! i IN MEMORIAM FLORIAN CAJORl Digitized by the Internet Archive in 2008 with funding from IVIicrosoft Corporation http://www.archive.org/details/advancedarithmetOOIymarich Advanced arithmetic BY ELMER A. LYMAN PROFESSOR OF MATHEMATICS IN THE MICHIGAN STATE NORMAL COLLEGE, YPSILANTI, MICHIGAN 5»r>ertf^o«2 CONTENTS PAGE Notation and Numeration 7 Addition . . . . / 12 Subtraction 17 Multiplication 22 Division 28 Factors and Multiples « ....... 32 Casting out Nines . . o 40 Fractions 46 Approximate Results 57 Measures 62 Longitude and Time 78 The Equation 86 Powers and Roots 89 Mensuration 99 Graphical Representations . . . ' . . . . 124 Ratio and Proportion lol Method of Attack 140 Percentage . . .152 Commercial Discounts 157 Marking Goods . . . 162 Commission and Brokerage 164 6 6 CONTENTS pAot": Interest „ . . . . 167 Banks and Banking . . . 186 Exchange , . . . 193 Stocks and Bonds , . . 200 Insurance 207 Taxes and Duties 216 The Progressions . 219 Logarithms 224 Exercises for Review 235 ADYAKCED ARITHMETIC NOTATION AND NUMERATION 1. Our remote ancestors doubtless did their counting by the aid of the ten fingers. Hence, in numeration it became natural to divide numbers into groups of tens. This accounts for the almost universal adoption of the decimal scale of notation. 2. It is uncertain what the first number symbols were. They were, probably, fingers held up, groups of pebbles, notches on a stick, etc. Quite early, however, groups of strokes I, II, III, llll, •••, were used to represent numbers. 3. The earliest written symbols of the Babylonians were cuneiform or wedge-shaped symbols. The vertical wedge (I) was used to represent unity, the horizontal wedge (— ') to represent ten, and the two together (f*— ) to represent one hundred. Other numbers were formed from these symbols by writing them adjacent to each other. Thus, yrr =1 + 1 + 1 = 3, -^-^rr = 10 + 10 + 1 + 1 = 22, ^]^ =10x100 = 1000, \\1^ ^) = 5 X 100 + 10 + 2 = 512. 7 8 NOTATION AND NUMERATION To form numbers less than 100 the symbols were placed adjacent to each other and the numbers they represented were added. To form numbers greater than 100 the sym- bols representing the number of hundreds were placed at the left of the symbol for one hundred and used as a multiplier. 4. The Egyptians used hieroglyphics^ pictures of objects, or animals that in some way suggested the idea of the number they wished to represent. Thus, one was repre- sented by a vertical staff (I), ten by a symbol shaped like a horseshoe (o), one hundred by a short spiral (^), one hundred thousand by the picture of a frog, and one million by the picture of a man with outstretched hands in the attitude of astonishment. They placed the symbols adja- cent to each other and added tlieir values to form other numbers. Thus, ^oi = 100 -f 10 + 1 = HI. The Egyp- tians had other symbols also. 5. The Greeks used the letters of their alphabet for number symbols, and to form other numbers combined their symbols much as the Babylonians did their wedge- shaped symbols. 6. The Romans used letters for number symbols, as follows : 1 5 10 50 100 500 1000 I V X L c D M Numbers are represented by combinations of these sym- bols according to the following principles : (1) The repetition of a symbol repeats the value of the number represented by that symbol; as, 111 = 8, XX = 20. (2) The value of a niuuber is diminished by placing a symbol of less value before one of greater value ; as, NOTATION AND NUMK RATION 9 IV = 4, XL = 40, XC = 90. The less lunubei' is sub- tracted from the greater. (3) The value of a number is increased by placing a symbol of less value after one of greater value, as XI = 11, CX = 110. The less number is added to the greater number. (4) The value of a number is midtipUed by 1000 by placing a bar over it, as C = 100,000, X = 10,000. 7. Among the ancients we do not find the character- istic features of the Arabic, or Hindu system where each symbol has two values, its intrinsic value and its local value^ i.e. the value due to the position it occupies. Thus, in the number 513 the intrinsic value of the symbol 5 is five, its local value is five hundred. Written in Roman notation 513 = DXIII. In the Roman notation each symbol has its intrinsic value only. 8. The ancients lacked also the symbol for zero, or the absence of quantity. The introduction of this symbol made place value possible. 9. With such cumbersome symbols of notation the an- cients found arithmetical computation very difficult. In- deed, their symbols were of little use except to record numbers. The Roman symbols are still used to number the chapters of books, on clock faces, etc. 10. The Arabs brought the present system, including the symbol for zero and place value, to Europe soon after the conquest of Spain. This is the reason that the nu- merals used to-day are called the Arabic numerals. The Arabs, however, did not invent the system. They received it and its figures from the Hindus. 10 NOTATION AND NUMERATION 11. The origin of each of the number symbols 4, 5, 6, 7, 9, and probably 8 is, according to Ball, the initial letter of the corresponding numeral word in the Indo-Bactrian al- ]3habet in use in the north of India about 150 B.C. 2 and 3 were formed by two and three parallel strokes written cursively, and 1 by a single stroke. Just when the zero was introduced is uncertain, but it probably appeared about the close of the fifth century a.d. The Arabs called the sign 0, sifr (sifra = empty). This became the English cipher (Cajori, ''History of Elementary Mathe- matics"). 12. The Hindu system of notation is capable of unlimited extension, but it is rarely necessary to use numbers greater than billions. 13. In the development of any series of number sym- bols into a complete system, it is necessary to select some number to serve as a base. In the Arabic, or Hindu system ten is used as a base; i.e. numbers are written up to 10, tlien to 20, then to 30, and so on. In this system 9 digits and are necessary. If five is selected as the base, but 4 digits and are necessary. If twelve is selected, 11 digits and are necessary. The following table shows the relations of numbers in the scales of 10, 5, and 12. (t and e are taken to represen.t ten and eleven in the scale of 12.) Bask 12 21 48 10 1 2 3 4 5 6 7 8 9 10 11 100 5 1 1 2 2 3 3 4 4 10 5 11 6 12 7 13 8 14 9 20 t 21 e 22 10 41 19 143 400 12 40 84 NOTATION AND NUMERATION 11 Ux. 1. Reduce 431^ to tlie decinjal scale. Note. 431, means 431 in the scale of 5. Solution. 4 represents 4 x 5 x 5 = 100 3 represents 3x5 =15 1 represents 1 =1 .. 431, = 116,, Fx. 2. Reduce 46^2^^ to the scale of 8. Solution. 4632 579 = 579 units of the second order and none of the first order. 72 3 = 72 nnits of the third order and 3 of the second order. 9 = 9 units of the fourth order and none of the third order. 1 1 = 1 unit of the fifth order and 1 of tlie fourth order. .-. 4632io = llOSOg. EXERCISE 1 1. What number symbols are needed for the scale of 2? of 8? of 6 ? of 11 ? Write 12 and 20 in the scale of 2. 2. Reduce 2345 and 54(3^ to the decimal scale. 3. Reduce 7649^^ to the scale of 4. 4. Compare the local values of the two 9's in 78,940,- 590,634. What is the use of the zero ? Why is the num- ber grouped into periods of three figures each ? Read it. 5. If 4 is annexed to the right of 376, how is the value of each of the digits 3, 7, 6 affected ? if 4 is annexed to the left ? if 4 is inserted between 3 and 7 ? 6. What is the local value of each figure in 76,345 ? What would be the local value of the next figure to tlie right of 5 ? of the next figure to the right of this? 7o For what purpose is the decimal point used ? 8. Read 100.004 and 0.104 ; 0.0002 ; 0.0125 and 100.0025. ADDITION 14. If the arrangement is left to the computer, numbers to be added shoukl be written in columns with units of like order under one another. 15. In adding a column of given numbers, the computer should think of results and not of the numbers. He should not say three and two are five and one are six 329 and four are ten and nine are nineteen, but sinij)ly five, six, 764 ten, nineteen, writing down the 9 as he names tlie last nuni- 221 ber. The remaining columns should be added as follows: 9642 three, seven, nine, fifteen, seventeen, writing down the 7 ; 7823 nine, fifteen, seventeen, twenty-four, twenty-seven, writing 18779 down tlie 7 ; nine, ei/z/i^e^??, writing down the 18. Time in 211 looking for errors may be saved by writing the numbers to be carried underneath the sum as in the exercise. 