. If the salary of an officer is $1700, how much does he receive in % % %, % year ? How much in %, 3 / 4 , 4 / 5 , */„ year ? 43. A laborer earns $20% a week. How much in % year ? In 1% yr. ? (Count 52 weeks.) 44. My neighbor pays $700 rent per year. How much is that per month ? 45. I buy some lots containing respectively %, %, 13 / 25 , 4: / 50 , 6 3 / 10 , 19 13 / 2 o acres. What is the cost of the whole, at $48 per acre ? 46. How many square feet in the surface of a stone slab 2 7 / s feet wide and 4% feet long? 47. Four boxes of hardware, weighing respectively 3y 4 cwt., 4% cwt., 4y 8 cwt., and 4 3 / 10 cwt., cost 16^ freight per cwt. What is the freight on each box, and on the 4 boxes ? Find the sum to be paid for 48. 5y 2 lb. sugar, at 9%0 49. 18 yd. calico, at 9^ 6 3 / 4 lb. coffee, at 32 \j4 20 % yd. alpaca, at 42 %^ 2 4 / 5 lb. rice, at 8 4 /^ 19% yd. shirting, at 170 14 3 / 8 lb. flour, at 4%^ 25 yd. ribbon, at W*fc 3 3 / 4 lb. butter, at 23 %# 10 doz. buttons, at $% 2% lb. cheese, at 11 %# 3 cloaks, at $18% 3% doz. eggs, at 20^ 10 yd. velvet, at $3 3 / 4 2% lb. starch, at 12 %^ jtt% yd. velveteen, at $1 % FRACTIONS. 155 Division in Fractions. Note. — In the first two exercises the square is the unit. In the third exercise the linear inch is the unit. 1. a. How many times 1 in 3, 2%, 2, 1% ? What part of 1 is contained in % ? What part of 2 in */,? In l 1 /, ? etc. b. How many times % in 1 ? In 2 ? In 3 ? How many times 1% in 3 ? 2% in 3 ? ( 5 / 2 *« 6 / 2 ?) e. How many times % in l 1 /, ? In 2 1 /, ? I 1 /, in 2% ? 2. «. How many times 1 in 1%? In 2y 3 ? What part of 1 is in % ? What part of 2 ? What part of 2 . is in %! In iy 3 ? #. How many times y 3 in 1 ? In 2 ? How many times % in 2 ? In 3 ? How many times 1 % in 3 ? 2 y 3 in 3 ? c. How many times y 3 in 5 / 3 ? In 2% ? How many times iy 3 in 2% ? % in 2 2 / 3 ? 1% in 3 ? 1% in 2% ? Inches / k > V / A Halves j \ / V J i Thirds y k / ^ y k J v y k I k. Fourths ) \ y k j \ y V / k y k y k / k I V Sixths Twelfths i k / V i k y \ > k y V / k / k / k. y k / k / k 7 k J V 12 Note. — The points of arrows divide the inches into halves, thirds, etc. By following the shafts downward equivalents in smaller parts are found. 3. a. How many times 2 in 2 3 / 4 ? What part of 3 in %, 1% ? I, How many times y 6 in 2 ? In 2y 8 ? In 3 ? How many times % in 1 ? In 7 / 12 ? In 2 ? How many times % in 3 ? What part of 2 is % % ? What part of 3 is 1% % % ? c. How many times % in % ? 5 / 12 in % ? % in 3 / 4 ? % in % ? What part of % is in %, % ? What part of % in % ? etc. 156 STANDARD ARITHMETIC. ^ I — — — — — — t J ) 4. Divide % by % Illustration. — To find how many times 3 / 5 is contained in 3 / 4 , the 5ths and 4ths are subdivided into parts of the same size by dividing each 5th into four and each 4th into five equal parts, thus making 12 / 20 and 15 / 2 oj whence we see that 3 / 5 is contained in 3 / 4 as many times as 12 is contained in 15 = 15 / 12 = 17 4 . 149. The arithmetical solution is performed on the same prin- ciple as the solution by diagram, the folding of paper, etc., thus : Indicating the division by writing the divisor under the dividend (see Art. 72, § 3), we reduce both fractions to 20ths, 20 being the least common multiple of 4 and 5, and divide the numerator of the dividend by the numerator of the divisor. But, since the common denominator does not affect the quotient ( 16 /i2)> the work, printed in italics, by which it is obtained may be omitted. The written process would then appear as at the right. 3/X5. 15/ — f». 15/ 3/ X4_12/ / 5X f ~ 1 20 3/X5/: 4X/3: 45/ in =iV« =i% 150. Mule. — Invert the divisor and multiply the numerators together for the numerator of the quotient, and the denominators for the denominator of the quotient. Note. — Since integers and mixe'd numbers may be expressed in the form of im- proper fractions, this rule applies to ail cases of Division in Common Fractions. 151. The following analysis leads to an equivalent rule : 3 / 5 is contained in 1, or 5 / 6 , five thirds times, and in 3 / 4 it is contained 3 / 4 of e / q times; hence we have 3/ y 5/. • /4 X /8" .15/ ;=iy* 3/^3/ /4 • /5~ /4" /3- l\%; 152. An inverted fraction shows how many times the fraction is contained in one, and is called the reciprocal of the fraction. Hence, for dividing one fraction by another, we have the liule.— Multiply the reciprocal of the divisor by the dividend. FRACTIONS. 157 153. In dividing a fraction by an integer, a part of the written work required by the rule may be omitted, as follows : Second, when the numerator of the fraction is not divisible by the integer, as in Example.— 2. Divide 5 / 7 by 3. (What part of 3 in 5 / 7 ?) The, process of division, under the rule, would bo /7 * 6 ~ /3 X jt~ l%V but we can multiply the denominator directly by 3 without writing out the second step ; hence, we need to put down only 1 54-. Hence, dividing the numerator or multiplying the de- nominator of a fraction divides the value of the fraction. 155. In dividing a mixed number by an integer, by a fraction, or by a mixed number, the written work may be as follows : Example.— 3. Divide 379% by 6. Explanation. — Six is contained in 379 3 / 4 63 times, with a re- 6)379% mainderof l 3 / 4 . 637/ l 3 / 4 = 7 / 4 ; 7 /4-J-6= 7 / 24 , which, being annexed to 63, gives the entire quotient, 63 7 / 24 . Note. — In the two following examples we multiply both divisor and dividend by the denominator of the divisor in order to get rid of the fraction in the divisor. This makes the division more convenient and does not alter the value of the quotient. The process then becomes the same as in the preceding solution. First, when the numerator of the fraction is divisible by the in- teger, as in Example.— 1. Divide «/ 7 by 3. (What part of 3 in 6 / 7 ?) The process of .division, according to the rule, would be 2 /7-^--/$ x /7-/7 ? but this is equivalent to dividing the numerator directly by the integer, thus, 6 /-3 If 6 ' 4. Divide 379 % by %. %) 379% 3 3_ 6 )1137% 568% 5. Divide 349% by 2%. 2%)349% 9) 1398 % 155 % 158 STANDARD ARITHMETIC. Divide ORAL EXERCISES. 1. 2. 3. 4. V 4 by V. V. by Vn % by •/» V, by V, 6 % by % Ve by V» % by Vi, 'A by % % by %i % by >/ M % by >/ tt 7t by V 33 % by % 'A by % 7s by % V. by »/„ Question. — In which of the above columns do the quotients increase as you descend ? Why do they increase ? In which do they decrease ? Why ? 6. Divide % by % ; % by % ; % by % ; % by »/„ ; % by •/„. 6. » % by >/, ; % by % ; % by %; % by %,; % by.'/, . 7. " % by %,; % by '/„; % by % ; '/„ by % ; % by %,. 8. How many times may y s of a pound be taken from 6 / 7 lb., V, lb., %ib., Vioib.? 9. 10. ll. 12. % +4= V»+2= % + 4= %-«-? 7. -3= 7.3-9 = %+»*? '7,7-3 Ve -7= % -2= 7,1- v= %*4 '/n+3= % -3= «/, 5 -16= % +8, % -6= '713-4 = 7, * 8= % -8: 13. Divide 6% by 4% ; 5% by 2% ; 3 8 / n by iy i3 ; 2% by % 14. Divide 9% by 4y 7 ; 2% by 9% ; 7% by 2% ; 6% by 3%. 15. 2 3 / 4 -r-7= 16. 5 6 / 7 -f-9 = 17. 9 4 / 5 -8= 18. 8%-4-5= 6%-$-8= 3%-*-7= 8%-r-8= 3%-5= 9%-S-5= 7%+$= 7%+6= 4%-s-8= 19. Divide 5 by % ; 7 by % ; 6 by %; 12 by % 20. 18+%= 21. 14-4-% = 22. 39-^%!= 23. 27^% 3 = 36-%= 41-y i9 = 23- 5 / 13 = 2o-y 10 = 84-h%= 35-%= 49-% = 19+%,= 17+%= 71+%= 21+%= 23-% 7 = 24. Divide 23 by 4%; 17 by 5%; 48 by 3%; 59 by 7%. 25. Divide 25 by 3%; 24 by 7%; 39 by 8 2 / 3 ; 72 by 2%. FR ACTIONS. 159 ORAL AND SLATE EXERCISES. Note.— Let the oral . exercises be carried as i ; ar as time and the ability of the pupils will permit. 1. 1-4-6= 2 . 4^3= 3 4+%= 4. 5+%= 5. S-»-l%= 1*8= 7-5-4= 3+y t = 6-7,= 5*-8%= 2-^-3 = 6-r-5 = 8+V,= 8-=-%= 27-=- 7%= 2-f-4= 14^8= 5+%= 4+%= 15-3%= 5-4-7= 8^-5 = 7+%= 10+%= 39+9%= 6. %-*-4= 7. % +y; = 8. 2% +% = 9 . 15%+4%= %+$= % +■/• = 3% +% = 17%-h3%= Ve-3 = % - s /t = 3% +7n= ' 16%-=-5%= %-6 = 14/ . 7/ — /15 • /15 — 5% +%> 297,-9%= 10. How many times $4% in $26% ? In $37% ? In $45%? 11. 3% hours is what part of 19%, 27 2 / 7 , 38% hours ? 12. What part of 18 % 6 , 24%, 30%, 49 % lb. is 6% 6 lb.? 13. %-% = 74-78 = 1/ . 1/ _ h~ /36 — A— ho— 17. %+?/ 14. 10/ _^2/ — /21— /3 — 12/ . 4/ — /24— /6 — 15. 18. % is.%-%; %; 20/ _i_5/ _ /35 • /7 — 35/ _i_5/ — /42 • /6 — 18/ _i_6/ _ /33~ /ll — /72~=~ /18 > /75"^" /15 > /§"*" A j A - ! - As > A8" 5 " A 5 A"*" A 3/ • 5/ . 3/ _;_2/ . 2/ j_4/ . 3/ * 5/ /8 • /6 ) /5 • /5 > /7— /9 5 /7 • /12' A— As— 16- % +% 5 = /«**- A* 33 Ai"^ - A3— %+y. = 25/ • 5/ /36~ /72 — A— A5 = A t" As 1 ^ A— As = /l8"*"7M = /«"*" Ae 5 5/ _i_5/ /36 — /9« 20. IV3-5-V, = 21. ll'/ 3 +l% 6 = 22. 3% -=-% = 23. 5V, 2 -4-4% = 3%-=-% 2 = »%+»% = 2Vu-%= 3% +2% = n+ 5 / w = 45%H-3 2 / 2s = 1%+%= 5% -M% = 4%+%5= 25V 2 +4%= l%+% 3 = 6%-=-3% 6 = !%+%«= 27%-=-2%= 2V 15 +%,= 7% +4%,- 24. Divide % by 2%; % by 3%; % by 1%; % by 2% 5 % by 3%. 25. Divide % by 7%; % by 3% % by 3%. % by 4%; % by 5%; 160 STANDARD ARITHMETIC. Applications. — l. If 4 yards of ribbon cost $%, what will 1 yard cost ? ' Analysis.— If 4 yards cost $ 5 /?, one yard will cost 1 / 4t of $ 5 / 7 = $ 5 /2 8- 2. A farmer sold 5 dozen eggs for § u /&. How much was that per dozen ? 3. In six days a man plows 5 / 14 of a field. At this rate, how mnch does he plow in 1 day ? 4. If a weaver earns $9 3 / 20 per week (6 days), what does she earn per day ? 5. If % lb. of coffee cost $ 3 / 8 , what will 1 lb. cost ? Analysis.— If 4 /s lb. cost $ 3 / 8 ," V# will cost l / A x$ 3 / 8 = $ 3 / 32 , and 5 / 6 lb. will cost 5x 3 / 32 = $ 15 / 32 . 6. If a traveler can make % of his journey in 3 / 7 of a month, what time will the entire journey require ? 7. If % of a bar of gold weighs 5 / 12 lb., what is the weight of the bar ? 8. How many are % of 2% dozen ? % of 2% gross ? 9. A garden, containing 148 4 / 5 □ yd., is to be divided up into beds of 12% □ yd. each. How many such beds will there be ? 10. A quantity of grain, weighing 110 */ 4 cwt., is to be put into bags, each holding 1% cwt. How many bags are required ? 11. How many yards of cloth at $ 3 / 8 a yard can be bought for $2, $5, $7, $9, $4, $23 ? 12. How many bushels of potatoes at $ 2 % 5 per bu. can be bought for $6, $8, $11, $13 ? 13. How many times may l 8 / 9 quarts be drawn from a can holding 17 quarts ? From one holding 22% quarts ? 14. If a laborer can mow a field in 7 14 /i 9 da}^s, how much of it can he mow in 1 day ? 15. Divide $22,500 among the 7 members of a family so that each of the 4 older ones may receive a third more than one of the younger. FRACTIONS. 161 Complex Fractions. If a slip of paper, a melon, or other object, is cut into 3 equal parts, one and a half of these parts are 1% thirds, 2% parts are 2V 8 thirds, etc. Show by the use of objects what is meant by — 3 , -~, etc. o 156. A complex fraction is a fraction having a fraction or mixed number for its numerator and an integer for its denom- inator. Reducing Complex to Simple Fractions. Example. — l. Reduce 2 1 / -£* to a simple fraction. Analysis. — If each fourth of an object be divided Written Work, into 3 equal parts, 4 fourths, or the whole, will con- %!/ 2 1 Ax3 = 7/ tain 4 times 3, or 12, and 2 1 / 3 will contain 2 l / 3 times —r- = ~7~ y Q /l 2 3, or V, of them, hence 2 l / 3 fourths are equal to 7 / 12 . ' Illustration. — The mode of demonstrating the foregoing analysis by means of objects is sufficiently indicated by the analysis. Reduce to simple fractions : . 2% 3V, U 8 3.^ 7 . 5% 2 % 6 'T «•? 7 % 8 ^ 7 - 7 8 - 8 11 3 / 10. A 15 n. ^ 12 »a 2356'/, 13, ~3872~ 14. 483 V n "^286 15. % 989 157. Expressions in which fractions or mixed numbers occur 4 as denominators, as —p-, are not properly fractions, though they 5 /2 are commonly classified as such. They only indicate division (see Art. 72), and are reduced by performing the division indicated, or by multiplying divisor and dividend of the complex expression by the least common multiple of the denominators of their frac- tional parts (see Art. 85). 162 STANDARD ARITHMETIC. SLAT E EXERCISES. Reduce to simple fi actions : ,: ay, 5% 6% 88% R 68% • 3% Z - 7% 3l 3% * 51%, 6 - 97% * 18% A 33% 8 " 9% 9 " 68% la 83'/, The reduction of the following complicated expressions will afford exercises in addition, subtraction, multiplication, and division of fractions. 1- % + (2 % of 2 %) + (% X 2 %) + (6 %»+<%,) 2. 4«/.x6%+-^ 3. ^kr 4- " /8,X 7 2%-l% l 8 % 5 of 73j %+%+% „ fi3/ . 3-!%, _ 1 1/ , 1/ , 1/ '« %-% 6+1% /2%+/3%+/4% «• 1%,X6% of22/wv _l% . o V ^l+ 15 % 3 %-!%+< « of 3 /•> * 12 9 " 3 / 3 • %+% ,„ %+%+"/« . % of 2%-hl "/„ ,> 15%-(4%Xl%) %+%+"/» 7» of 4%-f-9% "• (V,+%,)+8»/» 158. Applications. — To find what part one given number is ot another. Example.— l. What part of 8 is 3 ? Analysis. — 3 is 3 / 8 of 8 because it is 3 of the 8 equal parts into which 8 can be divided. Illustrate by the use of counters. What part of 2. 7 is 5 ? 16 is 12 ? 18 is 15 ? 21 is 14 ? 30 is 20 ? 3. 39 is 26 ? 42 is 28 ? 48 is 36 ? 72 is 48 ? 32 is 24 ? Example.— 4. What part of 5 is % ? Analysis. — 5 = 15 / 3 , and 2 / 3 is 2 / 15 of 15 / :J because it is 2 of the 15 equal parts into which 15 thirds can be divided. Note. — Illustrate with objects. By cutting 5 wholes into thirds, and taking 2 of the fifteen, we see how nearly this problem is like the preceding. FRACTIONS. 163 What part of 5. 4 is % ? 12 is % ? 15 is % ? 18 is % ? 25 is % ? 6. 39 is % ? 16 is % ? 21 is 3 / 7 ? 28 is % ? 36 is 12 / 13 ? Example.— 7. What part of % is % ? Analysis.- V 5 = 12 /i 5 and */ 9 = "/ lBt "/i, is »•/,', of 12 /is because it is 10 of the 12 equal parts into which 12 fifteenths can be divided. Illustrate with objects. What part of . 8. %is 3 / 8 ? %is%? %is%? %isy 7 ? 5 /nisy 5 ? 9. 3% is 1%? 2% is 1%? 4% is 2%? 4% is »!/<| 5 2 / 3 is4y 3 ? 10. 6% is 5%? 5% is 4 3 / 7 ? 11 is 5%? 5 9 / 10 is 2%? r/ 2 is6y 3 ? 159. In problems such as the preceding, the results may be reached by taking the number representing the part for the numerator and the number representing the whole for the de- nominator of a fraction, and reducing as suggested in Art. 157, or, by dividing the former by the latter. Thus : l 2 / 3 X 3 _ 5 2. What part is 5% of 7%? Solution: l. What part of 4 is 1% ? Solution: 4X3 5 3 / 4 Xl2 12 69 7%X12 94 SLATE EXERCISES. What part of 3. 42 is 3 / 7 ? 14 is 4 / 7 ? 30 is % ? 38 is % ? 45 is % ? 4. 48 is %f 33 is % ? 52 is % ? 48 is % ? 55 is n / 12 ? 5. y 9 isy 3 ? 5is2y 5 ? 4y 12 is3y 6 ? i%isiy 7 ? 5y 2 is4y 3 ? 6. 4% is 2 2 / 10 ? 7% is 4%? 8% is 5%? 8% is 7% ? 10 10 / 15 is 5%? 7. 12 is 7%? i2y 3 is ey 3 ? n% is y 5 ? y 5 is y 35 ? y. is % 8. Vs is % ? V* is V„ ? Vit is Vm ? 1 Vn is % 164 STANDARD ARITHMETIC. 160. From a known fractional part of a number to find the number. Example. — l. 6 is % of what number ? Analysis. — If 6 is 3 / 4 of a number, one fourth of the number is x / 3 of 6 = 2, and four fourths is 4 times 2 = 8. Example. — 2. 18 / 19 is 6 / 7 of what number ? Analysis. — If 18 / 19 is 6 / 7 of any number, 1 / 7 of the number is a / 6 of 18 / 19 = 3 / 19 of the number, and 7 / 7 or the entire number is 7 x 3 /j 7i9 = i7i 3. % is % of what number ? 6. % is 11 / 1S of what number? 4. 7 is 3 / 8 of what number ? 7. 5y 8 is y 3 of what number ? 5. 4y 2 is 9 / 10 of what number ? 8. 6 2 / 3 is 5 / 6 of what number ? Aliquot Parts. 161. An aliquot part of a number is any integer or mixed number that will exactly divide it without a remainder. Aliquot Parts of a Dollar. 50* =*y 8 12%*=$% 87%*=*% 33%*=$y 3 m =iyio 75* =•% 25* =$y 4 $ /st^tftB 62%*=$% 20* =$y 5 6%*=l% 8 60* =*% 16%*=$% 5* =$y 80 37%*=$% 162. To find the cost of a number of articles when the price is an aliquot part of a dollar. Example. — l. What will 20 doz. eggs cost @ 25* a doz. ? Analysis.— At $1 a doz., 20 doz. would cost $20, but at 250 (= J / 4 dol.) a doz., 20 doz. will cost l / 4 of $20, or $5. 2. What is the cost of 28 readers @ 25* ? % 50* ? 3. 144 lb. of beef @ 12 l / 8 * ? @ 16%* ? @ 8%* ? 4. 240 lb. of raisins @ 20* ? @ 16%* ? @ 12%*? 5. 48 pr. of socks @ 50* ? @ 33%* ? @ 37 y 8 * ? 6. 160 lb. of sugar @ 5* ? @ 6%* ? @ 8%* ? @ 12%* ? FRA CTIONS. 165 7. 24 pr. of gloves @ 87*/** ? @ W ? @ 62%$* ? 8. 35 handkerchiefs @ 600 ? @ 40^ ? 9. 16 baskets @ 87># ? @ 62 %£ ? @ 87%* ? 10. 12 pr. of slippers @ $1.25 ? @ $1.33 % ? @ $1.16% ? 11. 230 bu. of wheat @ 62 7,* ? @ 75^ ? @ 87 %^ ? 12. 84 chairs @ $1.25 ? @ $1.33% ? @ $1.37% ? 163. To find the number of articles that can be bought for a given sum, when the price of one is an aliquot part of a dollar. 13. At 33 y 3 0, how many lb. of butter can be bought for $7 ? Analysis.— If tt l f 9 f will bv:y 1 lb., $1 will buy 3 lb., and $7 will buy 7 x 3 lb. = 21 lb. Thus we have the convenient Rule.— Multiply the number of articles that can be bought for $1 by the number of dollars. 14. At 16%*, how many books can be bought for $9? At 33%?*? 15. At 25^, how many pr. of cuffs can be bought for $7.50 ? 16. At 33%#, how many handkerchiefs can be bought for $6 ? @ 50^ ? @ 250 ? 17. For $3.25, how many qt. of milk can be bought @ 6%0 ? @50? @8%0? 18. Find the cost of 19. Find the cost of 7 thermometers @ .50 13 printing-presses $2.75 9 thermometers ® .75 5 velocipedes @ $7.75 5 pr. opera-glasses <& $9.12V 2 7 canoes @ $55.75 3 pr. roller-skates @ $1.75 3 1 / 4 bushels potatoes @ .90 72 scroll-saw blades @ .20 16 cwt. rice @ $6.06V 4 13 pr. pincers ® . .35 32 bushels wheat @ $1.16 g / 3 3 magnets © .75 12 sacks salt @ $1.33V 3 18 pocket compasses ® $1.16 2 / 3 9 lb. candy @ .37Va 16 yards silk ribbon @ .87 1 /. 3 qt. ice cream @. .400 3 1 / 2 lb. copper- wire @ .62V. 2 bicycles @ $97.50 1 / 8 lb. fine copper-wire @. $1.87V» 64 lb. wool @ .87Vf 166 STANDARD ARITHMETIC. Miscellaneous Examples. ORAL EXERCISES. 1. Multiply the numerators of % % % 3 / 8 , 3 / 4 , % % 7 / 8 , by 3 ; by 4 ; by 5 ; by 6. How do these multiplications affect the value of the fractions ? Note. — Frequent illustrations should be given with diagrams and counters. 2. Divide the numerators of % % %, %, %, %, %* 8 / 18 , 12 / 32 , by 2. How are the values of the fractions affected by these divisions ? 3. Multiply the denominators of %, %, 5 / 6 , %, 5 / 8 , 7 / 8 , by 4 ; by 5 ; by 10. Are the fractions increased or diminished by these multiplications ? 4. Divide the denominators of 4 / 4 , %, 4 / 12 , 8 / 12 , 8 /m> %4> %2> by 2 ; by 4. How are the values of the fractions affected by dividing their denominators by integers ? 5. State four ways in which the value of a fraction may be changed, and give examples to illustrate each. 6. What fraction is % of 6 % 7 ? Vio> Vis? %• °^ ^ ne same ? 7. By how much is % greater than 1 / 12 ? Find the difference with the aid of slips of paper or parts of other objects. Tell what you do, and state the result. 8. Add together % % % % and % of 60. 9. Which is the greater, % of 40, or % of 30 ? % of 72 or 9 / n of 77 ? (Illustrate with counters.) 10. How many times greater is y 6 than y l5 ? y 6 than %^ ? % than % ? % than % ? 11. Add 3 to each of the terms of 3 / 7 , and tell how much the value expressed is increased or diminished. 12. Invert 3 / 7 thus, 7 / 3 ; add 3 to each term. By how much is the value p,f this fraction increased or diminished ? 13. From 75 subtract 7 /i 5 pf 75, and find 2 / 5 of the remainder. 14. What is the difference between 3 / 5 of % and % of 2 / 3 ? FRACTIONS. 1G7 15. Sixteen is % of what number ? 14 is % of what number ? 16. A train starts from Indianapolis at 7% a.m.; it reaches Cincinnati 6% hours later. At what o'clock does it arrive ? 17. Add together % inches, % foot, and % yard. 18. I bought 16 oranges for 24^. .How much was that per dozen ? 19. Man ordinarily spends y 3 of his time in sleep. How many hours at that rate does he sleep in a fortnight (14 days) ? 20. If I read 2 / 5 of a book in a day, how long at the same rate shall I be in reading the whole of it ? 21. Eight times % of 18=? 9 times % of 16=? 7 times % of 15=? 22. I had $% ; spent $y i0 for ink, $y, for writing-paper, and $y 2 o for pens. How much money had I left ? 23. Eead the following fractions in the order of their value, beginning with the smallest : 7 I^ 7 / 6 , y^, 7 / 8 , 7 / 12 . 24. Eighteen and two fifths yards are cut from a piece of cloth measuring 27 3 / 5 yd. What fraction of the piece remains ? 25. Eive ninths is 2 / 3 of what number ? 7 / 12 is 3 / 7 of what number ? 26. How many copper wires y 30 of an inch in diameter must be laid side by side to cover y 8 inch ? 2 / 3 , 5 / 6 , 9 / 10 inch ? 27. In an orchard y 6 of the trees are apple-trees, y i2 pear, y 9 plum, y 3 peach, and 22 are cherry-trees. How many in all ? 28. Five little girls held a fair for the benefit of the Fresh-Air Fund. After paying out for expenses y l0 of the whole sum re- ceived, they contributed $36 to the Fund. How much did they receive ? 29. Early June peas are 18^ a can at retail. How much do I save per can by buying them at wholesale, $1.90 per doz. ? 30. The sum of two numbers is 2 7 / 8 . One of the numbers is 1% What is the other ? 168 STANDARD ARITHMETIC. 31. Johnny weeded 1 / 7 of his garden on Monday ; % on Tues- day ; y 3 on Wednesday, and in the remaining days of the week finished the task in equal portions. What part did he do each of those days ? 32. Edith earns a cent for each extra % hour she practices her music. Last week she earned $y 4 . How many extra half- hours did she practice ? 33. A woman weaves 6 yards cloth in 2 days, of 12 hours each. What part is the work of 1 hour? If she is paid $% a yd., what are her day's wages ? 34. A painter bought 18% quarts of turpentine at y 6 of a dol- lar per qt. He sold it at y 4 of a dollar per qt. What did it cost him, and what was his profit ? 35. Divide the sum of 2y 3 and S% by their difference. 36. Twelve yards of goods 3 / 4 yd. wide will make me a dress. How many yards will I need of silk that is y g yd. wide ? 37. On the 4th of July Mr. Brown divided % of $4.00 among his children. To the eldest he gave J / 4 of the %, and to each of the others 45^. How many children had he ? 38. John and Will together mow the lawn. John mows y 4 in 1 hour, and Will '/§. How long does it take both to mow it ? 39. If a man can do 5 / 9 of a piece of work in 4 days, in what time will he do the entire job ? Analysis. — If he does 5 / 9 in 4 days, he will do 1 / 9 in 1 / 5 of 4 = 4 / 5 day, and 9 / 9 , or the whole, in 9 x 4 /« = 36 / 6 = 1 1 / 5 days. 40. If 4 /n of an acre of ground yields 40 bushels of tomatoes, how many bushels per acre ? 41. If % of a bushel of Bermuda potatoes costs 90^, what is the price per bushel ? 42. If 4 / 7 of a yard of velvet costs $5, what does 1 yard cost ? 43. Mr. Jackson sold 7 / 13 interest in his shop for $5,600. What was the whole business valued at ? FRACTIONS. 169 44. If a horse can trot % of a mile in 1 minute, in what time, at the same rate, can he trot 1 mile ? % of a mile ? 45. If I had % and 1 / i more tulips in my garden I should have 57. How many have I now ? 46. If a man earns $y 3 per hour, a woman $y 5 , and a boy $y i2 , what do all receive for 1 hour's work ? 47. How many hours must the boy work to receive an hour's wages of a man ? How many the woman ? 48. In what time, working together, can the woman and the boy earn an hour's wages of the man ? 49. What number diminished by % and % of itself leaves a remainder of 30 ? 50. Seven tenths of a certain number less % of it is 15. What is the number ? 51. My age is % of my brother's ; his age is % 2 of father's, who is 72 years old. How old am I ? 52. A plague carried off % of a flock of sheep in one week, % of the remainder the next, and 28 were left. What was the original number of sheep ? 53. A contractor was to receive $60,000 for a building, but forfeited y 40 that amount because it was not finished within the specified time. How much did he lose ? 54. If 2 leaps of a dog are equal to 3 leaps of a hare, how many leaps of the dog are equal to 27 of the hare ? 55. What number is reduced to 64, when % of it are taken away ? SLATE EXERCISES. _ 1. My study measures 14% feet in length, and 12% feet in width. How many square feet in the floor ? 2. A money-bag contains 37 half-dollars, 49 quarter-dollars, 37 twenty-cent pieces, 39 dimes, 63 nickels. How much money in all ? 170 STANDARD ARITHMETIC. 3. An employer pays $95 % to his workmen, each one re- ceiving $13 %. How many are there ? 4. If 56 laborers earn each $y 3 per hour, how much do they all earn in 6 days and 6 hours, reckoning 8 hours to the day ? 5. Twenty-nine and four fifths yards were sold from a piece of cloth measuring 42% yd. What was the value of the re- mainder at $l i y 20 a yd.? 6. A merchant buys 52 % bushels of beans at %\}J % a bu.; 35% bushels of peas at $1% a bu.; 28% bushels of cranberries at $2y 3 a bu. — (1) Find the cost of each item. (2) Find the total cost, and the whole number of bushels. 7. He made a profit of iy 8 on every quart of beans, 2y 3 ^ on every qt. of peas, and 2y 6 ^ on every qt. of cranberries. — (1) Find the profit on each item. (2) Find the total profit. 8. A farm of 276% acres rents for $2013 % What is the rent per acre ? 9. I bought 45 government bonds at $105 1 / 2 , and sold them at $106 7 / 8 . Find the gain. 10. A house-painter earns $3% a day of 10 working hours. One week in which he worked extra time his pay amounted to $25. How many extra hours did he work that week ? 11. What will 7% yards of lace cost at $% per yd. ? At $% ? At $%? At $y 12 ? 12. A person standing exactly under the equator is carried by the rotation of the earth 24,899 miles a day. How many miles is he carried in 1 hour, 2 h., 3 h., 5 h., 6 h., 8 h., 12 h. ? (What part of a day is 1 hour ? Do the work with as few figures as possible.) 13. If 7% lb. coffee cost $2y i0 , what will 11% lb. cost? 10% lb.? 4% lb.? 12% lb.? 14. Mr. A. left by will % of his estate to his wife, % of the remainder to his eldest son, y 3 of what was then left to his eldest daughter, and $20,000 to each of his two other children. What was the value of the estate ? FRACTIONS. 171 15. A grocer bought two tubs of butter, weighing together 70 u / 12 lb. One tub when empty weighed 7y 4 lb., and the other 8y 3 lb. How much butter did he buy ? 16. Find the sum and the difference of (3 3 / 4 -i-5 2 / 5 ) and 17. Find the sum and the difference of (5 4 / 5 x3 6 / 7 ) and (7%X3%). 18. If it takes a workman % of a day to do % of a piece of work, how much of it can he do in 5 / 8 of a day ? How much in 3 / 7 of a day ? 19. If % of an acre of land is sold for $45 3 / 20 , what is the re- mainder worth at double the rate ? 20. If y i5 of a box of merchandise is worth $7%, what is % worth ? 21. A grocer mixes 57 % lb. of tea, at $ 6 / 10 a lb., with 42 1 / 2 lb. of tea at $ 7 / 10 a lb. What is the value of a lb. of the mixture ? 22. A farmer sold 5 / 8 of his wheat at ll 1 /™ a bushel, and re- ceived $796 % for it. How many bushels did he sell, and how many did he have at first ? 23. Mr. Hill, having $500 to pay expenses, made a journey that lasted 6 weeks. On reaching home he had $46% left. — (1) How much did he spend ? (2) What was the average ex- penditure per week ? 24. I bought a house and paid down y 3 of the price, and in one year thereafter I paid % of the price. The two payments amounted to $43,780. What was the price of the house ? 25. A clerk has a monthly income of $75, and spends $54 2 / 5 per month. How much does he save a year ? 26. By how much would he have to diminish his expenses, per month, to save $20 1 / 2 per year more than he now does ? 27. A laborer borrowed from his employer $66 % , agreeing to pay it by having $2 45 / 100 deducted from his wages every week. How many weeks at that rate did it take him to pay his debt ? 172 STANDARD ARITHMETIC. 28. If % of 7 lb. of coffee costs $ 7 / 8 , how many lb. can be bought for $l 23 / 25 ? $2% ? $5% ? 29. What is the sum of the area of 5 fields, containing severally 93 %, 24%, 86 »/ M , 56%, and 89 % 4 acres? 30. What is the cost of 23% lb. flour at $%• ? 15% lb. oat- meal at $% 5 ? 3% lb. raisins at $% 5 ? 17% lb. nails at 4^? 1 doz. fire-shoyels at 12 %£ apiece ? 31. What number multiplied by 3 7 / 8 will give 2 ; what num- ber divided by it will give % ? 32. What number multiplied by % of ll 3 / 4 will produce 1. 33. In a school of 100 pupils, of whom 3 / 5 are boys, 7 boys and 4 girls are absent. What part of the boys are present ? What part of the girls ? 34. One third of the eighth part of what number is equal to 9%? 35. How many cubic feet in a box 4% ft. long, 3% ft. wide, 7% ft. deep. ? (See problems, page 103.) 36. If one faucet empties a cistern in 6 hours, and another in 9 h., in what time will both together empty it ? What part of the contents will the two faucets discharge in 1 hour ? 37. In what time will both empty it if the first begins to run after the second has run for 2 hours ? 38. A can set the type for a certain book in 6 days, B in 8, C in 9, and D in 12 days. In what time can they do it working together ? (What part will they all do in a day ?) ^ 39. How long must a room 4% yards wide be to contain as many square yards in the ceiling as a room 7% yards long and 5 y g yards wide ? 40. I can walk 20 miles in 5 hours, and my friend can do it in 6 hours. Starting at the same time from points 20 miles apart and walking toward each other, how far are we apart in 1 hour, and in what time from starting would we meet ? CHAPTER X. DECIMAL FRACTIONS. 164. The last chapter presented a mode of writing fractions in which the number of parts are indicated by one number and their names by another. This chapter shows how both the num- ber and name of certain fractional parts may be represented by the decimal system. Note. — Exercises on the following diagram are designed to familiarize the pupil with the relations of such parts. Bundles of jackstraws will also serve for illus- tration. 174 STANDARD ARITHMETIC. Illustration. — If a square sheet of paper were ruled into 10 long slips, and each of these were subdivided into 10 small squares, and the small squares into 10 short slips, and each short slip into 10 tiny squares, as shown in the two slips below : 1. How many long slips would there be ? How many small squares? How many small slips ? How many tiny squares f 2. What part of the whole diagram is a long slip? A small square? A small slip? A tiny square ? 3. What part of a long slip is a small square ? A short slip ? A tiny square ? What part of a small square is a tiny square ? etc. ? etc. Note. — The questions given above are only suggestive of exercises designed to make the pupil familiar with decimal parts and their relations. 165. The division of anything into ten equal parts, and the subdivision of these into ten smaller equal parts, and so on, are Decimal Divisions, and the parts are Decimal Parts. Note. — The dime is a decimal part of a dollar, the cent a decimal part of a dime, the mill a decimal part of a cent. 166. A Decimal Fraction is one or more of the decimal parts of a unit. DECIMAL FRACTIONS. 175 Decimals expressed in Figures. 167. The first illustration (page 173) represents 421 sheets of paper, and 2 tenths, 3 hundredths, 4 thousandths, 5 ten thou- sandths of a sheet ; and as each figure of 421 indicates by its place whether it represents units, tens, or hundreds, so the figures 2, 3, 4, and 5 may be made to indicate by their places whether they represent tenths, hundredths, thousandths, or ten-thousandths. But to show that they represent parts and not wholes, that they are decimals not integers, a point, called the decimal point (.), is placed before them, and the number is written thus : 421.2345. 168. Tlie cipher is used in decimal, as in integers, to mark vacant places. Thus, if the two long slips were omitted in the illustration, the number represented would be expressed by 421.0345. If there were no long slips nor small squares it would be written 421.0045, etc., etc. EXERCISES ON DIAGRAM. 1. How many sheets and how many and what parts of a sheet are represented by 4.2? 2.05? 3.82? .35? .23? 1.01? 2.71? .182? .19? 41.41? 3.00? .4321? 10.1? 7.15? 6.01? .101? 17.208? 15.001? 21.0021? .0053? Give first the descriptive names of the parts, as long slips, small squares, etc., then use the proper arithmetical terms, tenths, hundredths, etc., thus: 4 sheets and 2 long slips, or, 4 sheets and 2 tenths of a sheet. 2. Illustrate by diagram, on slate or blackboard, what is meant by .01, by .25, by .35, by 3.7, by 1.3, by 2.004, etc. 3. Is there any difference in value between 6.7 and 6.70 ? Be- tween 3.7 and 3.07 ? Between 5.16 and 50.16 ? Between .81 and .8100 ? (In stating the differences, tell what parts of the diagram are repre- sented in each case.) 4. Tell how many long slips, small squares, etc., must be cut from a sheet of paper to have .357 of a sheet ? To have .5642 ? To have .045? etc. 176 STANDARD ARITHMETIC. Without the aid of words, express in figures the number of sheets and parts of sheets described below, and read, using the proper decimal terms : 1. 207 sheets 7 long slips and 8 tiny squares ; 3 small squares 5 short slips and 5 tiny squares. 2. 10 sheets and 1 tiny square ; 75 sheets and 1 short slip ; 6 sheets and 6 small squares. 3. 17 sheets 7 hundredths and 9 thousandths of a sheet ; 13 sheets and 3 ten thousandths. 4. 24 sheets 3 tenths and 6 thousandths ; 8 thousandths and 7 ten thousandths. Note. — We can represent 1 / 3 , e / 6 , 3 / 7 , or other common fraction of a decimal part, by writing the common fraction after the decimal, thus : .2 1 / 3 is read 2 x / 3 tenths. 5 3 / 7 small squares would be expressed by .05 3 / 7 , which is read 5 3 / 7 hundredths. .6 3 / 7 is 6 tenths and 3 / 7 of a tenth. Without the aid of words, express in figures : 1. 3 sheets 8 % short slips ; 7 sheets 6 y 3 small squares. 2. 13 sheets 8 2 / 7 slips ; 23468 sheets 5 % small squares. 3. 25 sheets 4% hundredths ; 81 sheets 9% thousandths. 4. 86 sheets 5% tenths ; 4000 and 7 3 / 7 ten thousandths. 5. Which has the greatest value, the 1 / 2 in 2 1 / 2 , .2y 2 , or .02y> ? Note. — We may represent entire decimal parts by fractions written in the com- mon fractional form, thus: 3 small squares may be represented by 3 /i o- 6. Tell how many whole sheets, long slips, etc., are repre- sented by the following figures : 1 5 / 10 , 2y i00 , %, 3 8 / 100 o, VlOO, Viooo, 20%* 120y i0 , 99 9 / 100 , 37y 100 , 4 3 / 10 . 7. Write the following fractions in decimal form : 17 / 10 (= 1.7), 131/ 3468/ 2426/ 1769/ 4432/ 1286/ 316/ K 7/ 98 29/ /ioo> /ioo> /ioooj /loot)? /io> /loooj /ioo> ° /m ^° /ioooo> OJ.36/ K46/ 71243/ -\(\£5f 111/ 19/ °* / 100 , O /i000> /lOOOOj 1KJ ^= flOt /iooo> /10« Note. — Any fraction having for a denominator 10, 100, 1000, etc., is properly a decimal fraction, because it represents parts obtained by the division of the unit into tenths, tenths of tenths, etc., etc. But the term decimal is used alone only when there is no denominator expressed. DECIMAL FBACTIONS. . 177 Definitions. 169. A Decimal point, or sign (.), is a period prefixed to a decimal to distinguish it from an integer. 170. A Pure Decimal consists of decimals only. 171. A Mixed Decimal is one that consists of an integer and a decimal. 172. A Complex Decimal is one consisting of a decimal with a common fraction annexed. Decimal Table. 173. The following table will facilitate the learning of the several orders. The correspondence between the names of the places to the right and left of units should be noticed. Table. S5 tf I. Ill , I ill 3ii 9 ff Names. ^ § 'SIS J . .? gj'f §^ £ J» K i^sfl lS J5 f> §^ f r§ v2 22 J ^ 537 290 32 7.03 214 516 Integers. Decimals. Reading Decimals in Terms of the Lowest Order. 1. 2 long slips and 3 small squares, make how many small squares ? 2. 2 tenths and 3 hundredths, make how many hundredths ? 3. 5 long and 4 short slips, make how many short slips ? 178 STANDARD ARITHMETIC. 4. 2 tenths, 3 hundredths, and 4 thousandths, make how many thousandths ? 5. 2 long slips, 3 small squares, 4 short slips, and 5 tiny squares = how many tiny squares ? 6. 2 tenths, 3 hundredths, 4 thousandths, and 5 ten-thou- sandths = how many ten-thousandths ? Hence for reading decimals we have the 1 74-. Rule. — Read the decimal as if it were a whole number, and give it the name of the right hand order. Thus, .3567 is read 3567 ten-thousandths; .169 as 169 thousandths; .354789 as 354789 millionths. ORAL EXERCISES. 1. Eead, .1; .6; .9; .45; 11.4; 13.47; 51.67; 6.15; 8.24; 98.34; 100.1; 345693.71. 2. Eead, 100.73; 27.02; 50.57; 6.67; 41.01; 120.03; 200.01. 3. Eead, 1.111; .567; .004; 75.123; 3.004; 1.012; 6.953. 4. Eead, 92.009; 9.00012; 13.8947; 57.625341; 1.06777893. 5. Eead, pronouncing separately the order of each digit in the fractional parts : 61.43 (61 and 4 tenths, 3 hundredths). 10.9; 738.5423; 4.02; 5.063; 31.02803; 39.417356; 10.1324; 12.11; 26.103; 17.1101; 29.922; 30.87203456; 9.39485762. 6. Eead the following as mixed decimals, that is, the units first then the decimal : (Read 6.12 thus, six and 12 hundredths). 14.013; 6.57; 3.0154; 46; 1044; 9.999; 20.02; 35.04; 46.34256; 50.148735; 83.4283; 87.87328; 7.5983. 7. Eead the following as improper fractions, that is, read integer and fraction together as one number, giving to the whole the name of the lowest decimal order. (Read 7.04 as 704 hundredths.) 18.164; 516.2; 5.005; 29.092; 5.79; 13.579; 1357.9; 1.010; 263.4501; 63.4; 63.04; 63.004; 63.0004; 5.00013. (The last is read, five hundred thousand thirteen hundred thousandths.) DECIMAL FRACTIONS. 179 Suggestion. — Ask yourself whether it is true that '7.04 is equal to 704 hun- dredths. Turn to the illustration, on page 173, and study this out for yourself. How many small squares in 7 sheets of card-board ? How many in 7 sheets and 4 small squares ? 8. If . 04 were written in the form of a common fraction, what would the numerator be ? What the denominator ? Answer like questions with regard to the decimals in exercises 1 and 2. Note 1. — Observe that when the denominator is written, the decimal point and the ciphers preceding the first significant figure are omitted in the numerator : thus, 3/ .03 = 3 /ioo> not ,03 /ioo- ( ,03 /ioo is equivalent to the complex fraction -r^Sf-) Note 2. — Observe, also, that the denominator of a decimal, when written, con- tains as many 0's as there are figures in the decimal. Writing Decimals. For the writing of decimals, the following rule will be found serviceable. Skill is to be obtained only by practice. 175. Rifle. — Place the decimal point, then, after considering how many places are needed to give the last figure of the deci- mal its proper order; write each figure in the order to which it belongs. Example.— Write 375 hundred thousandths. Remembering that hundred thousandths is the fifth decimal order, and observ- ing that 375 contains only three figures, we perceive that two orders must be filled with ciphers, thus : .00375. SLATE EXERCISES. Write in figures : 1. Three and fifteen hundredths ; thirty-one thousandths. 2. One and one thousandth ; twelve and fifteen hundredths. 3. One hundred twenty eight and seventeen thousandths. 4. Seventy-eight ten thousandths ; seven hundredths. 5. Sixty-one hundred thousandths ; ten and one ten thou- sandth. 6. Fifty-four thousand and fifty-four ten thousandths. 7. Five thousand seventy-five millionths. 180 STANDARD ARITHMETIC. Addition of Decimals. 176. Rule.— Write the numbers to be added so that figures of the same order shall stand in the same column. Add as in integers, and place the decimal point in the sum directly under the decimal points in the numbers added. Examples. 1. 3.523 2. .9374 3. .12 4. 5.678934 23.42 13.21 5.2 2.16674 6.006 45.135 134.56 .00374 4.734 1.0006 42.03 17.00003 5. 6.6+77.77+888.888+26. 742+1.2+5.401+.002= 6. 4.1535+.92+12.3472+.006+11.3+2. 00046+9.07= 7. 100.2+59.012+8. +3.1205+69. +63.109+934563.4= 8. 604. 1+. 012+18. 069+9. 232+8. 01+2. 10004+3. 05 = 9. 10.901+12. +43.321986+.79342+4283.4132+6.7= 10. 11. 12. 13. 14. 14.3 +2.348 + 4.56 +17.01 +384.9000 = 15. 9.58 +8.71 +6.54 + .004 + 15.401 = 16. 73.374 + 9.234 + 3.042+ 9.345 + 3.789346= 17. 1.583 + 5.006 + 7.1 + 7.2003 + 100.007384= Test the accuracy of your results. (See note, page 32.) Applications. — l. Add the following sums of money : $28.36, $108.09, $27.50, $1.30, $38,742, $387,655, $998,999, $3.27. 2. Six marble blocks weigh respectively 5.73 cwt., 4.834 cwt., 7.938 cwt., 6.4 cwt., 15 cwt., and 387.1 cwt. Find the total weight. 3. A train on the Pennsylvania R. R. ran 56.3 miles in the first hour, 62.34 miles in the second, 59.247 in the third, 60.7304 in the fourth. How many miles altogether ? 4. A draper bought 2 pieces of buckskin, each containing 56.34 yards ; 2 pieces of rep, each containing 96.05 yards; and 1 piece of broadcloth, containing 27.2 yards. Find the number of yards in the 5 pieces. DECIMAL FRACTIONS. 181 Subtraction of Decimals. 177. Rule, — Write the subtrahend under the minuend, so that figures of the same order shall stand in the same column. Subtract, as in integers, and place the decimal point in the remainder directly under the decimal points of the minuend and subtrahend. 10. Examples.— l. 94.324 2. 73.6 3. 5.4 4. 9.7 5. 6.01 7.86 • 19.79 4.38 6.543 3.4 6. 7384.02 7. 9.004 8. 3.28764 9. 15.60003004 56.934 7.2043 1.00009 .794569376 1.01 11. 4.003 12. 15. 13. 70. 14. 50009. .09 2.006 6.3785 16.7345 5.0009 15. 8.452-3.1052= 21. 73845.009-1.23456 = 16. 92.8245—9.86543= 22. 9384.708-2.3457= 17. .0052— .0041 = 23. 342.5703-.1994= 18. 3.004— .0097= 24. 6534.70045-3.7634= 19. 121.12-8.943= ' 25. 897.309—3.1073= 20. 423.4567382—413.05 = 26. 328.00019—6.0004= Applications. — l. From a lot containing 10,000 □ yards, 437.296 □ yds. are sold. How large is the remaining part ? (The sign a is used for the word " square.") 2. From 17.256 tons of coal 5.625 tons were nsed. How much was left ? 3. Mr. Smith's property amounted to $47,300.75 when he died. Accounts, to the amount of $340.95, were presented and paid. How much was left to the heirs ? 4. The French meter is 39.37079 inches. How much longer than a yard is the meter ? 5. Find the difference in height of two flag-staffs, the one measuring 38.75 ft., the other 53.9 ft. 6. Find the difference between .57 and .7; between eight hundred fifty-two ten-thousandths and 1. 182 STANDARD ARITHMETIC. Multiplication of Decimals. Example.— l. Multiply .75 by 3. (Find 3 times .75.) Process. Analysis. — 3 times 5 hundredths = 15 hundredths = 1 tenth ^5 and 5 hundredths; 3 times *l tenths = 21 tenths; 21 tenths + 1 3 tenth = 22 tenths = 2 units and 2 tenths. Repeat the analysis, using the terms small squares, long slips, 2.25 and sheets, respectively, for hundredths, tenths, and units. The Multiplier a Decimal. Example. — 2. Multiply .75 by .3. (Find 3 tenths of 75 hundredths.) Analysis. — 3 tenths of 5 hundredths =15 thousandths = 1 Process. hundredth and 5 thousandths ; 3 tenths of 7 tenths = 21 hun- # 75 dredths; 21 hundredths + 1 hundredth = 22 hundredths = 2 o tenths and 2 hundredths. — '— Repeat the analysis, using the descriptive terms short slips, etc. .225 Thus, 3 / 10 of 5 small squares = 15 short slips = 1 small square and 5 short slips, etc. Example. — 3. Multiply .75 by .03. (Find 3 hundredths of .75.) Process. Analysis. — 3 hundredths of 5 hundredths = 15 ten thou- -,- sandths (see diagram, page 174) ; 15 ten thousandths (tiny squares) = 1 thousandth and 5 ten thousandths ; 3 hundredths of 7 tenths •0<3 = 21 thousandths ; 21 thousandths + 1 thousandth = 22 thou- . 0225 sandths = 2 hundredths and 2 thousandths. Repeat the analysis, using the descriptive terms tiny squares, etc. 178. Thus, we find that if the order of the multiplier is units, the order of the product is the same as that of the multiplicand. If the multiplier is tenths, the order of tbe product is one degree lower ; if it is hundredths, the order of the product is two degrees lower, etc. 179. Hence, in the product of two decimals there are as many decimal places as there are in the multiplicand, plus the number of decimal places in the multiplier. 180. Rule. — Multiply as in whole numbers, and from the right of the product point off as many figures for decimals as there are decimal figures in the multiplier and multiplicand to- gether. If there be not so many figures in the product, supply the deficiency by prefixing ciphers. DECIMAL FRACTIONS. 183 ORAL EXERCISES. 4. 5. 6. 7. 8. .2x3= 14x.6 = .42x3 = .2x.3 = .2x.003 = .4x6= 13 x. 5 = .26x5= .3x.6= .9x.005 = .5x5= 15 x. 4= .31x7= .4x.5= .5x.008= .7x2 = 17x.3 = .02x9= .5x.2= .7x.004= .8x4= 16 x. 2= .63x8: .6x.4= .3 x .007= SLATE EXERCISES. Note. — It is well for pupils to accustom themselves to estimate results; for instance, if it is required to multiply 5.65 by 7.001, they should be able to say at a glance that the product will be about 39, that is, a little more than 5 */i times 7. 9. 10. 11. 12. .3x.093= .5 x .934= .52 x .213= 1.5 x. 3= .8x.075 = .7x.825= .73 x. 332= 2.4 x. 5 = .5 x .084= .9x.738= .84 x. 252= 1.5 x. 7= .9x.063= .6x.225= .95 x. 163= 3.2 x. 6= .7x.052= .3x.367= .62 x. 421 = 1.6 x. 9= 13. 736.045 x. 843 18. .0009 x. 0543 23. 84.008 x 1000.4 14. 93 x .0067 19. 9.00134x8.004 24. 258.01 x 3030.1 15. 4.709 x .7635 20. .195 x. 00027 25. .98x36.0007 16. 84.008 x 100.001 21. .1825 x 18.24 26. .357x88345.4 17. 17827.032 x 8.754 22. .75 x .30052 27. 28.601 X 3.425 28. Multiply .004: .71; .70014; 1.04 b} ' .0091. 29. " .05 ; .17; .999; .7534 by . 0008. 30. 1000; 100; .001; .64; .01 by 2.847. Applications. — l. My age is 1.075 times my brother's ; if he is 30, how old am I ? If he is 25, how old am 1 ? 2. What is the area of a lot which is 9.34 yd. wide and 48.5 yd. deep ? (How many square yards in it ?) 3. Find the area of a field .876 miles by .0056 miles ? 4. At $5.87 per acre, what is the rent of a farm of 47.9 acres ? 5. If I buy 2 cwt. 66 lb. sugar at $13.09 per cwt., and sell it at $.12 per lb., what do I gain or lose on the whole ? 184 STANDARD ARITHMETIC. Division of Decimals. Example. — l. How many times .18 in 54 ? p rocess< Analysis. — In 54 units there are 5400 hundredths, and 18 1 . _ hundredths are contained 300 times in 5400 hundredths. - — - — - Illustration. — 18 of the small paper squares represented on oOU. page 174 can be taken 300 times from 54 sheets. 2. How many times .18 in 5.4 ? p Analysis. — In 5.4 there are 54 tenths = 540 hundredths. ' In 540 hundredths, 18 hundredths is contained 30 times. I— '- — Illustration. — Show that 18 small squares are contained 30 30. times in 5 sheets 4 long slips. 3. How many times .18 in .54? Ans., 3. 4. How many times .018 in 54 ? In 5.4 ? In .54 ? 1 8 1 Let it be observed that in every case the dividend must be reduced to an order at least as low as that of the divisor. Evidently, if we are to ascertain how many times 18 short slips there are in any number of sheets, long slips, or small squares, we must first ascertain how many short slips there are. Hence, in division of decimals, there must always be as many decimal places in the dividend as in the divisor. 5. How many times 1.08 in .05778 ? Process, Explanation. — Beginning with tenths, we count off as . 035 many decimal places in the dividend as there are in the di- 1 ao\ AKiiywo visor, and separate them from the places to the right by a '* ' short vertical line. This marks the point below which no in- 54:0 teger can be obtained in the quotient (no quantity can be 378 contained any whole number of times in a quantity less than 094. itself). Here also the decimal places must begin, for, though . one tenth of the divisor be not contained in the next partial 540 dividend, the place must be marked by a cipher in order that 54 } figures of lower orders may have their proper places. (82. Rule — 1. Annex ciphers to the dividend, if necessary, till the right hand order is the same as that of the right hand figure of the divisor. 2. Divide as in simple division. Place the decimal point imme- diately before the quotient figure that is obtained from the order of the dividend next lower than the lowest order of the divisor. Note. — There must always be as many decimal places in the quotient as there are in the dividend more than in the divisor. DECIMAL FRACTIONS. 185 ORAL EXERCISES. 1. 2. 3. 4. .4-f-8= 2-f-.5 = .64-4-.08= .164-. 8= .64-5 = 5-4-.8= .494-.07= .14-4-.7= .8-4-4= 4-4-.6 = .364-. 03= .084-.4= .24-2= 5-s-.7= .844-.04= ,09-h. 3 = SLAT E EXERCISES. 1. 7.32-T-6 6. .964-32 11. 14-.0037 16. 10004-.09 2. 123-^-6 7. .16-k4 12. 104-.001 17. .0045-4-9 3. 127-f-6 8. .58-^-3.1 13. .5^-1000 18. .03-S-1.004 4. 4-^-. 008 9. .63684-8 14. 45.984-10 19. .03754-.03 5. 4.5-S-67.8 -25 10. l-r-,0025 15. 10004-.5 20. 79864-3.75 21. 1.6875 j 26. 789.7-r ■1000 31. 604.56^-1000 22. 134.25-; -7.5 27. 1.6875- 4-6.75 32. 1220.6744-19 23. .045 6 -f- .04 28. 2.0005- 4-7.24 33. 144.6955-4-8.5 24, 733.264 -33 29. 128.1754-7.5 34. 12.345 -4-.00015 25. 1139-4-9250 30. 7.024-r -2.0005 35. 15.63386-4-4.367 36. 549.90254-2.345 37. 994.8015-h22.33 38. 600.26234-66.77 39. 7.006652-4-1.234 40. 1220.6744-64.246 41. .00134094-.583 42. .0000026-*-. 004 43. 5941.86234-66.77 44. 37.873565-4-8.765 45. .0897688 4- ..0202 46. 245.86776454-405 47. 20.34407403-i-.21 48. 12345.432 14-11 1.11 49. 1.33709774^-. Ill 11 50. 72.01440072-4-8.0008 Applications. — l. The circumference of a circle is 3.14 times the length of the diameter. Find the diameter of a circle whose circumference is 51.339 yd. ? 2. The area of a rectangle is 3414.012 □ yd., its width is 125.7 yd. What is its length ? 3. 56.325 cwt. of certain goods cost $49.45335; what is the cost of 1 cwt. ? Of 1 pound ? 4. 36.35 yd. of cloth cost $117.95; what does 1 yd. cost at the same rate ? 186 STANDARD ARITHMETIC. Reducing Common Fractions to Decimals and Decimals to Common Fractions. Exercises on Diagram, page 11 U. Express in decimals and also in lowest terms of common frac- tions the parts of the diagram 1. In 2, 3, 4, etc., long slips. 2. In 8, 25, 32, 20, 75 small squares. 3. In 2, 8, 14, 25, 125, 175 short slips. 4. In 8, 16, 32, 125, 1875, 625, 3125 tiny squares. 183. Changing Decimals to Common Fractions. — 5. Express .6 in the lowest terms of a common fraction. Process. — . 6 = 6 / 10 = 3 / 5 . 6. Express .4, .8, .16, .72, .75, .375, .875, .4375, .04, .0016 in lowest terms of common fractions. Note. — The learner will be able to write out his own rule for the foregoing process. 7. Express in integers and common fractions 1.2, 15.25, 8.6. 8. Express .4% in the lowest terms of a common fraction. Process.—. 4% = -^ = % = 7 / 15 . (See Art. 156.) 9. In like manner find the equivalents of .3%, .23%, .7 9 /i 6 , ,324y 7 in common fractions. 184. Changing Common Fractions to Decimals. — Any frac- tional part of an object must contain a like part of the decimal divisions of the object. Thus, 1 / 2 the diagram, page 174, contains 1 / 2 of ten long slips = 5 long slips = .5 ; 50 small squares = 50 hundredths, etc., etc. a / 4 of the diagram contains 1 / 4t of the decimal divisions, as : 2 1 / 2 long slips = .2 1 / 2 ; or, 25 small squares =.25 of the diagram. 3 / 8 of the diagram contains 3 / 8 of the long slips. 3 / 8 of 10 long slips ss 3 3 / 4 long slips = .3 3 / 4 . 3 / 8 of 100 small squares = 37 1 / i small squares = .37 V2 I an ^ 3 / 8 of 1000 short slips = 375 short slips = .375 of the diagram. DECIMAL FEAGTIGNS. 187 185. Hence, to convert a common into a decimal fraction, we take such part of the decimal divisions of the unit as is indicated by the common fraction. 10. Find decimals equivalent to the common fractions, %, 3 / 5 , 3/ 9/ 11/ 12/ 21/ 8/ 27/ 101/ 333/ /8> /16> /32> /25> /32> /125? /64> /125> /625« 11. Write in integers and decimals equivalents for 3y 2 , 563%, 5%, 7%, 9%, 16%. 12. Find equivalents for % %, %, %, %, 5 / 12 , % 4 , 16 / 2l , *%, 3 7s, 7 5 / 6 , 8%, 4 2 / 7 , 81 3 / 14 , 9% in pure or mixed decimals. Suggestion. — The question should be raised here, why it is that in Examples 10 and 11 all the common fractions are exactly reducible to decimals, while those in 12 are not. Thus the learner may discover for himself the condition under which exact decimal results are possible. 186. Bule,—1. To reduce common fractions to decimals, annex ciphers to the numerator of the common fraction, divide by the denominator. Continue the process till the division is complete, or until the result is sufficiently exact. 2. Point off as many decimal places in the quotient as there are decimal ciphers annexed to the numerator of the common fraction. If there he not so many places, ciphers must be pre- fixed to the significant figures to supply the deficiency. Note. — The further the division is carried, the more exact is the result. In most cases sufficient accuracy is reached in the third or fourth place of decimals. Repetends. — In the process of division, if a remainder is re- peated, the figures of the quotient will be repeated in the same order as after its first occurrence. 187. A figure or set of figures thus repeated is called a Repeating or Circulating Decimal, or simply a Repetend. 188. The sign of a repetend is a dot (•) written over the re- peating figure, or a dot over the first and last figure, if it contains more than one. Note.— At this point the pupil needs to learn no more of this subject than how to indicate a repetend when it occurs, and that he may discontinue the work of divis- ion on the first recurrence of any particular remainder. (See Appendix.) Examples. — 1-7. Reduce the following common fractions and indicate the repetends : % % %, % %* 8 / 15 , %. 188 STANDARD ARITHMETIC. SLATE EXERCISES. Express equivalents in pure and mixed decimals : 1. 9.7%, 7.7%, 1.6%* 5. .0000% X .9%. 2. $28%, $17.07%, .053 31 / 32 . 6. 10.111% X .033. 3. 15 % , 9.60%, 105. 00%. 7. 5.009 X .08%. ool/ 17 1 / 4 ' 4V? 7 ' 2 °- 03/5 * 8 ' 108V4 X % of 9 ^' 9. Find the sum of % and .54; the difference of % and .54; the product of % and .54; the quotient of % divided by .54. Find the sums of 10. 4%, 524.2%, 6.2%, 7, and .573%. 11. 3% miles, 5% miles, 4.7 miles, 7.11 miles, and 99.9% miles. 12. 4.79 lb., 9% lb., 10 9 / 20 lb., 38.59% lb., 141.1 lb. 13. .125 rod, .1875 rod, % rod, .5% 6 rod, 1.8% rod. 14. 1927.96 1 % 5 acres, .00% 5 a., 50.267 a., 1.709 a. Find the differences between 15. 1.79% and .777%; 11.111% and 11.110% 6 . 16. 1.001% and 10.100%; 7.9753% and 6.428104%. 17. What number divided by 1.25 will give the product 11 X 1.1X.001% ? 18. What was paid for 100 bbls. flour, each 196 lb., at $6.66% per 100 lb. ? For 100 bbls. pork, each 200 lb., at $.08% a pound ? 19. How many wagon loads in a freight car containing 2% 6 tons sheet copper, 3.75 tons sheet lead, '57s tons sheet iron, 7.9375 tons tin plate, 1% tons being a wagon load ? 20. From a sheet of lead weighing 1560.625 lb., circular discs were cut, weighing, respective^, 13% lb., 17% lb., 98.875 lb., 59.625 lb., 137% 6 lb., 122% 2 lb., 121% lb. What was the weight of the remnants (scraps) ? DECIMAL FRACTIONS. 189 189. To find cost when number and price per hundred or thou- sand are given. Per C is used for per hundred and per M for per thousand. (See page 18.) Example. — l. What is the cost of 480 lemons at $3.60 a hundred? Written Work. $3.60 4.80 28800 1440 $17.2800 Written Work. $7.35 17.3 Explanation. — In 480 there are 4 hundred and 80 hundredths of a hundred ; therefore we find 4 and 80 hundredths times the price of 1 hundred. Note. — Ciphers at the right of a multiplicand or multiplier may be omitted in computation, in- asmuch as they do not affect the value of the result. Hence the work may stand as at the right. $3.6 4^ 288 144 $17.28 2205 5145 735 $127,155 2. What must be paid for 17300 bricks at $7.35 per M ? Explanation. — 17300 = 17.3 thousand; hence, to find the cost, at $7.35 per thousand, we multiply $7.35 by 17.3. 3. Find the cost of 7854 railroad ties at $95.50 a thousand. 4. Find the cost of 1478 feet of lumber at $45 per M. 5. Mr. Smith bought 50000 shingles at 70^ a bundle of 250, and 38750 ft. of pine flooring at $18. 75 a thousand. What did they cost ? 6. We need 45350 bricks; the price being $6.90 a thousand, how much will they cost ? 7. Mr. Wick bought 280 melons at $7.40 a hundred. What did they cost him ? 8. Find the cost of 2750 laths at 45^ per C ; of 1950 pick- ets at $12 per M. 9. What is the cost of 1500 ft. of copper wire at $2.85 per hun- dred yards ? 10. How much will the steel rails necessary to lay one mile of road cost at the rate of $49.30 for 100 ft. of rail ? (5280 ft. = 1 mile.) 9 13 190 STANDARD ARITHMETIC. Mule.— Find the number of hundreds by pointing off two fig- ures, and of thousands by pointing off three figures, on the right of the given number (representing the quantity), and by this mul- tiply the price per hundred or thousand, as the case may be. 190. To find the cost when the number of pounds and the price per ton (2000 lb.) are given. Example. — l. What will a load of hay weighing Written U/n-I/ ° 2Y2 w 2386 P ounds cost at l19 ' 75 P er ton ? — Explanation. — There are 2 thousand and 386 thousandths of a l.lyo thousand pounds in the load, and one half as many, or 1.193, 19.75 times 2000 pounds, or tons. Hence the value of the hay is 1.193 5955 times $19.75, the price of 1 ton. 8351 11-14. Find the cost of 10737 3500 lb. of hay at $16 a ton. 1193 4835 lb. of salt at $25 a ton. 23.56175 9350 lb. of silver ore at $43.50 a ton. ' 380 lb. of straw at $9 a ton. 15. Find total freight charges on machinery shipped from New York to Buffalo in the following quantities, @ 7 / 8

on every dollar collected. How much did he get ? 28. In a city of 240768 inhabitants, it was found that .125 of the number could not read, and only .875 of those able to read could write. How many were there who could not read ? Who could not write ? 29. How much must be paid for the use of $750 per year at $5% a hundred ? For % % % % % %, % year? (Express results in decimals.) 30. The use of $750 cost me $37.50 a year. What did I pay per hundred ? 31. The use of $1200 for 10 years cost Mr. Lund $630. What did the use of $100 cost him per year ? DECIMAL FRACTIONS. 193 32. Mr. Smith paid $45 a year at $4.50 per hundred for the use of a certain sum. What was that sum ? 33. Mr. Cain borrowed a sum of money at $3.25 a hundred per year, and in 5 years paid $162.50 for the use of it. How great a sum was it ? 34. Find the cost of 6 gal. 3 qt. vinegar, at $. 125 a gal. (3 qt. = what part of a gal. ?) 35. Find the cost of 16 gross 6 doz. lead-pencils, at 55^ a doz. (A gross is 12 dozen.) 36. Find the cost of IS 1 /* yd. ribbon, at $.2325 a yd. 37. Find the cost of 6.25 doz. cabbage-heads at 3s. Dinner- Plates " £.50 u 5 14 5 Candlesticks " .#5 u itf " 5 Pitchers 4i .75 a M " 1 •*/# <&>2. (7w/?s «?i doz. Knives Rec'd Pay't, " 4.75 Edward Willis. 50 196 STANDARD ARITHMETIC. 3. 1884. Cleveland, Jan. Mk. J. P. KlNGSLEY, BoH of Adam Johnson. 11, 188 5. Nov. u it U u 8 tc 5 7 a 9 yd. Cashmere % %.75 X U v d - Velvet " Jf - 50 6 yd. Lawn " .1*"% 1*1 2 yd. Silesia " JO %yd. Cashmere " .65 Kec'd Pay't, Adam Johnson, i?y W. Wright. $ 4. Knoxville, Tenn., Aug. 15, 1885. Me. George Cttetiss, 1885. In aceH with James Akden and Company. July # « a tt u u u u 6 u it July i5 a a a a Dr. r«r " .40 % Rec'd Pay't by note at 30 d., Wm. Powees. 198 STANDARD ARITHMETIC. Rule paper in proper form, and make out bills for the fol- lowing transactions : 7. Mrs. Cole bought of E. P. Dale, of Boston, Feb. 5, 1884, 2 cans of String Beans, @ 100; % bu. Potatoes, @ $1.00; 2 lb. Tea, @ 600 ; Feb. 9, 1 lb. Crackers, 200 ; 1 doz. Eggs, 320 ; 7 lb. Graham Flour, @ 40 ; Feb. 16, 3 cans Tomatoes, @ 120 ; 4 lb. Prunes, @ 160 ; 1 doz. Oranges, 500. Receipt the bill as clerk for Mr. Dale. 8. Charles Martin bought of Joseph A. Snow, of Pittsburg, Pa., Feb. 2, 1884, 2% lb. Mutton Chops, @ 220; % pk. Ap- ples, @ 400 ; 6 lb. Beef, @ 200 ; Feb. 6, % pk. Sweet Potatoes, @ 300 ; 2 bunches Lettuce, @ 120 ; 2 qt. Turnips, @ 50 ; Chicken, 4% lb., @ 200; Feb. 9, 2 lb. Steak, @ 250; % pk. Apples, @ 700; 1 qt. Onions, 100; Feb. 16, 7% lb. Beef, @ 200 ; 2 cans of Peas, @ 180 ; % doz. Oranges, @ 500 ; Feb. 20, 9Vt lb. Ham, @ 180; Feb. 26, 2% lb. Lamb Chops, @ 220; 1 doz. Oranges, 500. Mr. Snow had bought of Mr. Martin 3 pt. of Cream, @ 120 a qt., daily through the month. Make out a receipted bill, using Bill 4 (page 196) as a model. 9. Alfred E. Robie bought of John Turner, of New Haven, Conn., April 2, 1885, 2 1 / 2 lb. Sausage, @ 140; % doz. Lemons, @ 250; 2 lb. Dried Apples, @ 100; Apr. 4, 3y 4 lb. Veal Chops, @ 200 ; Apr. 9, % pk. Spinach, @ 700 ; 2 lb. Mutton, @ 140 ; Apr. 14, x / f pk. Apples, @ 700 ; 2 qt. Sweet Potatoes, @ 100 ; Apr. 18, 6 3 / 4 lb. Beef, @ 200 ; y 2 doz. Bananas, @ 400 ; 2 doz. Pickles, @ 70 ; 2 qt. Bermuda Onions, @ 200 ; Apr. 23, 3V 2 lb. Steak, @ 220 ; Apr. 28, 2 lb. Rhubarb, @ 100 ; 3 bunches Radishes, @ 70. 10. Mrs. James Bird bought of John Burns, of New Orleans, La., the following articles : Feb. 17, 1883, % doz. Linen Nap- kins, @ $1.75; 2*/ 4 doz. Damask Towels, @ $4.50; 3 Bath Towels, @ $2.40 a doz.; Feb. 21, 1883, 2 Table-cloths, @ $5.50; 1 Piano-cover, @ $5.00; 7 yd. Cambric, @ $.12y 2 ; 2 pr. Lace Curtains, @ $2.50 a pair. DECIMAL FRACTIONS. 199 11. Kobert M. Miles bought of Lane & Bowers, of Philadel- phia, Pa., Nov. 21, 1885, 1 Suit for $28 ; 3 Shirts, @ $1.25 ; 1 pr» Shoes, $5.50 ; 6 pr. Socks, @ 35^ ; 1 Umbrella, $2.50 ; 2 pr. Gloves, @ $1. 75 ; and 4 pr. Cuffs, @ 35^. Payment was made by note at 3 months. 12. Mr. George Ross bought of Kobert James, of Albany, N. Y., on Mar. 13, 1881, 60 yd. Brussels Carpet, @ $.85 ; 40 yd. Moquette Carpet, @ $1.55 ; 35 yd. Canton Matting, @ $.55 ; 3 Curtain-poles, @ $4.50 ; 3 pr. Nottingham Lace Curtains, @ $5.50. 13. Albert Halsted, in Qjc with George Eeese : Aug. 7, 1881, 1% days' work, @ $3.25 ; 44 ft. Pine Lumber, @t 1.06%; 1 lb. Nails, $.07; work on Bookcases as per contract, $13.00; 65 ft. Pine Lumber, @ $.06%; % lb. Nails, @ $.07. Cr. by cash, $5.00. 14. Mr. Robert Holden, of Brooklyn, New York, bo't of Stan- ley, White & Co., of New York city/ Mar. 11, 1884, 3 doz. 8 in. Thermometers on polished walnut, @ $10 ; 1% doz, 8 in. Parlor Thermometers, @ $4 apiece ; 5 doz. tin-case Thermometers, @ $5 ; 9 Aneroid Barometers, @ $5 ; 15 pr. Opera-glasses, @ $4.25 ; 3 Microscopes, @ $15 ; 1 large first-class Microscope, $350 ; 2 Amateur Photographic Cameras, @ $25. Paid by note at 3 mo. 15. Mrs. H. R. Otis bo't of Richard Hayes, June 11, 1880, 1 pr. Ladies' Kid Button Shoes, $6; June 13, 2 pr. Ladies' Patent- Leather Oxford Ties, @ $4.50 ; 1 pr. Misses' Kid Button Shoes, $3.50 ; 2 pr. Infants' Black Kid Button Shoes, soft soles, @ $.45 ; 1 pr. Child's Pebble Spring-heel Button Shoes, $2. 16. James R. Baldwin bo't of Robert Price, Dec. 19, 1884, 1 copy "Little Men," $1.35; 1 "Modern Explorers," $10; 1 "Three Vassar Girls in South America," $1.30; 1 "Rose in Bloom" and "Eight Cousins," $2 ; 13 vol. Shakespeare, @ $1 ; 3 vol. "Diamond Edition Poetry," @ $.90; 5 vol. "Companion Edition Poetry," @ $1.25 ; 6 vol. Hawthorne, @ $1.3o ; 1 vol. "Sports and Pastimes for American Boys," $1.25. 200 STANDARD ARITHMETIC. Suggestions for Original Problems. 1. Pupils will find suggestions for original problems in the Miscellaneous Exercises ; or, it may be required that they con- struct problems of their own after models dictated by the teacher. 2. Having obtained reliable information from parents and others in regard to prices, trade customs, etc., they can make out bills, and furnish items for bills to be made by the class. 3. They may draw diagrams showing the forms and dimen- sions of lots to be fenced, dictate the kinds of fences to be built, prices of boards, posts, labor, nails, etc., and require the whole cost. They may give, in like manner, the information necessary to reckon the cost of digging cellars, building walls, laying board, stone, or brick walks, etc., etc. Pupils may often obtain from each other such information as may be needed. 4. Let illustrations, like the one on page 174, be required, showing .33, 1.27, etc., etc., of given squares. 5. Let pupils obtain where they can, the data necessary to enable them to calculate the cost of papering, carpeting, plaster- ing, the schoolroom. 6. Pupils who have a little constructive skill may make paper boxes, and require their classmates to calculate their contents — how many quarts of blackberries or vinegar they will contain, etc. 7. Try the experiment of ascertaining the height of some tall tree or steeple, by measuring the length of its shadow, and the length of the shadow cast at the same moment by a stick or post, the length of which above ground can be easily measured. 8. Give the dimensions of a pile or load of wood, and ask, How many cords ? or of a wood-shed, and ask, How many cords can be piled in it ? or the length of a pile of wood, and ask how high it must be to contain some required number of cords. 9. Give the dimensions of a box containing a gross of such crayons as are used at the blackboard, and ask the length and width of a case which will exactly contain a gross of such boxes. CHAPTER XI. MEASURES. (99. The length, breadth, and height of objects are their dimensions. A line has only one dimension — length. A surface has two dimensions — length and breadth. A solid or space has three dimensions — length, breadth, and height or thickness. Measures of Extension. 200. Measures used to ascertain how long a line is, or in calculating the size (extent) of a surface or solid, are called Measures of Extension. These are the Linear, Square, and Cubic Measures. Linear or Line Measure. 201. In measuring length or distance, linear or line measure is used. The standard unit is the yard. Table. 12 Inches (in.) = 1 Foot (ft.). 3 Feet = 1 Yard (yd.). 16V 2 Feet ) «--,-, (or5V 2 yard s) r = 1Rod ( rd -)- 320 Rods = 1 Mile (mi.). Equivalents. 1 mile = 320 rods = 1760 yards = 5280 feet = 63360 inches. Notes. — 1. For measuring cloth the yard is divided into halves, fourths, eighths, and sixteenths. In the United States custom-houses it is divided decimally. 2. A Furlong = 1 / 8 mile. — The rod is also called a Pole or Perch. 3. A Pace is variously estimated from 3 to 3.3 feet. 4. A Line = ] L inch. 202 STANDARD ARITHMETIC. 202. The mile given in the table is the mile used in land measurements. Its length is fixed by law, and is called the statute mile. It is thus distinguished from the geographical mile of the following table, used on shipboard and at sea. Table. 6 Feet = 1 Fathom. 120 Fathoms s= 1 Cable Length. 1.15 + Common Miles = 1 Geographical or Nautical Mile. 3 Geographical Miles ) . _ « M ~ ox f = 1 League (at sea). or 3.45 + Statute " ) & J A Knot corresponds to one geographical or nautical mile, and is used to esti- mate the speed of vessels at sea. Note. — In the absence of a more exact instrument the hand was formerly used as a measure. From this we have the Palm (breadth of four fingers) = about 3 inches ; the Hand (the breadth of palm and thumb, used in measuring the height of horses at the shoulder) = 4 inches ; the Span (the distance between the tips of the thumb and the little finger, when the hand is extended against a flat surface) = about 9 inches, or 1 / 4 of a yard. ORAL EXERCISES. How many 1. Feet in 3%, 4%, 7%, 4.4, 11%, 33% yd.? 2. Feet in 25, 16, 30, 39, 14% in. ? 3. Yards in 1% 1? 2%, 5, 8%i rods.? 4. Rods in % %, % mi. ; in 121, 49% yd. ? 5. Inches in 1%, 6%, 3%, 5%, 7% 2 ft.? 6. Feet in 2%, 3%, 10%, 6% fathoms? Surveyors' Measure. 203. Gunter's Chain, used in measuring roads and the bound- ary lines of land, is 4 rods (= 66 ft.) in length. It has 100 links, each 7.92 inches long. Table. 7.92 Inches = 1 Link (li.). 100 Links m 1 Chain (ch.). 80 Chains = 1 Mile (mi.). MEASURES. 203 5 V a yards. 1 CI Square or Surface Measure. 2 04-. There is no measure which is di- rectly applied to a surface to find its extent. Even if there were such a measure, it would be difficult to apply it. Suppose, for instance, that we wished to ascertain how many square yards there arc in a plot of ground 5 1 / 2 yards long and 5 l f t yards wide. If we had a square- yard measure we might perhaps mark off 25 square yards and the fractions of a yard, as in the diagram. But it would be much easier to measure the length and breadth with a yard- stick, and then compute the number of square yards in the surface. 