16. Checks. If the columns of figures have been added upward, check by adding downward. If the two results agree, the work is probably correct. Another good check for adding, often used by account- ants, is to add beginning with the left-hand column. Thus, the sum of the thousands is 16 thou- sands, of the hundreds 26 hundreds, of the tens 16 tens, and of the units 19 units. EXERCISE 2 16000 or 16 2600 26 160 16 19 19 18779 18779 1. What is meant by the order of a digit? Define addend^ sum. 12 ADDITION 13 2. Why should digits of like order be placed in tlie same column ? State the general principle involved. 3. Wliy should the columns be added from right to left ? Could the columns be added from left to right and a correct result be secured? What is the advantage in beginning at the right ? 4. In the above exercise, why is 1 added (''carried") to the second column? 1 to the third column? 2 to the fourth column ? 17. Accuracy and rapidity in computing should be re- quired from the first. Accuracy can be attained by acquir- ing the habit of ahvays checking results. Rapidity comes with much practice. 18. The 45 simple combinations formed by adding con- secutively each of the numbers less than 10 to itself and to every other number less than 10 should be practiced till the student can announce the sum at sight. These com- binations should be arranged for practice in irregular order similar to the following : 122598145764234 123138976869464 6 6 1 1 3 4 9 5 2 7 1 2 5 3 5 9 7 2 8 7 5 9 8 7 9. 6 8 r O 4 7 5 4 2 2 6 4 3 1 2 1 8 3 1 3 7 9 8 5 9 8 6 5 3 6 7 9 3 4 8 7 19. Rapid counting by ones, twos, threes, etc., up to nines is very helpful in securing both accuracy and rapidity. Ex. Begin with 4 and add 6's till the result equals 100. Add rapidly, and say simply 4, 10, 16, 22, • • •, 94, 100. 14 ADDITION 20. It is helpful also to know combinations, or groups 12 3 4 5, that form certain numbers, inus, n o rr ^ r ^nd y 8 7 b o 8 7 6 6 5 5 4 12 2 3 4 3 3, etc., are groups that form 10, and 112 112 3 99998887 9 8 7 6 8 7 6 7 are groups that form 20. 23454566 21. Such groups should be carefully studied and prac- ticed until the student readily recognizes them in his work. He should also familiarize himself with other groups.. The nine-groups and the eleven-groups are easy to add, since adding nine to any number diminishes the units' figure by one, and adding eleven increases the units' and the tens' digits each by one. EXERCISE 3 1. Begin with 8 and add 7's till the result is 50. 2. Begin with 3 and add 8's till the result is 67. Form the following sums till the result exceeds 100 ,- 3. Begin with 3 and add 7's. 4. Begin with 7 and add 8's. 5. Begin with 5 and add 9's. 6. Begin witli 8 and add 5's. 7. Beofin with 5 and add 6's. 8. Begin with 6 and add 3's. A DDITJON 15 Add the following' coliniiiis, beginning' at tlie bottom, and check the results by adding downward. Form such groups as are convenient and add them as a single number. In the first two exercises groups are indicated. 9. 10. 11. 12. 13. 14. 15. 16. re 7 5 25.4 2122 275 5427 47.683 3 1 r9 [1 4 76.1 7642 267 6742 72.125 1 1 34.59 8321 979 8374 94.467 '8 r4 6 43.33 9789 231 9763 53.2124 5 8 67.27 2432 486 2134 91.576 .2 ■'- 4 81.2 5765 523 bm'o 14.421 '9 4 2 28.3 1297 752 3249 32.144 ^' i I 2 9 32.99 6423 648 1678 67.6797 o 1 7 16.25 1678 486 2432 19.045 8 8 4 53.11 3212 529 5469 54.091 '9 8 7 3 91.5 7679 926 8761 86.2459 4 2 85.4 2144 842 2332 27.654 2 4 1 74.1 1576 236 5467 98.346 1 5 5 22.22 4467 574 1023 84.6211 In commercial operations it is sometimes convenient to add numbers written in a line across the page. If totals are required at the right-hand side of tlie page, add from left to right and check by adding from right to left. Add : 17. 23, 42, 31, 76, 94, 11, 13, 27, 83, 62, 93. 18. 728, 936, 342, 529, 638, 577, 123, 328, 654. 19. 1421, 2752, 7846, 5526, 3425, 1166, 7531, 8642. 20. 46, 72, 88, 44, 39, 37, 93, 46, 64, 73, 47. 21. 1728, 3567, 2468, 5432, 4567, 2143, 9876, 6789. 16 ADDITION Find the sum of the following numbers by adding the columns and then adding the results horizontally. Check by adding the rows horizontally and then adding the columns of results. 22. 7642 6241 5331 3124 4724 8246 9372 3623 2793 51096 23. 793 864 927 531 642 876 927 426 459 24. 7942 8349 2275 3673 9527 2136 3411 4212 6524 7641 5675 7987 3171 1234 2892 6425 25. 26 72 126 467 354 987 54 86 13 34 45 56 67 67 87 43 98 87 765 453 342 465 783 5 21 5 43 350 9 11 321 24 8 25 196 961 649 378 452 36 77 66 555 444 888 999 111 222 Exercises for further practice in addition can be readily supplied by the teacher. The student should be drilled till he can add accu- rately and rapidly. Accuracy, however, should never be sacrificed to attain rapidity. Expert accountants, by systems of grouping and much pi-actice, acquire facility in adding two or even three colunms of figures at a time. Elaborate calculating machines have also been invented, and are much used in banks and counting offices. By means of these machines, columns of numbers can be tabulated and the sum printed by simply turning a lever. SUBTRACTION 22. In subtraction it is important that the student shoukl be able to see at once what number added to the smaller of two numbers of one figure each will produce the larger. Thus, if the difference between 5 and 9 is desired, the student should at once think of 4, the number which added to 5 produces 9. 23. Again, if the second number is the smaller, as in 7 from 5, the student should think of 8, the number which added to 7 produces 15, the next number greater than 7 which ends in 5. 24. The complete process of subtraction is shown in the following exercise : 8534 "^ ^^^^^ - ^^^ ^^^ carry 1. (Why carry 1 ?) .go^ 3 and are 3. ■ 6 and 9 are 15, carry 1. 2907 p 1 o Q b and 2 are 8. 25. The student should think " What number added to 5627 will produce 8534 ? " After a little practice, it is unnecessary to say more than 7 and 7, 3 and 0, 6 and 9, 6 and 2, writing down the underscored digit just as it is named. 26. Check. To check, add the remainder and the sub- trahend upward, since in working the exercise the numbers were added downward. LTMAX'S ADV. AR. —2 17 18 SUBTRACTION 27. The above method of subtraction is important not only because it can be performed rapidly, but because it is very useful in long division. It is also the method of " making change " used in stores. 28. There are two other metliods of subtraction in common use. The processes are shown in the following exercises : (1) 643 = 600 + 40 + 3 zr 500 + 130 + 13 456 = 400 + 5 + 6 = 400 + 50+6 187= ~ 100+ 80+7 6 from 13, 7 ; 5 from 13, 8 ; 4 from 5, 1. (2) 643 = 600 + 40 + 3, 600 + 140 + 13 456 = 400 + 50 + 6, 500+ 60+6 187= 100+ 80+7 6 from 13, 7 ; 6 from 14, 8 ; 5 from 6, 1. EXERCISE 4 1. Define the terms subtrahend^ minuend^ difference. 2. How should the terms be arranged in subtraction ? Where do we begin to subtract ? Why ? 3. Is the difference affected by adding the same number to both subtrahend and minuend ? Is this principle used in either (1) or (2) ? 4. If a digit in the minuend is less than a digit of the corresponding order in the subtrahend, explain how the subtraction is performed in both (1) and (2). 29. Arithmetical Complement. The arithmetical com- plement of a number is the difference between the number and the next higher power of 10. Thus, the arithmetical complement of 642 is 358, since 358 + 642 = 1000. The arithmetical complement of 0.34 is 0.66, since 0.66 + 0.34 = 1. SUBTRACTION 19 EXERCISE 5 1. Name rapidly the complements of the following numbers: 75, 64, 82, 12, 90, 33, 25, 0.25, O.lG, 125, 500"^, 5000, 1250, 625. 2. Name the amount of change a clerk must return if he receives a five-dollar bill in payment of each of the following amounts: il.25, ^3.75, $^2.34, 83.67, 10.25, $0.88, -S4.91, 11.85. 3. Name the amount of change returned if the clerk receives a ten-dollar bill in payment of each of the follow- ing amounts: 17.34, $3.42, $9.67, 15.25, 12.67, $6.45, $4.87, $0.68, $3.34. Determine in each of the following exercises what num- ber added to the smaller number will produce the larger. The student will notice that in some cases the subtrahend is placed over the minuend. It is often convenient in business to perform work in this wa}^ 4. 5. 6. 7. 8. 9. 9 36 75 246 8937 5280 5 42 31 167 9325 3455 10. 11. 12. 13. 7621 2339 9654327 4680215 6042 5267 6098715 9753142 14. Show that to subtract 73854 from 100000 it is necessary only to take 4 from 10 and eacli of the remain- ing digits from 9. 15. Subtract 76495 from 100000, and 397.82 from 1000, as in Ex. 14. 20 SUBTRACTION 6, 9, 15 and 2; 17. 5, 12, 15 and 0; 15. 4, 6, 8 and 9 ; 17. 4, 6, 7 and 1 ; 8. 16. Show that to subtract 3642 from 5623 is the same as to add the arithmetical complement of 3642 and sub- tract 10000 from the sum. 17. From 8757 take the sum of 1236, 2273 and 3346. 