205. The square inch, foot, yard, rod, and mile are derived from corresponding linear measure. Table. 144 sq. Inches = 1 sq. Foot. 3074 sq. Yards = 1 sq. Rod. 9 sq. Feet = 1 sq. Yard. 160 sq. Rods = 1 Acre. 640 Acres == 1 sq. Mile (or Section of Land). Equivalents- d mile, acres. d rods. □ yards. n feet. d inches. 1 = 640 = 102400 = 309 7600 = 2 78 78400 = 4014489600 The sign □ is used for the abbreviation " sq." In written exercises, either can be used. Note. — The acre has no corresponding denomination in linear measure. A square, measuring 208.'71 + feet on each side, contains 1 acre. ORAL EXERCISES. How many 1. Square yards in 12, 1881, 26, 100, 66 □ ft. ? 2. Acres in Vie, % % % D mi - ? 3. Square feet in a board 6 ft. 6 in. long, 2 / 13 ft. wide ? 4. A board 18 in. long contains half a d ft.; how wide is it ? 5. How many □ rods in %, %, 8 / 8 , 3 /ie of an acre ? 6. How many acres in a half section of land ? In a quarter ? 204 STANDARD ARITHMETIC. / / Cubic Measure. 206. To measure a block of marble, or to find how much a box, a bin, or a room will contain, we have to ascertain its length, breadth, and height or thickness, by a linear measure, as a foot-rule, a yard-stick, or a tape-line ; and, with the aid of the dimensions thus found, to calculate the contents of the block, or bin, or room, in Cubic Measure, that is, we calculate how many times the room, or the space occupied by the block, would contain some known cubic unit, such as a cubic inch, cubic foot, etc. 207. A Rectangular Solid is a solid haying six rectangular faces. 208. A Cube is a rectangular solid having six equal square faces. (See also page 103.) The figure at the left represents the outlines of a cubic foot, with a layer or course of cubic inches at the bottom. With this figure before the pupil let him answer the fol- lowing questions : 1. How many cubic inches in the course represented? 2. How many such courses are needed to complete the foot ? 3. How many cubic inches in a cubic foot? In l / 12 ? 3 / 4 ?y 24 ?etc. On inspection of the figure at the right, an- swer the following questions : 1. How many cubic feet in a cubic yard? 2. What is the length of each edge of a cubic foot ? 3. Can you lift a cubic foot of granite ? 4. How many cubic feet in a / 3 of a cubic yard? 5. How many cubic feet in */ 9 of a cubic yard ? 6. How many cubic inches in a cubic foot ? 7. How many cubic inches in a cubic yard ? In */ 3 of a cubic yard? 209. Thus the cubic inch, foot, and yard are derived from the corresponding linear measures. Table. 1728 Cubic Inches = 1 Cubic Foot. 27 Cubic Feet = 1 Cubic Yard. Equivalents. 1 Cubic Yard = 27 Cubic Feet = 46656 Cubic Inches. Note. — Higher denominations than these are seldom referred to. MEASURES. 205 Wood Measure. 210. Wood cut in "lengths" of 4 feet is called "cord wood." A pile of cord wood four feet high and eight feet long, or equal bulk of other material, is called a Cord. 211. One foot in length of such a pile is called a cord foot. Table. 16 Cubic Feet = 1 Cord Foot. 8 Cord Ft. or 128 Cubic Ft. = 1 Cord. ORAL EXERCISES. How many 1. Cubic feet in % 2% 1%, 3% cu. yd. ? 2. Cubic feet in % % % 2.625 cords ? 3. Cubic inches in an iron bar 13 y 2 in. long, 3y 3 in. wide, y 2 in. thick ? 4. Cubic inches in a brick 8 by 4 by 2 1 / 2 inches ? 5. Cubic yards in a wall 6 ft. high, 9 in. thick, and 20 yd. long ? (6 ft. = 2 yd., 9 in. = */ 4 yd.) 6. Cord feet in 3%, 7.125, 4.375 cords? Measures of Capacity. 212. For measuring fruits, berries, roots, grains, and other dry commodities, we use Dry Measure. The standard unit is the Bushel = 2150.42 cubic inches. Dry Measure. Table. 2 Pints (yt.) = 1 Quart (qt.). 8 Quarts = 1 Peck (pk.). 4 Pecks = 1 Bushel (bu.). Fquivalents. 1 Bushel = 4 Pecks = 32 Quarts = 64 Pints. Charcoal and coke are frequently measured by the chaldron, of 36 bushels. 206 STANDARD ARITHMETIC. 213. For measuring liquids, such as water, wine, vinegar, milk, etc., we use Liquid Measure. The standard unit is the Gallon. = 231 cubic inches. Liquid Measure. Table. 4 Gills (si.) = 1 Pint (pt.). 2 Pints = 1 Quart (qt.). 4 Quarts = 1 Gallon (gal.). Equivalents. 1 Gallon = 4 Quarts = 8 Pints =s 32 Gills. 2 1 4-. Comparison of Dry and Liquid Measures. The Dry Quart contains 67.2 cubic inches. The Liquid Quart contains 57.75 cubic inches. Notes. — 1. Thus it will be seen that the retailer who uses the liquid instead of the dry quart, in measuring berries and small fruits, cheats his customers out of a little more than one quart in seven. 2. Barrels, tierces, hogsheads, puncheons, pipes, butts, tuns, etc., have no stand- ard capacity. The quantity of liquid contained in them is usually found by actual measurement, called gauging. 3. When the barrel is spoken of as a measure of the capacity of vats, cisterns, etc., 3172 gallons are meant. In measuring beer, the barrel has 36 gallons, and V/z barrels (or 54 gal.) make a hogshead. ORAL EXERCISES. How many 1. Quarts in 2% 3%, 6%, 4.25, 5.5 gal.? 2. Pints in 2, 10, 15, 180 gi.? 3. Gallons in 9, 14, 27, 17, 30, 111, 63 pt. ? 4. Quarts in 7, 20, 31, 15, 50, 25, 35, 45 gi.? 5. Quarts in % 2%, l 7 / 8 bu.? 6. Pints in 3 1 /* 8%, 6.25, 9%, 10.125 qt.? 7. Bushels in 10, 1.6, 23, 2.8, 17 pk.? 8. Pecks in %% 4%, 3%, 5.75 bu. ? 9. Quarts in 5, 8% 10 % 33 y 3 , 13.825 pt.? MEASURES. 207 Measures of Weight. 215. For weighing gold, silver, the precious stones, etc., Troy Weight is used. The standard, unit is the Troy pound = 5760 grains. Troy Weight, Table. 24 Grains (gr.) = 1 Pennyweight (pwt.). 20 Pennyweights = 1 Ounce (oz.). 12 Ounces = 1 Pound (lb.). Equivalents. 1 Pound = 12 Ounces == 240 Pennyweights = 5760 Grains. Practical illustrations of Troy weight are to be found in the United States coins : The gold dollar weighs 25.8 grains ; the silver dollar, 41 2^2 grains ; the small silver coins, 385.8 grains to a dollar (that is, 10 single dimes, or 4 quarters, or 2 half- dollars, weigh 385.8 grains). The nickel 5^ piece weighs 77.16 grains ; the 3^ piece, 30 grains, and the bronze 1^ piece, 48 grains. Gold and silver are bought and sold by the ounce, weights of these metals never being expressed in pounds. The carat, very nearly equal to 3 x /5 Troy grains, is used in weighing diamonds and other precious stones. The word carat is also used in expressing the number of parts of pure gold in articles of jewelry, etc. If 18 parts out of 24 are pure gold, and the remaining 6 parts are alloy, the metal is said to be of 18 carats, etc. How many 0RAL exercises. 1. Pennyweights in 3%, 5.3, 6%, 9.2, 4% oz.? a. Ounces in 1%, 3%, 4%, 7% lb.? 3. Pounds in 16, 30, 27, 9, 9y 3 , 23.3 oz.? 4. Grains in % % 1% pwt.? In 1.5, 2% oz.? 5. Ounces in 45, 56, 90, 18, 50 pwt.? 6. Ounces of pure gold in 44 oz. of watch-cases, 18 carats fine? 7. Of how many carats is a mixture of 27 oz. gold and 13 % oz. alloy ? 8. How many pwt. of alloy must be put with 25 pwt. of pure gold to make a mixture of 20 carats ? 208 STANDARD ARITHMETIC. Apothecaries' Weight. 216. Apothecaries' Weight is used only by physicians in pre- scribing and by apothecaries in compounding medicines. When sold by weight, avoirdupois weight is used. Table. 20 Grains (gr.) = 1 Scruple 3. 3 Scruples = 1 Dram 3 . 8 Drams = 1 Ounce | . 12 Ounces = 1 Pound ft. Equivalents. & 1 = l 12 = 3 90 = 3 288 = gr. 5760. Note. — 1. It should be observed that in this weight the signs precede the num- bers to which they belong. 2. The grain, the ounce, and the pound are of the same value as the corresponding denominations in Troy weight. u _ ORALEXERCISES. How many 1. Pounds in ? 36 ? I 40 ? \ 27 ? I 75 ? 2. Grains in 3 I 1 /, ? 3 % ? 3 % ? 3 3.75 ? 3 .1 ? Avoirdupois Weight. 217. For the common purposes of trade, Avoirdupois Weight is used. The standard unit is the pound of 7000 grains. Table. 16 Ounces (oz.) = 1 Pound (lb.). 100 Pounds = 1 Hundredweight (cwt). 20 Hundredweight = 1 Ton (T.). The term cental is beginning to be used for hundredweight. Equivalents. 1 Ton = 20 Hundredweight = 2000 Pounds = 32000 Ounces. Formerly 112 lb. were reckoned a hundredweight, and 2240 lb. a ton. This weight is still used in weighing iron, coal at the mines, ores, and goods on which duties are paid at the United States custom-houses. 218. Comparison of Troy with Avoirduoois Weight. Avoirdupois: 1 lb. = 7000 grains. 1 oz. = 437y 2 grains. Troy: 1 lb. = 5760 grains. 1 oz. = 480 grains. MEASURES. 209 ORAL EXERCISES. How many 1. Ounces in 1% 2%, 5%, 4 5 / 16 , 8%, 6% lb.? 2. Pounds in % 1%, 7.1, 6%, 6 1 /*, 12 % cwt? 3. Pounds in 12, 46, 22, 33.6, 29, 176 oz.? 4. Cwt. in 2%, 3.%, 4%, 11.2, 9.9 T.? 5. Pounds in 1.3, 2%, 3%, 5%, 7% T.? 219. Weight being very commonly employed in estimating quantities of grains, roots, etc., the weight of the bushel, as fixed by law in many States, for some of the more important commodi- ties, is given below.* The general usage is found in the second column. In the third, exceptions are noted so far as known. (See Haswell, Ed. 1885, and Report No. 14, H. R., 46th Congress, 1st Session.) Commodities. Lb. per bu. Exceptions. Barley 48 56 32 56 60 60 60 Ariz, and Wash., 45 ; Cal , 60 ; Md. and Penn., 47 ; N. H. and Del., not reported. Ariz., 54 ; Cal., 52 ; N. Y., 58. Iowa, Mont., and Mo., 35 ; Md., 26 ; Neb. and Ore., 34 ; Me., N. H., and N. J., 30 ; Wash., 36 ; Ky., 33 x / 3 ; Del., not reported. Cal., 54 ; La., 60 ; Del. and Me., not reported. Ohio, 58 ; Ariz., Cal., Del., La., Md., Penn., not re- ported. Conn., 56 ; R. I., not reported. No exceptions reported. Shelled corn Oats Rye Potatoes Wheat Pease Usage in regard to the following articles is not so uniform as in case of those given in the foregoing list : Corn in the ear. Variously estimated from 68 to 70 lb. Corn meal. Del., 44 lb. ; 111., 48 lb. ; most other States, 50 lb. Beans. Me., 64 lb. ; N. Y., 62 lb. ; many others, 60 lb. Clover seed. Mont., 45 lb. ; N. J., 64 lb. ; Penn., 62 lb. ; in almost all others, 60 1b. Timothy seed. Wis., 46 lb. ; N. Y. and Mont., 44 lb. ; Dakota, 42 lb. ; in many others, 45 lb. Mineral coal. Ky. and Penn., 76 lb. ; Ind., 70 lb. ; in most others, 80 lb. 210 STANDARD ARITHMETIC. The following standards are generally accepted : 100 lb of grain or flour = 1 cental. 196 lb. of flour s= 1 barrel. 100 lb. of dry fish = 1 quintal. 200 lb. ot beef or pork = 1 " 100 lb. of nails as 1 keg. 220. Measures of Value. United States or Federal Money. For the United States or Federal Money table, see Art. 89. The gold eoins are the $1, $f*/ a (quarter-eagle), $3, $5 (half-eagle), $10 (eagle), and $20 (double-eagle) pieces. The silver coins are the $1, 50^, 250, and 100 pieces. The 50 and 30 pieces are made of nickel; the cent of bronze. Other coins are occasionally found in circulation, but are no longer coined, such as the trade-dollar, the 200, the 50 and the 30 silver pieces, and the 20 piece of bronze. Canadian Money. 22 (. The unit of Canadian currency, like that of the United States, is called a dollar. It is divided into 100 cents, and the cent into 10 mills. The legal coins are — Gold : the British sovereign, worth $4.8665, and the British half-sovereign; Silver: the 50^, 25^, 10^, and 5^ pieces ; Bronze : 1$. The silver and bronze coins have the same values as the corresponding coins of the United States. English or Sterling Money. Table. 4 Farthings (far.) = 1 Penny (d.). 12 Pence = 1 Shilling (s.). 20 Shillings = 1 Pound (£). Equivalents. 1 Pound = 20 Shilling = 240 Pence = 960 Farthings. The coins of Great Britain are — Gold : the sovereign = $4.8665, and the half- sovereign ; Silver: the crown (5 shillings) = $1,216 + , and the half-crown; the florin (2 shillings) = $.486 ; the shilling = $.243 ; the sixpenny, fourpenny, and threepenny pieces; Copper: the penny, half-penny, and the farthing ( 2 / 4 penny). The guinea = 21 shillings, though no longer coined, is frequently mentioned as if in common use. MEASURES. 211 222. Money of Other Countries. a, French money : 1 franc (fr.) = 100 centimes (c.) = 19.3^ in United States money. The Gold coins of France are the 100, 50, 20, 10, and 5 franc pieces ; the Silver coins are the 5, 2, 1, 1 / 2 , and 1 / 5 franc pieces; the Bronze, 10, 5, 2, and 1 centime (pronounced sonteem) pieces ; Copper, 10 and 5 centimes. The values of these coins are indicated by their relations to the franc. b. German money: 1 mark (reiehsmark) \yX> m.) = 100 pfennigs (pf.) = 23.8^ in U. S. money. The Gold coins of the German Empire are the 20 and 10 mark pieces ; the Silver coins are the 5, 3, 2, 1, and ] / 2 mark and 20 pfennig pieces ; the Nickel, 10 and 5 pfennig ; the Copper, 2 and 1 pfennig. The Thaler (silver) = $.'746, and the Groschen (silver) = 2 1 / 2 $, are also in common use. For the values of other foreign coins, see Appendix. 223. The following approximations are sufficiently exact for general estimates : One U. S. Dollar may be counted as equal to 5 Francs (France, Belgium, and Switzerland), or to 5 Lire (Italy), or to 5 Peseta (Spain), or to 4 Shillings (England), or to 4 Marks (Germany). ORAL EXERCISES. How many 1. Pence in 2, 3%, 9%, 14%, 21 s.? In % crown ? 2. Farthings in 8%, I*%» 23%, 3 pence? 3. Pounds in 50, 15, 75, 105, 130, 244 shillings ? 4. Shillings in 30, 6, 45, 33, 81, 108 pence ? 5. Shillings in £7%? £15%? £22%? 2 2 / 3 guineas? 6. Centimes in %, 1%, 5%, 17%, 28% francs? 7. Marks in 175, 210, 1728, 3042 pfennigs ? 8. Dollars may be counted as equal to 18 roubles ? 42 roubles ? 166 roubles ? 9. Dollars may be counted as equal to 34, 78, 92, 118 s. ? To 1 guinea? To 75, 130, 195 peseta? 10. Dollars may be counted as equal to 250000 francs ? To £340000 ? To 3000 marks ? To 1500 roubles ? 212 STANDARD ARITHMETIC. Definitions. 224. A point has position, without length, breadth, or thick- ness. 225. A line is the path of a point in motion. If the point moves without change of direction, the path is a straight line. If the point changes its direction continually while moving, the path is a curved line. 226. If the moving point passes around a fixed point, so that its distance from the fixed point does not vary, the path of the moving point is the circumference of a circle, and the fixed point within is the center of the circle. 227. For the measurement of angles (see Art. 56), the circum- ference of the circle is conceived to be divided into 360 equal parts, called degrees. The angle in this figure is an angle of 90 degrees — one fourth of the 360 equal parts into which the circumfer- ence is supposed to be divided. 228. An angle /of 90 degrees (written 90°) is a right angle. An angle of less than 90° is an acute angle. An angle greater than 90° is an obtuse angle. 229. Two lines which form an angle of 90° are said to be perpendicular to each other. (See also Art. 57.) Note. — A degree, being 1 / 360 part of any circumference, is very minute, if the circle is a small one; but a degree of the circumference of the earth is 69.16 miles in length. A degree of the sun's circumfer- ence is about 7444 miles long. Compare these with one degree on the protractor, as here represented. A protractor is an instru- ment used for the measurement of angles. 230. For more exact measurements, the degree (°) is divided into minutes ('), and the minutes into seconds ("), according to the following table : MEASURES. 213 Circular Measure, Table. 60 Seconds ( ") = 1 Minute ('). 60 Minutes = 1 Degree (°). 360 Degrees = 1 Circumference. ORAL EXERCISES. How many 1. Degrees in % % */«, % V» Vs* circumference ? 2. Minutes in 3%° ? 2%° ? 5%° ? 4%° ? 7%° ? 3. Degrees in 180' ? 150' ? 3600' ? 420" ? 1800" ? 6000" ? 4. Minutes in 900" ? 720" ? 342" ? 1275" ? 3333" ? 231. Time Measure. Table. 60 Seconds (sec.) = 1 Minute (min.). 60 Minutes = 1 Hour (h.). 24 Hours = 1 Day (d.). 7 Days = 1 Week (wk.). 365 Days 5 Hours 48 Min. 46.4 Sec. = 1 Solar Year (yr.). The year given in the table, which is a little less than 365 l / 4r days, is the time it takes the earth to go around the sun, and hence the time required for a complete change of seasons. But to count this quarter of a day with every year would be extremely inconvenient. It is much easier to count one additional day every fourth year, and hence this is generally done, but not always, for in a hundred years we should thus gain nearly a day too much ; so the hundredth years (centennial years) are commonly counted as ordinary years ; but here again we have to say not always, for we should thus lose nearly a day in 400 years. Hence the centennial years divisible by 400 are counted as leap years. The following is the rule by which leap years may be known for several thousands of years to come : 232. All years divisible by 4, except centennial years not divisible by 400, are leap years. 233. There are 12 months in a year. The number of days in each is given in the following 10 214 STANDARD ARITHMETIC. Table. Months. Days. 1st. January (Jan.) 31 2d. February (Feb.) 28 or 29 3d. March (Mar.) 31 4th. April (Apr.) 30 5th. May (May) 31 6th. June (June) 30 Months. Days. 7th. July (July) 31 8th. August (Aug.) 31 9th. September (Sept.) ... 30 10th. October (Oct.) 31 11th. November (Nov.) 30 12th. December (Dec.) .... 31 The 29th day of February is the day added to make a leap year. The following lines are used to aid the memory in recalling the number of days in the several months : " Thirty days hath September, April, June, and November ; All the rest have thirty-one, Except February alone, Which has but 28 in fine, Till leap year gives it 29." The long months may be distinguished by observing that their names are the only ones that contain the letter c, or that have a for their second letter, and, except June, the only ones that have u for their second letter. (See whether this is true.) ORAL EXERCISES. How many 1. Days in 2%, 4%, 14 wk.? 2. Hours in %% 4%, 3% d.? 1 wk.? 3. Minutes in 1%, 8%, 2%, 4 3 / 5 , 12, 24 h.? 4. Weeks in 9, 30, 23, 17, 60, 90, 365 d.? 5. Days in Aug.? Apr.? Dec.? Jan.? Sept.? Feb.? July? Nov.? Mar.? Oct.? June? May? 6. Days in the year 1886? 1894? 1896? 1900? 1800? 2000? 234-. Miscellaneous Measures. 12 Units 12 Dozen 20 Units 5 Score Counting. = 1 Dozen (doz.). = 1 Gross (gro.). = 1 Score. = 1 Hundred. 24 Sheets 20 Quires 2 Reams 5 Bundles Paper. = 1 Quire (qu.) t = 1 Ream (r.). as 1 Bundle. = 1 Bale. MEASURES. 215 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. MISCELLANEOUS ORA How many Inches in 2 % 3.75 ft.? 21. □ ft. in3Va, 5.75 □ yd. ? 22. Ou. in. iniy 8 , 1.5 cu. ft? 23. Pints in 11.75, 9% qt. ? 24. Pecks in 3%, 5.125 bu.? 25. Quarts in 15, 20.5 pk. ? 26. Grains in 7, 7.66% pwt.? 27. Seconds in 3y 4 , 5y 8 min. ? 28. Minutes in .125, 5 / 6 h. ? 29. Degrees in .16 2 / 3 circum. ? 30. Dozen in 3y 4 , 4.375 gross? 31. Pwt. in 11.5, 9%oz.? 32. D ft. iniy 8 , 2.5 □ yd.? 33. □ in. in .375, l 3 / 8 □ ft. 34. Cu. ft. in 1.88 y t OIL yd.? 35. Feet in ?% 7. 5 fathoms? 36. Feet in 3 y 4 cable lengths ? 37. Yards in 1.5, 2.6 rd.? 38. Rods in 7.7, 11.5 yd.? 39. Feet in 9.6, 15.6 in.? 40. L EXERCISES. How many Lb. pork in 2 % 3.7 bl? Pence in 14 5 / 6 s. ? Pounds in 32 oz. avoir. ? Pounds in 8.4 oz. Troy ? Gallons in 2.6, 17 qt.? Gills in 27, 1.3 pt.? Ounces in 36, 96 pwt. ? Lb. Troy in 3, 18 oz. □ ft. in 288, 72 p in. ? Feet in 10.8, 8.4 in.? Feet in 7.2, 10% in.? Cu. yd. in 10.8 cu. ft.? Bushels in 23, 97 pk. ? Pecks in 23, 97 qt. ? Pints in 7.2, 39 gills ? Fath. in iy 8 cable lengths ? Feet in 15 % 14 hands ? Quarts in 73, 95 pt. ? Pecks in 3.2, .96 qt.? Lb. pork in 5, iy 4 bl.? Definitions. Compound Denominate Numbers. 235. Number when applied to specified objects is said to be concrete. 236. Number when not applied to specified objects is said to be abstract. ______ 237. To measure a quantity is to find how many times it contains some known quantity used as a standard of comparison. 216 STANDARD ARITHMETIC. 238. A known quantity, fixed by law or custom as a standard of comparison, is called a Unit of Measure. Note. — A yard is a standard fixed by law for measuring length or distance. A hand is a standard fixed by custom for estimating the height of horses. 239. Units of measure have special denominations or names by which they are designated, and hence they are called Denom- inate Units. (Denomination means name.) 240. A Denominate Number is a number of denominate units. 241. A Simple Denominate Number is one that consists of units of only one denomination. 242. A Compound Denominate Number is one that consists of units of two or more denominations. 243. Changing the denomination in which a quantity is ex- pressed is called Reduction. 244. Reduction Ascending is changing an expression of quantity from a less to a greater unit of measure. 245. Reduction Descending is changing an expression of quantity from a greater to a less unit of measure. SLAT E EXERCISES. Example. — l. Eeduce 3 gal. 2 qt. 1 pt. 3 gi. to gills. Analysis. — Since there are 4 qt. Process, in one gallon, there must be 3 times 3 gal. 2 qt. 1 pt. 3 gi. 4 qt. = 12 qt. in 3 gal. ; 12 qt. + ^ 2 qt. = 14 qt. Since there are 2 pt. in 1 qt., there must be 14 14 qt. times 2 pt. = 28 pt. in 14 qt. ; 28 2 r Caution. — Say , pt. + 1 pt. = 29 pt. Since there 2g ^ \ 12, 14 ; not 4 times / are 4 gi. in 1 pt., there must be 29 * 1 3 = 12, and 2 are L times 4 gi. in 29 pt. = 116 gi. ; 116 ^ ' 14, etc. / gi. + 3 gi. = 119 gi. Hence, in 3 119 gi. gal. 2 qt. 1 pt. 3 gi. there are 119 gi. MEASURES. 217 Example. — 2. Keduce 119 gi. to higher denominations. Analysis. — Since there is 1 pt. in 4 gills, there arc as many pints in 119 gi. as there are times 4 gi. = 29 times, with 3 gills remaining. Since there is 1 qt. in 2 pt., there are as many quarts in 29 pt. as there are times 2 pt. sr 14 times, with 1 pt. remain- ing. Since there is 1 gal. in 4 qt., there are as many gallons in 14 qt. as there are times 4 qt. = 3 times, with 2 qt. remaining. Hence, in 119 gi. there are 3 gal. 2 qt. 1 pt. 3 gi. Note. — Analysis here supersedes the necessity for any rule. Reduce Process. 4)119 gi. 2)29 pt. + 3 gi. 4)14 qt. + 1 Pt. 3 gal. + 2 qt. 1. 375.96 inches to yards. 2. 2480 oz. to hundredweight. 3. 23 h. .48 min. to seconds. 4. 29738.7 inches to rods. 5. 5. 33 y 3 days to minutes. 6. 96 oz. of lead to pounds. 7. 96 cu. ft. to cu. yd. 8. 1 gro. 10 doz. to units. 9. 3 mi. 173 yd. 2 ft. to in. 10. 19.3 pecks to bushels. 11. 5238 far. to shillings. 12. 29 wk. 6 d. to hours. 13. 495 sheets to quires. 14. 69472 lb. to hundredweight. 15. 13 mi. 1.537 yd. to yards. 16. £6 17 s. 10 d. to farthings. 17. 5620 hours to weeks. 18. 593 yd. 1.8 in. to inches. 19. 27 gro. 11 doz. to units. 20. 25 T. 7 cwt. to ounces. 21. 157 quires to reams. 22. 187.6 quarts to pecks. 23. 5 bu. 3 pk. 7 qt. to pints. 24. 10 wk. 3 d. 10 h. to sec. 25. 27 lb. 9 oz. Troy to grains. 26. 705 quarts to bushels. 27. 5. 934 feet to yards. 28. 5638 d. to pounds sterling. 29. 14 h. 36 min. .5 sec. to sec. 30. 893 units to gross. 31. 14 mi. 18 rods .6 yd. to yd. 32. 19 cwt. 46 lb. 9 oz. to oz. 33. 3 wk. 5 d. .19 h. to min. 34. 484 pecks to bushels. 35. 276457 ounces to tons. 36. 4563 shillings to pounds. 37. 654 dozens to gro. 38. 2 s. 6 d. 3 far. to farthings, 39. 8349250 seconds to days. 40. 5 gal. 3.5 qt. to pints. 41. 14 pk. 1 qt. to pints. 42. 628 pints to pecks. 43. 12 bu. 1.75 pk. to pints. 44. 1 mi. 58 yd. .8 in. to in. 218 STANDARD ARITHMETIC. Reduce 45. 945. 6 ounces to lb. avoir. 46. 25 yr. 79 d. to days. 47. 2572 gills to quarts. 48. 1 mi. 13.62 yd. 2 ft. to in. 49. 2500 inches to feet. 50. 17 cwt. 95 lb. to pounds. 51. 93 reams 2 quires to quires. 52. 976.3 far. to pounds. 53.. 42345 ounces to tons. 54. 2 cwt. 75 lb. to ounces. 55. 2 qt. 1 pt. .3 gi. to gills. 56. 7892.8 minutes to days. 57. 29650 seconds to hours. 58. 19 gr. 8 doz. 10 units to units. 59. 71 bu. 3 pecks to quarts. 60. 593 pints to gallons. 61. 25971 yards to miles. 62. 3000 gills to gallons. 63. 79 tons 2 cwt. to pounds. 64. 930780 minutes to weeks. 65. 738 reams to sheets. 66. £345 18 s. 8 d. to pence. 67. 7453 sheets to reams. 68. 284 gal. 3 qt. 1 pt. to gills. 69. 7 gr. 6 doz. 11 units to units. 70. 127 T. 15 cwt. 25 lb. to lb. 71. 4 reams 22 sheets to sheets. 72. 9256.35 feet to miles. 73. 13 gal. 1 pt. 2 gi. to gills. 74. 57289 hundredweight to T. la How Analysis. 2. How Analysis. 3. How Analysis 4. How 5. How 6. How 7. How 8. How Reduction of Denominate Fractions. Reduction Descending. ORAL EXERCISES. many ounces in 7 pounds avoirdupois ? — 7 lb. = 7 x 16 ounces, or 112 ounces. many cents in '%, %% %, %o, n / 5 o, 3 /io dollar ? — $1 = 1000 ; V, dol. = V, of 1000 = 500. many inches in % % % % % % % *<>ot ? — 1 foot = 12 in. ; 2 / 3 foot = 2 / 3 of 12 in. = 8 in. many inches in % % % % %, 7 / 10 yard? many gills in % % % % %, % quart ? many pints in %, 5 / 6 , % %, 3 / 10 , % gallon ? many grains in %, % % % %, % oz. Troy ? many ounces in % % %, % % % lb. Troy? MEASURES. 219 SLA1E EXERCISES, On pages 216 and 217 the pupil learned to change integral numbers from higher to lower and from lower to higher denominations. Here he will find the same prin- ciples applied to the reduction of fractional expressions from one denomination to another. Examples. — (l.) Reduce 3 bushels to pints. (2.) Reduce % bu. to pints. (3.) Reduce . 75 bu. to pints. These problems differ from each other only in this, that in the first the number of bushels to be reduced is expressed by an integer, in the second by a common fraction, and in the third by a decimal fraction. They are all solved by multiplica- tion, and the reasons for multiplication are the same. (1.) 3bu. (2.) %bu. (3.) .75 bu. A J J 12 pk. 12 / 4 = 3pk. 3pk. 8 _8 8 96 qt. 24 qt. 24 qt. 2 2 2 192 pt. 48 pt. 48 pt. 4. How many units in 2% gross ? Analysis.— 2 gr. = 2 x 144 units = 288 units. z / 3 gr. = 2 / 3 of 144 units = 98 units. 2S8 + 96 = 384 units. 5. How many □ inches in y 4 , y 8 , 2 / 5 , 2y g , 3 3 / 4 n yards ? 6. How many □ feet in % % iy 4 , 2 3 / 8 , 7, 6 a rods ? 7. How many inches in $% 3 3 / 4 , 6% 9% feet ? 8. How many cu. inches in 2 ! / 4 , 6 3 / 7 , 5 5 / 12 , 3 7 / 23 cu. feet ? 9. How many pints in 3 3 / 4 , 3% 4 5 / 7 , 6 3 / 5 bushels? 10. How many pennyweights in 3 5 / 8 , 5 7 / 12 , 9 3 / 5 pounds ? 11. How many feet in % % %, 1%, % % rods ? 12. How many grains in 7 / 8 , 3 4 / 9 , n / 12 , 5 7 / 16 pounds avoir. ? 13. How many hours in 6 5 / 9 , 3 7 / 8 , 2 7 / 16 , 5 n / 12 months ? 14. How many feet in 7 / 8 , 4 / 9 , 5 / 12 , 11 / 16 mile ? 15. How many gills in % 1% 3 4 / 5 , 2 5 / 48 gallons? STANDARD ARITHMETIC. Reduction Ascending. ORAL EXERCISES. 1. What part of a lb. is 2, 4, 5, 10, 12, 15 ounces avoir. ? Analysis. — 2 oz. are 2 / 16 of a pound avoirdupois because they are 2 of the 16 equal parts (ounces) into which a pound can be divided. 2. What part of a bushel is 6, 9, 13, 17, 21 quarts ? 3. What part of a month is 2 % days ? 2 1 / Analysis.— 1 d. = 1 / 30 mo., 2y 2 d. = -—^- 2 mo. = 5 /eo, or J / 12 month. 4. What part of a shilling is % penny ? SLATE EXERCISES. Examples. — (l.) Keduce 1 pt. to the fraction of a bushel. (2.) Eeduce % pt. to the fraction of a bushel. (3.) Reduce .2 pt. to the decimal of a bushel. These are similar problems, and are all solved by division, the divisors being the same in each case. (1.) 2 )1 pt. (2.) 2 )% pt. (3.) 2) 0.2 pt. 8 )% qt. 8 )V 10 qt . 8)0.1 q t. 4 )% 6 pk. 4)% pk. 4)0.0125 pk. y M bu. y 320 bu. 0.003125 bu. 246. From the above illustrative examples it may he seen that the process of reducing fractions is the same as that of reducing integers from one denomination to another. 4. How many days in 34%, 37%, 48y l0 , 50% hours? This question if given in full would be, " How many days and what fraction of a day in," etc. 5. How many gallons in 75% pints? In 83%, 92%, 102 "/„ quarts? In 56%, 48% gills? 6. What part of a hundredweight is 3 / 5 , 5 / 8 , 7 / 10 , 2 3 / 7 , 4%, 7% 2 , 19% pounds? 7. Reduce to acres : %> , 1%, 845%, 98374% □ yd. MEASURES. 221 To Integers of Lower Denominations. SLATE EXERCISES. Example. — l. Find the value of the fractional part of 2 17 / 36 lb. Troy in integers of lower denominations. Common Fractions. 2|"/t.n>.Troy. 12 204/ _ 5|24/ 20 ; pWt. =8gr. Decimal Fractions. 2»V,.lb.=2.47»/,lb. 2|.47 2 / 9 lb. 12 5.|G6 2 / 8 oz. 20 480/ _ , lf/ /36 d| /86 24 288/ 13.|33V 8 pwt. 24 8gr. Answer to loth. — 2 lb. 5 oz. 13 pwt. 8 gr. 2. How many pounds and ouncas in %, %, 3 / 8 , 5 / 9 , 7 / 10 , "/«, %n "As, 9 %i cwt.? Analysis.— 2 / 3 cwt. = 2 / 3 of 100 lb. = 66 2 / 3 lb. 2 / 3 lb. = 2 / 3 of 16 oz. = l0 2 / 3 oz. Hence 2 / 3 cwt. = 66 lb. 10 2 / 3 oz, 3. How many □ yards and n feet in 3 / 8 , %, %, 4 / 9 acre ? 4. How many hours, minutes, and seconds in 1 / 21 , 4 / 7 , %, 3 /i , Vic 7 Ao, 2 %7 day ? 5. How many feet and inches in 7 / 8 , 4 / 7 , 6 / n , 7 / 18 yard ? 6. How many pence and farthings in %, 7s? 9 /io> "/» 12 /i3> 14 /i 7 , 6% shillings? 7. How many weeks, days, and hours in 2 3 / 4 , 5%, 11 / 19 , 21 / 31 , 3 / 17 common years ? 8. How many cubic yards and feet in 1%, 9 5 / 7 , 7y l2 , 3y 8 , 11 13 / 120 cords ? 9. How many pennyweights and ounces in 11 %, 7y 8 , 5 2 / 13 , 67y 13 , 88 B /» 25y i6 pounds Troy? 10. How many grains and scruples in 3 T/g, 3 9y 8 , 3 3y 6 , 3 7y 480 ? in i iy 18 , i % i y 33 ? in ib 7y 52 , & iy n , s> sy 21 ? 22'2 STANDARD ARITHMETIC. Change to integers 1. 7s ton. 2. .1125 ton. 3. .1325 pint. 4. 3 / 9 CWt. 5. .125 hhd. 6. .99 bl. pork. 7. .38675 oz. Troy. 8. .9975 ft. 9. y l2 d yd. 10. .7846 acre. 11. 4 / 5 mile. SLATE EXERCISES. of lower denomination : 12. 3 / 10 day. 23. 13. .8 cord of wood. 24. 14. .98974 qt. 25. 15. .7859 cwt. 26. 16. .7775 oz. Troy. 27. 17. %, ton. 28. 18. 9 /n bl. of flour. 29. 19. .375 ream. 30. 20. y 6 cu. yd. 31. 21. y 8 cu. ft. 32. 22. 3 / 7 □ mile. 33. % acre. Vis rod. %se lb. avoir. 5 / 6 degree. 3 / 5 of 5 / 7 lb. avoir. 3 / 5 of 1 cwt. 56 lb. y 7 of 1 mi. 160 rd. % of % lb. Troy. .735 bl. of beef. .9x.87 1b. Troy. .1755 yd. To Fractions of Higher Denominations. SLATE EXERCISES. Eeduce 5 d. 14 h. 24 min. to a fraction of a week. Explanation. — Reducing the two periods of time to be compared, to the same denomination, we have 1 wk. 7 74. 24 168 h. 60 10080 min. 5 d. 14 h. 24 rain, is or 4 / 5 of a week. 10080 /5 247. Second Method. — The lowest denomination may be re- duced to a fraction of a unit of the next higher ; then this frac- tion, together with the integer to which it now belongs, may be reduced to a fraction of the next higher, and so on till the entire compound number is reduced to the required fraction, as follows : 5 d. 14 h. 24 min. _24 134 h. 60 8064 min. Analysis. — 5 d. 14 h. 24 min. = 8064 min. One week = 10080 min. 8064 8064 of 10080 mm. ii min 10080 Hence, MEASURES. 223 Suggestion. — The work is conveniently arranged by writing the several denom- inations in a column, beginning with the lowest, and writing the resulting fractional quotients on the right of a light vertical line, as below. In the progress of the work this line is disregarded. It is useful only to prevent mistakes. Analysis. — 1 min. = 1 / 60 h. 24min. = «Veo = V«h. Pr0C2SS - 2 / 5 h. + 14h. = 14 2 / 5 h. 60)24 min. 24)14 h. 7)5 d. 2^— lh.= 7 24 d 14V 5 h. = ^d. 5 d. 14 h. 24 min. = 4 / 5 wk. 14 */, 72, A 20 d. = 3 A d. 3 / 5 d.+ 5d. = 5 3 / 6 d. 1 d. = V 7 wk. • ■■/• 5 3 / 5 d. = -f wk. 5^ Wk. = »/„ = V6 Wk. The operation by decimals is as follows : p Explanation. — Here the process of reasoning is precisely fin\9A ' *^ e same as ^ or s i m ^ ar reductions in integers, thus: Since there / min. . g j jj j n go mm ^ there are as many h. in 24 min. as there are 24 )14.4 h. times 60 min. in 24 min. = .4 times, etc. etc. But this process 7) 5.6 d. differs from the process of reduction in integers in the addition o w fr of the higher denominations as we come to them, while in inte- gers there are no such additions to make. SLATE EXERCISES. What part of 1. 1 ton is 7 cwt. 79 lb. 11 oz. ? 9. 4 □ miles is 347 acres ? 2. £5 is 1100 d. ? 10. 59° is 13° 13' 13" ? 3. 1 cwt. is 79 lb. 9% oz.? 11. 13 cu. ft. is 578 cu. in.? 4. 3 acres is 1700 □ ft.? 12. 3 oz. is 5% pwt. ? 5. 1 yr. is 89 d. 1 h. 12 min.? 13. 7 days is 37 min. 37 sec? 6. £1.7835 is £1 15 s. 8.04 d.? 14. 1 yr. is 89 d. 17 h. 8 min.? 7. 6.75 bu. is 3 pk. 3 qt. ? 15. 52 d. 16 h. is 49 d. 9 h. ? 8. 5 d ft. is 289 d in. ? 16. 1 cwt. is 13 lb. 16 oz. ? 224 STANDARD ARITHMETIC. What part of 17. 25 cu. ft. is 864 cu. in. ? 32. 1 ton is 6 cwt. 7 lb. ? 18. 6 d. 1 far. is 4 d. 20 h. ? 33. % lb. Troy is 12 gr. ? 19. 13 cords is 13 cu. ft.? 34. 17 h. is 19 min. 13 sec? 20. 1.25 ton is 17% oz. ? 35. 17 h. is .1175 d. ? 21. 13 yd. is 13 in.? 36. 777 oz. Troy is 3 lb. 11 oz.? 22. 13 gal. is 3 qu. 1 pt. 1 % gi. ? 37. % lb. avoir, is 21 gr. ? 23. 36 gal. is 27 gal. 2 qt. 1 pt. ? 38. % lb. avoir, is % lb. Troy ? 24. 1728 cu. in. is 445 cu. in. ? 39. 3 cwt. 99 lb. is % ton 33 lb. ? 25. 1 lb. Troy is 7 oz. 6 pwt. ? 40. 69 cwt. is 69 lb. 26. 1 lb. Troy is 11 oz. 7 pwt. ? 41. 2 n mi. is 345 a. 17 n rd. ? 27. 1 ton is 47.73 lb. ? 42. 5 cords is 7.125 cord feet ? 28. 3 / 4 mi. is 527.3994 yd. ? 43. K> 3 is 3 3 3 1 32 gr. 16 ? 29. 2 / 3 acre is 420.1883 n yd. ? 44. 180° is 5° 18' 22" ? 30. 1 oz. Troy is 1 oz. avoir. ? 45. 1 ch. is 3 rd. 3 li. 5 in. ? 31. 1770 oz. is y 3 cwt.? 46. 1 yr. is 5 h. 46.4 sec? Addition. (Compound Denominate Numbers.) Example. — 1. What is the sum of 13 gal. 2 qt. 1 pt. 3 gi.; 14 gal. 2 qt. 2 gi.; 7 gal. 3 qt. 3 gi.; 9 gal. 1 qt. 1 pt. 2 gi.; 6 qt. 1 pt. 1 gi. ? Explanation. — Numbers of the same denomination are written in the same column for convenience of ad- dition. The sum of the column of gills is 11 = 2 pt. 3 gi. The 3 gi. is written under the column added, and the 2 pt. are added with the column of pints. The sum of the pints is 5 = 2 qt. 1 pt. The 1 pt. is written under the column of pints, and the 2 qt. are added to the quarts, and so on. Jy Q 3 Note. — The pupil will perceive that the process at the left is like that of simple addition, except that instead of the divisor being always 10, as in simple numbers, it varies with the de- nomination. It is always as many units of the denomination of the column added as are required to make a unit of the next higher. Written Work. Gal. qt. P t. gL 13 2 1 3 14 2 2 7 3 3 9 1 1 2 6 1 1 2. MEASURES. 