8757 1236 2273 3346 1902 18. From 53479 take the sum of 23, 1876 and 41253. 19. From 7654 take the sum of 3121, 126 and 2349. 20. From 764295 take the sum of 45635, 67843, 125960 and 213075. 21. A clerk receives a twenty-dollar bill in payment of the following items: 12.25, 111.50, 10.13, 10.75. How much change does he return ? 22. Find the value of 2674 + 1782 - 1236 + 8420-4536 by adding the proper arithmetical complements and sub- tracting the proper powers of ten. 30. To find the balance of an account. Dr. First National Bank, Ypsilanti, in acct. with John Smith Cr. 1904 1904 Aug. 3 Balance 1 486 87 Aug. 4 By check 500 00 Aug. 22 To deposit 290 00 Aug. 10 By check 57 30 Sept. 30 To deposit 198 75 Sept. 1 By check 235 75 Oct. 24 To deposit 773 40 Sept. 21 By check 11 80 Nov. 20 To deposit 110 Oct. 15 By check 97 30 Nov. 3 By check 1 000 00 Nov. 25 Balance 956 87 2 859 02 2 859 02 Nov. 25 Balance O.^P) 87 SUBTRACTION 21 The preceding form represents the account of Jolm Smith with the First National Bank from Aug. 3 till Nov. 25. The items at the left of the central dividing line are the amounts that the bank owes Mr. Smith. This side is called the debit side of the account. The items at the right represent the amounts withdrawn by ]\Ir. Smith. This side is called the credit side of the account. The diiference y)etvveen the sums of the credits and the debits is called the balance oi the account. It is evident that the debit side of the above account is greater than the credit side. Therefore, to balance the account, add the debit side first, and then subtract the sum of the credit side from the result, as in Ex. 17 above. The difference will be the balance, or the amount left in the bank to the credit of ]Mr. Smith. The work can be checked by adding the balance to the credit column. The result should equal the sum of the debit column. EXERCISE 6 Find the balance of each of the following accounts : 1. Dr. Cr. 2. Dr. Cr. 3. Dr. Cr., 234 50 246 84 798 34 125 00 500 00 450 00 212 60 55 30 351 00 97 30 100 00 60 00 75 00 198 30 1250 00 527 30 888 80 131 60 210 60 927 50 1100 00 500 00 2681 50 975 25 69 00 659 75 10 46 50 00 100 00 235 67 564 90 1000 00 ! 75 00 750 25 34 68 104 69 100 00 566 66 1200 00 195 75 275 80 302 00 625 30 259 00 4. On May 1 R. F. Joy had a balance of f 1376.2-t to his account in the bank. He deposited on May 1, ^189; June 27, 1166; July 28, 175; Aug. 5, $190.60; Aug. 10, i 192.22. He withdrcAv by check the following amounts : June 1, 1153; June 10, 1300; July 3, #25; July 27, 1575.50. What was his balance Aug. 15? MULTIPLICATION 31. The multiplication table should be so well known that the factors will at once suggest the product. Thus, 7 X 6, or 6 X 7, should at once suggest 42. 32. The student should also be able to see at once what number added to the product of two numbers will produce a given number. Thus, the number added to 4 x 9 to produce 41 is 5, or 4 x 9 and 5 are 41. It is a common practice in multipHcation to write the multiplier first as 2 X I 5 = ^ 10. In this case the sign ( x ) is read " times." If the multiplier is written after the multiplicand, as in $5 x 2 = $10, the sign (x) is read "multiplied by." The multiplier is always an abstract quantity (Why?), but the multiplicand may be either abstract or concrete. 33. The following examples show the complete process of multiplication : Ex. 1. Multiply 2743 by 356. Solution. In multiplying one number by 2743 2743 another it is not necessary to begin with the ■ 356 356 units' digit of the multiplier. We may begin 164.58 with either the units' digit or the digit of the 13715 highest order. In fact, it is frequently of de- §229 cided advantage to begin with the digit of 976,508 highest order, especially in multiplying deci- mals ; but care should be taken in placing the right-hand figure of the first partial product. Since 3 hundred times 3 units = 9 hundred, the 9 must be put in the third or hundreds' place, etc. MULTIPLICATION 23 Rv. 2. Multiply 3.1416 by 213.34. Solution. In beginning the multiplicatiou we see that 20x0.0000 = 0.012. Hence the 2 is written in the thou- •^■HIC sandths' place. The work is then completed as indicated ^^^-'^^ in the annexed example. It will readily be seen that the 02.832 rest follows after pointing off the first partial product 18.8496 correctly. .91:218 The advantage of beginning with the digit of the .12.5064 highest order is seen in approximations (see p. 59), o.> 71 0-4.4. where considerable work is thereby saved. 34. Check. Multiplication may be checked by usijig the multiplicand as the multiplier and performing the multiplication again. However, the check by "casting out the nines " (p. 41), is more convenient. EXERCISE 7 1. Define multiplier^ multiplicand, product. 2. Explain why multiplication is but an abridged method of addition. 3. Can the multiplier ever be a concrete number ? Explain. 4. How should the terms be arranged in multiplication ? Does it make any difference in what order we multiply by the digits of the multiplier ? flight we begin to multiply with the 5 in Ux. 1 and with the 6 in Ux. 2 ? 5. How is the order of the right-hand figure of each partial product determined ? 6. How does the 2^1'esence of a zero in the multiplier affect the work ? 7. In multiplying 3.1416 by 26.34, can we tell at once how many integral places there will be in the product ? Can we tell the number of decimal places ? 24 MULTIPLICATION 8. How many decimal places will there be in each of the following products : 21.34 x 5.9 ? 98.65 x 76.43 ? 321.1 X 987.543 ? 1.438 x 42.345 ? 35. The following short methods are useful : 1. To multiply any numher hy 5, 25, 16 J, 33|^, 125. Since 5 = y-, to annex a cipher and divide by 2 is the same as to multiply by 5. The student in a similar manner should explain short processes of multiplying by 25, 16 1, 33|^, 125. 2. To multiply any numher hy 9. Since 9 = 10 — 1, it is sufficient to annex a cipher to the number and subtract the original number. Ex. Multiply 432 by 9. 432 X 10 = 4320 432 X 1 = 432 432 X 9 = 3888 3. To multiply any 7imnher hy 11. Since 11 = 10 -f 1, it is sufficient to annex a cipher to the number and add the original number. Ex. Multiply 237 by 11. 237 X 10 = 2370 237 X 1 = 237 237 X 11 = 2607 This result can readily be obtained by writing down the right-hand figure first and then the sums of the first and second figures, the sec- ond and third, etc., and finally the left-hand figure. 4. To multiply any mmiher hy a 7iumher differing hut little from soine poioer of 10. Annex as many ciphers to the number as there are ciphers in the next higher power of 10, and subtract the product of the number multiplied by the complement of the multiplier. MULTIPLICATION 25 Ex. Multiply 335 by 996. 900 = 1000 - 4. 335 X 1000 = 33r)000 In practice written 335 335 X . 4 = 1310 1»M0 335 X 996 = 333G0O 333(j(j0 5. To midtiply any number hy a number of two fiyures ending ivith 1. Multiply by the tens' figure of the multiplier, writing this product under the number one place to the left. Ex. Multiply 245 by 71. 245 X 1 = 245 245 X 70 = 17150 245 X 71 = 17395 6. To multiply any iiumher hy a number between tivelve and twenty. Multiply by the units' figure of the multiplier, writing the product under the number one place to the right. Ex. Multiply 427 by 13. 427 X 10 = 4270 427 X 3 .:= 1281 427 X 13 = 5551 7. To square a number ending hi 5. 352 = 3 X 400 + 25, 452 z= 4 X 500 + 25, 55--2 = 5 x 600 + 25, etc. 8. To midtiply by a number when the multiplier contains digits ivhich are factors of other parts of the multiplier. Ex. Multiply 25631 by 74221. Since 7 is a factor of 42 and 21, mnltiply by 7. placing 25631 the first figure in the partial product under 7. (Why V) "4001 Then multiply this product by 6 (42 := 6 x 7), placing the first figure under 2 in hundreds' place. (Why?) I'-p-^.o Then multiply the first partial product by 3 (21 = 3x7), ^^* ^o .0-, placing the first figure under 1. (Why?) The suui of — — II- these partial products will be the product of the numbers. 190L3ob4ol 26 MUL TIPLICA TION EXERCISE 8 Name rapidly the products of the successive pairs of digits in each of the following numbers : 1. 75849374657. 3. 67452367885. 2. 265374867598. 4. 98765432345. 5. In each of the following groups of digits add rapidly to the product of the first two the sum of all that follow : 567, 432, 7654, 3456, 9753, 3579, 8642, 2468, 7896, 5436, 3467. 6. Multiply 1264 by 125 ; by 121 ; by IJ. 7. Multiply 76.26 by 16f ; by 331 8. Multiply 2348 by 25; by 21; by 50; by 0.5. 9. Multiply 645 by 9; by 11 ; by 17 ; by 41. 10. Multiply 8963 by 848. 11. Multiply 37439 by 4832. 12. Show that to multiply a number by 625 is the same as to multiply by 10000 and divide by 16. 