225 SLATE EXERCISES. T. cwt. lb. oz. 3. Mi yd. ft in. 11 18 77 11 4 1678 2 11 32 11 31 10 2 1123 1 10 43 17 63 13 3 1456 2 9 Ed. yd. ft. in. 5. A. d rd. d yd □ ft d in. 5 3 2 8 2 115 20 3 31 8 1 9 7 218 32 G 15 15 4 1 10 1 25 31 8 25 10 1 2 3 3 34 27 7 100 39 4 2 1 6 6 15 75 21 7 2 27 36 39 5 1 15 75 22 63 es. — 1. In E) i, 4, the sum of the ya rds is 10 , or 1 rod 4 7 1 3* L ; we write Oz. pwt. gr. 44: 5 16 15% 16 1 14 83% 15 17 0% 2 14 2 4 21% 20 3 19 8% 18 14 12 22^24 83 the 4, and for the 1 / 2 yd. we add 1 ft. 6 in. 2. In Ex. 5, the sum of the a yards is 112, or 3 □ rods 21 1 / 4 d yd. ; write the 21 □ yd., and for the '/ 4 d yd. add 2 □ ft. 36 d in. 3. The fractions of the lowest denomi- nation being added together, and reduced, the resulting integer, if any, is added to the given integers of that denomination. Examples.— 7. Find the sum of 13 cwt. 21 lb. 13 % oz. ; 3 cwt. 18 lb. 9 7 / 10 oz.; 25 cwt. 31 lb. 15% oz. 8. Add 58 gal. 3 qt. 1 pt. 3% gi.; 45 gal. 3 qt. 1 pt. 1% gi.; 38 gal. 1 qt. 1 pt. 3% gi.; 26 gal. 3 qt. 3y 3 gi. 9. Add £7305 14 s. 8% d.; £8737 13 p. 4% d.; £513 6 s. 5 d.; £67 5 s. 10% d. 10. Add 37 cu. yd. 15 cu. ft. 1084 en. in.; 9 cu. yd. 13 cu. ft. 1556 en. in.; 86 cu. yd. 22 cu. ft. 695 cu. in.; 24 cu. yd. 8 cu. ft. 924 cu. in. 11. Add 17 tons 11 cwt. 99 lb. 15 oz.; 7 cwt. 97 lb. 13 oz.; 7 tons 7 cwt. 7 lb. 7% oz.; 11 tons 11 cwt. 11 lb. 11 oz.; 179 cwt. 1780 lb. 11797 oz. ; 137 tons 19 cwt. 89 lb. 15 oz. 226 STANDARD ARITHMETIC. Subtraction. Example. — l. Mr. Jones had £4 4 s. 2d., out of which he paid Mr. Smith £1 3 s. 6 d. How much did Mr. Jones have left ? Explanation. — To pay the 6 d. Mr. Jones obtains change Written Work. (12 d.) for a shilling, and putting this with the 2 d. he has £ s . d. 14 d. 14 d. — 6 d. = 8 d. Having taken 1 s. from 4 s. there 4 4 2 remain but 3 shillings, which he pays to Mr. Smith, and has 1 Q p no shillings left. He then pays £1 out of the £4, and has £3 s. 8 d. left of the £4 4 s. and 2 d. which he had at first. 3 8 On comparing this process with the one represented on page 42, the pupil will see in what they are alike and in what respect they differ. SLATE EXERCISES Find the differences 2. £ s. d. far. 3. Bu. pk, 173 8 5 324 2 75 9 7 3 235 3 5. Ed. yd. ft. in. 15 3 2 3 8 4 1 9 6 4 6 1 6 qt. 3 7 P t. 1 4. Mi. yd. ft. 17 1375 2 7 938 2 □ rd. 35 13 d yd. d ft. d in. 14 6 81 25 7 108 21 18 7 117 2 36 6 4 2 21 19 1 9 Notes. — 1. In Ex. 5, when we come to the yards, we say, 4 from 8 1 / 2 yd. leaves 4 l / 2 yd. ; set down 4, and for the 1 / 2 yd. add 7. Cwt. lb. oz. i ft. 6 i n . to the feet and inches respectively. 15 33 11/4 2. In Ex. 6, when we come to the sq. yd., 8 98 14 V* we say, 26 from 44 x / 4 sq. yd. leaves 18 1 / 4 ; set — rj 1Q3 , down 18, and for the J / 4 sq. yd. add 2 sq. ft. b o4 1/6/4 an( i 35 sq . i n . to the remainder. 8. Tr. wk. d. h. min. sec. 9. Bu. pk. qt. pt. 14 2 20 31 52 169 2 1 1% 9 1 6 23 56 58 71 3 7 1% 10. Gal. qt. pt. gi 11. Mi. rd. yd. ft. in. 15 3 1 3% 26 230 4 2 10 7 3 13% 19 309 5 2 ll 7 /. MEASURES. 227 Multiplication. Example. — l. Seven bins of equal dimensions are full of wheat. On careful measurement one of them is found to contain 12 bu. 3 pk. 5 qt. How much is there in the 7 bins ? Explanation. — Seven times 5 qt. = 35 qt. = 4 pk. 3 qt. Written Work. 3 qt. being written under quarts in the multiplicand, the 4 Bu. pk. qt. pk. are added to seven times 3 pk. Seven times 3 pk. = 21 \<% 3 5 pk. 21 pk. + 4 pk. = 25 pk. = 6 bu. 1 pk. Seven times 12 „ bu. = 84bu. 84 bu. + 6 bu. = 90 bu. Hence seven times 12 bu. 3 pk. 5 qt. = 90 bu. 1 pk. 3 qt. 90 1 3 Note. — The pupil should obtain the result also by addition, and thus the rela- tion of addition and multiplication will be more deeply impressed on his mind. (See also page 55.) Explanation.— 12 times 5 / 8 in. = 60 / 8 = 1 4 / 8 2. Y d. ft- in. or <7i/ 2 i n . 12 x 7 in =84 in. 84 in. + 1 l / 2 5 2 7 % in. = 91 y g in. = 1 ft. 7 l / 2 in. 1 l / 2 inches being 12 written under the inches of the multiplicand, the ^ :j ^J7 rest of the process is similar to that explained ' 2 above. 3. Multiply 7 gal. 3 qt. 1 pt. 3 gi. by 156. If the multiplier is large, it is sometimes convenient to multiply by its factors, as in this case by 13 and 12. An advantage of this method is that all the written work is preserved as a part of the process. One process may be used to test the other. Examples.— 4. Multiply 3 lb. 8 oz. 18 pwt. 8 gr. by 35. (Employ factors of 35.) 5. Multiply 27 gal. 3 qt. 1 pt. 3 gi. by 36 ; by 236. 6. Multiply 17 wk. 4 d. 13 h. 27 min. 36 sec. by 9 ; by 79. 7. Multiply 23 cu. yd. 6 cu. ft. 459 cu. in. by 8; by 72. 8. Multiply 6 lb. 8 oz. 15 pwt. 19 8 / 13 gr. by 42 ; by 84. 9. Multiply ft, 9, I 11, 3 7, 3 2, gr. 17, by 17 ; by 36. 10. Multiply 1 ton 13 cwt. 73 lb. 9 oz. by 65 ; by 47. 11. Multiply 17 h. 47 min. 39 sec. by 25 ; by 124. Gal. qt 7 3 pt. gi. 1 3 13 103 2 3 12 1243 18 pwt. 8 1 gr. by 228 STANDARD ARITHMETIC. Division. Example. — l. If £13 8 s. 7 d. is equally distributed among 12 boys, how much does each receive ? w ... yj. , Explanation. — This example is taken from ' an old English work. It is accompanied with a 1 17.25 ? 230 STANDARD ARITHMETIC. Adding and Subtracting Denominate Fractions. l. Add % gal. and % qt. 2. Subtract % h. from % d. First reduce the fractions to integers, then proceed as above. Operation. Operation. % gal. = 2 qt. 1 pt. 2 / 9 d. = 5 h. 20 min. »/i qt. = 1 pt. % h. = 50 min. 3 qt. pt. 4 h. 30 min. Examples.— 3. Add % wk., % d., and % h. 4. Add Y, cw k, % lb., and % oz. 5. Add 2% bu., % pk., and y 3 qt. 6. Add 7 / 9 gal. and y 10 qt. 7. Add £%, % s., and 8 / 10 d. 8. Subtract y l6 h. from 6 / 7 d. 9. Subtract 2 3 / 4 sq. rd. from V/ A acre. 10. Subtract % oz. from 2 / 5 lb. Troy. 11. Subtract % pwt. from % oz. 12. Add Yie cwt. 10 3 / 5 lb. and 7 2 / 5 oz. 13. Add 3 / 5 of a ton, % of a cwt., and % of a lb. 14. 5 Y, miles - 5% fur. + 33 % rods = ? 15. 5 / 32 n mile -f 7 /io acre + 0. 75 p rod = ? 16. 4 / 7 of a wk. + 3 / 5 of a day + 5 /e of an h.+ y 4 of a min.= ? 17. Subtract 4 / 7 lb. avoirdupois from 4 / 5 lb. Troy. (Find the result in grains.) 18. From 2 17 / 36 lb. Troy take 19 / 96 oz. Troy. 19. Take 47 / 6 4 cwt. from 1355 / m2 T. 20. From 11% wk. subtract 8% d.; 8 6 / 7 h. 21. Find the sum of 4 / 7 cwt., 8 5 / 6 lb., and 3% oz. 22. Find the difference between 3 7 / n miles and 35 29 /33 rd. 23. Add 3 / 5 wk., % d., 5 / 7 h., and 2 / 3 min. 24. Add 4 / 5 of a pound avoirdupois and 3 / 7 of a pound Troy. Yr. mo. d. 1879 5 7 1868 6 12 MEASURES. 231 To find Difference of Time between Dates. Example. — l. How many years, months, and days from June 12, 1868 to May 7, 1879 ? Solution. — May 7, 1879, is the 7th day of the 5th month of 1879, and June 12 is the 12th day of the ^6th month of 1868. Hence, by subtracting 1868 years 6 months and 12 days from 1879 years 5 months and 7 days, we find the time elapsed from the earlier to the later 10, 10, 25, date. We consider 30 days a month, irrespective of what calendar months may intervene between the two dates. 24-8. This method, though usually employed in business, does not obtain the exact time elapsing from one date to another. A more accurate method is to find first the number of entire years between the dates, then of entire months, and lastly of the days. The difference between the results of these methods may be seen from solutions of the following problem : 2. A sum of money was borrowed Sept. 18, 1867, and paid March 16, 1870. How long was it in the hands of the borrower ? First Method. 1870 yr. 3 mo. 16 d. 1867 " 9 " 1 8 " 2 yr. 5 mo. 28d7 Second. Method. From Sept. 18, 1867, to Sept. 18, 1869 = 2 yr. " Sept. 18, 1869, to Feb. 18, 1870 = 5 mo. " Feb. 18 to March 16 = 26 d. Total : 2 yr. 5 mo. 26 d. 249. The first method is based on the supposition that there are 360 days in the year, and 30 days in each month. The second method takes no account of the number of days in the several years nor in the entire month, but reckons a year from a given day of one year to the corresponding day of the next, and a month from a given day of one month to the same day of the next. In reckoning the odd days, however, it takes into ac- count the number of days in the month preceding the last 232 STANDARD ARITHMETIC. 250, To find the exact number of days between two dates, we must add together the number of days in the several years, allow- ing 366 days for a leap year, then the number of days in the odd months, according to the calendar, and finally the number of re- maining days. Third Method. m , j From Sept. 18, 1867, to Sept. 18, 1868 = 366 days, wnoie years. -j „ Sept. 18, 1868, to Sept. 18, 1869 = 365 " Remaining j Sept. 12, Oct. 31, Nov. 30, Dec. 31, days. ( Jan. 31, Feb. 28, Mar. 16. = 179 " The exact time in days = 910 days. Examples.— Find the interval of time between the following dates by the first method : 1. Feb. 3, 1845, and Dec. 17, 1852. 2. Oct. 19, 1871, and Nov. 1, 1873. 3. Apr. 2, 1876, and Jan. 31, 1881. 4. Sept. 30, 1872, and July 2, 1879. 5. How many years, months, and days from the Declaration of Independence to the surrender of Cornwallis, at Yorktown, 1781, Oct. 19 ? 6. Find the preseut age of the American Republic, born 4th of July, 1776. 7. Washington was born Feb. 22, 1732, and lived 67 yr. 9 mo. 22 d. At what date did he die ? 8. Find the exact number of days of your own life. 9. A person born Dec. 8, 1845, died, aged 36 years, 1 mo. 18 d. What was the date of his death ? 10. General Grant died July 23, 1885, at the age of 63 yrs. 2 mo. 26 d. What was the date of his birth ? 11. Abraham Lincoln died Apr. 15, 1865, and General Gar- field Sept. 19, 1881. By what length of time did the death of each precede that of General Grant ? MEASURES. 233 Longitude and Time. 251. The line which may be supposed to be drawn from pole to pole through any place on the surface of the earth is called the meridian (mid-day line) of that place. All places having the same meridian have their noon at exactly the same moment. Since the earth revolves upon its axis from west to east, the sun seems to come from the east to each meridian successively, and thus to go around the earth from east to west in 24 hours. The circumference of the earth is divided into 360 degrees (360°), and inasmuch as it revolves once in 24 hours, 15° must pass under the sun in one hour ; y 60 of 15° = 74° = 15' of circum- ference in 1 minute of time, and y 6 o of 15' = *// = 15 ff of circum- ference in 1 second of time. Hence we have the following table, showing the correspondence of longitude and time. Table of Longitude and Time. 360° of Longitude correspond to 24 hours in time. 15° of " " 1 hour in time. 15' of " " 1 min. in time. 15" of " " 1 sec. in time. Note.— If three clocks, all keeping correct time, be placed at the distance of 15° longitude from each other, the one farthest east will, at any moment, be found one hour faster, and the one farthest west one hour slower, than the clock midway between them. ORAL EXERCISES. 1. If, in traveling, I find my watch, which is a reliable time- keeper, growing faster and faster as compared with the time in the places through which I pass, should I conclude that I am traveling eastward or westward ? 2. If I find my watch an hour fast, how many degrees have I gone, and in which direction ? If I find it half an hour fast ? 15 minutes ? 3. How many degrees of the earth's surface pass under the sun's vertical rays in 2, 4, 7, 13, 19, 21 hours ? 234 STANDARD ARITHMETIC. 4. I start from Cincinnati., and, on arriving at another city, compare my watch with a well-regulated clock, and find it faster than my watch ; have I traveled east or west ? I find it 20 min- utes faster ; how many degrees have I traveled ? 5. When it is 12 o'clock noon at Omaha, what is the time at places lying 15, 7 1 / 2f 3 % degrees west ? East ? 6. What difference in longitude makes a difference of 1 hour 30 min. in time? Of 1% min.? Of V/ 2 sec? 7. Suppose the sun is rising at 4 o'clock A. m. on the first me- ridian (Greenwich), on what meridian is it noon ? 8. What is the difference of time between Greenwich (on the first meridian) and a place lying under the 74th meridian ? SLATE EXERCISES 9. What is the difference of longitude between Washington and Cleveland, the difference in time being 18 min. 32 sec. ? Explanation. — One second in time corresponds to a difference of 15" of longitude, hence 32 sec. in time 18 mm. 32 sec. correspond to 32 times 15" in long. cs 480" = 8' in long. 15 One min. in time corresponds to 15' in long., hence to oqi ' a 18 min. in time correspond to 18 times 15' in long. = 270' of longitude. 270' + 8' = 278' == 4° 38' of longi- tude. Note. — Let it be kept in mind that, inasmuch as there are 360° in the circum- ference of the earth, and only 24 h. in the day, there are more degrees in any differ, ence of longitude than hours in difference of time ; more minutes of longitude than minutes in time, and more seconds of longitude than seconds in time. (How many times as many ?) 10. When it is noon at Washington, it is only 11 o'clock, 17 min. and 44 sec. A. m. at Chicago. Find (a) the difference in time ; (I?) the difference in longitude. To find Difference in Time. To find Difference of Longitude. a. 12 h. min. sec. h. 42 min. 16 sec. 11 17 44 15 42 min. 16 sec. 10° 34' 0" MEASURES. 235 252. The names of a few important cities are given below, with the longitude of each from Greenwich (see Hasweli, ed. 1885): Cities. Longitude. Cities. ] Jongitude. . Albany, N. Y. 73° 45 24" W. New Orleans, La. 90° 3' 28" W. Berlin, Germ. 13° 23' 45" E. New York, N. Y. 74° 24" W. Boston, Mass. 71° 3' 30" W. Paris, France. 2° 20 0" E. Calcutta, India. 88° 20 0' E. Philadelphia, Pa. 75° 9 3" W. Chicago, 111. 87° 37' 47' W. Rome, Italy. 12° 27' 0" E. Cincinnati, 0. 84° 29' 45" W. St. Louis, Mo. 90° 12' 4" W. Cleveland, 0. 81° 40' 30" W. St. Petersburg", Russ. 30 ° 19' 0" E. London, Eng. 0° 0" San Francisco, Cal. 122° 23 19" W. (Greenwich.) Washington, D. C. 77° 3«" W. 11. When it is noon at St. Louis, is it before or after noon at Washington ? At San Francisco ? 12. When it is noon at San Francisco, is it before or after noon at St. Louis ? At Washington ? 13. When noon in St. Louis, about what time in Calcutta ? 14. When it is midnight at Paris, what is the local time at Cleveland ? Solution. {1st step.) Cleveland 81° 40' 30" Paris 2 20 84° 0' 30" 15)84< {2d step.) ' 0' 30' i 5 1 i. 36 min. 2 sec. 12 h. 5 {3d step.) rain. 36 sec. 2 6 23 58 Ans.— 23 min. 58 sec. past 6, r. m. Explanation. — First step. — To find the dif- ference of longitude we add the longitude of Paris to that of Cleveland, since Paris lies east and Cleveland west of Greenwich. Second step.— Since 15° of longitude pro- duce a difference of 1 hour in time, 84° pro- duce a difference of 5 hours in time and some- thing more, for there are 9° more than 5 times 15° in 84°. The remainder, 9° = 9 x 60' = 540', which being added to the 2' in the next term, we have 542', etc. (The pupil should be able to take the remaining steps of the analysis with- out aid.) Third step. — Since Cleveland is west from Paris, the time of Cleveland is earlier than that of Paris, henco we subtract 5 h. 36 m. 2 sec from 12 h. 236 STANDARD ARITHMETIC. Rules for Computations in Longitude and Time. I. For finding difference of longitude, difference in time be- ing given : 253. Rule, — Multiply the difference in time by 15, and the hours, minutes, and seconds of time will give respectively °s, 's, and "s of longitude. II. For finding difference in time, difference of longitude be- ing given : 254. Rule, — Divide the difference of longitude by 15, and the °s, 's, "s of longitude will give respectively hours, minutes, and seconds of time. SLATE EXERCISES. Examples. — Using the longitudes given in the above table, Find the difference in time between 1. Albany and Chicago. 6. St. Louis and San 10. New York and New 2, Berlin and Paris. Francisco. Orleans. 3. Greenwich and St. 7. Home and Paris. 11. Washington and Petersburg. 8. Washington and Philadelphia. 4. Boston and Cleve- Calcutta. 12. San Francisco and land. 9. Cincinnati and San Calcutta. 5. Boston and St. Louis. Francisco. For oral exercises, let the differences in time be estimated. Rough estimates are as frequently required as exact computations. 13. When it is 12 o'clock noon at Greenwich, what is the time at each of the cities mentioned in the table (Art. 252) ? 14. Suppose it to be 8 o'clock p. m. (Post meridian or after noon) at Berlin, what time is it at the other cities ? 15. Suppose it to be 7 o'clock A. m. (Ante meridian or before noon) at New York, what time is it at the other cities ? 16. The difference in time between two places is 1 b. 15 min. 2G sec. ; what is their difference of longitude ? 17. Find the distance in geographical miles between two places on the equator that are 3 h. 2 min. 12 sec. apart. (1° = 60 geo. mi.) MEASURES. 237 Applications and Review. 1. If 1 cwt. costs $16.16, $18.25, $19.50, $25.25, $36.36, what is the cost of 328 lb. ? 2. If 1 gallon costs $2 4 / 5 , $3.50, $1%, $3.30, $4%, $2%, what is the cost of 5 y 4 gallons ? 3. If 1 pound costs 7y 2 francs, what is the cost of 3%, 2.5, 1%, 4.6, 5% pounds? 4. If 1 cwt. costs $127.64, what is the cost of %% 6%, 9%, 17y 8 lb. avoirdupois? 5. What will a lot, measuring 57y 2 Xl00 feet, cost, if the price of 1 n foot is $25%, $37%, $48 % $57%? 6. What will be the cost of 5%, 8%, 15 7* 23% oz., if 1 oz. cost 37%**? 7. What does a family spend for meat in a month of 30 days, at 43 3 / 4 ^ a day ? In a year of 365 days ? 8. If 5 gal. cost $1.15, $2.35, $4.25, $6.75, $7.45, $15.25, $20.20, what is the cost of 1 / 2 gallon ? (One step.) 9. If 4% lb. cost $2.32, $3.48, $4.28, $5.44, $6.64, $7.22, $9.04, what is the cost of %> lb.? 10. If 3% dozen cost $14.48, $28.36, $30, $16.75, $27.50, what is the cost of 1 gross ? 11. If 6 reams of writing-paper cost $7.20, $18.50, $20.25, $30.75, $50.15, what is the cost of 18 quires ? 12. If 9 acres cost $100.75, $140.25, $225.50, $350.40, what is the cost of 67.5 acres ? 13. If 7 barrels cost $48.25, $36.70, $64.83, $94.24, what is the cost of y 2 barrel ? 14. If 8 cords cost $24.25, $25.75, $26.40, $38.85, what is the cost of y 4 cord ? 15. If 1 cwt. costs $568.25, what will 20 lb. cost? 25 lb. ? 33y 3 lb. ? (Pursue the shortest method.) 11 238 STANDARD ARITHMETIC. 1. If % gal. costs 520, what will 2, 5, 7, 17, 46 gal. cost ? Analysis.— If 1 / 6 gal. costs 520, 1 gal. will cost 5 x 520 =$2.60, and if 1 gal. costs $2.60, 2 gal. will cost 2 x $2.60 = $5.20. 2. If Vb bu. costs 800, what will 5, 9, 13, 23 bu. cost ? 3. If y s dol. is paid per hour for labor, what is paid for 19, 23 h.? For 2% days, 10 h. per day ? For 3 days ? 4. If y i0 gal. costs 480, what will 7, 18, 32, 21 gallons cost ? 5. If y i3 lb. Troy costs $%, what will 9, 11, 19, 35 oz. cost ? 6. If 8 oz. avoirdupois cost $28.30, what will 17, 73, 85, 99 lb. cost ? 7. If % quire costs % fr., how many francs will 8, 13, 23, 45 quires cost ? 8. If % lb. costs $.07, $.09, $.11, what is the cost of 6 lb. ? 9. If % doz. pens cost y 10 dol., what will 6, 9, 17, 28 doz. cost ? 10. If 1 qt. costs 10& 120, 180, what will 18 bu. cost ? 11. If % doz. costs $.60, $.40, $.30, $.20, what will 1 gross cost? 12. If % doz. costs $y 6 , $y 3 , $y 8 , $y 9 , what will 1 gross cost ? 13. If l 3 / 4 thousand shingles cost $6 2 / 3 , what will 28 M cost ? 14. If 6 oz. Troy cost 9 / 10 dol., what will 5, 8, 30 lb. cost ? 15. If 1% bu. cost $2 7 / 8 , what will 19, 31, 84, 73 bu. cost ? 16. If 4% bl. cost $105, what will 8, 7%' ll 3 / 4 bl. cost? 17. If 3% cords cost $24 % what will 9, 12, 19 cords cost ? 18. If 3% oz. cost $.70, what is the cost of 5, 8, 17 oz.? 19. If 1% pk. cost $1%, what is the cost of iy 4 , 2%, 3 3 / 4 bu. ? 20. If 3% cwt. cost $46%, what is the cost of 3 3 / 4 , 6 1 /? cwt.? 21. If 1% doz. cost $l 3 / 4 , what is the cost of 2%, 7% doz.? 22. If 4.5 yd. cost $12.30, what is the cost of 7.7, 9.13 yd.? 23. If 3.48 lb. cost $1.24, what is the cost of 2.36, 9.81 lb.? MEASURES. 239 Square and Cubic Measures. Examples. — l. How many square yards of oil-cloth will cover a floor 14 ft. long and 12 feet wide ? Analysis.— The area of the floor is 14 x 12 ft. = 168 □ ft., = 18 2 / 3 □ yd., hence 18 2 / 3 □ yd. of oil-cloth will be required to cover the floor. 2. How many acres in a roadway 100 rods long and 18 yards wide ? 3. How many bricks will it take to pave a sidewalk 32 ft. long and 6 ft. wide, there being 4% bricks to the □ foot ? 4. How many yards of paper will be needed to paper a room 14 ft. long, 12 ft. wide, and 9 ft. high, if the paper is 18 in. wide, no deduction being made for doors and windows ? 5. How many rolls of paper of 8 yd. each will be needed to paper a room 18 ft. long, 15 ft. wide, and 10 ft. high, if the paper is 18 in. wide, and one roll is saved by the windows and doors ? Explanation. — Rolls of wall-paper 24 ft. long would make 2 strips each 10 ft. long, the strips not being pieced ; but, if the 4 ft. left were used under and over the openings, we would need to know how many times the surface of a roll is contained in the surface of the walls. (For paper-hangers' method, see p. 266.) 6. How many □ yd. of plastering are required for a room 20 ft. long, 15 ft. wide, and 10 ft. 6 inches high ? 7. How many shingles will it take to cover both sides of a roof, the rafters of which are 16 ft. long and the ridge-pole is 23 ft. long, if each shingle has an area of 162 □ inches, but % of it is covered by other shingles ? 8. How many cords of wood in a pile 36 ft. long, 10 ft. 6 in. high and 4 ft. wide ? Analysis. — Length x width x height = no. cubic ft., and 128 cubic feet are equal to 1 cord. 9. How many cords of wood in a pile 50 ft. long, 11 ft. 3 in. high, and 5 ft. wide ? 10. How many cubic feet in a room 16 ft. long, 14 ft. wide, and 9 feet high ? 66 ft. 240 STANDARD ARITHMETIC. 11. What is the capacity in gallons of a vat 10 ft. long, 3 ft. wide, and 4 feet deep ? 12. How many cu. ft. in a square tank, 2y 3 yd. wide and long, and 8 ft. 6 in. deep ? 13. How many qt. of milk can be put into a can containing 1496 1 / 2 cu. in.? Remember, a gallon fills the space of 231 cubic inches. 14. How many loads of earth must 4i ft. 8 in. ' be removed in digging a cellar to the depth of 6 ft., and of other dimen- sions as given in the diagram ? (A load is estimated to be 1 cubic yard.) 15. How much will it cost to pave the floor of this cellar at 14^ per □ foot ? How may bricks will it require if laid on edge 7 to the □ foot ? 16. How many square inches in the largest circle that can be cut from a card-board 2 ft. square ? (See note, page 243.) 17. A tank is 5 ft. 6 in. long, 5 ft. 3 in. wide, and 6 ft. 8 in. deep. How many gallons will it hold ? How heavy is the water Contained in it, if 3 ft. deep ? (A cubic foot of pure water at 62° weighs 997.68 oz.) 18. If a horse can draw 1600 lb. on a given road, how many cubic feet of lead can 2 horses draw on the same road, a cubic foot of lead weighing 709.5 lb.? How many men whose average weight is 165 lb. 12 oz. ? 19. A cubic foot of ice at the temperature of 32° weighs 57.5 lb. How many tons can be stored in an ice-house that is 80 ft. long, 30 ft. 9 in. wide, and 20 ft. deep ? 20. How many paper boxes 3 in. long, 2 in. wide, and 2 in. deep, can be packed in a box 3 ft. long, 2 ft. wide, and V/ g ft. deep ? 21. How many cubic inches in a can holding 16 gal. 1 pt.? MEASURES. 241 22. How many square feet in the floor of your school-room to each pupil present ? 23. At 3y 8 # a d yd., how much will it cost to have 3 ceil- ings kalsomined, each measuring 15 by 14 1 / 2 ft.? 24. At the same rate, what will it cost for kalsomining the ceiling and walls of a room 16 ft. long, 15 ft. wide, and 10 % ft. high, allowing 8y 8 □ yd. for doors and windows ? Analysis. — The ceiling contains 16 x 15 ft. = 240 □ ft. Two walls contain each 16 x lO 1 ^ ft. = 336 □ ft. Two walls contain each 15 x lO 1 /^ ft. = 315 □ ft. 99 □ yd. - 8 l / 2 □ yd. = 90 1 /* a yd. 891 □ ft. or 99 □ yd. At S 1 /^ what will 90 Va yd- cost? 25. What will it cost to paint a room 24 ft. long, 20 */, ft. wide, and 12 ft. high, at 10 1 / 2 each ? 20. A merchant bought cloth at $1.14 a m.; for how much per m. must he sell it to gain 1 / 3 of the cost ? 21. If I buy 6.328 Ml. at 5^ a 1., and sell it at 2 96 /24o> an d 100 /240- H ere 24 ° IS tne common standard of comparison, and the several offers are respectively equivalent to 105, 96, and 100 out of 240 bushels produced, whence we see that the first offer is the best. But the offer of V bushels out of 16, or 7 / 16 of a crop, is equivalent to the offer of 7 / 16 of each 100 bu., or at the rate of 43 3 / 4 bu. out of 100. Comparing all the offers with this standard, we find that they are equivalent to 43 3 / 4 , 40, and 41 2 / 3 bushels per hundred, respectively ; whence we readily see how the several offers compare with each other. 2. In like manner compare the value of two iron ores, one of which produces 52 tons of metal from 65 tons of ore, and the other 42 tons of metal from 56 tons of ore. Suggestion. — What common fractional part of each ore is metal ? How many tons of metal can be produced from 100 tons of each ore ? Note. — Because of its simplicity and convenience, 1 00 has been adopted as a standard of comparison in almost every department of business and by all civilized nations ; hence we hear of a boy's spelling a certain per cent, of the words dictated, that is, at the rate of so many in a hundred, and in like manner of the merchant gaining or losing a certain per cent, of the money he lays out for his goods, of an increasing per cent, of children who are near-sighted, etc., etc. 272 STANDARD ARITHMETIC, Definitions. 273. Per Cent, is an abbreviation of the phrase per centum, and signifies by the hundred. Caution. — The abbreviation cent, in the phrase per cent, has no reference to the cent of our decimal currency. 274. A Rate per cent, is a rate per hundred. 275. The sign % is annexed to the rate, and stands for the phrase per cent. Thus, 7 % is read seven per cent., .07 % is read seven hundredths of one per cent., 1 / 3 % is read V3 °* * P er cent., .00 1 / 3 % is read 1 / 3 of one hundredth of 1 per cent. 276. Any per cent, of a number is equivalent to so many hundredths of it. Illustration. — If 100 marks be made, 4 in line and 25 I I I I in column, the following questions may be asked : till What common fractional part, what decimal I I I I part, what per cent, of the whole number, are and 22 more Uneg in the top line ? etc. One mark is what common fractional part, what decimal part, and what per cent., of 25 marks ? Additional marks being made at the foot of a column, it may be asked : What per cent, is added to the marks of the column ? that is, How many additional marks would be made in all if 1 were added to each 25 in the hundred ? What fractional part, and what per cent, of all the letters in the italicized lines below, are contained in the first word, in the first two ? etc. In each part of eighty-four f In watchful ? etc. Manuscript, importance, regulation, house-plant, county maps, blind-mouse, eighty -four, be watchful, cash profit, spring-halt. What per cent, of all the letters in either line are a's, b's ? etc. What per cent, of the letters in the word Oconomowoc are o's ? What percent, are c's ? etc. What per cent, of the letters in Ohio are vowels ? Are consonants ? What # of the numbers from 1 to 100 are primes ? From 101 to 200 ? etc. PERCENTAGE. 273 SLATE EXERCISES. The pupil who desires to become expert in computations of percentage should be able to convert per cents, into corresponding decimal and common fractions almost at sight. Write the decimals and the common fractions that are equiv- alent to the following expressions (all common fractions should be given in lowest terms) : 1. 100 200 300 4. 800 80 160 2. 250 500 750 5. 140 120 50 3. 40 400 600 6. 150 240 650 Example. — What common fractional part of a number is equiv- alent to 12 % per cent, of it ? 101/ 12 %0 is equivalent to .12% = -^ 12%__25/ _1/ 100 ~" /200 "" /8 Or, since the numerator, 12 1 / 2 , is an aliquot part of the denominator 100, the reduction can be made directly by dividing both numerator and denominator by 12 %> thus: 12%.+ 13% 1/ 100 ~ 12% /8 In like manner the following rates per cent, are all reducible to simple common fractions. What decimal and what common fractions are equivalent to 1. 2%0 18%0 31%0 5. 41%0 58%0 m^i 2. 37%0 43%0 56%0 6. 83%0 91%0 6%0 3. 62%0 68%0 81%0 7. 87%0 93%0 6%0 4. 8%0 16%0 33%0 8. 3%0 13%0 23%0 What per cents, are equivalent to the following fractions : 9. % % % % % 14. .04 .40 ' .41% .14 10. % % % % % 15. .75 .075 .91% .12% n. %* % % % % 16. .00% .187% .83% .18% 12- % % %o Vio Vn 17. .0075 .005 .58% .37% 13. % 5 %of% %of% 18. .625 .00625 .56% .00% 274 STANDARD ARITHMETIC. Applications. — Example. — l. I bought a horse for $280 and sold it so as to gain 25$. How much did I gain ? Analysis. — At 1$ (1 to a hundred) I would gain $2.80, and at 25$ I would gain 25 times as much. 25 x $2.80 = $70 Ans. Or, since 25$ = .25 or T / 4 of a number, to obtain 25$ of 280 we may take 1 / 4 , of it, */ 4 of $280 = $70. For exercises, see Case I, pages 276-279, Example.— 2. If the horse was bought for $280 and sold so as to gain $70, what per cent. was gained ? Analysis. — At 1$ I would gain $2.80, and to gain $70 the rate of gain would have to be as many times 1 $ as $70 is times $2.80, which is 25. 25 x 1$ = 25$ Ans. Or, the gain $70 is 70 / 280 or 1 / 4 of the price paid. */ 4 of any number = 25$ of it. For exercises, see Case II, pages 280, 281. Example. — 3. If the horse was sold so as to gain $70 at the rate of 25 $, what was the price paid f Analysis.— If $70 is 25$ of the price, 1$ is l / t8 of $70 = $2.80, and 100$ or the whole price = 100 times $2.8 = $280 Ans. Or, if 70 is 25$ or 1 / 4 of the cost, the cost must be 4 times $70 = $280. Written Work. 2.80 = 1$ 25 1400 560 $70.00 = 25$ Written Work. 2.80)70.00(25 560 1400 1400 25X1 Written Work. 25)70.0(2.8 = 50 200 200 25$ 1* 100 X 2.8 = $280 For exercises, see Case III, page 281 ; and Case IV, page 282. 277. The three principal cases of percentage are presented in the foregoing examples. They are : I. To find a required per cent, of a number. II. To find what per cent, one number is of another. III. To find a number from a given per cent, of it. A fourth case is added, which differs from the third in nothing except that the rate per cent, to be operated with is derived from the given rate by adding it to or subtracting it from 100. (See Case IV.) PERCENTAGE. 275 Definitions. 278. The result of taking any per cent, of a number is called a Percentage. 279. The number of which a percentage is given or on which it is to be computed, is called the Base. 280. The base plus the percentage is called the Amount. 281. The base minus the percentage is called the Difference. Rules. I. To And the Percentage, the Base and the Rate % being given. Mule.— Multiply 1 % of the base by the rate %. II. To find the Rate #, the Base and Percentage being given. Mule. — Divide the percentage by 1 % of the base. III. To And the Base, the Rate % and the Percentage being given. Mule.— Divide the percentage by the rate % and multiply the quotient by 100. Formulas. 282. The process of finding a given per cent, of a number may be indicated by signs, as follows : I. Rate % x 1% of the Base = Percentage. Since the rate per cent, is thus presented as the multiplier, 1$ of the base as the multiplicand, and the percentage as the product, we derive from this formula two others, as follows : II. Rate = Percentage -i- 1 # of Base. III. 1 £ of Base = Percentage -f- Rate #. Note. — It is best that the pupil should not become accustomed to depend on the formulas as he should not rely on rules to direct his solutions ; but if he uses a formula at all, it is best that he should refer exclusively to the first. If he under- stands that, it will suggest the others readily enough. 283. Let the learner accustom himself to compute percentages in the short* est way possible. Thus, since 25$ is equivalent to .25 or */ 4 of a number, we may obtain 25$ of it either by multiplying the number by .25 or by l / 4 , that is, taking x / 4 of it. 12 72$ is equivalent to .12 x /2 or */• of a number, etc. 276 STAtfDA&D ARITHMETIC. Case I.— To find the Percentage, the Base and the Rate % beiny given. EXERCISES. It is supposed that the majority of pupils will be able to note the results in the following exercises without the aid of written solutions. As far as practicable, the work should be purely mental. Find 1. 20# of 5; of 25; of 35; of 45; of 75. 2. 25# of 4; of 28; of 36; of 44; of 144. 3. 4# of 25 ; of 75; of 125 ; of 175 ; 75$ of of 350. 4. 12%$ of 1600 ; 62%$ of 4000; 8000. 6. 16%$ of 1860 ; 33%$ of 2424; 18% of 1600. 6. 9$ of 900; 7$ of 800 ; 12$ of 1200. 7. 8%#of 24; of 72; of 1440 ; of 84; of 90. 8. 37%$ of 40; of 84; of 4000 ; of 96; of 14. 9. 66%$ of 36; of 69; of 36000 ; of 53 ; of 71. 10. 6%$ of 64; of 32; of 64000 ; of 76 ; . of 80. 11. 31%$ of 80; of 20; of 80000 of 60; of 75. 12. 87%$ of 12; of 94; of 12000 ; of 86 ; of 63. 13. 8$ of 37%; of 62%; of 87%; of 6%; of 4%. 14. 9$ of 22%; of 33%; of 66%; of 77%; of 44%. 15. 50$ of %; of% 6 ; of 8 /.; of % ; of "/ 21 . 16. 6$ of 150; of 375 ; of 245 ; of 180 ; of 65. 17. 16% of 12; of 42; of 54; of 66; of 72. 18. 37%$ of 32; of 48; of 1.6; of 2.4; of 5.6. Note. — By taking any common per cent, of two or more bases separately, and adding the percentages thus obtained, the same result is reached as by taking a like per cent, of the sum of the bases. Thus, by taking 20$ of 5, 25, 35, 45, and 75, we obtain 1, 5, 7, 9, and 15, the sum of which is 37; and by taking 20$ of 185, which is the sum of the bases 5, 25, 35, 45, and 75, we obtain the same result, 37. In this way each pupil may test the correctness of his results when a common per cent, of two or more bases is required, as in the several lines of the foregoing exercises. No answers are given to these exercises. SLATE EXERCISES. &i i 19. Find 41%* of $97.68. m Solution. 