13. Subtract 5 x 12631 from 87642. The work should be done as follows : 5x1 and 7, 12. 87642 5 X 3 and 1 and 8, 24. 12631 5x6 and 2 and 4, 36. 24487 5x2 and 3 and 4, 17. 5x1 and 1 and 2, 8. 14. Subtract 3 x 2462 from 9126. 15. Subtract 6 x 42641 from 768345. MULTIPLICATION 27 1 784 2 1568 '.] 2:i52 4 ai;J6 5 3920 6 4704 7 5488 8 6272 9 7056 16. Subtract 2 x 8(1478 from 291872. When the same miinher is to be used as a tiiultii)lier several times, work may be saved l>y t'oruiiiig a table of its multiples. 'J'lius, 5764 X 784 = 3i:]6 (4) 4704 (6) 5488 (7) 3920 (5) 4518976 The partial products in each case are taken from the table. 17. Use the above table and multiply 5764, 74591, 84327, 23145, each by 784. 18. Form a table of multiples of 6387, and use it to find the product of 7482, 3.1416, 742896, 342312, 67564584, 897867, 65768798, 56024.85, each by 6387. 19. Multiply 2785 by 9998, and 1728 by 997. 20. Multiply 78436 by 25 x 125. 21. Multiply 32.622 by 0.0125. 22. Multiply 486.72 by 0.25 x 0.25. 23. Multiply 320.4 by 5 X 1.25. 24. Multiply 5763 by 16f x 33f ' DIVISION 36. In division the student should be able to see at once how many times a given digit is contained in any number of two digits with the remainder. Thus, 7 is contained in 46, 6 times with a remainder 4. The student sliould think simply 6 and 4 over. Ex. 6 )354279 59046 remainder 3. The whole mental process should be 5 and 5, 9 and 0, and 2, 4 and 3, 6 and 3. Two interpretations arise from considering division as the inverse of multiplication. Thus, since 4 x <|6 = ,|24. (1) $24 ^ 4 = |6, separation into groups. $24 has been separated into 4 equal groups. (2) $24 -^$6 = 4, involving the idea of measuring, or being con- tained in. $6 is contained in $24, 4 times. 37. The following examples show the complete process of long division. 346 4541)1571186 1^6-'^ It assists in determining the order 20888 ^^ ^'^^ digits in the quotient to write 18164 them in their proper places above the dividend. 27246 27246 28 DIVISION 29 38. The work in long division may be very much abridged by omitting the partial products and writing down the remainders only. These remainders are ob- tained by the method used in Ex. 13, p. 26. Ux, Divide 764.23 by 2.132. The work will be simplified by multiplying both niimljers by 1000 to avoid decimals. The first remainder, 124(), is obtained as follows : 358 2132)764230 3 x 2, and 6, 12. 12463 3 X 3, 9 and 1, 10 and 4, 14. 18030 3 X 1, 3 and 1, 4 and 2, 6. 974 3 X 2, 6 and 1, 7. Then bring down 3 and proceed as before to form the other remainders. 39. Check. Division may be checked by multiplying the quotient by the divisor, the product plus the remainder should equal the dividend. The check by ''casting out the nines " (p. 42) may be used. EXERCISE 9 1. Define dividend^ divisor, quotient^ remainder. 2. Explain the two interpretations arising from consider- ing division as the inverse of multiplication. 5 x 8 10 = "5^ b'^. Give the two interpretations as applied to this example. 3. How is the order of the right-hand figure in each partial product determined ? 4. Explain why the sum of the partial products plus the remainder, if any, must equal the dividend if the w^ork is correct. 5. Explain why the quotient is not affected by multi- plying both dividend and divisor by the same number. 30 DIVISION 40. If the same number is used as a divisor several times, or if the dividend contains a large number of places, work may be saved by forming a table of multiples of the divisor. Thus : Ux. Divide 786342 by 4147. 1 4147 189 2 8294 4147) 786342 3 12441 37164 4 16588 39882 5 20735 2559 remainder 6 24882 7 29029 8 38176 9 37323 EXERCISE 10 1. Divide 987262, 49789 and 314125 each by 4147. 2. Divide 896423, 76425, 9737894 each by 5280. 3. Divide 44.2778 by 63.342. Find the value of : 4. 32.36-8.9. 5. 1.25 -^ 0.5 and 12.5-0.05. 6. 144-1.2 and 14.4-12. 7. 625-25 and 62.5^2.5. 8. 1125-50 and 11.25 -j- 9.5. 9. 5280-12.5 and 580-125. 10. 750-2.5^0.5. 41. In addition to the checks on the fundamental pro- cesses given above, it is well wlien possible to foryn the habit of estimating results before beginning the solution of a DIVISION 31 problem. Thus, in iiiultijjlyiiig 11*.] by 12] it is evident that the result will be about 12 x 20= 240. In using this check the student should form a rough estimate of the result, then solve the problem and com- pare results. A large error will be at once detected. EXERCISE 11 Solve the following, first giving approximate answers, then the correct result : 1. Multiply 15.3 x 3|f (about 15 x 4). 2. Divide 594 by -f^^ (about 594 - i). 3. Divide 32.041 by 0.499 (about 32.041 - i). 4. How much will 21 horses cost at S145 each? 5. Multiply 30.421 by 20.516. 6. At 12 J ct. a dozen, how much will Q^ doz. eggs cost ? 7. At 37 1 ct. a pound, how much will 11 lb. of coffee cost ? 8. How many bushels of potatoes can be bought for $5.25 at 35 ct. a bushel? 9. At $1,121 a barrel, how many barrels of salt can be bought for 122.50? 10. How far will a train travel in 121 ]n\ at the rate of 45 mi. an hour ? 11. How much will 8| T. of coal cost at $7.25 a ton ? 12. The net cost of printing a certain book is 49 ct. a copy. How much will an edition of 2500 cost ? 13. At the rate of 40 mi. an hour, how long will it take a train to run 285 mi. ? FACTORS AND MULTIPLES 42. A factor or divisor of a number is any integral number that will exactly divide it. 43. A number that is divisible by 2 is called an even number, and one that is not divisible by 2 an odd number. Thus, 24 and 58 are even numbers, while 17 and 83 are odd numbers. 44. A number that has no factors except itself and unity is called a prime number. Thus, 1, 2, 3, 5, 7, etc., are prime numbers. 45. Write down all of the odd numbers less than 100 and greater than 3. Beginning with 3 reject every third number ; beginning with 5 reject every fifth number ; beginning with 7 reject every seventh number. The numbers remaining will be all of the prime numbers between 3 and 100. (Why?) 46. This method of distinguishing prime numbers is called the Sieve of Eratosthenes, from the name of its inventor, Eratosthenes (276-196 B.C.). He wrote the numbers on a parchment and cut out the composite numbers, thus forming a sieve. 47. A number that has other factors besides itself and unity is called a composite number. 48. Numbers are said to be prime to each other when no number greater than 1 will exactly divide each of them. Are numbers that are prime to each other necessarily prime numbers ? FACTORS AND MULTIPLES 33 49. An integfral nninbcr tluit will exactly divide two or more numbers is called a common divisor, or a common factor of these, numbers. Thus, 2 and ^3 are conunon divisors of 12 and 18. 50. The greatest common factor of two or more num- bers is called the greatest common divisor (g. c. d.) of the numbers. Thus, 6 is the g. c. d. of 12 and 18. 51. A common multiple of two or more numbers is a number that is exactly divisible by each of them. Thus, 12, 18, 24, aud 48 are common multiples of 3 and 6, while 12 is the least common multiple (1. c. m.) of 3 and G. 52. It is of considerable importance in certain arith- metical operations, particularly in cancellation, to be able readily to detect small factors of numbers. In proving the tests of divisibility by such factors, the two following principles are important. 1. A factor of a numher is a factor of any of its multiples. Proof. Every multiple of a number contains that number an exact number of times; therefore, it contains every factor of tlie number. Thus, 5 is a factor of 25, and hence of 3 x 25, or 75. 2. A factor of any ttvo numhers is a factor of the sum or difference of any two multiples of the numhers. Proof. Any factor of two numbers is a factor of any of their mul- tiples by Principle 1. Therefore, as each nudtiple is made up of parts each equal to the given factor, their sum or difference will be made up of parts equal to the given factor, or will be a multiple of the given factor. Thus, 3 is a factor of 12 and of 15, and hence of 5 x 12 + 2 x 15, or 90. 3 is also a factor of 5 x 12 — 2 x 15, or 30. ltman's adv. ar. — 3 34 FACTORS AND MULTIPLES 53. Tests of Divisibility. 1. An^ number is divisible by 2 if the number represented by its last right-hand digit is divisible by 2. Proof. Any number may be considered as made up of as many lO's as are represented by the number exclusive of its last digit plus the last digit. Then, since 10 is divisible by 2, the first part, which is a multiple of 10, is divisible by 2. Therefore, if the second part, or the number represented by the last digit, is divisible by 2, the whole number is. Thus, 634 = 63 X 10 + 4 is divisible by 2 since 4 is. 2. Any number is divisible by 4 if the number represented by the last two digits is divisible by 4. Proof. Any number may be considered as made up of as many lOO's as are represented by the number exclusive of its last two digits plus the number represented by the last two digits. Then, since 100 is divisible by 4, the first part, which is a multiple of 100, is divisible by 4. Therefore, if the number represented by the last two digits is divisible by 4, the whole number is. Thus, 85648 = 856 x 100 + 48 is divisible by 4 since 48 is. 3. Any nu?nber is divisible by 5 if the last digit is or 5. The proof, which is similar to the proof of 1, is left for the student. Note. is divisible by any number, and the quotient is always 0. 4. Any number is divisible by 8 if the number represented by its last three digits is divisible by 8. The proof is left for the student. 5. Any number is divisible by 9 if the sum of its digits is divisible by 9. Proof. Since 10 = 9 + 1, any number of lO's = the same number of 9's + the same number of units; since 100= 99 + 1, any number of lOO's = the same number of 99's + the same number of units ; since FACTORS AND MULTIPLES ' 35 1000 = 9f)f) + 1, any iiiinil)er of lOOO's = the same nuin})er of ODO's + the same number of units; etc. Therefore, any number is made up of a multiple of 9 + the sum of its digits, and hence is divisil^le by 9 if the sum of its digits is divisible by 9. Thus, 7:]Gl> = 7 x 1000 + 8 x 100 + G x 10 + 2 = 7(999 + 1) + ;5(99 + 1 ) + 0(9 + 1) + 2 = 7 X 999 + 8 X 99 + 6x9 + 7 + 13 + 6+2 = a multiple of 9 + the sum of the digits. Therefore, the number is divisible by 9 since 7 + 3 + 6 + 2 = 18 is divisible by 9. 6. Any number is divisible by 3 if the sum of its digits is divisible by 3. The proof, which is similar to the proof of Principle .5, is left for the student. 7. Any even number is divisible by 6 if the sum of its digits is divisible by 3. The proof is left for the student. 8. A7iy number is divisible by 11 if the difference between the sums of the odd and even orders of digits, counting from units^ is divisible by 11. Proof. Since 10 = 11 — 1, any number of lO's = the same number of ll's— the same number of units; since 100 = 99+1, any number of lOO's = the same number of 99's + the same number of units ; since 1000 = 1001 - 1, any number of lOOO's = the same number of lOOl's — the same number of units ; etc. Therefore, any number is made up of a multiple of 11 + the sum of the digits of odd order — the sum of the digits of even order, and hence is divisible by 11 if the sum of the digits of odd order — the sum of the digits of even order is divis- ible bv 11. 36 • FACTORS AND MULTIPLES Thus, 753346 = 7x100000 + 5x10000 + 3x1000 + 3x100 + 4x10 + 6 = 7(100001-1) + 5(9999 + 1) + 3(1001-1) + 3(99 + l)+4(ll-l)+6 = 7x100001 + 5x9999 + 3x1001 + 3x99 + 4x11 . -7+5-3+3-4+6 = a multiple of 11 + the sum of the digits of odd order — the sum of the digits of even order. Therefore, the number is divisible by 11 since 5+3 + G — (7 + 3 + 4) = is divisible by 11. 9. The test for divisihility hy 1 is too complicated to he useful, EXERCISE 12 1. Write three numbers of at least four figures each that are divisible by 4. 2. Write three numbers of six figures each that are divisible by 9. 3. Is 352362257 divisible by 11 ? by 3 ? 4. Without actual division, determine what numbers less than 19 (except 7, 13, 14, 17) will divide 586080. 5. Explain short methods of division by 5, 25, 16|, 331 125. 6. Divide 3710 by 5 ; by 25; by 125; by 121. 7. Divide 2530 by 0.5; by 0.025; by 1.25. 8. Prove that to divide by 625 is the same as to mul- tiply by 16 and divide by 10000. 9. State and prove a test for divisibility by 12 ; by 15; by 18. 10. If 7647 is divided by 2 or 5, how will the remainder differ from the remainder arising from dividing 7 by 2 or 5 ? Explain. FACTORS Ay I) MULTIPLES 37 11. If 2()72T is divided l)y 4 oi* 2"), liow will llic reniiiiii- der differ from the remainder arising- from dividing- 27 l^y 4 or 25 ? Explain . 12. Explain how you can hnd the remaindei- iirising- from dividing 26727 by 8 or 125 in the sliortest possiljle way. 54. Relative Weight of Symbols of Operation. In the use of the symbols of operation ( + , — , x , ^), it is impor- tant that the student should know that the numbers con- nected by the signs x and -r- must first be operated upon and then tliose connected by + and — ; for the signs of multiplication and division connect factors, while the signs of addition and subtraction connect terms. Factors must be combined into simple terms before the terms can be added or subtracted. Thus, 5 + 2 X 3 - 15 -- 5 + 4 = 12, the terms 2x3 and 15 -f- 5 being simplified before they are combined by addition and subtraction. 55. The ancients had no convenient symbols of operation. Addi- tion was generally indicated by placing the numbers to be added adja- cent to each other. Other operations were written out in words. The symbols + and — were probably first used by Widman in his arithmetic published in Leipzig in 1489. He used them to mark excess or defi- ciency, but they soon came into use as symbols of operation, x as a symbol of multiplication was used by Oughtred in 1G31. The dot ( • ) for multiplication was used by Harriot in 1631. The Arabs indicated division in the form of a fraction quite earlv. ^ as a symbol of divi- sion v;as used hy Rahn in his algeV)ra in 1059* Robert Recorde intro- duced the symbol = for equality in 15.")7. : was used to indicate division by Leibnitz and Clairaut. In 1631 Harriot used > and < for greater than and less than. Rudolff used y/ to denote square root in 1526. 56. Greatest Common Divisor. In many cases the g. c. d. of two or more numbers may readily be found by factoring, as in the following example : 377 3 348 1 29 12 38 FACTORS AND MULTIPLES Ex. Find the g. c. d. of 3795, 7095, 30030. 3795 = 3 X 5 X 11 X 23, 7095 = 3 X 5 X 11 X 43, 30030 = 2 X 3 X 5 X 11 X 91, and since the g. c. d. is the product of all of the prime factors that are common to the three numbers, it is 3x5x11 = 165. 57. Euclid, a famous Greek geometer, who lived about 300 B.C., gave the method of finding the g. c. d. by division. This method is useful if the prime factors of the numbers cannot be readily found. Ex. Find the g. c. d. of 377 and 1479. 1479 , . o. The g. c. d. cannot be greater than 377, and since .3 ,Q 377 is not a factor of 1479, it is not the g. c. d. of the oj^Q two numbers. Divide 1479 by 377. Then, since the g. c. d. is a common factor of 377 and 1479, it is a factor of 1479-3 x 377, or 348 (Principle 2, p. 33). Therefore, the g. c. d. is not greater than 348. If 348 is a factor of 377 and 1479, it is the g. c. d. sought. But 348 is not a factor of 377. Therefore, it is not the g. c. d. sought. Divide 377 by 348. Then, since the g. c. d. is a factor of 377 and 348, it is a factor of 377 - 348, or 29 (Principle 2, p. 33). Therefore, the g. c. d. is not greater than 29, and if 29 is a factor of 348, 377, and 1479, it is the g. c. d. sought. (Why ?) 29 is a factor of 348. Therefore, it is a factor of 377 and of 1479. (Why?) Therefore, 29 is the g. c. d. sought. 58. Least Common Multiple. In many cases the 1. c. m. of two or more numbers may readily be found by factoring, as in the following example. Ex. Find the 1. c. m. of 414, 408, 3330. 414 = 2 X 3 X 3 X 23, 408 = 2 X 2 X 2 X 3 X 17, 3330 = 2 X 3 X 3 X 5 X 37. FACTORS AND MULTIPLES 39 The ].c. 111. must contain all of the i)riine factors of 414, 408, :>:}80, and each factor must occur as often in the 1. c. m. as in any one of the numbers. Thus, 8 must occur twice in the I. c. m., 2 must occur three times, and 28, 17, -5, 87 must each occur once. Therefore, the 1. c. m. = 2 x 2 x 2 x 3 x 8 x 5 x 28 x 17 x 87 = 5208120. 59. When the numbers cannot readily be factored, the g. c. d. may be used in finding the 1. c. m. Since the g. c. d. contains all of the factors that are common to the numbers, if the numbers are divided by the g. c. d., the quotients will contain all the factors that are not common. The l.c.m. is therefore the product of the quotients and the g. c. d. of the numbers. Ex. Find the 1. c. m. of 14482 and 32721. The g. c. d. of 14182 and 82721 is 18. 14:482 ^ 18 = 1114:. .-. the 1. c. m. of the two numbers is llUx 82721 = 86451194. EXERCISE 13 1. Find the 1. c. m. and g. c. d. of 384, 2112, 2496. 2. Find the 1. c. m. of 3, 5, 9, 12, 14, 16, 96, 128. 3. Find the g. c. d. and 1. c. m. of 1836, 1482, 1938, 8398, 11704, 101080, 138945. 4. Prove that the product of the g. c. d. and 1. c. m. of two numbers is equal to the product of the numbers. 5. What is the length of the longest tape measure that can be used to measure exactly two distances of 2916 ft. and 3582 ft. respectively ? 