1$ of $97.68 = = $.9768. 41 ! y 3 = 4i%x $.9768 = $40.70. Find 20. 31$ of 6.25 ; of 0.75; of 4.55; of 16%; of 19%. 21. 29$ of 752; of 8.61; of 58.4; of .378; Of 2 %3- 22. 105$ of 100 ; of 20 ; of 120 ; of 140 ; of 160. 23. 117$ of 49; of 51 ; of 79; of 117 ; of %. 24. 25. 26. 27. 28. Find 20$ of 16%$ of 37%$ of 18$ of 34$ of $7.80 $18.37% $24.85 $13.12% $42.75 5.30 15.25 19.23 9.56 83.16 2.70 32.18% 37.82 12.21 19.88 6.20 17.75 19.06 15.66% 71.21 1.18 12.62% 18.71 5.18% 88.45 2.27 52.40 15.01 1.87% .18 4.38 64.80 18.27 2.25 .24 1.07 51.13 1.65 12.68% 1.31 Caution. — Let it be remembered that 20$ of an y number = l { 5 of it, that 16 2 / 3 % — x / 6 , etc. Shorten the work as much as possible. No answers are given to Examples 24-28. The correctness of results may be tested by comparing the sum of the percentages with the like per cent, of the sum. (See Note, page 276.) Each column may be conveniently divided into two or more shorter ones. Loss and Gain. Example.— l. A lot is bought for $1285 and sold at a gain of 15 $. How much is gained, and what is the selling price ? Analysis.— 1% of the cost is $12.85, and 15$ of the cost is 15 times $12.85 = $192.75, which is the sum gained. The gain $192.75 being added to the cost of the lot, we obtain the selling price, $1477.75 $1477.75 Selling price $12.85 = : 1? {, of cost m 15 6425 1285 $192.75 = : 15 * 1285 278 STANDARD ARITHMETIC. 2. If the lot is bought for $1285 and sold at a loss of 15 $, how much is lost, and what is the selling price ? Here the first step of the solution is the same as that of the preceding problem, but inasmuch as there is a loss of 15$ in this case, $192.75 is deducted from the purchase price to find the selling price. 284. When the purchase price and gain or loss per cent, are given to find the selling price, the following is the more convenient process : $2 88 = 1 io 3. If I buy goods for $288 and sell them 145 at a gain of 45$, what do I receive for 1440 them ? 1152 Analysis. — If the goods are sold at a gain of 45$, 288 they are sold for 145 % of what they cost me. 1 % of the 4417 60 == 145/ cost * s $ 2 - 88 > an d 145 f° ls 145 times $2.88, which is equal /0 to $417.60 Ans. If the goods were sold at a loss of 45 %, the selling price would be 55 % of what they cost me. 1 % of $288 = $2.88, and 55$ = 55 x | 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. To $75 add 33%* 1.25 " .48 " 1.26 " 1.54 " .81%" .72 " .84 " .72 " .96 " 1.44 " 37% i 62%* 18* 25* 16%* 45* 37%* 12%* 116%* 15. From 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. = $158.40 Ans. .64 take 18%; 20* 15* 45* 83% .18 2.65 1.92 .94 1.57 9.85 6.88 3.04 5.75 9.58 14* 23* 56%* 43%* 23%* 87%* 26. Find 12%*, 18%*, 25*, 6%*, and 37%* of $147.36, and add together the percentages. (Why is the sum equal to the base ?) 27. If the sum of 18%* and 31%* of 1876 be subtracted from 1876, how many will remain ? (Solve orally.) 28. How many will remain if 2%* of 897,659 be subtracted from 12 %* of the same number ? (Solve orally.) PERCENTAGE. 279 Applications. — l. A little boy who has 8 apples gives 25 £ of them to his brother, 12 7>$ to his sister, and 50$ to his mother. What per cent, and how many has he left ? 2. Charles sold his sled, which had cost him $1.75, at 20$ below cost. How much did he get for it ? 3. A lot of damaged calicoes are to be sold at 75 $ below the marked price. What prices must be asked for those that are marked 8^, 10^, 12 %& 160, 20^, 30^ ? 4. A grain dealer bought wheat for $9384, and sold it at a gain of 47 2 $- What did he receive for it ? 5. If a man owes $2500, and agrees to pay it in 4 instalments, the first to be 50$ of the whole, the second 25$, the third 15$, the fourth 10 $, what will each instalment be ? 6. A man having 1000 bushels of apples, sold 5 $ of them at $1.25 per bushel; 8$ of the remainder at $1 per bu. ; 50$ of what was then left at 75^ per bu., and the rest at 60^ per bu., thus receiving 10$ more than he paid ; how much did he pay for the whole quantity ? 7. Mr. Brooks bought a farm, which was in very poor condi- tion, for $1586 ; and, after two years of careful cultivation, which paid for itself with some improvements, he sold it for 65$ more than he paid for it. What did he sell it for ? 8. The number of inmates in a workhouse 5 years ago was 110 ; this number has since increased 180$. How many inmates are there now ? 9. A merchant bought goods for $297.70, and paid an addi- tional sum equal to 7$ of the purchase price for cartage, freight, etc. What must he sell them for to gain 40$ on the whole cost ? 10. In a mixture of alcohol and water 85$ is alcohol. How many gills of alcohol in 3 gallons of the mixture, and how many gills of water ? 11. 560 bushels of wheat, bought at $1.10 per bu., were sold at a profit of 10$. What did the wheat sell for ? 280 STANDARD ARITHMETIC. Case II.— To find the Rate & the Base and Percentage being given. ORAL EXERCISES. What per cent, of 1. 10 is 1 ? 5 ? 10 ? 20 ? 30 ? 40 ? 50 ? 60 ? 70 ? 80 ? 2. 50 is 9 ? 12 ? 15 ? 18 ? 30 ? 45 ? 50 ? 100 ? 125 ? 3. 200 is 25? 75? 125? 250? 12%? 87%? 16%? 62%? SLATE EXERCISES. 4. $212.62% is what per cent, of $486 ? Solution. $212.62% -v- $4.86 = 43.75. 43.75 X 1$ = 43%# Ans. Analysis. — 1 % of $486 = $4.86, and $212.62 : / 2 is as many per cent, as $212.62 l / t are times $4.86, which is 43 3 / 4 . 43 3 / 4 times 1 % = 43 3 / 4 % Am. What per cent, of 6 * 225 is 9 ? 6. % is .03 ? 7. 5% is 5.5? 8. .25 is .0175 ? 9. 6.45 is .32%? io. 1 is % ? 11. 45 is .3 ? 12. .1879 is 18.79? 13. 55 is 167%? 11.25 ? .045? 1.1? .27? .25 %? %5? .25? 187 % : 29.25 ? .0.6? 5.22%? .3? .451%? %o? .36? 281.85? 33.75? .075? 27.? .295? 38.25? .09? .825? .337% ? .580%? 1.29? %? %? .15 ? .05 ? 319.43? 394.59? 2000? 660.27%? 550.22? 112%? . 14. What per cent, is 26%, 29%, 33%, 36%, of 175? 15. What per cent, is 49.5, 56.25, 58.50, 63, of 225? 16. What per cent, is .024%, A% .06%,, .09%, of % ? 17. What per cent, is .4%, 4.9%, 4.67%, 1.3%, of 5%? * Answers: 4$, 5%, 13$, 15$, and 17$. The sum of thes^ rates is 54, and 54$ of 225 is 121.5, which is equal to the sum of the given percentages. In like manner the pupil may test for himself the correctness of his answers to the remain- ing questions PERCENTAGE. 281 Applications. — 1. A boy buys an old pair of skates for 50^ and sells them for 25^. He then buys a pair for 25^ which he sells for 50^. What per cent, did he lose on the first pair ? What per cent, did he gain on the second ? 2. If a dealer buys a hat for $3, and sells it for $4, what $ does he gain ? If he buys it for $4 and sells it for $3, what per cent, does he lose ? 3. One hundred pounds of beef were sold for $6, having been bought @ 4^ a lb. What per cent, profit ? 4. A retail dealer in boots and shoes sold 50 pairs of boots for $300. They cost him $5 a pair. What rate per cent, did he gain ? 5. A merchant bought goods for $500. What per cent, would he gain by selling them for $530 ? For $525 ? For $550 ? For $540 ? For $560 ? For $575 ? For $600 ? For $1500 ? 6. The price of a single ticket from Glenwood to New York city is 30^, but 20 coupon tickets can be bought for $5. What per cent, is saved by buying coupon tickets ? What per cent, is lost by buying single tickets ? 7. 10$ of a flock of sheep were killed by dogs; 6%$ of the rest were lost ; 33V 3 $ of the remaining number were sold, and 28 then remained. What was the original number ? 8. At harvest time a farmer sold 60 bushels of wheat, which was 25$ of the quantity he seut to mill, and what he sent to mill was 40$ of what he kept over till the next spring. How many bushels had he at first ? 9. When a merchant sold his goods for $261, he gained twice as much as he would have lost had he sold them for $207. What was his gain per cent. ? (How many times the loss is the difference between $261 and $207?) 10. A grocer sold butter at 12$ profit. Had he sold it for 2$ more per pound, he would have gained 20$. What did 50 pounds cost him ? 282 STANDARD ARITHMETIC. Case III.— To find the Base, the Percentage and Bate % being given. ORAL EXERCISES. 1. Ten apples are 1 / 2 of how many apples? 50$ of how many ? 2. Eight bushels are % of how many bushels ? 16$ of how many ? 3. 25 tons are % of how many tons ? 25 $ of how many ? 4. 9 is % of what number ? 20$ of what number ? EXERCISES. 5. 234 is 56%$ of what number ? Analysis.— If 234 is 56 1 / i % of any oo u ion. number, 1 % of the number is such part 234 -f- 56 / 4 = 4.16 = 1$. f 2 34 as is found by dividing 234 by 416 = 100$ or the number. 56 y 4 , which is 4.16 and 100$, or the number itself is 416 Ans. Or, since 56 ^^ of any number equals 9 / 16 of it, 234 is 9 / 16 of the number sought. If 234 is 9 / 16 of the number, the number itself is 16 times 1 / 9 of 234 = 416 Am. Find the number of which 6. 3 is 10$ 19. 45 is 5$ % . 7. 20 is 20$ 20. 22 is %$ 32 ' % 1S 16 /s * 8. 18 is %$ 21. 2 is 80$ 33. 99.9 is 1.75$ 9. 56 is 2y 3 $ 22. 100 is 66 2 / 3 ^> 34. .001 is 8%$ 10. 75 is 1$ 23. 210 is 105$ 35. 81 is 9$ 11. 125 is 95$ 24. 65 is 14%$ 36. 195 is 200$ 12. 40 is 62%$ 25. 16 is 33%$ 37. 95 V* is 8y 3 $ 13. 7 is 12V 2 $ 26. 35 is 41%$ 38. % is 0.9$ 14. 11 is 87%$ 27. 525 is 25$ 39. 2001 is % $ 15. 20 is 33%$ 28. 11% is %$ 40. 6.25 is 37%$ 16. 14% is 14%$ 29. 232 is 29$ 41. 7 is 2% 17. 19 % is 62%$ 30. 38 is 3%$ 42. 999 % is 100^ 18. 5 is 20$ 31. 12% is 12% 43. 87% is 50$ PERCENT A GE. 283 Case IV.— To find the Base, the Rate % and the amount or dif- ference being giren. I. The retail price of a certain article is 68^. How much can the retailer pay for it to realize a gain of 33 % ? aQ j 1 oqi/ d f f Explanation. — If when the article sells for 680 b Sff — 166 / 3 ff Ot COSt there ig tQ be a gain of 33 1/3 per cent ^ 6g ^ mugt be 133 Y 3 ) 68 133 1 / 3 % of the cost; hence this problem is similar 3 3 to that of Case III, which is to find the base, the per- iod ATTu centage and rate % being given. ±vv )&.yj± 0r ^ if when thc artic]e sel i s for 68 tuere is t0 be .51 = 1 # a gain of 33 x / 3 # = >'/i of the cost, 680 must be 4 / 3 gi _ cos f of the cost; hence the cost must be 3 times */ 4 of 68. 3 times J / 4 of 68 = 510 Am. What number increased by 2. 10 i of itself equals 110 ? 7. % % of itself equals 9.06 ? 3. 75 * of itself equals $420 ? 8. % # of itself equals $81.72 ? 4. 62 %# of itself equals 89.37V 2 ? 9 1 /* „ . 10 c oiw #u-i# , i qokp \ 9. ^?<£ of itself equals $90. 342? 5. 21.5$ of itself equals 32.562? 25^ ^ 6. 83 y s of itself equals $87.12 ? 10. 43 3 / 4 $ of itself equals $1.38? II. I am charged $2.50 for a book, which the bookseller says is 33 y 3 io less than it cost him. What was the cost ? Explanation.— If when the book sells for $2.50 „„ 2/ . 9 ^ n there is a loss of 33 V, & the $2.50 must be 66 2 / 3 % bb /3/*- oU of the cost; hence this problem also is similar to 3 3 that of Case III. 200 )7.50 Or, since $2.50 is 66 2 / 3 % or 2 / 3 of the cost, AQ7* 1 e/ the cost must have been 3 times l / s of $2.50 — $3.75 Am. 3.75 = 100$ What number diminished by 12. 5$ of itself equals $6.65 ? 16. 87y 2 $ of itself equals 10 ? 13. 5 i of itself equals 19 ? 17. 16 2 / 3 $ of itself equals 95 5 / 18 ? 14. 20 of itself equals 80 ? 18. 5 5 / 8 $ of itself equals 67.95 ? 15. 9$ of itself equals 9y l0 ? 19. %$ of itself equals 216.38 ? Note. — No special rule is needed for Case IV, thc process of solution being the same as that of Case III. 284 STANDARD ARITHMETIC. Applications. — l. William buys a penknife for 20^ and sells it to James for 25^, and James sells it to Fred for 20^. What per cent, does William gain, and what per cent, does James lose ? 2. If the 25 minutes of school time given to recesses are 8y 3 $ of the daily session, how many hours in the session ? 3. If a book is marked to be sold at 25$ above cost, but it is sold at 20$ below the marked price, what was the gain or loss per cent. ? 4. If 80 pounds of coffee are exchanged for 120 pounds of sugar, what $ is the coffee worth per pound more than the sugar ? 5. What per cent, do I gain by selling an article for $3 for which I paid $2.25 ? What per cent, do I lose by buying an article for $3 and selling it for $2.25 ? 6. A drover sold a horse for $226, and thus gained 25 $. What did he pay for him ? 7. The assets of a business man are $135,700, which sum is 43$ of his debts. What is his indebtedness ? 8. A fruit dealer sold a lot of oranges for $337.50. which allowed him a profit of 127 2 $. What did he pay for them ? 9. A city lot was sold for $25,500, the gain on the cost being 325$. What was the cost ? 10. A grocer sold 300 bushels of potatoes for $285, which was 16 2 / 3 $ less than he had paid for them. How much did they cost him per bushel ? 11. A. sold goods at a gain of 18$. His profit was $29.70. How much did he sell them for ? 12. By selling a lot of goods for $380, I gain 3 times the per cent, that would be gained by selling them for $340. What per cent, is gained in the latter case ? ($380 — $340 = 2 times the gain.) 13. In the schools of a village yesterday there were 1235 pupils present, which was 95$ of the whole number belonging. How many belonged to the schools ? PERCENTAGE. 285 Trade Discount. 285. A discount is a deduction from a price, from the amount of a bill, or other account. 286. In some branches of business it is customary to have fixed price lists of certain kinds of goods, and, when a rise or fall of prices occurs, instead of changing every price on a long list, the rate of discount is changed. 287. The fixed price is called the List Price, and the dis- count is called Trade Discount. The Net Price is the list price minus the discount. Example. — l. If penknives of a certain quality are sold at $18 per doz., with a discount of 33 l / 9 jf s what is the net price ? 2. How much must be paid on a bill of $5560 for books if 20 discount is allowed on account of the great number of books sold, and a second discount of 5$ is made for cash ? Each successive discount is made from the results of preceding discounts. Find the net prices : List prices. Discounts. 8. $5.37 250 and 330 9. $4.82 400 " 300 10. $6.72 300, 100, and 50 11. $3.98 400, 200, " 100 12. $4.97 500, 100, " 10 Note. — To find a single direct rate of discount equivalent to two successive dis- counts, deduct from the sum of the two rates either per cent, of the other. Thus : GO and 10 off = 60 + 10 - 10# of 60 = 70 - 6 = 64. List prices. 3. $5.40 Discounts. 250 and 100 4. $6.56 5. $8.35 6. $7.80 400 " 200 600 " 50 500 " 300 7. $6.75 100 " 100 13. A bill of hardware at list prices amounts to $276.98, the discounts are 400, 12*40, and 100. What is due on the bill ? 14. What is the difference on a bill of $780, between a direct discount of 250 and successive discounts of 100, 100, and 50 ? 16. If the list price of a certain size and quality of slates is $12 per gross, shall I gain or lose by buying 15 gross of Mr. Brown, whose discounts are 250 and 100, instead of from Mr. Green, whose discounts are 200 and 100 and 50 ? 13 286 STANDARD ARITHMETIC. Insurance. Insurance is security against loss by fire, water, accident, etc. Life Insurance is a contract for the payment of a specified sum at the death of the insured or at the end of a specified time, though he may be still living. The Premium is the sum paid for insurance. It is usually computed at a given rate per cent, on the sum insured. The Policy is the written contract between the insurer and the insured. The insurer is called an Underwriter, because his name is written under the policy. ORAL EXERCISES. 1. What will be the premium, if I insure my house for $2000 at 10? Aty>0? At 20? At%0? 2. What is the premium on an insurance of 1600, $400, $800, $1900, $2400, $100,000, at 10 ? At 20 ? At y 3 ? At iy 2 ? At iy 4 0? At 2%0? At %0? 3. A vessel is insured for $45960 at y 4 0. Find the premium. WRITTEN EXERCISES. 4. A match factory is insured at 4y 2 0; the premium being $217.50, for how much is it insured ? 5. A barn was insured at the rate of 3 / 4 0; the premium was $19.50. What did the owner receive when it was burned ? 6. At a rate of 5 0, a shipper pays $213. 95 for the insurance of 3 / 4 of the value of his goods. What was their value ? 7. Find the the sum of the premiums paid for the following insurances : $4000 at 5 / 8 for 1 year, $3200 at iy 4 for 2 years, $5000 at iy 2 for 3 years, $2500 at 2y 2 for 4 years, $3500 at 20 for 3 years, $2200 at 7 / 8 for 1 year, $5400 at %0 for 1 year, $3600 at 2y 4 for 5 years, $4700 at iy 4 for 2 years. (The rates here given are not annual, but for the times specified.) Note. — Great risks are commonly distributed in small amounts to many different companies. (Why ?) 8. A building is insured in 19 companies for $2500 each, in 9 others for $5000 each, and in 4 others for $3500 each. What was the total annual premium at 3 / 5 ? PERCENTAGE. 287 9. The goods in the building just mentioned were insured as follows : in 1 company for $10000, in 1 for $9000, in 16 for $2500 each, in 7 for $3500 each, in 4 for $1500 each, and in 1 for $1000. What was the total premium paid annually at 75^ (per $100) ? (75^ per $100 = .75$ or 3 / 4 #.) 10. What is the rate at which a factory is insured for $5250, if the premium is $6,5.62% ? 11. The Ohio Mutual Insurance Company insured my house for $5800 for a period of 3 years at 1 % $. What was the premium ? 12. The cargo of steamer Gallion, bound for Liverpool, is insured at 1 / 2 $. For what sum is it insured, the premium being $1500 ? 13. My house cost me $8400. I insured it for 3 / 4 of its value, at % $ per year. My books and furniture were insured for $3000 at the same rate. What did I pay annually for insurance on both ? 14. If you have your life insured for $5000 at $15.50 on $1000 annually, what premium do you pay ? 15. When 30 years of age a man insures his life for $8500, at the annual rate of $22.70 on $1000. If he dies when 60 years old, how much more do his heirs receive than he had paid for in- surance ? 16. Suppose the man above mentioned had been insured from his 20th to his 60th birthday, how much would the sum of his annual premiums have fallen short of the sum insured, the rate being 1%0? 17. A manufacturing company paid $214.80 premium for in- surance on % of the cost of its buildings and machinery, at 60^ per $100. What was their cost ? 18. If in 1 year an insurance company takes the insurance of 1000 dwellings at 2 / 3 i on an average valuation of $3000, and pays to its agents 15 # of the amount received for premiums, what balance remains for profit and to meet the expenses of the company if 1 of the houses is totally destroyed by fire ? 288 STANDARD ARITHMETIC. Commission and Brokerage. An Agent is a person who is authorized to transact business for another. The person for whom he acts is his Principal. Commission is the allowance made to an agent for transacting the business of another. It is usually reckoned at a certain rate per cent, on the sum of money invested or realized, sometimes at a certain rate per bushel, barrel, bale, etc., bought or sold. Agents are known by various names, as, commission merchants, brokers, col- lectors, correspondents, etc., according to the nature of their business. A Broker effects bargains and contracts for and generally in the name of others. The broker does not take possession of the property bought or sold. The name of broker is erroneously applied to those who deal in stocks, bonds, etc., on their own account. The commission allowed to a broker is called Brokerage. A Commission Merchant buys and sells on account of others, but in his own name. He has the merchandise in which he deals within his immediate control. His commission is usually greater than that of a broker. A Consignment is a quantity of merchandise sent by one party to another. The one that sends it is called the Consignor; the one to whom it is sent is called the Consignee. The Gross proceeds of a consignment is the whole amount for which it is sold ; the Net proceeds is the sum due the consignor after deducting commission and all other charges. The calculations in commission and brokerage are simple applications of the rules of percentage. ORAL EXERCISES. 1. If my commission for selling an article for $450 is 4$, how much do I receive ? How much at 4t,%£ ? At 5$ ? At 15 # ? At 25$? 2. An agent sold a piano at $350, and received $35 commis- sion ; what rate per cent, is that ? What rate if he had received $14? If $17.50? 3. What will be the fees of a collector of taxes on $1,200,000 if allowed 1%*? Ifl 1 /**? If 1 3 / 4 ^ ? 4. Find the commission on $200, $220, $250, $300, $580, at y 4 & y 8 fo. 5. A broker buys 5 tons of currants at $8.50 per cwt. What is his brokerage at 2 fo ? PERCENTAGE. 289 WRITTEN EXERCISES. 6. An agent purchases 5 tons of raw sugar at &%# a pound, and charges 2y g # commission. How much money must be sent to him to cover the cost and commission ? 7. I sell through my broker 7 tons of Brazil nuts at $7.50 per cwt. How much do I receive if the broker charges 1 fo for selling ? 8. A broker charged $74.25 for effecting a loan of $3300. What was the charge per cent. ? 9. A fruit broker sold $680 worth of apples, and after deduct- ing 5fo commission and 20$ for freight and other charges, in- vested the balance in oranges. How much did he invest in oranges if he charged 2$ for buying ? Explanation. — Charges and commission, together amounting to 25$ of the whole sum received for the apples, being deducted from $680, there is a remainder of $510, with which the broker is to buy oranges and pay himself 2% on the pur- chase price. If now the broker were to buy a dollar's worth of oranges at a time, and each time to pay himself 2^, it is plain that he would expend for oranges only as many times $1 as there are times $1.02 in $510. 10. John Wells & Co. sell $150 worth of eggs for W. Smith, charging him 2 1 / 2 # commission. They invest the proceeds in groceries, and charge 2$ for buying. How much do they invest ? n. A shoe manufacturer forwarded 50 dozen pairs of shoes to his agent in New York, who sold them at $42.60 per doz., charging 5 io commission. He purchased leather with the proceeds, charging 2fo for buying. What was his total commission ? How much did he pay for the leather ? 12. A cotton dealer in New Orleans ships $10,000 worth of cotton to his broker in New York, with instructions to purchase dry goods and hardware with the proceeds. The broker charges 2y 2 $ for selling the cotton and 2f for buying. How much does he invest, and what is his total commission ? 13. A commission merchant having sold a consignment for $3578, retains $95.70 to pay charges amounting to $6.25 and his own commission. What rate per cent, commission did he charge ? 290 STANDARD ARITHMETIC. 14. A commission merchant sold 500 lb. of butter at 180 per pound, and invested the proceeds in oats at 42^ a bushel. He charged 4%$ for selling and 1%# for buying. What was his total commission, and how many bushels of oats did he buy ? 15. Sold a consignment of merchandise for $5000. What was the balance due the consignor after the deduction of $110.50 freight, $250 duty, cartage, and storage, $75.40 insurance, and 5$ brokerage ? Stocks. 288. 1. Here the pupil needs to know what stocks are, and the meaning of some of the more common expressions used in relation to them. To this end the following illustration will serve better than mere definitions. 2. Suppose that the citizens of a town desire to have gas-light in their streets and houses, and that about $50,000 will be needed to construct the necessary works. 3. No one person or private company will be willing to risk so much money in the experiment, but if 500 persons will take a share in the enterprise and each put in $100, the plan can be carried out. 4. A subscription is started, and it is found that many are willing to put in $100, and that some arc ready to take two or more shares, and possibly one or two who will take a hundred shares each. Thus it is found that the $50,000 can be raised. 5. The subscribers then obtain a charter, or legal authority to act as a company, and appoint a Board of Directors, each subscriber having one vote in the elec- tion for each share of stock he has taken. 6. As soon as the company is ready for business the Board of Directors calls for the payment of the subscriptions, either at once or by instalments as the money is needed, until they are all paid. 289. A certificate of stock is now given to each stock or shareholder, showing the number of shares he has taken and the price paid per share. The latter is called its face or par value. 290. If the company is prosperous and pays in dividends (division of profits) more than the money could earn in other ways, the stock will be at a premium, that is, worth more than its par value ; but if the dividends are small, a share will be worth less than $100 ; then it will be said to be below par, or at a discount. This is an illustration on a very small scale. The stocks of all the incorporated companies in the United States amount to more than a thousand millions of dollars, and there are many men engaged in buying and selling them in all the great cities. PERCENTAGE. 291 291. A stock broker is one who buys and sells stocks for others. For buying or selling stocks in the New York Stock Exchange the regular charge is x / 8 of 1% on their par value. 292* A stock. Jobber buys and sells stocks on his own account. The following report of the number of shares of certain stocks sold and the highest prices paid for them at the New York Stock Exchange, November 16, 1885, is taken from a long list to be found in the papers of the following day : Sales. Highest. Sales. Highest. Adams Express 10 142 l / t Central Iowa 1700 22 1 / 4 Atlantic & Pacific. ... 850 10 3 /s Central Pacific 900 47 1 / 4 Alb. & Susq 15 140 C, C, C. & 1 800 64 l / 2 Atch., T. & S. Fe 100 88 Chic, B. & Q 1500 135 l J t Canadian Pacific 1900 55 % Chicago & N. W.. . . 15,580 115 */ 4 Note. — In the following problems it is supposed that all the transactions take place through brokers, in behalf of outside parties. Hence the person for whom stock is bought must pay the price of the stock plus the brokerage, and the seller will receive the price for which it is sold minus the brokerage. Examples. — At the above quotations : 1. How much did the buyers pay for the several stocks sold above par, including brokerage ? Adams Express. Solution. — 10 shares at 142 l / 2 per share = $1425 Brokerage at 1 / 8 £, 1.25 Cost of stock, $1426.25 Am. 2. How much did owners receive for the several stocks sold below par, brokerage being deducted ? 3. What was the brokerage, at the usual rate, on the purchase of 1500 shares of Chic, B. & Q.? 4. How much did the buyer pay and how much did seller re- ceive for 1900 shares of Canadian Pacific ? 5. If I give my broker orders to sell 800 Central Iowa, and buy 50 Adams Express, what balance will he put to my credit after deducting brokerage on both sale and purchase ? 6. How many shares of C, C, C. & I. could be bought for $10,381, including brokerage ? What balance would remain ? 292 STANDARD ARITHMETIC. Taxes. 293« A tax is a sum of money assessed on the income, person, or property of individuals for public purposes. 294-. A tax on property is called a Property-Tax, that on the income is called an Income-Tax, that upon the person, a Boll-Tax or capitation-tax. 295. Fixed property, such as lands, houses, etc., is called Real Estate. Movable property, such as furniture, money, cattle, merchandise, etc., is Personal Property. 296. The persons elected or appointed to estimate the value of property to be taxed are called Assessors or Appraisers. Example. — l. The sum of money to be raised by taxation in a certain city is 8562,600, the total appraised value of the property is $44,800,000, and there are 25,000 persons subject to a poll-tax of $1 each. How much will Mr. Hunter have to pay, whose property is valued at $2560 ? An*. $31.72. Solution.— First Step. — Subtract the $25,000 to be received on the polls from the sura to be levied ; the remainder will be the tax on property. $562,600 — $25,000 as ? Second Step. — Divide the tax on property by the total appraised value of the property to find the tax on $1. $537,600 -J- $44,800,000 = ? The rate of taxation being thus found, the tax on Mr. Hunter's property is readily ascertained. To the property-tax must be added his poll-tax. After the rate is determined, as above, the computation of the tax to be paid by each individual is greatly facilitated by a table like the following: TABLE SHOWING TAXES AT THE RATE OF 12 MILLS ON $t $1 pays $0,012 $10 pay $0.12 $100 pay $1.20 $2 pay $0,024 $20 ' 1 $0.24 $200 "* $2.40 $3 " $0,036 $30 « ' $0.38 $300 " $3.60 $4 " $0,048 $40 ' ' $0.48 $400 " $4.80 $5 " $0,060 $50 ' ' $0.60 $500 " $6.00 $6 " $0,072 $60 ' ' $0.72 $600 H $7.20 $7 " $0,084 $70 ' ' $0.84 $700 " $8.40 $8 " $0,096 $80 ' ' $0.96 $800 " $9.60 $9 " $0,108 $90 ' ' $1.08 $900 " $10.80 The tax on $2, $3, etc., being found by multiplying the rate by 2, by 3, etc., the rates for $10, $20, etc., are found by removing the decimal points of the first column one place to the right, and for $100, $200, etc., two places, etc. PERCENTAGE. 293 2. Find the amount of taxes Mr. A. has 4nAA . ._ . AA to pay on property assessed at $2475. _ Explanation. — From such a table as the above we -^ 94. would take $24.00, the tax on $2000, $4.80, the tax on ~ $400, $.84, the tax on $70, and .06, the tax on $5, and 5 = 0.06 adding these together we would find Mr. A.'s tax on his $2475 = $29. 70 property to be $29.70. 3. My real estate is estimated at $4500, my personal property at $1345, and I have to pay $2 poll-tax. How much tax will I have to pay, the rate being 12% mills on the dollar ? 4. Find the amount of taxes my neighbor will have to pay on $9876, and $1 poll-tax. Same rate. 5. Find the amount of taxes my three neighbors across the street pay, at the same rate, on $2732, $3695, $8351 ; each paying $1 poll-tax. 6. Find how much a non-resident must pay on his real estate, which is listed at $6129. (No poll-tax.) 7. A person has to pay $100.20 taxes at 12 mills on the dollar, there being no poll-tax ? What is the assessed value of his prop- erty ? 8. Suppose my property, real and personal, to be listed at $1500, and that I have to pay 3 mills on the dollar for state purposes, 2 mills for county purposes, 2 mills for township pur- poses, 5 mills extra for school purposes, and 2% milts for cor- poration (village) expenses ; how much in all would I have to pay ? 9. In a state of Europe 1 $ is required to be paid on incomes from $100 to $300, l'%# on incomes from $300 to $500, and 2%# on incomes from $500 to $800. Mr. A.'s income was $450, Mr. B.'s $175, Mr. C.'s $760. What income-tax did each have to pay ? 10. If my property is valued at $2500, and the rate of taxa- tion for school purposes is 5 mills on the dollar, what does the tuition of each one of my three children cost me if all of them attend the public schools ? 294 STANDARD ARITHMETIC. 11. Allowing 5/o for taxes uncollectible, and 2/c for collection, what sum must be levied that $50,000 may be realized for the building of a school-house ? $1 must be collected for every 980 needed for the school-house, because 20 out of every hundred go to the collector, and $1 must be levied for every 950 supposed to be collectible, since those who do not pay wiil keep back 50 of every hundred levied. 12. The people of Abdera wish to levy a tax which will net them $18,979, after paying the expense of collection, which will be 3$. The assessed value of the real and personal property is $1,260,000, and there are 323 polls, each taxed $2. How much will $1 be assessed ? 13. Make out a table similar to that on page 292. From the table (Ex. 13) find how much 14. Mr. W. M. Hart pays on $6000 17. Mr. H. Kidd pays on $10000 15. Mr. John Handy u " $5583 18. Mr. L. B. Pease u " $7534 16. Mr. E. G. Eliot " u $5354 19. Mr. R J. Luck u " $5821 20. For the purpose of building a town-house, a tax of $15,961.60 is to be levied on property valued at $1,856,000. What will be the tax on Mr. Burns' property, which is valued at $8650 ? 21. A bridge costing $18,135 was built by the proceeds of a tax levied upon the property of a town, the rate of taxation being 50^ on $100 (5 mills on $1), the cost of collection being 2y g #. What was4he assessed valuation of the property ? 22. If the assessed value of the real and personal property of a city is $80,000,000, and a special tax is desired for the construc- tion of sewers, what must be the rate of levy to realize $188,160 for the purpose, if 2$ be allowed for collection and 4$ of the levy be uncollectible ? Note. — The answers given to problems such as the preceding ones are based on the method of analysis given under Example 11, but, since the amount of tax un- collectible can never be known beforehand, the sum to be assessed for any given use can be determined with sufficient exactness by adding to the sum needed the estimated percentage of taxes uncollectible and the percentage charged for collec- tion. In States where the collector is paid a fixed salary, the cost of collection would not be taken into account. PERCENTAGE. 295 Miscellaneous Problems in Percentage. 1. Of 480 persons in a village, 30 moved away within one year. What per cent, of the whole number remained ? 2. If two hundred pounds of wheat make 150 lb. of flour, what per cent, of the weight of the wheat is the weight of the flour ? 3. Twenty pounds of coffee lose 4 4 / 5 lb. in weight by roast- ing ; what # ? 4. A village of 1253 inhabitants has 200 children attending school. What per cent, of the whole population in school ? 5. A person paid $22 y 2 tax on his income at the rate of 1%$. What was his income ? 6. A house was sold by an agent for $5600. The agent's com- mission was iy 2 #. How much did the owner receive ? 