6. Find the number of miles in the radius of the earth, having given that it is the least number that is divisible by 2, 3,4, 5, 6, 9, 10, 11, 12. CASTING OUT NINES 60. The check on arithmetical operations by casting out the nines was used by the Arabs. It is a very useful check, but fails to detect such errors as the addition of 9, the interchange of digits, and all errors not affecting the sum of the digits. (Why?) The remainder cunsing from dividing any 7iumber hy 9 is the same as that arisijig from dividing the sum of its digits Jy 9. ' • Thus, the remainder arising by dividing 75234: by 9 is 3, the same as arises by dividing 7 + 5 + 2 + 3 + 4 by 9. The student should adapt the proof of Principle 5, ix 34, to this statement. 61. The most convenient method is to add the digits, dropping or " casting out " the 9 as often as the sum amounts to that number. Thus, to determine the remainder arising from dividing 645738 by 9, say 10 (reject 9), 1, 6, 13 (reject 9), 4, 7, 15 (reject 9), 6. There- fore, 6 is the remainder. After a httle practice the student will easily group the 9's. In the above, 6 and 3, 4 and 5, could be dropped, and the excess in 7 and 8 is seen to be 6 at once. 62. Check on Addition by casting out the 9*s. Ux. Add 56342, 64723, 57849, 23454 and check the work by casting out the 9's. 40 CASTING OUT NINES 41 Since each number is a nuiltipl(,' of 9 plus some remaiiultT, the numbers can be written as indicated 56342 = 9 X 0200 + 2 rem. j,, the annexed sohition. 64723 = 9 X 719L+ 4 rem. 57849 = 9 X 0127+ 6 rem. But 12 = 9 + 3. 23454 = 9 X 2000 + rem. .^ 202308 = 9 x 22-184 + 9 + 3 202308 = 9 X 22484 + 12 rem. ^ ^ ^ ^2485 + 3. Thus, the excess of 9's is 3 and the excess in tlie sum of the ex- cesses, 2, 4, 6, and 0, is 3, therefore the work is probably correct. 63. The proof may be made general by writing the numbers in the form 9 a: + r. This can be done since all ', ^ , numbers are multiples of 9 plus a remainder. " „ ' ,, Hence, by expressing the numbers in this form "_ and adding w^e have for the sum a multiple of 9 plus the sum of the remainders. Therefore, the Z~. '. ~ ~ excess of the 9's in the sum is equal to the excess in 9(x + x' + x" -\--") ^ ^ , ,, . the sum of the excesses. + r + r' + r"+ •■•) -^ 64. Check on Multiplication by casting out the g's. Since any two numbers may be written in the form 9 a: + ;- and 9 x' + r', multiplying 9 x + r by 9 x' + r', ^x -^r ^re have ^1 xx' + Q(^x'r -\- xr') + rr'. From ^ X + f' this it is evident that the excess of 9's in 9 xr' + rr' the product arises from the excess in vr'. 81 xx' + 9 x'r Therefore, the excess of 9's in any product is ^1 xx' -\- Q(x'r -\- xr')-\-rr' equal to the excess in the product found hij multiplying the excesses of the factors together. Ex. Multiply 3764 by 456 and clieck by casting out ^^® ^'^- 3761 X -150 = 1710384. The excess of 9's in 3764 is 2 ; the excess in 456 is 6 ; the excess in the product of the excesses is 3 (2 x 6 = 12 ; 12 — 9 = 3); the excess in 1710381, the product of the numbers, is 3. Therefore, the work is probably correct. 42 CASTING OUT NINES 65. Check on Division by casting out the p's. Division being tlie inverse of multiplication, the dividend is equal to the product of the divisor and quotient plus the remainder. Therefore, the excess of 9's in the dividend is equal to the excess of 9's i7i the remainder plus the excess in the product found by multiplying the excess of 9's in the divisor by the excess of 9's in the quotient. Ex. Divide 7456r3 by 428 and check by casting out ^^^® ^'^- 74563 -f- 428 = 174 + /2V or 74563 = 174 x 428 + 91 The excess of 9's in 74563 is 7 ; in 174, 3 ; in 428, 5 ; in 91, 1. Since 7, the excess of 9's in 74563 = the excess in 3 x 5 + 1, or 16, which is the product of the excesses in 174 and 428 plus the excess in 91, the work is probably correct. EXERCISE 14 1. State and prove the check on subtraction by casting out the 9's. 2. Determine without adding whether 89770 is the sum of 37634 and 52146. 3. Add 74632, 41236, 897321 and 124762, and check by casting out the 9's. 4. Multiply 76428 by 5937, and check by casting out the 9's. 5. Determine without multiplying wliether 2718895 is the product of 3785 and 721. 6. Show by casting out 9's that 18149 divided by 56 = 324^V 7. Show that results may also be checked by casting out 3's ; by casting out ll's. MisCELLAXEOrs EXEIiCISE 43 8. Is 734(;r)7 divisible ])y 9? by 3? by 11? 9. lY'i'forin till' following operations and clieck : 91728 X 762 ; 849631 - 2463 ; 17 x 3.1416 ; 78.04 - 3.1416. 10. Does the proof for casting out the 9\s hold as well for 4, 6, 8, etc. ? May we check by casting out the 8's '/ Explain. MISCELLANEOUS EXERCISE 15 1. What is the principle by which the ten symbols, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, are used to represent any number ? 2. Why is the value of a number unaltered by annexing zeros to the right of a decimal ? 3. How is the value of each of the digits in the number 326 affected by annexing a number, as 4, to the right of it ? to the left of it ? 4. How is the value of each of the digits of 7642 affected if 5 is inserted between 6 and 4 ? 5. Write 4 numbers of 4 places each that are divisible by ia) 4, (5) 2 and 5, (c) 6, (c?) 8, (^ 9, (/) 11, Qg) 16, (A) 12, (0 15, (y) 18^A-) 3, (0 50, (m) 125, (n), both 6 and 9, (o) both 8 alPfe, (p) both 30 and 20. 6. Determine the prime factors of the following num- bers : (a) 3426, (5) 8912, (c) 6600, (i) 6534, (^ 136125, (/) 330330, (^) 570240. 7. Mr. Long's cash balance in the bank on Feb. 20 is 1765.75. He deposits, Feb. 21, .fl50; P^eb. 25, -^350.25; Feb. 26, $97.50; and withdraws, Feb. 23, ^f 200; Feb. 24, !?^123.40 and .^112.50; Feb. 28, 1321.75. What is his balance March 1 ? 44 MISCELLANEOUS EXERCISE 8. Form a table of multii)les of tlie multiplier and multiply 7642, 98856, 24245, 6420246, each by 463. 9. Form a table of multiples of tlie divisor and use it in dividing 86420, 97531, 876123, 64208, each by 765. 10. Use a short method to multiply 8426 by 16| ; by 331 ; by 945 ; by 432. 11. Find, without dividing, the remainder when 374265 is divided by 3 ; by 9 ; by 11. 12. Evaluate 45 + 32 x 25 - 800 - 125 + 180 x 33i. 13. Determine by casting out the 9's whether the following are correct : (a) 786 x 648 = 509328 ; (b) 24486 - 192 = 127 + 102 rem. ; {c) 415372 - 267 = 1555 + 187 rem. ; (d) 16734 x 3081 = 52557454. 14. Perform each of the operations indicated in Ex. 13. 15. Subtract from 784236 the sum of 7834, 5286, 23462 and 345679. 16. What are the arithmetical complements of 12000, 1728, 3.429, 86, 0.1, 125? 17. Light travels at the rate of 186000 mi. per second. Find the distance of the sun from the earth if it takes a ray of light from the sun 8 min. 2ijfec. to reach the earth. 18. A cannon is 2 mi. distant from an observer. How long after it is fired does it take the sound to reach the observer if sound travels 1090 ft. per second ? 19. Replace the zeros in the number 760530091 by digits so that the number will be divisible by both 9 and 11. 20. Show that every even number may be written in the form 2n and every odd number in the form 2n-\-l where ?^ represents any integer. MISCELLANEOUS EXERCISE 40 21. Show that the product of two consecutive numbers must be even and the sum o(hl. 22. Show that all numbers under and including 15 are factors of 360360. 23. Find, without dividing, the remainder after 364257 lias been divided by 3 ; by 9 ; by 11. 24. Evaluate 10 + 144 x 25 - 2180 - 15 + 5 x 3. 25. Write 4 numbers of 5 places each that are divisible by both 9 and 11. 26. Write 4 numbers of 6 places each that are divisil)le by both 3 and 6. 27. Write 4 numbers of 4 places each that are divisible by 4, 5, 6. 28. Evaluate 47 x 68 + 68 x 53. 29. Evaluate 346 x 396.84 - 146 x 396.84. 30. Evaluate 27x3.1416-41x3.1416 + 49x3.1416 ^Qb X 3.1416. 31. If lemons are 20 ct. a dozen and oranges are 25 ct., how many oranges are worth as much as 12| doz. lemons ? 32. A farmer received 6 lb. of coffee in exchange for 9 doz. eggs at 12| ct. a dozen. How much was the coffee worth per pound ? 33. Two piles of the same kind of shot weigh respectively 1081 lb. and 598 lb. What is the greatest possible weight of each shot '/ FRACTIONS 66. Historically the fraction is very old. A manuscript on arithme- tic, entitled " Directions for obtaining a Knowledge of All Dark Things," written by Ahmes, an Egyptian priest, about 1700 B.C., begins with fractions. In this manuscript all fractions are reduced to fractions with unity as the numerator. Thus, the first exercise is f = ^ + ^^. 