7. A real estate agent collects rents as follows : For Mr. Williams, $2384.20 John Jones, $936.18 Mr. Cook, $786.15 " Mr. Johnson, 856.75 Henry .Jones, 1852.00 Mr. Doan, 385. What is the amount of his commissions at 3 per cent.? 8. One and a quarter per cent, of the inhabitants of the king- dom of Prussia are annually called into military service. How many men do the city of Breslau, with 240,000 inhabitants, and the city of Hildesheim, with 23,000 inhabitants, have to furnish ? 9. A farmer bought a team of horses, but could pay only $155 in cash, 37 y g # remaining unpaid. What was the price of the horses ? 10. An inspector of coal mines, having a salary of $2400 a year, pays $560 rent for house and barn. iy 2 # taxes on an assess- ment of $480, and y 2 $ for insurance of books and furniture valued at $1250. What $ of Ijis salary does he pay for rent, taxes, and insurance, respectively ? 11. On one occasion the price of a barrel of petroleum fell from 90^ to 78^. What per cent, was the decline ? Shortly after the price rose again to 90^. What per cent, was the advance ? 296 STANDARD ARITHMETIC. 12. Twenty pounds flax, when spun, make 17% lb. of yarn. What per cent. ? 13. Eight pounds of beef are reduced 1 lb. in weight by boil- ing and 1% lb. by roasting. What per cent, of weight is lost by each process ? 14. If a single railway fare to the city is 30^, what per cent, would I save by the purchase of 100 tickets for $20 ? 15. The Union Steel Screw Co. declared a dividend of 17% $ upon its stock. What did stockholders receive who had respect- ively $900, $2000, $4700, $2300, and $1100 worth of stock ? 16. A merchant sends out bills for collection as follows : $184.75 $57.61 $384.21 $728.13 136.54 98.13 17.86 564.21 19.81 156.22 918.54 . 1986.54 5.78 7.61 12.32 .95 846.00 387.60 50.65 18.70 The collector receives 6$ on all sums less than $100, 4$ on amounts from $100 to $500, and 2$ on all sums greater than $500. What will be his commission if all are collected ? 17. Church bells commonly contain 80$ of copper, 5.6$ of zinc, 10.1$ of tin, and the rest is lead. At that rate, how much of each is contained in the great bell at Moscow, which weighs 443,772 pounds ? What per cent, is lead ? 18. A carpet dealer reduced the price of certain goods 12y 2 $, which amounted to 12^ on the yard. What did the goods sell at per yard before and after the reduction ? 19. Twelve quarts of good milk will give iy B qt. of cream. What per cent, of the milk is cream ? 20. The liabilities of a bankrupt merchant are $7200, his assets only $3200. How much will his creditors get, to whom he owes respectively $2572, $856, $782, $1025, $1912, and $53 ? 21. The daily wages of a workman were increased 25^, or 6%$. What did he get before and after the increase ? PERCENTAGE. 297 22. Charles has a salary of $750, his brother $850. What $ does his brother receive more than he ? By what per cent, is Charles's salary less than his brother's ? 23. If eight pounds of imperial tea may be had for $9, and a single lb. of the same kind costs $1.20, what is the per cent, saved by buying 8 lb. at a time ? 24. A gentleman is insured for $5000. His premium is $96.25. How much does he pay on a thousand ? What per cent. ? 25. The butcher estimates a beef to weigh 980 pounds, of which 57$ is salable as meat and 6y 4 $ tallow. What is the weight of the meat and tallow together ? 26. Colonel A. has to pay $1200 rent, Major B. $1000. The owner of the two houses raises the rent $100 on each. What per cent, is the major's rent raised more than that of the colonel ? 27. The weight of a cubic foot of American pine, when green, is 44.75, when seasoned, 30.7. Ash u * " 58.18 u " 50. Beech " " " 60 u " 53.37. Cedar " " " 32 " " 28.25. English oak " " u 71.6 " " 43.5. What per cent, does each lose in weight by being seasoned ? 28. What per cent, income does Mr. Abel have more than Mr. Bain, if A. has $25X)0 and B. $2000 ? If A. has $3000 and B. $2500 ? If A. has $2000 and B. $1500 ? If A. has $1750 and B. $1250 ? If A. has $600 and B. $100 ? 29. Find what per cent, the lower income is of the higher in each of the above-mentioned cases. 30. Anna bought 8 yd. of tape for 5 ? At 3f ? 4tf ? 5f ? 7^ ? Sfo ? 9f ? Interest at 6 in 30 years, and one which will yield $111 at 2 1 / 2 fo in 40 years. SLAT E EX E R C 1 S E S . Find the principal : Find the principal : Eate. Time. Interest. Eate. Time. Interest. 8. H, 6 months, $6.25. 13. 5fe, 3 months, $61.00. 9. ±7iX, 6 M $4.50. 14. 4%& 3 " $20.25. 10. H, 6 " $27.00. 15. 6#, 3 " $25.50. 11. H, 1 month, $1.50. 16. t% *, 1 month, $3.00. 12. H, 1 " $1.80. 17. 60, 1 " $7.50. 18. How many times greater is the principal than the annual interest, when the rate is 2 i ? 2 % from March 8, 1869. I received as partial payments (1) $140 on Sept. 10, 1870 ; (2) $50 on July 20, 1872. What amount was due on set- tlement, Oct. 15, 1873 ? 6. A note of $300, dated Jan. 1, 1878, had the following in- dorsements : Aug. 1, 1878, $50 ; Dec. 1, 1878, $50 ; July 1, 1879, $100. What amount was due on Jan. 1, 1880, interest at 7$ ? 7. On the 1st of June, 1879, H. R. Fox borrowed of Charles Lever $800, and gave his note for that sum with interest at 7$. Sept. 1, 1879, Fox made a payment of $240, and a new note was made out for the balance. What was the face of this note ? (Write this new note out in proper form, dating it at Richmond, Va.) 8. -$575. Cleveland, O., Feb. 1, 1879. Eight months after date I promise to pay C. F. Cutter & Co., or order, five hundred seventy-five dollars, ivith interest at 7f c , for value received. R. W. Cane. Indorsements: Oct. 1, 1879, $300; July 1, 1880, $150. What balance was due on settlement, Dec. 1, 1880 ? 9. Aug. 1, 1873, F. Critland gave to Robert Ingham a note for $143.50, with interest at 7$. He made payments as follows : Dec. 17, 1873, $37.40; July 1, 1874, $7.09; Dec. 22, 1875, $13.13 ; Sept. 9, 1876, $50.50. What amount was due Dec. 28, 1876 ? 10. A note for $2800, dated June 30, 1884, has the following indorsements : Dec. 23, 1884, $50 ; May 16, 1885, $90 ; June 3, 1885, $10 ; April 23, 1886, $150 ; May 30, 1886, $22. What bal- ance remained due at the last payment ? (Why the original principal ?) 326 STANDARD ARITHMETIC. The Mercantile Rule. 330. The following method of finding the balance due on a mercantile account or other debt running for a year or less is very commonly adopted when partial payments have been made. The principle on which it is based will be readily understood from the state- ment of the rule. 331. Rule,— l. Compute the amount of the debt from its date to the time of settlement. 2. Compute the amount of each payment from its date to the date of settlement. 3. Subtract the sum of the amounts of the several payments from the amount of the debt. The difference will be the balance due. The answers given to the following problems are based on the common method of finding differences of time (compound subtraction — 3 GO days to the year) though other methods may.be used. SLATE EXERCI SES. 1. On a debt of $420, contracted March 15, 1885, payments were made as follows : June 1, 1885, $150 ; Sept. 6, 1885, $130 ; Oct. 14, 1885, $75. What was the balance due Dec. 24, 1885, at 7$ interest ? 2. On a note for $645, at 6 $ interest, dated Jan. 1, 1886, and maturing 9 months after date, the following indorsements were made : Mar. 4, 1886, $50 ; Apr. 2, 1886, $75 ; Aug. 10, 1886, $200. What was the balance due at time of payment ? 3. ^Tr. Thomas gave his note, dated Feb. 15, 1885, to Amos King, for $1940, payable Jan. 1, 1886, with interest at Sf c . Allien due, the note had the following indorsements : Aug. 1, 1885, $440 ; Sept. 6, 1885, $500 ; Oct. 1, 1885, $300 ; Nov. 15, 1885, $400. What was the balance due Jan. 1, 1886 ? 4. May 1, 1884, goods were purchased to the value of $1250, on which the following payments were made : Aug. 1, 1884, $250 ; Sept. 4, 1884, $300 ; Oct. 15, 1884, $450 ; Dec. 8, 1884, $120. What was the balance due Dec. 20, 1884 ? INTEREST. 327 Annual Interest. 332. Annual Interest is simple interest payable annually. Iu some States annual interest, if not paid when due, draws interest at the same rate as the principal, in others at the legal rate, whatever may be the rate paid on the principal ; but in some it is illegal to collect interest on unpaid annual interest. Example. — l. Mr. Hart gives his note for $2000, payable in 4 y^ars with interest annually at 5 $. What will be the amount due when the note matures, provided Mr. Hart pays interest as agreed upon ? Solution. — The interest on $2000 at 5% is $100 per year. Hence Mr. Hart should pay $100 each year till the close of the 4th, when he should pay the last year's interest ($100) with the principal, making together $2100; but 2. If Mr. Hart does not pay the interest annually, as agreed, his account at the close of the 4 years would stand as follows : The principal, $2000 Total annual interest ($100 per year for 4 yr.) = $400 Interest on 1st ann. int. (3x5x$l) = 15 " " 2d " (2 x 5 x $1) = 10 " 3d " (1 x 5 x $1) = 5 430 Amount due at the close of the 4th year, = $2430 This amount differs from the amount of the same principal at simple interest only by the $30 interest on the deferred payments. Applications. — l. The annual interest on $2000 invested at 6 $ for 6 years remaining unpaid, what is the amount due ? 2. I invested $550 for 3 years at 5 $ interest, payable annually. What w T as due at the end of the three years, interest for the first only having been paid ? 3. At the end of 3 years, what is due on a debt of $500, with interest payable annually ? 4. Find the amount due in 8 years on $320, invested at 7^ interest, payable annually, the interest for the first 3 years having been paid when due. 5. The interest on a note of $400, payable after 6 years, with annual interest at 5*/tA has not been paid. What is the note worth at the time of its becoming due ? 328 STANDARD ARITHMETIC. 6. A note for $1000, dated July 1, 1865, at 7$ interest, pay- able annually, was paid January 1, 1868. What was the amount due at maturity ? 7. No interest having been paid on a note for $500, dated June 1> 1878, with interest payable annually, find the amount due September 1, 1880. Miscellaneous Problems, 1. A gentleman has at interest $10,640 at 5#, $37,500 at 6/ c , and $26,000 at 6y>$. What income does he derive from these sums per annum ? 2. Mrs. Stone has 24 government bonds of $1000 each, bear- ing 4$ interest. What is her income per quarter ? 3. $1850 yields $55 % int. in 6 months. What rate is paid ? 4. A principal of $500 was increased by $35 interest per year. W^hat was the rate per cent. ? In how many years will the prin- cipal be doubled by simple interest ? 5. A sum of money was invested at 3%$ interest. After 3 years 4 months the amount was $10,887.50. Find the principal. 6. What is the principal, if after % year the amount is $413, the rate being 6%^ per annum ? 7. Mr. A. had a mortgage on his house, and paid 6y 4 $ inter- est. In 4 years the amount was $5262.50. What was the debt ? 8. A lady inherits $6480, and desires to derive $32.40 per month from it. At what rate must it be invested ? 9. In what time will $462.50 produce $37 interest at 4^ ? 10. In what time will $723 produce $60 y 4 interest at 5^? 11. What sum will yield $35 interest at 7$ in 1 yr. 4 mo.? 12. What principal will pay $21.59 interest at 5f in 8y 2 months ? 13. Mr. A. borrowed a sum of money at 5y 2 $, and after V/j yr. paid the amount $4330. What was the principal ? INTEEEST. 329 14. What principal gives 183 ?/ 8 interest per month at 5$ ? 15. A gentleman draws $2940 per year ; 4 / 5 of his money bears 4% Vs 5$. Find the principal. 16. A principal of $6000 has by simple interest grown to $8820 in a number of years ; y 3 of the time it brought 3 fo, l / i of the time 5$, and the remainder of the time 4$ per annum. How long did the principal stand ? 17. The discount at 6$ on a note due Nov. 1, and sold on May 1, was $13 y 2 . What was the face of the note ? 18. I had a note for $250, due in 2 1 / 2 months, and sold it at a discount of 1$ per month. How much did I get for it ? 19. If a person wishes to get the same interest for $1200 in 4 yr. which he receives for $1000 at 4$ in 6 years, what rate must he charge ? 20. In how many years will $820 at 6y g # produce $278. 82 y, interest ? 21. In how many years will a principal of $5000 grow to be $8000, if put out at the rate of 6$ ? 22. The sum of $3360 is at 4 x / 2 ^ interest, and has thus far yielded $1058.40 interest. How long has the principal been standing ? 23. What is the rate of discount if a note of $300, due June 20th, is sold April 20th for $297 y 8 ? 24. What is the face of a note, sold 1 month before it fell due at 9$ discount, the discount amounting to $15% ? 25. Mr. M. buys a house by paying 13 / 20 of the purchase money, and securing the remainder, $3500, by a mortgage at 4y 2 $. At the end of 1 year he pays the remainder. How much does the house cost him ? 26. Three principals, of which the first, $600, has been at 6$ interest for 2 i / 2 years ; the second, $425, at 4$ for 3 years ; and the third, $550, for 2 years, together yield $190.50 interest. At what rate was the third principal invested ? 330 STANDARD ARITHMETIC. 27. What principal will bear as much interest in 6 years at 5$ as $840 in 8 years at 4$ ? 28. What interest will you get on $960 in 7 years, if $280 in 5 years yields $63 interest at the same rate ? 29. If I borrow $480 at 5$ per annum, and the interest for the first year is deducted at once, what per cent, do I really pay ? 30. $400 were at 5^ interest for 3y 2 years, and $350 at 4$ for a longer time. The principals with interest, when collected, amounted to $890. How long was the second principal standing ? 31. A gentleman borrows $60, and promises to pay $70 in 3 months. What rate per cent, will he pay ? 32. A guardian put his ward's money, $17,500, at interest : % of the money at 4$, and the remainder at 6y 8 #. How much could he lay by annually for the future benefit of his ward, after deducting $650 per year for expenses ? 33. For what time does Mr. B. pay interest, if he pays $84 on $5600 at the rate of 4%$ ? 34. Mr. Frank buys a house and pays % of the price in cash. The remainder, $5400, is secured by a mortgage, and paid in 2 years together with 7y 2 $ interest. How much was paid, in- cluding the interest ? 35. A man was asked what money he had at interest. He said : One half of it yields 47§ A the other half 5$, and I receive on the whole $114 a month. How much had he at interest ? 36. Mr. Conklin having bought a piano for $350, he rented it at once for fifteen months at $4y 2 , payable monthly. Then he sold it for $325. How much did he gain by the transaction, taking into account the interest on the cost and interest on pay- ments received for rent ? 37. I paid $600 per year rent for a factory which I afterward bought for $12,000. I gave $5000 cash (which was worth 6$) and a 4$ mortgage for the balance. How much per year did I save ? INTEREST. 331 Compound Interest. 333. Compound interest is interest on interest. Payment of compound interest, or interest on interest, can not be enforced by law. 334. Interest is usually compounded at specified intervals, as annually, semi-annually, etc., by adding interest to principal, and computing interest on the amount. Example. — l. Find the compound interest of $500, at 6$, for 3 yr. 5 mo. $500.00 Principal. $530 30.00 Interest 1st year. - -$E20W Amount. _$3U» Int. 2 d yea, 31.80 Interest 2d year. $561.80 Amount. $561.80 33.708 Interest 3d year. $33.7080 Int. 3d year. $595,508 Amount. 14.8877 Interest 5 mo. " **ia o*~h * j. o * $5.95508 Int. 2 mo. $610.39o7 Amount 3 yr. 5 mo. -^ttttt^ T * , Laa f\ • • i • • i $11.91016 Int. 4 mo. $500 Original principal. 2 97754 " l " $110.3957 Compound int. 3 yr. 5 mo. $14.8877 Int. 5 mo. 2. What is the compound interest on $7325 for 2 yr. 2 mo. at 7 $ ? (Carry the work to four decimal places.) 3. Find the compound interest on $3333, at 3y 3 # semi- annually, for 1 yr. 7 mo. 4. What amount was due March 25, 1886, on $1512, borrowed Jan. 25, 1885, with compound interest at 1 1 / 2 $ quarterly ? 5. What is the amount of $4615, at compound interest, for 2 yr. 5 mo. at 8 $ ? 6. Find the amount of $3500, at compound interest, from Oct. 29, 1884, to Nov. 15, 1885, at 2 ^ quarterly. 7. How much greater, at compound than at simple interest, would be the amount of $1568 in 3 yr. 8 mo. at 6 planation for eaeh item. ^— — ^ The use of $1500 in parts, as Am% 4*/ l4 mo . = 4 mo, 8 d. specified in the example, that is, $200 for 3 mo., $450 for 4 mo., etc., is thus found to be equivalent to the use of one dollar for 6400 mo. But the use of $1 for 6400 mo. is equivalent to the use of $1500 for Visoo of 6400 mo. =4 4 /is mo., or 4 mo. 8 d. Am. Hence, to find the average term of credit for several sums of money, due at different times, by the Method of Products. 339. Utile,— Multiply each item of the debt by its term of credit, and divide the sum of the products by the sum of the items ; the quotient will be the average term of credit. Notes. — 1. In computing terms of credit, it is customary to reject the cents in any item if less than 50 ; and, if 50 or greater, to reckon them as one dollar. 2. Less than 1 / 2 day in a result is rejected; a 1 / 2 day or greater fraction is counted as 1 day. EQ UA TION OF PA YMENTS. 335 The Interest Method. 34-0. The time in which the sum of several items of indebted- ness would become justly due is the time in which the use of the sum would balance the use of the items for the several terms allowed for their payment. Thus, in the foregoing problem, The use of $200 for 3 mo. at 6f would be worth $3.00 " " $450 " 4 " u " " $9.00 " " $500 " 4V 2 " as 12 days is times 6 days, work in 12 days ) times ( work in 6 days ) When solved, it would lead us to the conclusion that it would take 16 men 12 days to do a work which requires only 8 men 6 days, which is absurd. If we allow double the time for any work, we do not need twice as many men to do it, but only half as many. The statement then should be : The number of men ) ( the number of men ) required to do the [• p a J" 5 \ required to do the [ as 6 d. is of 12 d. work in 1 2 days ) ( work in 6 days ) Men. Men. Days. Days. Or, using the shorter form : ? : 8 :: 6 : 12. We reason that, since 6 days is one half of 12 days, the number of men re- quired to do the work in 12 days is only one half as many as would be required to do it in 6 days. Ans., 4. In all the problems preceding this, more required more ; that is, more goods required more money, more worh required more men, etc., etc. But, as we see in this example, there are problems in which more requires less and less requires more. 340 STANDARD ARITHMETIC. ORAL EXAMPLES. 2. If a man can perform a journey in 6 days, traveling 12 hours a day, how many days will be required if he travels only 6 hours a day ? 4 hours ? 3 hours ? 3. The owner of a livery stable sends 16 horses to pasture for 8 days. How many could he send for the same money for 16 days ? 4 days ? 4. Six masons can perform a certain work in 24 days. How long will it take 12 men ? 18 men ? 5. Six horses consume a certain quantity of oats in 12 days. How long will it feed 36 horses ? 6. Fifteen farm hands will mow the harvest of a farm in 8 days. How many will do it in 2 days ? In 24 days ? SLATE EXERCISES. 7. Fred has to walk 250 steps from his house to school, each of his steps measuring 18 in. His brother's steps measure 27 in. each. How many steps does his brother take ? 8. A messenger who traveled 12 miles an hour reached his point of destination in 3 hours. How long would it have taken him had he traveled only 8 miles per hour ? 6 miles ? 4 miles ? 9. If he had traveled 4% miles an hour, how many hours would it have taken him. 10. Thirty sailors can subsist on their provisions 4 months. At the same rate, how long will the same provisions last 20 sailors ? 11. Suppose the same provisions would last 30 men 4y 2 months. Find how many months and days they would last 20 men. 12. A carter agrees to transport 7% cwt. 6 miles for a certain sum. How far will he carry 9 cwt. for the same money ? 13. A bridge was built by 15 workmen in 4 weeks and 4 days. How many would have built it 8 days sooner ? 14. A certain number of trees was felled by 28 men in 4 weeks and 3 days. How many could have done it in 12 days ? PROPORTION. 347 Ratio and Proportion. 343. A comparison of two numbers is made by showing how many times, or what part/ one number is of another. 344-. In any comparison of numbers there must be at least hvo numbers, or quantities, compared. 345. Two numbars or quantities thus compared are together called a couplet. The first is called the antecedent (the one going before) ; the second, the consequent (the one coming after). 346. The antecedent and consequent are called the terms of the couplet. 347. The consequent is the standard of comparison. 348. The relation of two numbers, that is, the quotient ob- tained by dividing the antecedent by the consequent, is called the ratio of those numbers. 349. Proportion is an equality of ratios. For instance, the following is a proportion : The cost of 9 yd. m as many the cost of 3 yd. yd. are yd. $1*0 time8 12 7 8 a8 9 time8 3 The arithmetical form for the statement of which is : Cost. Cost. Quantity. Quantity. 37 V0 : l&ftf :: 9 yd. : 3 yd. 350. The double colon :: is the special sign of a proportion. Note. — The colon ( : ) is a sign of division, the line between the dots being omitted. The sign of equality ( = ) is often used instead of the double colon ( : : ). Thus the statement of a proportion frequently appears in this form : Cost. Cost. Quantity. Quantity. 37V*0 f 1*70 = 9 yd. - 3 yd. 351. A proportion must contain at least two couplets. The first and second terms make the first couplet; the third and fourth terms, the second couplet. 352. The first and fourth terms of a proportion are called the extremes; the second and third are called the means. 348 STANDARD ARITHMETIC. EXERCISES IN FINDING RATIOS Find the ratios of the following couplets : The consequent being the standard of comparison, the question to be answered, in each case, is, How does the antecedent compare with the consequent ? How many times, or, What part of, the consequent is the antecedent ? 1. 18 : 9 = 4. 95 : 19 = 7. 13 : 4 = 10. 42 8 == 2. 36 : 18 = 5. 18 : 12 = 8. 16 : 15 = 11. 48 5 = 3. 72 : 12 = 6. 85 : 17 = 9. 51 : 3 = 12. 88 9 S3 Note. — The quotient, arising from dividing one number by another, may be expressed in the form of a fraction, thus, 15 -s- 3 = 15 / 3 , which, being narrowed down to lowest terms, is equal to 5. In writing out the foregoing exercises, the pupil may therefore adopt the following form: 15 -f- 3 = 15 / 3 = 5. He should recollect that 15-4-3, 15 : 3, and 15 / 3 , indicate the same thing, viz., that 15 is to be divided by 3. Questions. — When we compare a greater number with a less, is the ratio greater or less than a unit ? — Can a ratio be expressed by a mixed number ? By a fraction ? — Why is the ratio of 32 : 8 (read, 3^ to 8) not greater than that of 8 : 2 ? — Give other couplets having the same ratio as 32 : 8. As 15 : 3, etc. — If you double both numbers does the ratio increase ? Why not ? — Is the ratio of the halves of two numbers the same as the ratio of the num- bers themselves ? Why ? 13-28. Find the ratios of the following numbers : Note. — Express the ratio in the form of a fraction, and reduce to lowest terms. 5: 25 = (' / 2 5 = 7 5 ) 18 54 = 27 108 = 15: 25 = 9 : 72 =s 28 42 = 17 93 = 33 : 132 = 3:33 = 16 96 = 23 69 = 31 : 124 = 6: 72 = 19 104 = 14 98 = 13 : 117 = 29-52. Find the ratios of the following fractions : % Vs = 7 8 : %~ 0.6 :0.12 = 0.16: 0.4 = % V. = %.:!%« 0.5 :0.05 = 0.33: 0.3 = % Vs = »%:*%* 0.35:0.07 = 3.5 : 6.5 = V« %•« 23 3 / 8 :8V 3 = 27.2 :3.6 = 2.6 :10.4 = % % = 4% : 8% = 3.35:0.07 = 8.45:10.25 = V* V. = *y«t*% = 0.26 : 0.( )7 = 0.01 : 0.5 = PROPORTION. 34£ Fill the blanks in 1. 42 : — = 7 2. — : 19 = 3 3. 18 : % = - 4. 27 : - = % the following statements' : 5. 4: — = y 8 9. 10.4 6. %:- = % 10. 23% 7. 3 : 15 = — 11. 10 8. 36: — = 7% 12. — — =4 - = 2% :— =2 :3 =9 Suggestion. — To fill the blank in the first problem the pupil has to answer the question, " 42 is 1 times what number ? " The second, " What number is 3 times 19?" The third, " 18 is how many times l f t ? " The fourth, " 27 is 1 / a of what number ? " From the foregoing definitions and exercises we may derive the following Rules. 353. Mule, — 1. Multiply the consequent by the ratio; the product will be the antecedent. 2. Divide the antecedent by the ratio ; the quotient will be the consequent. 13-18. Prove the following proportions to be correct : 3:4:: 6: 8 3 : 8 :: 6 : 16 % : % :: % : 1% 2:8:: 6:24 1.05: 8.4:: 1: 8 0.05:7 :: 0.3:42 19-33. Fill the blanks in the following statements, determin ing the ratios from the complet3d couplets : 7 56 : — : 16 20 5 : — : 2 15 — : 6: 18 — 21 :: 56 8 21 /3 — : 6: 9 — : 3: : 16: 5: — : 8: 3 / 4 : 1 : — : %: 1 •: 60: 17:51 :: — : 9 — : 5: :3% 6% 64 18: 6: : 21 — 12 V, : % : : — 24 — 0.2 : — : : 6 108 48 10%:-: : 19 95 From the preceding principles and exercises we derive the following 354-. Rules for finding the Missing Term of a Proportion. Rule, — 1. Find the ratio of the complete couplet ; then, 2. If the antecedent of the incomplete couplet be wanting, multiply the consequent by the ratio ; or, 3. If the consequent be wanting, divide the antecedent by the ratio. 4. The result will be the term required. 350 STANDARD ARITHMETIC. EXERCISES IN PROPORTION 1. If 24 hats cost $44, what will 150 hats cost ? Note. — To avoid error in the statement of a proportion, an arrangement of the terms of the question such as the following is recommended : Complete couplet. Incomplete couplet. ? is the cost of 150 hats if $44 " " 24 " It matters not whether the complete or the incomplete couplet is placed first, nor whether the sign for the wanting term be the first or last of the proportion ; but it is essential, in all questions in which more requires more and less requires less, that, if the upper term be taken as the antecedent in one couplet, the same should be done with the other. In a problem in which more requires less, and less requires more, it is neces- sary to invert the terms of the complete couplet in the statement. For example 2. If it requires 7 men to build a wall in 27 days, in how many days would 9 men perform the same work ? Preliminary Arrangement. Days. Men. ? j 9 Note. — An arrow pointing downward may 27 v 7 be used to indicate terms to be inverted. Since 9 men require less time to do the work than 1 men, the terms of the complete couplet are inverted, and the statement is : Days. Days. Men. Men. w — : 27 :: 7 : 9 27 X | = 21 days Am. 3. How many tons of hay will 325 acres produce if, at the same rate, 13 acres produce 40 tons ? 4. What time would it require for 7 men to mow a field, if 3 men can mow it in 3y 3 days? 5. At Christmas 8 eggs were sold for 25^. What was the cost of 6 dozen ? 6. Farmer Black pays $52 % rent for 24 acres of land. At the same rate, what will he have to pay for 51 acres ? 7. Mr. H. agrees to do certain work in 15 days, thereby earn- ing $3.20 a day. How much will he earn a day if he does the work in 10 days ? In 13 % days ? PROPORTION. 351 8. In canning 5 lb. of raspberries 3 lb. sugar are needed. How many pounds sugar for 38 lb. of berries ? 9. If with the money I have, I can buy 84 lb. of coffee at 25^ a lb., how many lb. could I buy for the same money at 30^ a lb.? 10. If 3 yd. of calico cost 20^, what will % yd. cost ? Arrangement. Statement. Solution. ? is the cost of 4 / 5 yd. if . nn *L ,'.'"'' 2(¥ I I £ 5 yd . ?:20:: «/. :3 20 x ^ = « Vf '-A* 11. If wall paper be 20 inches wide, I shall need 7 rolls to paper a room. How many rolls will suffice if the paper be 24 inches wide ? If 30 inches wide ? 12. If $750 will yield $120 interest in a certain time, what interest will $600 yield in the same time ? 13. A man, whose step measures % yard, counts 1200 steps from his house to his office. How many steps will his son have to take, whose step measures 1 / 2 yd. ? 14. If each man on board ship consumes daily \ % / 4 lb. bread, their bread will last 57 5 months. How much will each man get per day if it is to last 6 y g months ? 15. The rate of two pedestrians is as 5 : 4. How many miles will the first travel in the same time in which the second travels 84% miles ? 16. At the rate of $180 for 3 / 10 acre, what will 5 acres cost ? 17. The heat produced by a cubic yard of beech-wood is to that produced by a cu. yd. of pine as 9 : 7. How many cu. yd. of beech-wood are needed to produce the heat of 50 cu. yd. of pine ? 18. If 1% yards of velvet cost 15 Vj, what will 9 yd. cost ? 19. A farmer sowed 3 bu. of buckwheat on 2% acres. How much would he need for a field containing 4% acres ? 20. % of a sum of money is $800. How much is % of it ? 21. If bread is 7^ a loaf when flour is sold at $6 a barrel, what should flour be worth when bread is sold at 8 U' 358. A cubic block, measuring 3 in. on the edge, contains 27 cubic inches, hence 27 is said to be the cube of 3. For a like reason, 64 is said to be the cube of 4, % of y 2 , etc. What are the cubes of 1. .1 % */. 5. .3 & sq. in. % 7« ft % % Note. — The cubes of integers and fractious should be frequently illustrated by the use of modeling clay, cubic blocks, etc. 359. To square a number, we multiply the number by itself ; that is, we use it twice as a factor to produce the square. Hence, the square of a number is also called its second power. 10 r 358 STANDARD ARITHMETIC. 360. To cube a number, we multiply the square of the num- ber by the number itself ; that is, we use the number three times as a factor to form the cube. Hence, it is said that the cube of a number is its third power. Definitions. 361. A power of a number is the number itself, or the product obtained by the use of the number two or more times as a factor. The number itself is called the root of the power. 362. The number of times a root is employed as a factor is indicated by an exponent, which is commonly a small figure written to the right and a little higher than the root. 7 2 indicates the second power of 7, hence 7 2 = 49. 7 3 indicates the third power of 7, hence 7 3 = 343. 363. The process of raising a number to any required power is called Involution. The process of involution is a process of simple multiplication ; therefore no rule is necessary, except that the root is to be used as a factor as many times as there are units in the exponent. EXAMPLES FOR PRACTICE. Raise the following numbers to the powers indicated: 1. 17 3 4. 38 2 7. 12 3 10. 33 3 13. 2.15 2 2. 3.2 2 5. 1.3 3 8. 0.1 3 11. 128 2 14. (5V 3 ) 2 3- CA) 3 6. (Ye) 2 9. (%) 3 12. 325 2 15. (7.iy 4 ) 3 Find the values of 16. 12 2 X 2 18. (7 2 X 3 3 ) + 10 20. (9 2 -4- 3 3 ) X 7 2 17. 9 2 X 2 2 19. 3 3 X (4 2 + 2) 21. - ( 5 Jx2) + 10 Raise to the second power : 22. 97 24. 35 26. 128 28. 826 30. 5287 23. 98 25. 47 27. 371 29. 981 31. 6520 SQUARES AND CUBES. 359 Write answers to the following questions : 1. How many places are there in the second power -of a num- ber expressed by one figure ? By two figures ? By three figures ? By four figures ? 2. Does the power always contain twice as many figures as there are in the root ? Does it ever contain more than twice as many ? May it contain less ? Suggestion. — To answer the foregoing questions correctly, and with confidence, the pupil should square the greatest and the smallest numbers that may be ex- pressed by one figure (1 and 9), by two figures (10 and 99), etc. 3. How many decimal places are there in the second power of a decimal expressed by one figure ? By two figures ? By three figures ? By four figures ? In the square of a decimal, should we ever have more than twice as many decimal places as there are in the root ? Should we ever have less ? Why not ? Raise to the third power : 1. 1 5. 10 9. 100 13. 1000 17. 10,000 2. 9 6. 99 10. 999 14. 9999 18. 99,999 3. .1 7. .01 11. .001 15. .0001 19. .00001 4. .9 8. .99 12. .999 16. .9999 20. .99999 Note. — The square of 999 may be conveniently found by subtracting 999 from 999,000. Why ? The same method may be applied to any other number expressed by 9's. Write answers to the following questions : 1. How many places in the third power of an integer expressed by one figure ? By two figures ? By three figures ? By four figures ? 2. How many times as many figures in the power as there are in the root ? Are there ever more than three times as many ? Are there ever less ? How many less may there be ? 3. How many places are there in the third power of a decimal of one place ? Of two places ? Of three places ? etc. In the cube of a decimal, should we ever obtain either more or less than three times as many decimal places as there are in the root ? Why not ? 360 STANDARD ARITHMETIC. 364-. Another method of raising numbers to the second and third power. It will be found useful to note carefully how the tens and units are combined in the process of raising a number, expressed by two or more figures, to its second or third power. For illustration, let us take the example : Kaise 43 to its second power. Analytical Method. 40 +3 40 +3 40 X 3 + 3X3 40 X 40 + 40 X 3 40 X 40 + 2 X (40 X 3) + (3 X 3) sq. of the tens. twice the tens by the units. sq. of the units. Little explanation is here necessary, except that, in the analytical method, instead of actually multiplying and adding, as in the common process, we only indi- cate the multiplications and additions by means of signs. This we do, that we may trace the tens and units separately through the process, to see where we may find them in the product. Thus, in this case we sec that 365. The square of Jf.3 is equal to the square of the tens, plus twice the product of the tens by the units, plus the square of the units. It may be shown that the same is true of any number. Raise the following numbers to the second power, using the ana- lytical method : 21. 26 25. 64 29. 76 22. 35 26. 78 30. 23 23. 45 27. 59 31. 54 24. 53 28. 82 32. 83 33. 