67. AVhile the Egyptians reduced all fractions to those with con- stant numerators, the Babylonians used them with a constant de- nominator of 60. Only the numerator was written, with a special mark to denote the denominator. This method of writing fractions lacked only the symbol for zero and the substitution of the base 10 for 60 to become the modern decimal fraction. Sexagesimal frac- tions are still used in the measurement of angles and time. 68. The Romans used duodecimal fractions exclusively. They had special names and symbols for J^, j\, • • •, ii, ^\, J^, etc. To the Romans, fractions were concrete things. They never advanced beyond expressing them in terms of uncia (J^), silicus (i uncia), scrupulum (^^ uncia), etc., all subdivisions of the as, a copper coin weighing one pound. 69. The sexagesiinal and duodecimal fractions prepared the way for the decimal fraction, which appeared in the latter part of the sixteenth century. In 1.58.5 Simon Stevin of Bruges published a work in which he used the notations 7 4' 6" 5'" 9"", or 7® 4® 6(2) 5® 00 for 7.4659. During the early stage of its development the decimal fraction was written in various other forms, among which are found the foUow- I II III IV 12 3 4 ing: 7 4 6 5 9, 7 4 6 5 9, 7 1 4659, 7 [4659 , 7,4659. The decimal point was first used, in 1612, by Pitiscus in his trigonometrical tables ; but the decimal fraction was not generally used before the beginning of the eighteenth century. 40 FR ACT IONS 47 70. Tlie primary conception of a fraction is one or several of tJie equal parts of a unit. i'lius, the fraction | indicates that 4 of tlie 5 e(|nal parts of a nnit are taken. 71. The term namin;/ the number of parts into which tlie unit is divided is called tlie denominator. The term numbering the parts is called the numerator. 72. A proper fraction is less than unity ; an improper fraction is ecjual to or greater than unity. 73. A number consisting of an integer and a fraction is called a mixed number. 74. Our conception of a fraction must, however, be enlarged as we proceed, and be made to include such ex- 2 5 3 3 pressions as -~^^ — -^ 3.14159 • • -, 1, etc. The more o.2o — i ^ general conception of a fraction is that it is an indicated operation in division where the numerator represents the dividend and the denominator the divisor. 75. Decimal fractions are included in the above defini- tion, as 0.5 means ^-^ or J. Using the decimal point (0.5) is simply another way of writing y^Q^. 76. General Principles. 3IuItipJ//ing the ^mmerator or dividing the denominator of a fraction by a number midti- plies the fraction by that number. Let Y be any fraction where 7 represents the number of parts into which unity is divided, and 5 the number of these parts taken. (1) ' = 3 X I;, since there are three times as many of the 7 7 7 parts of unity as before. 48 FRACTIONS 5 5 . . . (2) = 3 X -. since dividing the denominator by 3 divides by 7^37 3 the number of equal parts into which unity is divided and there- fore the fraction is 3 times as large as before. 77. Dividing the numerator or multipli/in;i the denomina- tor of a fraction hy a number divides the fraction by that number. (1) — — - = - -^ 4, since dividing the numerator by 4 divides by 4 the number of parts taken without changing the value of the parts. (2) = - -=- 4, since multiplying the denominator b}* 4 multi- y X 4 y plies by 4 the number of parts into which unity is divided and there- fore the fraction is \ as large as before. 78. Multiplyi7ig or dividing both numerator and de- nominator of a fraction by the same number does not cha)ige the value of the fraction. 5 X v* o (1) '- = -• Multii^lying both numerator and denominator by 5 X 3 3 5 both multiplies and divides the value of the fraction b}^ .5. The value of the fraction therefore remains unchanged. 9^5 9 C2) " = -• Dividing both numerator and denominator bv 5 ^ ^ 3 - .5 3 ^ both divides and multiplies the value of the fraction by 5. The value of the fraction therefore remains unchanoed. 79. A mixed number may be reduced to an improper frac- tion and an improper fraction may be reduced to a mixed number or an integer. Thus, 5| = '--^ — ^—-. Since 5x4 = the number of 4ths in 5 and 4 5x4 + 3 = the number of 4ths in 5 J, .-. o^ = ^^. Reversing the process, V = 23 - 4 = 5 + I = 5f. FRACTIONS 49 80. When the numerator and dejiominator of a fraction are prime to each other^ the fraction is said to he in its low- est terms. Ex. Express || in its lowest terms. 42 ^ 3 X 14 ^ 3 70 5 X 14 5* 81. Two or more fractions may he reduced to equivalent fractions having a common deno7ninator . Ex. 1. Reduce |, f , ^^2' ^^ equivalent fractions having a common denominator. The 1. c. m. of 4, 9, 12, is 36. 3 3x9 27 4 4x9 36 5 5x4 20 9 9x4 36 1 _ 12 1 X 3 12 X 3 _ 3 36 •*• §6? ih 3% ^^'^ fractions having a common denominator, equivalent to |, f, j\. Since 36 is the 1. c. ni. of 4, 9, 12, it is called the least common denominator. 82. Sometimes, instead of finding the l.c.m., it is more convenient to take as the common denominator the product of all the denominators and multiply each numerator by the product of all the denominators except its own. Ex. 2. Reduce |, ^ and | to fractions having a common denominator. 3 3x6x3 54 Since the common denominator is 4 x 6 x 3, 4 ~ 4 X 6 X 3 ~ 72 4 is contained in it 6 x 3 times and the first numerator will be 3 x 6 x 3, 6 is contained in the common denominator 4x3 times and the second numerator will be 1 x 4 x 3, 3 is contained 1 1x4x3 12 6 4x6x3 7 2 _ 2x4x6 _ 48 in the common denominator 4x6 times and the 3 4x6x3 72 third numerator will be 2 x 4 x 6. lyman's adv. ar. — 4 50 FB ACTIONS 83. Addition and Subtraction of Fractions. Since only the same kinds of units, or the same parts of units, can be added to or subtracted from one anotlier, it is necessary to reduce fractions to a common denominator before per- forming the operations of addition or subtraction. Ux. 1. Add y%, 3^g and -^j. The 1. c. m. of 12, 36 and 84 is 252. 5 ^ 5 X 21 ^ 105 7 ^ 7x7 ^ 49 1^ 1x3 ^ 3 12 12 X 21 252' 36 36 x 7 252' 84 84 x 3 252' 5 7 1 ^ 105 49 3 ^ 157 12 36 84 252 252 252 2.52' Ux. 2. Add 2|, If and 3lf . 2| + If + 3H - 2 + 1 + 3 + I + f + If = 6 + }- ft + j\% + j\% = 7t\V After a little practice the student should be able to abbreviate the work very much. &. 1 might be worked briefly, thus ; 5 7 1 ^ 105 + 49 + 3 ^ 157 12 36 84 252 252' . Ux. 2, thus : 9. + u , 313 _ 6 + l_0_8 + 80 + 39 _ . Ex. 3. From 22^^^ subtract 181|. 21f I - 18H =: 3|| = 3if 84. Multiplication of Fractions. The product of two numbers may he found hy performing the same operation on one of them as is performed on unity to produce the other. Thus, in 3x4 = 12, nnity is taken three times to produce the mul- tiplier 3, hence 4 is taken three times to produce the product 12. Again, in | x ^ = ^f , unity is divided into 3 parts and 2 of them are FRACTIONS 51 taken to produce the multiplier |, hence, 5 is divided into .3 parts, each h is — '- — 3x7 2 X 5 1.0 of which is ——- (Why?), and 2 of them are taken to produce the 8 X / ^ in-oduct ^ 3 X 7 21 When the multiplier is a common fraction, the sign ( x ) should be read " of." Thus, | x $5 means | of 1 5. Ex. 1. Multiply II by fJ. 1 1 23 y^\ ^ n x 2X ^1 42 69 12 X ^p e' 2 3 85. The student should use cancellation luhenever jjossible. He shoidd never midtiply or divide until all possible factors have been removed by cancellation. Note. Although a knowledge of the principles of multiplication and division of decimals has been assumed in examples given before, it is well to review these principles at this point to make sure that they are thoroughly understood. Ex. 2. Multiply 0.234 by 0.16. Solution. 0.234 x 0.16 = '^ x ^ 0.234 1000 100 Q ^g 234 X 16 1000 X 100 234 1404 •^'^^^ =0.03744. 0<03744 100000 The number of decimal places in the product is the same as the number of zeros in the denominator of the product, that is, it equals the number of decimal places in the multiplicand plus the number in the multiplier. The decimal point in (0.03744) simply provides a convenient way of writing xtwjo- 1^ ^^ better, however, to determine the position of the decimal point before beginning the multiplication. This can be done by considei-ing only the last figure at tlie right of the multiplier and multiplicand. Thus, we see that 0.004 x 0.06 = 0.00024. Hence, the order of the product will be hundred thousandths. 52 FRACTIONS 86. Tlie following simple truths, or axioms, are fre- quently used in arithmetic. (1) Numbers that are equal to the same number are equal to each other. Thus, if X = 5 and // = 5, then x = y. (2) If equals are added to equals., the sums are equal. Thus, if X = 5, X + 3 = 5 + 3. (3) If equals are sid^tracted from equals., the remainders are equal. Thus, if a; = 4, then a; - 2 = 4 - 2. (4) If equals are midtlplied by equals., the products are equal. Thus, if ^ = 3, then x = Q. (5) //' equals are divided by equals^ the quotients are equal. Thus, if 3 X = G, tlien x = 2. 