94 34. 72 35. 63 36. 49 40 s 1600 2 (40 xq; = 720 9* 81 Note. — The pupil should be able to perform these operations without the aid of the pencil. For the last he would say 1600, 720, 2320, 2401 the square of 49. If required to write out the work, he would write as in the margin. 49 2401 A like process may be used in raising numbers, expressed by three or more figures, to the second power. SQUARES AND CUBES. 361 Example. — l. Raise 493 to the second power. Tlie square of 490 is obtained by annexing two ciphers to the square of 49, as already found = 240100 Twice the product of 490 x 3 = 2940 The square of the units = 9 The square of 493 243049 Note. — The pupil will do well to familiarize himself with this process, not for its own sake, but that he may be the better prepared for the demonstration of the process of extracting the square root, which is exactly the reverse of this. In like manner raise the following numbers to the second power : 2. 528 3. 732 4. 236 5. 429 6. 523 366. If a number be divided into any two parts, it may be shown that the square of the whole number is equal to the square of the first part -f- twice the product of the first by the second + the square of the second. Example. — 7. Find the square of 16. Solution. — 16 = 7 + 9 ; according to the formula, therefore, 16 2 =49 + 126 + 81 =256. In the same way compute the squares of the following numbers : 8. 17 10. 35 12. 81 14. 126 16. 839 9. 23 11. 46 13. 94 15. 348 17. 476 "We shall find it useful to observe also how the tens and units of the root are combined in its third power. Process of computing the Third Power analyzed. Example.— What is the cube of 43 ? Solution. — Multiplying the square of 43, as already found, by 40 + 3, we have (40X40) +2(40X3) +(3x3) 4 + 3 (40X40X3) + 2 (40X3X3) + (3x3x3) (40X40X40) + 2 (40X40X3) + (40x3x3) (40X40X40) + 3 (40X40X3) + 3 (40x3x3) + (3X3X3) \ cu. of tens. 3 x sq. of tens by units. 3 x sq. of units by tens. cu. of units, r — Or, 64000 + 14400. + 1080 + 27 ) 362 STANDARD ARITHMETIC. Whereby we find that 367. The cube of 43 is equal to the cube of the tens + three times the square of the tens multiplied by the units + three times the square of the units multiplied by the tens + the cube of the units. The same may be shown to be true of any number whatsoever. In like manner find the cubes of 1. 36 2. 27 3. 92 4. 85 5. 73 6. 95 Note. — Test the accuracy of these results by the common process. 7. Calculate the cube of 47, and write the solution in the fol- lowing form : The cube of the tens 40x40x40= 64000 Three times the square of the tens by the units. 3 x 40 x 40 x 7 = 33600 Three times the tens by the square of the units. 3x40x 7 x 7= 5880 The cube of the units 7x 7x 7 = 343 103823 Write out the solution of the following examples in the same way: Find the third powers or cubes of 8. 37 10. 45 12. 65 14. 83 16. 28 9. 54 11. 23 13. 71 15. 92 17. 74 368. If a number be divided into any two parts whatsoever, it may be shown that the cube of the first part + three times the square of the first multiplied by the second + three times the square of the second by the first + the cube of the second is equal to the cube of the number itself. Example. — Find the cube of 82. Solution. — 82 = 28 + 54 ; according to the formula, therefore, we have 82 » = (28 + 54) 3 = 28 3 + 3 (28 2 x 54) + 3 (28 x 54 2 ) + 54 3 , or, Cube of the first 28 3 = 21952 Three times the square of the first by the second . . 3 x 28 2 x 54 = 127008 Three times the square of the second by the first . . 3 x 28 x 54 2 = 244944 Cube of the second 54 3 = 157464 Third power or cube of 82 == 551368 SQUARES AND CUBES 363 Definitions. 369. Evolution is a process of finding the root of a given number. Evolution is the converse of Involution. In the latter the root is given to find the power ; in the former the power is given to find the root. 370. Square root is indicated by the sign */ y thus, Vl6 = 4 is read, "The square root of 16 is equal to 4." The cube root is indicated by the same sign with the aid of a small figure 3 placed above it ; thus, V'27 = 3. 371. The sign of evolution (yO is called the radical sign, from the word radix, which means root. The figure which in- dicates the degree of the root to be extracted is called the index of the root. 372. The root of a number is indicated also by a fractional exponent, the denominator of which is the index of the root. Thus, 16* means the same as a/16. 27^ is only another ex- pression for a/27. To find the Square Root of a Number. Example. — l. Let it be required to extract the square root of 1849. It is to be remembered that the square of any number ex- pressed by tens and units is equal to The square of the tens -\- twice the product of the tens by the units + the square of the units. Let it also be remembered that No part of the square of tens can be found in tens or units' place, and that no part of the product of tens and units can be found in units 9 place. The process of extracting the root consists in obtaining the tens of the root from the square of the tens, and the units of the root from the remaining parts oi the power. 364 STANDARD ARITHMETIC. First Step. — To find the tens 9 figure of the root.— Since the square of the tens can not contain anything less than hundreds, the two figures at the right con- tain no part of the square of the tens, and are therefore disregard- 1849 | 40 + 3 = 43 ed for the present. The greatest 40 x 40 = 1600 ~~ square in 1800 is 1600, the square 249 root of which is 40. 80 x 3 = 240 We place 40 to the right of 3x3= 9 the given number and subtract 1600 (40 X 40) from 1849. Second Step. — To find the units' figure of the root. — Since no part of the product of tens by any integer whatsoever can be less than ten, the right-hand figure of the remainder can contain no part of the product of the tens by the units, and hence it is dis- regarded for the present. The 24 tens then being twice the prod- uct of the tens by the units, we obtain the units by dividing 24 by twice the tens. The quotient is 3, which we add to the 40. Multiplying twice the tens, or 80, by 3, we have twice the prod- uct of the tens by the units, and subtracting this we have re- maining only the square of the units. Subtracting the square of the units nothing remains, and thus 43 is found to be the square root of 1849. Notes. — 1. Instead of multiplying 80 and 3 separately by 3 1849 | 43 and adding the products, the work is somewhat shortened by mul- ] qqq ~~ tiplying the sum of 80 and 3 by 3. The work would then stand 831i249 as in the margin. 249 2. When the number whose root is to be found is expressed by five or six figures, and it is thus known that the root must contain three figures, the work may be commenced with the two left-hand periods as if they were the only ones, and when the root of these has been so obtained the operation may be completed as though the two figures of the root already found were so many tens, as they really are. 3. The only difficulty in the extraction of the square root is met with when, on multiplying twice the tens + the units by the units, the product is found to be too great. This difficulty arises from the trial divisor being sometimes considerably increased by the tens that come from the square of the units. SQUARES AND CUBES. 365 Thus, in the example here given, if the 54 tens expressed by the first two figures of the remainder were the product of only the tens by the units, we should obtain the exact number of the units by dividing it by twice the tens. But, in this case, we should have 9 for the quotient, which is evidently too great, since by forming the sum of the product of twice the tens + the units by the units we get 621, which is greater than the dividend. In such cases as this we have to diminish the units of the root, not forgetting to change the right-hand figure of the partial divisor at the same time, until we obtain a product not greater than the tens. The following example presents an extreme case of this nature : 1444 1 38 9 68)544 544 2. Required the square root of 321735969. The number being divided into periods of two figures each, the first is 3 (hundred mill- ions). The greatest square in 3 is 1. Sub- tracting this, and annexing the next period to the remainder, we have 22(1) for a partial dividend, and, doubling the root already found, we have 2 for the partial divisor. But the next term of the root can not be greater than 9. We try it, and obtain a re- sult too great. We next try 8, and again the product is greater than the partial dividend. Finally, by trying 7, we obtain a product less than the partial dividend. Having subtracted this, we proceed with the solution. Note. — In this example we have also a case which is of common occurrence, that is, the increase of the tens of the previous divisor by the doubling of the units. The pupil can avoid any confusion by observing that the en- tire part of the root already found is always doubled before the new figure of the root can be found. First Trial. 32i735969 | 19 1 29)221 261 Second Trial. 321735969 [ 18 1 28)221 224 Third Trial. 321735969 [ 17937 1 27)221 189 349)3273 3141 3583)13259 A like difficulty will be found in each one of the following : Find the square roots of 3. 310993225 4. 738534976 5. 27950824225 In the foregoing examples some of the remainders are large. A few are here given in which some very small remainders occur. 3(36 STANDARD ARITHMETIC. 6. Let it be required to extract the square root of 731864809. 731864809 | 27053 On bringing down the third period, we 4 . find that the dividend does not contain twice 47)331 the root already found ; in such a case we 329 write a cipher in the root and also one to the right of the divisor. We then bring down the next period and proceed as before. 5405)28648 27025 54103)162309 162309 Find the first powers or roots of the following squares : 7. 1855197184 8. 36125464489 9. 4901120064 Rule for extracting the Square Root.' 373. Mule. — 1. Separate the given number into periods of two figures each by placing a dot over the units' place and every second figure to the left, and in case of decimals to the right also, annexing a cipher, if necessary to complete a decimal period. 2. Find by trial the greatest square in the left-hand period, and place its root in the form of a quotient at the right. 3. Subtract the square of the root thus found from the first period, and to the remainder bring down the second period for a dividend. 4. Double the root already found for a trial divisor ; divide the dividend by it, and write the quotient for the second term of the root. 5. Annex the second figure of the root to the trial divisor. The result will be the complete divisor. Multiply this by the second term of the root, and subtract the product from the dividend. 6. Repeat the operation as in 4 and 5 until the periods are all brought down. Notes. — 1. When a partial divisor is not contained in a dividend, annex a cipher to the root already obtained, and also to the partial divisor. Bring down the next period, and proceed as in 4 and 5. 2. When the given number is not a perfect square, and hence a remainder occurs after the last period has been used, one or more periods of decimal ciphers may be annexed, and the operation continued as before. The figures in. the root corresponding to the decimal periods will be decimals. 3. It must be kept in mind that no period should contain an integer and deci- mal, and that, if there is an odd number of decimal places in the given number, the last period must be completed by annexing a cipher. SQUARES AND CUBES, 367 To find the Square Root of a Common Fraction. 374-. Mule, — 1. Reduce the common fraction to a decimal, and extract the square root. Or, 2. Extract the square root of the numerator and of the de- nominator. The result will be the terms of the root. Note. — If only the denominator of the fraction is a perfect square, the latter is the more convenient method. If the denominator is not a perfect square, it may be made so by multiplying both terms of the fraction by the denominator. EXAM PLES. Find the square root of 1. 36864 5. 244036 9. 579121 13. 966289 2. 81225 6. 258064 10. 734449 14. 1081600 3. 168921 7. 396900 11. 820836 15. 1177225 4. 212521 8. 499849 12. 950625 16. 1234321 Find one of the two equal factors of 17. 6838225 20. 296356225 23. 44502241 18. 9048064 21. 3196944 24. 61685316 19. 6885376 22. 19228225 25. 179586801 Extract the square root of 26. .0961 30. 28867 34. 3819.24 38. 5416.96 27. 15.21 31. 33489 35. 1.338649 39. 50.1264 28. 22.09 32. 4.2849 36. 226.8036 40. .00720801 29. .0004 33. 17.3056 37. .00001024 41. 290.225296 Extract the square root of 42. 5 46. 2 50. 20 % 54. 3% 43. .5 47. .6 51. 153% 55. 35% 44. .05 48. 26 52. 1% 9 56. 27% 45. .005 49. .02 53. 23.1 57. 36% Find the square root of 58. % fil 625/ •*• /6T6 64. % 67. % 59. % fi9 3136/ *>^' /5329 65. 17% 68. 5% 60. % fi o 84681/ . 66. 11% 69. 38% 368 STANDARD ARITHMETIC. 375. To find the Cube Root of a Number. Example. — l. Let it be required to find the root of which 42875 is the third power. It must be kept in mind that the cube of any number com- posed of tens and units is equal to The cube, of the tens -f- three times the product of the square of the tens by the units + three times the product of the tens by the square of the units ~\- the cube of the units. Since the cube of tens can never fall short of a thousand, we shall not find any part of it in the first three figures to the right, hence they are disregarded in finding the tens. Evidently the root of 42000 can net be so 42875 | 3 great as 40, since the cube of 40 is 64000 ; 30 X 30 X 30 = 27000 ~ but, the cube of 30 being only 27000, the T^ftT^ real root must be between 30 and 40, and hence 3 must be the tens' figure sought for. Subtracting the cube of 30 from 42875 we have 15875 remaining. But having taken the cube of the tens from 42728, the remainder must contain (1.) 3 times the square of the tens x the units + (2.) 3 times the tens x the square of the units + (3.) the cube of the units. Now, since we know that the tens' figure is 3, and hence that three times the square of the tens (3 times 30 x 30) is 2700 ; and since we know also that the remainder, 15875, contains the product of this 2700 by the units, we next try to find the units by dividing 15875 by 2700. But, since 15875 contains something more than 3 times the square of the tens by the units, our quotient is very likely to be too great. If it contained nothing more, it would be easy to find the units by division. Yet we may be sure that it can not be 9, nor 8, nor 7, nor 6, since 6 times 2700 alone is greater than 15875. The figure may be 5, but we can not be sure that even 5 is not too great, until we find that the sum of the three items is not greater than 15875. Let us try it, however, by completing the work as if it were the right figure, thus : 3 times the square of the tens x the units. . . 3 x 30 x 30 x 5 = 13500 3 times the tens x the square of the units. . . 3 x 30 x 5x5= 2250 The cube of the units 5x 5x5= 125 15875 The sum of all these parts being equal to the remainder, 15875, it is clear that 5 is the correct units' figure, and that 35 is the cube root of 42875. SQUARES AND CUBES. 369 The whole work of solu- tion may be put into this form : But, instead of multiply, ing three times separately by the factor 5, and adding the products, we may multiply the sum of the products of the other factors by 5 in one operation. The work would then stand as here jriven. 30 X 30 x 30 = 42875 27000 35 3x30x30x5 = 13500 3x30x 5x5= 2250 5x5x5= 125 15875 15875 30 x 30 x 30 = 42875 27000 ' 35 3 x 30 x 30 = 2700 3 x 30 x 5 = 450 5x5= 25 15875 3175 x 5 15875 Note. — If, on completing the work, we had found that 5 was too great, we should have had to take 4 as the units' figure, and possibly even that might have been found too great. In that case, we should have had to take 3, and try again. The finding of the square or cube root of a number is often a process of guessing, and testing the correctness of the guess. 2. Find the cube root of 22906304. Here the partial di- visor is contained 11 times in the first re- mainder, but inasmuch as we know that the next figure of the root can not be greater than 9, we try 9, and, finding it to be too great, we try 8, which proves to be the right figure. 2 8 = 3 x 20 2 = 1200 3 x 20 x 8 =480 82= 64 22906304 | 284 8 1744 14906 13952 3 x 280 s = 235200 3 x 280 x 4 = 3360 4 2 = 16 238576 954304 954304 3. Extract the cube root of 28372625. On bringing down the second period, it is found that the partial divisor (2700) is not contained in the 28372625 | 305 3 3 = 27 dividend, hence we place a cipher in the root, bring down the next period, and proceed again ac- cording to the rule (4 and 5). 3 X 300 2 = 270000 3 x 300 x 5 = 4500 5x5= 25 274525 1372625 1372625 370 STANDARD ARITHMETIC. Rule for extracting the Cube Root. 376. Mule, — 1. Separate the given number into periods of three figures each by placing a dot over the units' place and every third figure to the left, and, if there be decimals, to the right also. Annex one or two ciphers if necessary to complete a decimal period. 2. Find the greatest cube in the left-hand period, and place its root at the right for the first term of the root sought. 3. Subtract the cube of the first term of the root from the first period, and to the remainder annex the second period for a divi- dend. 4. Take three times the square of the root, already found, as a trial divisor, ascertain how many times it is contained in the dividend, and annex the result to the root as a trial term. 5. Find the sum of 3 times the square of the first term of the root, + 3 times the product of the first by the trial term, + the square of the trial term, for a complete divisor. Multiply the sum by the trial term of the root, and subtract the product, if not too great, from the dividend. 6. If the product be greater than the dividend, diminish the trial term of the root, and proceed as before. 7. The remainder, if there be any, with the next period, will form another partial dividend with which we proceed again, as directed in 4 and 5. Notes. — 1. When a partial divisor is not contained in a dividend, annex a cipher to the root obtained, annex two ciphers to the partial divisor, bring down the next period, and proceed as directed in 4 and 5. 2. When the given number is not a perfect cube, and hence a remainder occurs after the last period has been used, one or more periods of decimal ciphers may be annexed, and the operation continued as before. The figures in the root cor- responding to the decimal periods will be decimals. To extract the Cube Root of a Common Fraction. 377. Hule.—l. Reduce the common fraction to a decimal, and extract the cube root. 2. Or, if the numerator and denominator are perfect cubes, extract the cube roots of the terms separately. The results will be the terms of the root. Note. — If only the denominator of the given fraction is a perfect cube, the latter is the more convenient method. SQUARES AND CUBES. 371 -SLATE EXERCISES. Find the cube root of 1. 6859 4. 2406104 7. 49027896 2. 12167 5. 3869893 8. 66430125 3. 27000 6. 5545233 9. 929714176 Extract the cube root of 10. 1412467848 12. 3341362375 14. 3616805375 11. 1865409391 13. 2857243059 15. 4065356736 Find the cube root of 16. 830.584 18. 1.092727 20. .000175616 17. .970299 19. .002197 21. .007645373 Find the cube root of the following numbers, carrying incomplete roots to three or fve decimal places, as may be required : 22. 1. 24. .01 26. .001 28. % 30. % 23. 2. 25. .02 27. .002 29. 3 / 4 31. % The Extraction of Roots of Perfect Powers. 378. The Square Root. — The square root of a number being one of two equal factors, the square root of a perfect power may be found by resolving the power into its prime factors, separating them into two identical sets, and finding the product of one set. Example. — l. Let it be required to find the square root of 3136. The prime factors of 3136 are 2, 2, 2, 2, 2, 2, 7 and 7. These are separable into two sets of factors, each containing 2, 2, 2 and 7. The product of 2 x 2 x 2 x 7 = 56. Hence, 56 is the square root of 3136. In like manner find the 2. V484 3. a/1024 4. a/16384 5. V234256 379. The Cube Root. — On the same principle, the cube root of a perfect power may be obtained by resolving the power into its prime factors, separating them into three identical sets, and finding the product of one set. Example. — 6. Find the cube root of 13824. The prime factors of 13824 are 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3. These are separable into three sets of factors, each containing 3, 2, 2 and 2. The product of 3 x 2 x 2 x 2 is 24. Hence 24 is the cube root of 13824. In like manner find the 7. V3375 8. V9261 9. V35937 10. V250047 372 STANDARD ARITHMETIC. Constructive or Geometric Solution of the Problem of the Square Root. Problem. — Let it be required to find the side of a square which shall contain 3249 sq. in. Solution. — Suppose that we have 3249 pieces of card-board, each one inch square, and let it be required to arrange them so that they shall together make one complete square. The number of pieces on one side will be the length of the side in inches. First let us try one hundred pieces to the side. To make a square, we must have as many rows of square pieces as there are pieces in a row ; hence, with this beginning, we would need 10,000 pieces. But, inasmuch as we have not so many, we plan to make our square a smaller one. Let us try 60. This we would find would require 3600 pieces, which again is more than the given number. Trying 50, we find that we can complete a square of this size, and have some pieces left. The required square will therefore contain between 50 and 60 pieces, and hence 5 must be the tens' figure of the root. Having completed the square of 50 pieces to the No p j ece( , side, we ascertain, as in the margin, that there are 749 3249 I 5 • pieces left with which the size of the square is to be in- Kn KA _ okaa creased. Laying down a row on each one of two adjacent * 4y sides (50 to the side) — for we must increase the length and breadth equally — we require twice 50, or 100 pieces. If we add two rows to each side, we shall require 2 times 100 pieces, and. if three rows be added, we shall require 3 times 100 pieces, that is, without filling the corner. To fill the corner, we shall need as many more pieces in each row as there are additional rows. Thus, if there are three rows added to each side, we must extend each row of one side by the addition of 3 pieces. Thus we may proceed, making suc- cessive additions of one row at a time to each of two sides until all the pieces are taken, or until the largest possible square is made out of the 3249 pieces given. But the process can be somewhat shortened by ascertaining at once how many rows are to be added. This we can do very nearly by dividing 749 by the number of pieces in two rows, exclusive, of course, of the number of pieces necessary to com- plete the rows when the corner is filled. Dividing 749 by 100, the number required to make a row on each of the two sides, we ascertain that about 7 additional 3249 [ 57 rows can be made out of the remaining 107)749 { 7 5 x 5 = 25 pieces. But we can not complete the 749 107)749 square by adding these seven rows, un- 749 less we can add also seven squares to each row, for the corner must be filled in order that we may have a square. Hence, we add 7 to 100 = 107, and divide by that as a complete divisor. Uniting the two parts of the arithmetical operation, we have the work on the left; SQUARES AND CUBES. 373 Constructive or Geometric Solution of the Problem of the Cube Root. Problem. — Let it be required to find the edge of a cube which shall contain 175616 cu. inches. Solution. — Suppose that we have 175616 cubic blocks, each measuring an inch, and let it be required to make of them one cubic block as large as possible. If we lay down 100 blocks in one row, we will need a hundred rows for the first layer of our large cube. 100 in a row and 100 rows would require 10000 blocks, and the hundred layers necessary to complete the work would take 1000000 blocks ; clearly, then, our cube can not ba 100 inches in length, breadth, and thickness. If we take 90, 80, 70, or 60 for a first row, and try to complete a cube of so many inches, wc shall fall short of blocks. Such repeated trials with the blocks, how- ever, will hardly be necessary. A few trials with the slate and pencil would lead the beginner to discover that the number of blocks in a row must be somewhere between 50 and 60, and hence that the tens in the root can not exceed 5. To build up a cube 50 inches in length will take 125000 blocks (50 x 50 x 50). How many 175616 | 5» blocks shall we have left. The computation in 50 X 50 X 50 = 125000 the margin, which will be readily understood, 50616 tells us that we shall have 50616. In making additions to a cube for the purpose in view, it is most convenient to make them to three sides. Remembering that the block already constructed measures 50 inches long and wide and high, we know that we need 50 x 50 blocks for an addition of one layer to one side, and 3 times 50 x 50 = 7500 for an addition of one layer to each of the three sides. Can we add 7 such layers ? Evidently we can not, for 7 times 7500 blocks would be 52500. Can we add six? Six times 75C0 = 45000. This leaves us 50616 — 45000 = 5616 blocks. Will 5616 blocks be enough to complete the edges and the corner? To test this we first fill the upper front edge by placing one layer after another on the top of the addition already made to that side. For one layer we need 50 x 6, and for six layers we need 5616 50 x 6 x 6, or 1800 blocks ; for three edges we 3x50x6x6 = 5400 need 3 times 50 x 6 x 6, or 5400 blocks. Taking „-.« these from the number of blocks remaining, we have 216 still left with which to complete the work. Will 216 blocks be enough to fill out the corner? We can readily see that we need 6 rows of 6 blocks each for one layer upon the upper end of the addition made in the front right-hand edge, and that we must have 6 such layers to complete the block; 6 x 6 x 6 = 21 6, exactly the number of blocks we had left. Thus out of 175616 small cubic blocks, each measuring one inch in length, breadth, and thickness, we have constructed one large block measuring 50 + 6 inches on each edge. Hence 56 is the cube root of 175616. 374 STANDARD ARITHMETIC. Applications of Square and Cube Root. 1. 5041 slabs of marble 9 inches square will pave a square court. How many slabs on each side ? 2. How many rods long is the side of a square field containing 10 acres ? 40 acres ? 90 acres ? 490 acres ? 640 acres ? Note. — There being no linear unit corresponding to the acre, acres must be reduced to other denominations before we attempt the solution of such a problem as the above. 3. Being told that a certain cubic block of marble contained 91 y 8 cubic feet, I measured an edge and found it to be 4 ft. 5 % in. How much more should it have measured ? 4. What must be the inner dimensions, in feet and inches, of a cubic bin containing 25,000 bushels of grain ? 5. The product of three equal numbers is 3189506048. What are the numbers ? 6. A rectangular court that is twice as long as it is wide con- tains 31,250 square feet. How long and wide is it ? Suggestion. — If the rectangle were divided into two equal squares, how many square feet would there be in each ? 7. A block of stone three times as long as it is wide and high is represented to contain 823 7 / 8 cubic feet. How wide and high should it be ? (See suggestion under Ex. 6.) 8. What must be the dimensions in feet and inches of a square garden-lot, which shall be equal to two rectangular ones measur- ing respectively 8 by 10 and 8 by 18 rods ? 9. What are the dimensions of a cubic bin which will hold as much as three bins measuring respectively 12 by 18 by 9 ft., 18 by 27 by 6 ft., and 12 by 9 by 9 ft. ? 10. A bar of metal 5.75 ft. long, 3.9 in. wide, and .7 in. thick being melted and cast into cubic form, what was the edge of the cube ? 11. If one face of a cube contains 11 sq. ft. 16 sq. in., what are the contents of the cube ? SQUARES AND CUBES. 375 Right-Angled Triangles. 380. A right-angled triangle is a triangle that has one right angle. 381. The side opposite the right angle is called the hypote- nuse. Of the two sides forming the right angle, either one may be taken as the base, and the other as the perpendicular. The pupil should become thoroughly familiar with B ase> the following proposition, proved in geometry : 382. The square of the hypotenuse is equal to the sum of the squares of the other two sides. Example. — l. The base of a right-angled triangle is 4 inches ; the perpendicular is 3 inches. What is the length of the hy- potenuse ? Solution. — The square of the base is 9, the square of the perpendicular is 16. Hence, to obtain the square 3 2 = 9 of the hypotenuse, we add 9 and 16 = 25, the square 4 8 = 16 root of which is 5 = the hypotenuse. 3 s 4- 4 2 = 25 Note. — To ascertain whether this result is correct, V25 __ 5 hvD measure the distance from 3 on either arm of a carpen- ter's square to 4 on the other. Example. — 2. The hypotenuse of a right-angled triangle is 10 inches ; the perpendicular, 8 inches. What is the base ? Solution. — The square of the hypotenuse is 100, the square of the perpendicular is 64. To obtain the 1^2 -• ™ square of the base we subtract 64 from 100 (100 — ft 2 _ _ . 64 = 36). The sq. ** ~ b4: root of_36 is the 10 2 — 8^= 36 base, V36 = 6 Arts. Base = V36 = 6 Ans. Note. — Test cor- rectness of the answer by measuring from 6 on one N -. arm to 8 on the other arm of a carpenters' square. I 1 1 » 1 1 1 * 1 1 1 rrT| 383. The carpenters square is an instrument used in measuring and testing the work of the car- penter, stone-mason, etc. _ 3 .. \ Note.— Test cor- v \ rectness of the answer by measuring from 6 on one Base. Perpen- dicular. Hypote- nuse. a.i%- % — 4. 6 4% — 5. 2% 3% 'erpen- Hypote- icular. nuse. — n% T% 12% 11 376 STANDARD ARITHMETIC. EXERCISES. 1. If the width of a book is 9 inches, and the length 12, how many inches between the opposite corners ? 2. Two dimensions of right-angled triangles being given, as follows, find the third dimension of each : Base. 6. 9 7. — 8. 8% 9. If a room is 21 ft. long and 20 ft. wide, how long is the diagonal ? If it is 45 ft. long and 28 ft. wide ? 45 ft. long and 24 ft. wide ? 24 ft. long and 10 ft. wide ? 10. Find the length and width of a square box which shall con- tain as much as two boxes, one 2 ft. and the other 2 % ft. square, the three boxes being of the same height. If one is 4.2 and the other is 5.6 in. square. 11. What is the diameter of the largest circular saw that can be taken through a doorway 8y 2 ft. highland 6 3 / 8 feet wide ? If it is 7% ft. high and 5% ft. wide ? 12. On a level play-ground there is a rope, 11 y 4 ft. long, fast- ened to a ring at the top of a pole 9 ft. high. How far from the foot of the pole will the rope reach the ground ? 13. A horse is to be tethered in the center of a rectangular lot 240 ft. long by 238 ft. wide. How long must the rope be which will allow him to graze in the corners of the lot ? 14. The figure here represents a rectangular farm. The only dimen- sions given are 1984 rods, one of the longer sides, and 2434 rods, the di- agonal line. How many acres does the farm contain ? CHAPTER XIX. MENSURATION. I. Plane Surfaces. 384. If a straight-edge laid anywhere upon a surface touches at every point the surface is a plane surface. This is the practical test of a plane. It can be tried on the surface of a desk, table, floor, wall, or any other surface with an ordinary straight-edge rule. The carpenter uses the edge of his plane for the purpose. 385. A portion of a plane bounded by one or more lines is a plane figure. 386. A Circle is a plane figure bounded by a curved line, every point of which is equally distant from a point within, called the center. 387. The boundary line of a circle is called the circumfer- ence, n 388. A straight line drawn through the center and terminating at the circumference on both sides is called a diameter. 389. A straight line drawn from the center to the circumference is called a radius. A radius is one half of the diameter. (The plural of radius is radii.) With a narrow slip of paper measure the circumference of a dinner plate, then measure the distance across it, and you will find the circumference a little more than three times the length of the diameter. If the plate is ten inches in diameter, and you have taken the measurements carefully, you will find the slip of paper to be very nearly 2,\ l j 2 inches long. Thi3 result agrees very nearly with the great truth, proved in geometry, that 378 STANDARD ARITHMETIC. 39 0« The circumference of any circle is 3.14159 times the length of its diameter. Hence, having the diameter, 391. To find the circumference of a circle: Multiply the diameter by 3.14159. Conversely, having the circumference, 392. To find the diameter of a circle: Divide the circum- ference toy 3.14159. Note. — The improper fraction corresponding to 3.14159 may be remembered by referring to the series of figures, 113355 (the first three digits representing odd numbers written doubly). Taking the last three for the numerator and the first three for the denominator, thus, ff$ f we have the ratio of the circumference to the diameter. The reciprocal, $$£, represents the ratio of the diameter to the circum- ference. For mere approximations, the circumference may be said to be 3 77 times the diameter. SLATE EXERCISES. 1. If the diameter of an iron column is 3.5 in., what is the circumference ? If the girth of a tree is 5 ft. 9 in. what must be its diameter ? 2. If the equatorial diameter of the earth is 7925 miles, how long in miles and rods is the equator ? 3. The distance from the center of the hub of a wheel to the outer edge of the felly is 15 in. How long must the tire be ? 4. If the length of an oar from the thole-pin to the end of the blade is 5 ft., how many feet would the end of the blade travel in the water during 6000 strokes, each describing an arc of 60° ? (60° = l / 6 of the circumference.) 5. If the circumference of a circular pond is 628.318 rods, what part of a mile must I row to pass from shore to shore across the center of the pond ? 6. If a horse is tethered to the middle post of a fence, from which he can graze out into the field in a curved line 78.539314 ft. long, how long is the tether ? 7. What will be the circumference of the largest circle that can be drawn on a sheet of paper 12 in. wide and 18 in. long ? MENSURATION. 379 393. A plane figure bounded by three straight lines is a triangle. 394. The base of a triangle is the side on which it is sup- posed to rest. (Any side of a triangle may be taken for its base.) 395. The altitude of a triangle is the perpendicular dis- tance from the angle opposite the base, to the base, or to the base produced. {Produced— con- tinued in the same direction.) 396. Triangles take different names, according to the rela- tions of their sides. If the sides of a triangle are equal, it is an eqilateral triangle. If only two sides of a triangle are equal, it is an isosceles triangle. If no two sides are equal, it is a scalene triangle. If one of the angles is a right angle, it is a right-angled triangle, or a right triangle. Equihteral Triangle. Scalene Triangle. Eight-angled Triangles. 397. Parallel lines are straight lines that have the same direction but do not coincide, and can never meet, however far they may be produced. 398. A plane figure bounded by four straight lines is a quad- rilateral. ( Quadrilateral means four-sided.) 399. Quadrilaterals take different names from their angles and from the relation of the sides to each other. 400. A quadrilateral which has no two sides parallel is a trapezium. 401. A quadrilateral which has only two sides parallel is a trapezoid. 380 STANDARD ARITHMETIC. 4-02. A quadrilateral that has its opposite sides parallel is a parallelogram. 403. Parallelograms take different names from their angles and the relation of the sides to each other. 404. A parallelogram that has all its angles right angles, and all its sides equal, is a square. 405. A parallelogram that has all its angles right angles, and only its opposite sides equal, is called a rectangle. 406. A parallelogram that has its sides all equal, but whose angles are not right angles, is called a rhombus. 407. A parallelogram that has only its opposite sides equal, and whose angles are not right angles, is called a rhomboid. , 408. A straight line that joins the vertices of two angles, not adjacent, is a diagonal. To find the Areas of Quadrilaterals and Triangles. The Rectangle, including the Square. —We have already found that 'to compute the area of a rectangle, we must multiply the number of the proposed square units of measure which can be placed on one side of the rectangle by the number of corresponding linear units in the adjacent side. Example. — Let the figure represent cu .,,.,.,.,,,,,,,,,,,,£ a rectangle 14 inches wide and 19 inches long. What is the area ? * Solution. — 19 square inches can be placed on the side c d, and, since there are 14 such rows, there will be 14 times 19 sq. in. in the whole rectangle = 266 sq. in. MENSURATION. 381 The Rhomboid and Rhombus.— Example.— Let it be re- quired to compute the area of a rhomboid, 10 in. long and 6 in. wide. Solution. — 6 x 10 □ in. = 60 □ in. Am. If from either end of a rhomboid we cut a right-angled triangle, and add it to the other end, as indicated by dotted lines in the figure, we should form a rectangle equivalent to the rhomboid ; hence, ' ' 4-09. To find the area of a rhomboid : Multiply the length of one of two parallel sides by the distance between them. The rule for the rhombus is the same. , -~« It is to be observed that, to obtain the width of a rhombus / j ^^/ or rhomboid, we do not measure a side, but the perpendicular ,• L^^ / distance between parallel sides. ^— I ' The Triangle— Example.— Given the base of a triangle 14 yd. and the altitude 9 yd., to find the area. The triangle is one half of a parallelogram having the same base and altitude, as may be seen by the above diagram. Hence, 4-10. To find the area of a triangle: Find the area of a rectangle of the same base and altitude, and take one half of it. 411. The following rule is sometimes necessary : When the three sides of a triangle are given, to find the area : From half the sum of the three sides subtract each side separately. Multiply the half sum and the three remainders to- gether, and extract the square root of the product. The Trapezoid. —Example. — Given the > length of each of the two parallel sides of a / jV trapezoid, 6 and 10 feet, and the distance be- T i A tween them, 5 feet, to find the area. Solution.— 6 ft, + 10 ft. = 16 ft. x / 2 of 16 ft. = 8 ft. 5x8 sq. ft. = 40 sq. ft. Ans. By inspection of the figure it will be seen that, by the aid of dotted lines, we have constructed a rhomboid equal in area to the trapezoid whose area is required ; and, further, it is plain that the side of this rhomboid is equal to half the sum of the two parallel sides of the trapezoid. Hence, 4-12. To find the area of a trapezoid: Multiply one half the sum of the parallel sides by the distance between them. 17 382 STANDARD ARITHMETIC. 4-13. Th.8 Trapezium. — The surface of a trapezium may be found by dividing it into two triangles ; then having measured the length of the diagonal and the two perpendiculars, we calcu- late the area of each triangle separately. The sum of the areas of the two triangles is the area of the trapezium. Example. — In a trapezium A B C D, we measure the diagonal A C, and find it to be 24 feet ; also the perpendiculars, and find one to be 18, the other 9 feet. What is the area of the trapezium ? Solution. Area of the triangle A B C = 24 x 9 = 216 sq. ft., l / 9 of 216 = 108 sq. ft. Area of the triangle A D C = 24 x 18 = 432 sq ft., l fy of 432 = 216 sq. ft. Area of A B C +A D C,orthewho\etm.= 648 sq. ft., 1 / i =1*24 sq. ft. Am. SLATE EXERCISES. l. How many acres in a piece of woodland 220 yd. in length and 1 furlong in width ? 2. How many square miles in a township 5 miles and 40 chains square ? 3. How many square feet in a floor 20 ft. long and 5 yd. wide ? 4. Find the surface of a pane of glass measuring 37% in. long and 23 in. wide. 5. How many square yards in the four walls of a room 15 ft. 6 in. high and 80 ft. in compass ? 6. A rectangular pavement, 50 ft. 9 in. long and 12 ft. 6 in. wide, was laid with a central line of stone 5 ft. w r ide at $1.75 a running foot ; the sides were flanked with brick at 80^ per square yard. What did the paving cost ? 7. How many square feet in a surface 24 ft. long 20 ft. wide ? How many in another surface of half these dimensions ? MENSURATION. 383 8. I have a box without a lid ; it is 5 ft. long, 4 ft, wide, and 3 ft. deep, interior dimensions. How many square feet of zinc will it take to line the bottom and sides of this box ? 9. Find the area of a rhomboid whose length is 1 yd. 2 ft. 6 in., and whose width is 2 ft. 3 in. Draw this figure on your slate, with the scale reduced by 12. 10. What is the height of a rhomboid whose area is 12 A. and its length 13y 3 chains? 11. The four eaves of a pyramidal roof measure each 44 ft. 3 in., and the common peak of the four triangles has a perpendicu- lar distance of 24 ft. from the eaves. What is the area in slaters' squares (1) of one triangle ? (2) /of the roof ? 12. I have a triangular garden containing 233 % square yards. The perpendicular distance from the apex to the base is 20 ft. What is the length of the base ? 13. A triangular field, whose sides are unequal, contains 5 acres. The base-line measures y 4 mile. What is the altitude in chains ? 14. What is the area of a triangle whose three side3 are 13, 14, and 15 ft.? 15. What is the area in acres of a triangular field whose three sides measure respectively 47, 58, and 69 rods ? 16. The parallel sides of a trapezoid measure respectively 3y 3 ft. and 6 ft. 8 in. ; the perpendicular distance between them is 2 ft. What is the area ? 17. Find the area of a trapezium whose diagonal is 168, and one perpendicular 42, the other 56. 18. How many centares in a rhomboid one side of which meas- ures 50 meters, the perpendicular distance to the opposite side being 35 meters ? 19. What is the area of a square field, the diagonal of which measures 174 meters ? 384 STANDARD ARITHMETIC. Regular Polygons, having more than four sides, take different names according to the number of sides. The following are some of them : PeDtagon. Heptagon Octagon. Nonagon. Decagon. 414. By dividing a regular polygon into triangles by lines drawn from the center to the several angles, it will be readily seen that the area of the polygon is equal to the sum of the areas of the triangles ; hence, 415. To find the area of a regular poly- gon : Multiply the perimeter (the sum of all the sides) by one half the perpendicular dis- tance from the center to one of the sides. Example. — The side of a pentagon measures 5 ft., and the perpendicular distance from the center to the side is 4% ft. What is the area of the polygon ? Solution. — The area of each triangle =; 1 / 2 (4 T / 2 x 5), that is, one half the product of the base by the altitude, and the area of the whole polygon = 1 / 2 (4 l / 2 x 25) = 56 y 4 □ ft. Ans. 416. The Circle* — The calculation of the areas of polygons leads us by an easy step to the calculation of the area of a circle, for it is plain that, as the number of sides of the polygon is increased, the perimeter becomes more and more nearly equal to the circumference of the circum- scribed circle, and the perpendicular distance from the center to the sides of the polygon becomes more and more nearly equal to the radius of the circle. Hence, having the circumference and the radius, 417. To find the area of a circle : Multiply the circumfer- ence by one half of the radius. MENSURATION. 385 Another Method of finding the Area of a Circle. — If we have a square, and find the center of it by lines joining op- posite corners, and with this center inscribe a circumference exactly touching the sides of the square, the corners outside of the circle will contain .2146, and the circle itself .7854 of the surface of the square. Hence, we have another rule, 4-18. To find the surface of a circle: Mul- tiply the square of the diameter by .7854. S.ATE EXERCISES. 1. Required the area of a regular hexagon whose side is 73 ft. and the perpendicular is 63.2 ft. 2. How many acres in an octagonal section of land whose side is 1983.2 rods and perpendicular 7% miles ? 3. How many square yards are there in a circle whose diame- ter is 12 ft. 6 in. ? 4. Draw a square containing 81 square inches ; inscribe a circle in this square. What is the superficies of this circle in square inches ? 5. A cow is tethered to a post driven in the center of a lot 100 ft. square ; the tether is just long enough for her to reach the fence. How much of the surface of the field is she unable to crop ? 6. Find the radius of a circle whose area is 95.0334 square ft. 7. Find the area of a tent floor, with / A 7? semicircular ends, from the dimensions of the following diagram, in which the line A B equals 200 ft. and the line A G equals 90 ft. 8. What is the difference in length between a fence around a circular lot 123 ft. in diameter and a square lot of the same width ? 9. Find the difference in cost at 87 1 / 2 < f per rod between fenc- ing a square field of 10 acres and a rectangular field 32 rods wide of the same area. OZZD 386 STANDARD ARITHMETIC. II. Mensuration of Solids. 419. A Solid is a limited portion of matter haying length, breadth, and thickness. 420. A solid bounded by a curved surface, every point of which is equally distant from a point within, called the center, is called a globe, or sphere. A circle which divides the surface of a sphere into two equal parts is called a great circle of the sphere. \ Sphere. 421. The surface of a sphere is exactly equal to four times the surface of a great circle of the sphere. It would require just four times as much gold-leaf to cover a sphere as would cover one side of the section made by cutting the sphere into two equal parts (hemispheres). It will be recollected that the surface of a circle is found by multiplying the circumference by one half the radius ; hence, to find the surface of the sphere, we multiply its circumference by four times one half the radius, or, in other words, by the diameter. Hence, 422. To find the surface of a sphere: Multiply the cir- cumference by the diameter. Conversely, having the surface, 423. To find the diameter of a sphere: Divide the surface of the sphere by 4 to obtain the surface of a great circle. Divide this by .7854 to find the area of the circumscribed square. Ex- tract the square root to find the side of the square. The side of the circumscribed square is equal to the diameter of the circle. Applications. — l. If the earth were a sphere with a diameter of 7925 miles, what would its whole surface be ? 2. How many square inches of leather would it require to cover a ball 3 in. in diameter ? 3. The surface of a sphere is 11170.12 square feet. What is its diameter ? MENSURATION. 387 4-24-. Prisms. — A prism is a solid whose bases are equal and parallel polygons, and whose faces are parallelograms. t-25. Prisms take different names from the forms of their bases. It will be seen that cubes and rectangular solids are prisms. They are also called Parallelopipeds. A, a, a, a, are bases; b, b, b, b, are lateral faces; the edges between the faces are lateral edges (side edges) ; the edges between the faces and bases are basal edges (base edges). If the faces and edges arc perpendicular to the bases, the prism is said to be a right prism, otherwise it is said to be oblique. 426. The sum of all the lateral faces is the convex surface. The sum of bases and faces is the entire surface. The altitude is the shortest distance between the bases. Since the faces of prisms are all parallelograms, having a common altitude, if we have a side of the base and the altitude given, 4-27. To find the convex surface of a prirm : Multiply the perimeter (sum of all the sides) of the base by the altitude of the prism. To find the entire surface : Add the areas of the bases to the convex surface. Suggestion. — Suppose that you have a block of the shape of one of these prisms, and that you have fitted to it a piece of paper so as to exactly cover its convex surface. The paper will be a parallelogram, one side being equal to the height and the other side to the perimeter of the base. Hence the rule as riven above. ■388 STANDARD ARITHMETIC. 428. Cylinders.— When the number of sides of the bases of a right prism is so increased that the bases become circles, the prism becomes a cylinder. In this case the prism loses the lateral edges, and the faces become one. Other terms used for prisms apply also to the corresponding dimensions of the cylinder. Cylinder. 4-29, To find the convex surface of a cylinder : Multiply the circumference of the base by the altitude of the cylinder. 4-30. To find the entire surface: Add the areas of the bases to the convex surface. 431. Pyramids. — A solid that has a polygon for its base, and triangles, meeting in a point, for its faces, is a pyramid. 432. The vertex is the point in which the triangular faces meet. The altitude is the shortest distance from the vertex to the base. The slant height is the shortest distance from the vertex to one of the sides of the base. Other names applied to the parts of the prism apply also to the corresponding parts of the pyramid. pyramid. 433. Since the faces of a pyramid are equal triangles having the sides of the base for their bases, and the slant height for their common altitude, 434. To find the convex surface of a pyramid : Multiply the perimeter of the base by one half the slant height. Applications. — l. Find the entire surface of an octagonal prism, 12 in. high with 2 in. sides. 2. Find the convex surface of a piece of stove-pipe, 6 in. in diameter and 2 ft. in length. 3. Find the convex surface of a great pyramid, 764 feet square, and having a slant height of 451 feet. MENSURATION. 389 4-35. The Cone.— When the number of the sides of the base of a pyramid is so increased that the base becomes a circle, the pyramid be- comes a cone. From the explanation of the rule for finding the convex surface of a pyramid, the reason of the following rule is plain. Having the circumference of the base and the slant height given, 436. To find the convex surface of a cone: Multiply the circumference of the base by one half the slant height. 437. Frustums of Pyramids and Cones.— If a part of a pyramid or cone be cut away, so that the section is parallel to the base, the por- tion between the section and the base is called the frus- Frustum of a pyramid. turn of the pyramid or cone. Frustum of a cone. A face of a frustum of a pyramid is evidently a trapezoid, the surface of which is found by multiplying one half the sum of the parallel sides by the slant height of the frustum. Hence, having a side each of the upper and lower bases, 4-38. To find the convex surface of a frustum of a pyra- mid: Multiply half the sum of the perimeters of the upper and lower bases by the slant height. Also, having the circumference of the upper and lower bases, 4-39. To find the convex surface of a frustum of a cone : Multiply half the sum of the circumferences of the upper and lower bases by the slant height. Applications. — l. What is the convex surface of the frustum of a hexagonal pyramid, each side of the lower base being 4 ft., and of the upper base 3 ft., and the slant height 8 ft. ? 2. What is the convex surface of the frustum of a cone, the diameter of the lower base being 18 in., that of the lower base 8 in., and the slant height 8y 3 in.? 390 STANDARD ARITHMETIC. The Volume or Contents of Solids. Contents of Prisms and Cylinders.— To find the volume or contents of rectangular solids, we have learned to multiply the number of solid units which can be laid upon the base by the number of layers necessary to complete a parallelo- piped of the required height ; or, as it is commonly expressed, we multiply the base by the altitude. The rule is the same for all prisms, and also for cylinders. Hence, 4-4-0. To find the contents of prisms and cylinders : Hav- ing found the base of the prism or cylinder by the rules for plain surfaces, multiply the base by the altitude. The product will be the volume sought. Contents of Pyramids and Cones.— It is proved in geometry that the volume of a pyramid is exactly one third of that of a prism which has the same base and altitude, and that the volume of a cone is exactly one third of that of a cylinder having the same base and altitude. If a solid iron cylinder of any dimensions were turned down to a cone, as represented in the cut, the cone would weigh just one half as much as the parts cut away, that is, only one third the solid contents of the cylinder would remain; hence we have the rule : 4-4-1. To find the contents of pyramids and cones: Hav- ing found the base, multiply it by the altitude of the pyramid or cone, take one third of the product, and the result will be the contents required. Contents Of the Sphere.— As the mode of cal- culating the area of a triangle was applied to the cal- culation of the area of a circle, so the mode of finding the contents of a pyramid may be applied to the calcu- lation of the contents of the sphere. For, as the triangles into which a polygon can be divided may be regarded as being so increased in num- ber that their bases finally become the circumference of a circumscribed circle, so the number of faces of a solid similar to the one here represented may be regarded as being so increased that they become the faces of a solid differing so little from a perfect sphere that the solid may be regarded as a sphere composed of a great num. ber of pyramids, all the bases of which make up the surface of the sphere ; hence, 4-4-2. To find the contents of a sphere: Multiply the sur- face of the sphere by one third of the radius. MENSURATION. 391 Another Method for finding the Solid Contents of a Sphere. — The solid con- tents of any sphere are .5236 of a cube whose edges meas- ure the same as the diameter of the sphere. (A base-ball that just touches every side of the box containing it occu- pies a little more than one half, or more nearly .5236, of the space in the box.) Hence, having the diameter, 4-4-3. To find the solid contents of a sphere : Find the contents of a cube whose edges are equal to the diameter, and take .5236 of the result. Applications. — l. What is the solidity of a triangular prism whose length is 12 ft., and either of the equal sides of one of its equilateral ends is 3 ft. ? 2. How many gallons of water would a cylindrical boiler con- tain if 25 in. high and 12 in. in diameter ? 3. Find the cubic inches in the largest cone that can be cut from a cylinder 2 ft. 6 in. high and 14 in. in diameter. 4. A sphere 8 in. in diameter is placed in a cubic box whose interior dimensions are 8 in. How much vacant space is left ? 5. I have a cylindrical tank which contains 160 gallons ; it is 6 ft. 5 in. in diameter. How deep is it ? 6. Find the cubic feet in a log 30 ft. long and 2 ft. in diameter at the larger and 1 ft. 10 in. at the smaller end. 7. Find the cubic contents of the great pyramid mentioned in Problem 3, page 388. 8. How many cubic feet in a circular mound 48 ft. high, and having a diameter of 86 ft. at the top, and a circumference of 471.24 ft. at the bottom? 9. How many cubic miles in the earth, supposing it to be a perfect sphere 8000 miles in diameter ? 10. How many barrels of oil in a tank 60 ft. in diameter if the Oil is 5 ft. deep ? (40 gal. to the barrel.) 11. Find how many cubic meters in a sphere, the surface of which contains 5682 a meters/ 392 STANDARD ARITHMETIC. Duodecimals. 444. A scale of tivelfths, called a Duodecimal Scale, is sometimes used in the measurement of surfaces and solids. Example. — What is the area of a table top which is 3 ft. 9 in. long by 2 ft. 7 In. wide ? Ft. 12ths. Solution. — Write one dimension under the other, calling 2 7 the inches 12ths (of a foot). Then multiplying by 3, we have g c> 3 x 7 / 12 = 21 /i2 = 1 sq. ft. and 9 / 12 sq. ft. = the area of the narrow strip at the bottom and on the left of the arrow. Writing the 3 . 12ths under 9. 8. 3 I2ths, and adding the units to the product of 3 x 2 (area of the large squares above) we have 3 x 2 + 1 = 7 sq. ft. Writing the 7 under ft., we have 7 sq. ft. and 9 twelfths of a sq. ft., the entire area of that part of the figure on the left of the arrow. 9 11 Next, 9 /i2 x 7i /144 — 3 feet kt//Z , *i k\ 1 5 twelfths and 3 /i 44 , the area of the small rectangle at the lower H right-hand corner. Writing the 3 in the line below the first partial product, and to the right, as a lower denomination, and adding the 5 twelfths to 9 /i 2 x 2 (the area of the remaining part of the figure), we have 23 / 12 = 1 and ll /i 2 . Writing the X1 /i2 under the 12ths, and the 1 under the ft., we have in the second part of the product the entire area of that part of the figure which is represented on the right of the arrow. The sum of the two parts thus found is the entire area required. 445. In duodecimals, the unit is a linear foot, a square foot, or a cubic foot, according as it is used to represent the length of a line, the area of a surface, or the contents of a solid. One twefth of a foot is a prime ('), t /- i2 of a prime is a second ("), 1 / 12 of a second is a third ('"), etc. Hence, for linear, square, and cubic measure, wc need only the following Tabla : 12 "" = 1 '" 12 ' m 1 " 12 = 1 12 = 1 foot MENSURATION. 393 Addition and subtraction of duodecimals are performed as in compound numbers. For multiplication, we observe the following 446. Rule. — 1. "Write the terms of the multiplier under the corresponding terms of the multiplicand. 2. Multiply each term in the multiplicand, beginning with the lowest, by the highest term of the multiplier, then by the next lower, etc., observing that 12 of any lower denomination make one of the next higher. 3. Add together the partial products thus obtained. 35 ft. 9' 10 ft. 6' 350 ft. 90' 210' 54" SLATE EXERCISES. l. Find the area of a rectangle measuring 35 ft. 9' by 10 ft. 6\ Solution. 35 fL 9 ' Or, all reductions 10 fo» 6 may be made in the . 357 ft. 6' process of adding the 17 ft. 10* 6" partial products, as at m~K V~¥' the ri S ht - 375 ft. 4' 6" A t 14$ per u yard, find the cost of painting 2. 64 ft. 3' by 25 ft. 3' 4. 108 ft. 9' by 31 ft. 6' 3. 28 ft. 6' by 17 ft. 9' 5. 36 ft. 8' by 38 ft. 8' At 16$ a □ yard : At the prices given per □ yard : 6. 65 ft. 9' by 1 ft. 1' 6" 9. 198 ft. 9' by 2 ft. V at 100 7. 34 ft. 10' 6" by 2 ft. 3' 10. 283 ft. 9' by 3 ft. 4' at 110 8. 123 ft. 4' 6" by 1 ft. 8' 3" 11. 114 ft. 6' by 5 ft. 11' at 120 Find the solid contents of blocks of marble measuring 12. 3 ft. 2' by 2 ft. V by 1 ft. 6' 15. 7 ft. 2' by 4 ft. 5' by 3 ft. 6' 13. 4 ft. 9' by 1 ft. 9' by 2 ft. 3' 16, 10 ft. 1' by 3 ft. 2' by 4 ft. 3' 14. 5 ft. 3' by 5 ft. 2' by 15 ft. 9' 17. 7 ft. 8' by 8 ft. 7' by 6 ft. 5' 447. The process of division being the reverse of multiplica- tion, no rule is needed. 10 ft. 6' ) 375 ft. 4' 6" (35 ft. 9' Example.— Divide 375 ft. 367 ft. 6' 4' 6" by 10 ft. 6'. 7 ft. 10' 6" 7 ft. 10' 6" 394 STANDARD ARITHMETIC. Original Problems. 1. Drop a plumb-line from a window ; mark the distance from the ground, and find what else is needed to make a problem re- quiring the distance from the sill of the window to any distant object on the ground. 2. The masts used for electric lights in many cities will sug- gest good problems. 3. Several pieces of smooth, straight wire of uniform length, being bent into outlines of various geometric planes, will suggest many problems. How should one of these wires be bent to inclose the greatest space, in the shape of a triangle, of a square, of a circle, or what ? (A good foot-rule should be used to take the necessary dimensions.) 4. Card-boards being placed in their hands, the members of the class may be asked to make boxes of them so that there shall be only one piece in a box ; to make the convex surfaces of cylin- ders, cones, prisms, pyramids, etc. Ask how a circular card-board may be cut so that, when the cut edges are neatly joined edge to edge, the surface of the cone thus produced may contain % or other specified fractional part of the surface of the circle. 5. Ask the class to take the necessary measurements and to calculate the contents of a block of stone, of a large water-pipe, of a log of wood, or other suitable object in the neighborhood. 6. Ask the diameter and solid contents of the largest cylinder, the largest cone, or of the largest sphere that can be cut from a rectangular block of wood which you may bring into the class- room, leaving the members of the class to make their own meas- ments. 7. Construct a cubic box into which a sphere may be placed touching all the sides, and take the ball out and fill the box with sand ; weigh that, and ask the class what the weight of the sand would be if it had been filled in around the ball. Weigh, and see how nearly the two results correspond. CHAPTER XX. EXCHANGE.— DUTIES OR CUSTOMS.— BONDS. 448. Exchange is a method by which one person may make a payment to another person at a distance without transmitting money. Illustrations. — 1. A familiar illustration of exchange is found in the use of Postal-Notes and Money-Orders for paying small sums to persons at a distance. 2. If larger ones are to be paid, the Government leaves the business to private persons, usually bankers, who have credit and the necessary understanding with each other. Thus, if a person in New York buys merchandise of another at New Orleans, he does not send gold or bank-notes to pay for it, but goes to a New Yo:k bank and buys an order, called a draft, on a bank in New Orleans, for the amount desired. This he sends to his creditor, who takes it to the banker on whom it is drawn and gets the money for it. 3. Postal notes and money-orders are always payable at sight, that is, when pre- sented, but, on such orders as those just mentioned, time is often allowed for pay- ment. Hence, the banker in New York who gets "cash down" for a draft allows interest on the sum till the time set for its payment in New Orleans. 4. So long as New York is buying cotton and sugar from New Orleans, and New Orleans is buying manufactured goods from New York to about equal amounts, the sums to be paid in each city by the merchants of the other are about equal, and the number of drafts drawn in each city on the other is about the same ; but, if New York were buying $10,000,000 worth per month and New Orleans only $5,000,000 worth, money would have to be sent from New York to New Orleans to pay the balance. In this case, the banks of New York, in selling drafts on New Orleans, would charge something to pay for the risk and expense of shipping money to New Orleans ; and, on the other hand, the banks of New Orleans would be glad to sell drafts on New York for something less than their face, for thus they would be getting some of the balance due them. 5. The rates of premium on drafts for considerable sums can be but little greater than the charges made by the express companies for carrying the money, for, if a merchant in Chicago had to pay $10,075 for a draft of $10,000 on New York, and the charge for expressage were but $50, he would save $25 by sending the money directly. •-,-••• 396 STANDARD ARITHMETIC. Definitions. 4-49. A Draft is a written order directing or requesting the party to whom it is addressed to pay a certain sum to a certain person, or to his order, and to charge the same to the person who makes the request. 450. Drafts payable in the country in which they are drawn are called Domestic or Inland Bills; when payable in a foreign country, they are most commonly called Bills of Exchange. 451. The party making the order is the Drawer; the party ordered or requested to pay is the Drawee ; the party named to whom or to whose order the payment is to be made is the Payee. 452. The payee may transfer a draft to another party in the same way that he would transfer a promissory note made payable to his order. (See page 321, note 2.) 453. A Sight Draft is payable when presented. A Time Draft is payable at the time named in the draft. Three days' grace are allowed on time drafts and sometimes on sight drafts. 4-54. The following is the common form of domestic bills : Sight Draft. $500. Cleveland, 0., May 15, 1886. At sight, pay to the order of Henry James & Sons, five hundred dollars, and charge to the account of M. W. Chester. To Wilson & Hewitt, Boston, Mass. Notes. — 1. If time is allowed for the payment of a draft, it should be made to read Thirty days after sight, Sixty days after sight, etc., according to the time agreed upon. In this case it would be called a Time Draft. 2. When a time draft is received, the payee should at once present it to the drawee, who writes the word Accepted, with the amount, date, and his own name, across the face of the draft, if he is willing to honor it, that is, pay the sum called for. The time in which it matures is then reckoned from the date of accevt&nce. EXCHANGE. 397 SLATEEXERCISES. On May 14, 1886, exchange on New York was quoted as below in the several cities named : Chicago, 50 premium. Charleston, */ t and l f 4 ft premium. St. Louis, 25 and 50 " New Orleans, 100 " Savannah, 3 / 16 and l / A % " San Francisco, 15 and 20 " Boston, 17 and 20 premium. Note. — When exchange is quoted at a given sum, it means so much on $1000. 50^ on $1000 is equivalent to 1 / 2 o^ 1. Find the cost in Milwaukee of a sight draft on New York for $600, exchange being % fo premium. Analysis. — Since the draft is sold at a premium of 3 / 4 $, each dollar costs $1.00 3 / 4 , and $600 costs 600 times 1.00 3 / 4 =$604.50 Ans. 2. How much must be paid in Chicago for a sight draft on New York for $11,200, the discount being 3 / 16 $ ? Analysis. — Since the discount on the face of the draft is 3 /i 6 #, each dollar costs $.99 13 / 16 , and $11,200 costs 11,200 times $.99 13 / 16 =$11,179 Ans. What must be paid for a draft drawn at 3. New York on Cleveland for $1500 at */,£ discount? 4. Detroit " San Francisco " $500 " 1 . 21. $050, discount being 1 / 2 which would be considered sufficiently accurate. The answers given to the following examples arc found by the process given in the solution. EXCHANGE. 399 SLATE EXERCISES. Interest being 6%, find the face of a 30-day draft that can be bought for 27. $500, ex, being 1 / B ^ premium. 31. $1216, ex. being 3 / 4 $ premium. 28. $325, " " */ 4 # discount. 32. $1925, " " */ 8 # discount. 29. $90, " u lf " 33. $2500, " " V 8 ^P remium - 30. $720, u " i/g^ premium. 34. $1650, « " 8 / 5 $ discount. /U £/?e same rate of interest, find the face of a 60-day draft that can be bought for 35. $375, ex. being 7 / 8 $ discount. 40. $750, ex. being 5 / 6 $ premium. 36. $1465, " " i/*# " 41. $2000, " M 3 / 5 % discount. 37. $1390, " " 8 / 8 ^P remium - 42. $5650, " " %fa M 38. $1500, " " 9 / 10 ^ discount. 43. $560, " " 8/ 4 $ P rem hmi- 39. $695, " " 1 /s#P remium - 44. $225, " " 1$ discount. The rate of interest being 4%, find the face of a 90-day draft that can be bought for 45. $2600, ex. being */ 8 # discount. 43. $5000, ex. being 3 / 5 % discount. 46. $700, " " */*# premium. 49. $3750, " " B /s# premium. 47. $1950, " « 1V 8 ^ " 50. $9000, " M l'/^dis't. Applications. — l. A merchant in St. Louis ordered his broker in New York to purchase $5000 worth of merchandise for him. On shipping the goods, the broker draws on the merchant for the amount, with commission at 3 $. What should be the face of the draft, the premium on St. Louis being 1 / 2 per lb. ; 1000 lb. figs at 2