87. Division of Fractions. Division may be regarded as the inverse of multiplication. The problem is, therefore, to find one of two factors when the product and the other factor are given. Thus, 3x4= 12, .-. 12 -3 = 4, and 12 - 4 = 3. Axiom 5. To divide one fraction by another. Solution. Let || -^ | = 7 (a quotient). Then || = | x r/ (multiplying both members of the equation by f), and II X f = 7 (multiplying both members of the equation by f). ••. the quotient is obtained by nmltiphjing the dividend by the reciprocal, of the divisor. FRACTIONS 53 ^a?. Find the value of .] + 2 x | x J + [; ^ :^ x 2 - J. 2 5 3 4 (3 ii 7 5 2 5 )J 7 5 o Observe tliat in the above exercise the fractions connected by x or -f- are first operated upon, then those connected by + or — . 3 88. A fraetion of the form |- is called a complex frac- 6 tion and may be considered as equivalent to | -J- 1 and treated as a problem in division. In general, however, a complex fraction may be more readily simplified by multi- plying both terms by the 1. c. m. of the denominators of the two fractions in the numerator and denominator, ^^ 1 f_63xf_35 Rv. 2. Divide 38.272 by 7.36. 5.2 7.36)38.272 Solution. 38.272 - 7.36 = \%'J^^ - lU 36.80 — SAT? 2. y log _ 38212 y 100_ — 5^ x J- — ,0 '> , ,„^ — 1000 ^^73 6^— 736 -^TOOO" 1 -^10— ^— ' 1.472 1.472 The number of decimal places in the quotient will equal the num- ber of zeros in tiie denominator of the last product. This will be the same as the number of zeros in the denominator of the dividend minus the number of zeros in the denominator of the divisor, or, Avhat is the same thing, the number of decimal places in the dividend minus the number of decimal places in the divisor. If the number of decimal places in the dividend is less than the number of decimal places in the divisor, we may annex zeros to the dividend till the number of decimal places is the same in both divi- dend and divisor. The quotient up to this point in the division will 54 FB ACTIONS be an integer, and, in case it is necessary to carry the division farther, more zeros may be annexed to the dividend. The remaining figures of the quotient will be decimals. Ux. 3. Divide 52.36 by 3.764. 13.9 3.764)52.360|0 37.64 14.720 11.292 3.4280 3.3876 404 EXERCISE 16 1. Change | to 9tbs ; 2^ to 168ths. 2. Reduce to lowest terms each of the following frac- tinim • -9- 111 -7-2- 1128 3. Explain the reduction of 7| to an improper fraction. 4. Explain the reduction of -^^5^- to a mixed number. ^. ,., 4f I 0.5 0.75 5. Simplify ^, |, -J-, -^. ^6323 fi Arlrl 3668 1221 r,nfl 5 b. i\aa 10 9 8 9' 13 4 31 ^^^^^ 12 2 1- 7. From 75^^2 ^-^ke 12-\. 8. Multiply 21 - f by J of f X |. 9. Find the value of f of ^3_ _^ 2 >< _6_ of |f . 10. Find the value of 1-^f of f x|-|-^| of -|--fo-^f 11. Find the value of 5|-0.9 of 2.7 + 251x0.02-^. 12. rind the value or — i-^ 4is *'t -^T •7 1 FRACTIONS 55 13. By what must \ he multiplied to produce -SJ ? 14. Wliat number divided by -V^^- of | will give 4| as a quotient ? Simi)lifv: ^-l-i:3j^.G ^>ij7:. 15. Simplify: ^-^^^-^^ ^^- 1 _ 1 of 1 ^ 4- 3 V 4 -■■ 5 ^^ 13 + ? >^ Y 16. What fraction added to the sum of |, ^, and 5.25 will make 6.42 ? 17. Simplify: / ^^//^ ^^ • ^ "^ 5 + 0.5 of (Jg-0.9) 18. How is the value of a proper fraction affected by adding the same number to both numerator and denomi- nator ? How is the value of an improper fraction affected ? 19. A merchant bought a stock of goods for f 2475.50 and sold -|- of it at an advance of \ of the cost, \ of it at an advance of ^ of the cost, and the remainder at a loss of ^^ of the cost. Did he gain or lose and how much ? 20. A ship is worth $90,000 and a person who owns -f^^ of it sells \ of his share. What is the value of the part he has left ? 21. If 1 is added to both numerator and denominator of |, by how much is its value diminished ? 22. If 1 is added to both numerator and denominator of |, by how much is its value increased ? 89. Cancellation. Much time may be saved in solving problems by w^riting down a complete statement of the condition given and then canceling common factors if any are present. The student should do this at every stage in the solution of a problem, always factoring and canceling whenever possible, and never multiplying or dividing till all 2?osi5, the differ- 0.2 )."S<;47 ence cannot be greater or less than the correct value by a 0.0!M)yi5l fraction as large as 0.000001. Hence, the difference must lie between 0.090882 and 0.090880, and the value correct to five places is 0.09088. 98. Cases will arise where the second and third digits from the last are in doubt, as in addition. The student should determine how far the result may be relied upon in the following examples : (1) Subtract 0.371492 from 0.764237. (2) Subtract 0.11132 from 0.23597. (3) Subtract 15.93133 from 43.71288. 99. Multiplication. From the examples in addition given above the student will notice that it will be suffi- cient in most cases to carry out the partial products correct to two places more than the required result. Ex. Find the square of 3.14159 correct to four decimal places. Solution. The multiplication in full and the contracted form are as follows: 3^^^.^ 3^^^.^ 3.14159 • 3.14159 9.42477 9.42477 .3141.59 .314159 .1256636 .125664 .00314159 3142 .001570795 1571 .0002827431 283 9.8695877281 9.8696 After pointing off the first partial product we proceed as indicated in the above contracted form until the multiplication by 3 and 1 are 60 FRACTIONS completed. Multiplication by 4 would give a figure in the seventh place. • Instead of writing down the figures we add the nearest 10 to the next column. Thus, 4 times 9, 36, add 4 to the next column since 3.6 = 4 apj^roximately. 4 times 5, 20 and 4, 24. 4 times 1, 4 and 2, 6, etc. In multiplying by the next 1 it is not necessary to take the 9 in the multiplicand into account. So, also, in multiplying by the 5, the 5 and 9 in the multiplicand may both be ignored. And so on until the multiplication is completed. 100. Division. JSx. Divide 9376245 by 3724 correct to the units' place. Solution. The division in full and the contracted form are as follows : 2517 19376245 7448 2517 3724)9376245 7448 19282 18620 19282 18620 6624 3724 662 372 29005 26068 290 260 2937 30 The first two digits in the quotient are 2 and 5 and the second re- mainder is 662. It is not necessary to bring down any more figures to have a result correct to units since tens divided by thousands will give hundredths. The divisor may also be contracted at this stage of the work. Thus, cutting off the 4, 372 is contained once in the second remainder, 662. Cutting off the 2, 37 is contained 7 times in the next remainder, 290. This gives the units' figure of the quotient. 2517 It will be noticed that the next figure of the quo- 3704.^9376045 tient is greater than 0.5, therefore the result correct 19'>8"^ to units is 2518. ggo The work may be further abridged by omitting the Qon partial products and writing down the remainders only. oq APPROXIMATE RESULTS 61 101. Ex. Divide 62.473 by 411).GT89.* miQc-nn •^ O.1488o90 Solution. First shift the decimal point 410G7fSf>)0LU7;)O.<) four places in each so as to have an integral "JO.lOol 10 divisor, and then work as follows : 'I'he 1 ^37 17!*")!: and 4 are obtained without abbreviating ^OO.l^;} and the 8, 8, 5, 9, by cutting off 9, 8, 7, 24780 6, 9 in succession from the divisor. 3796 19 E.V. Divide 0.0167 by 423.74.* 0.00003941 * From Langley's "Treatise 42374)1.67000 on Computation," p. 68. 39878 1741 46 4 EXERCISE 18 1. Divide 100 by 3.14159 correct to 0.01. 2. Find the quotient of 67459633 divided by 4327 cor- rect to five significant figures. 3. Determine witliout dividing by what number less than 13, 339295680 is exactly divisible. Determine by casting out tbe 9\s whether the following are correct : 4. 959x959 = 919681. 5. 954x954x954 = 868250664. 6. 33920568-729 = 42829, 7. 1019x1019 = 1036324. 8. 6234751 - 43265 = 14.41+ a remainder 2645. 9. Find the sum of 23.45617, 937.34212, 42.31759, 532.23346, 141.423798 correct to two decimal places. 10. Subtract 987.642 from 993.624 correct to tenths. 11. Find the product of 32.4736 x 24.7955 correct to five significant figures. 12. Divide 47632 by ^. 13. :\Iultiply 23793 by 124- MEASURES 102. Measures of Weight. It is curious to note what ati important part the grain of wheat or barley has played in the establishment of a unit of weight, both among the ancients and the more modern Europeans. In England, as early as 1266, we find the pennyweight defined as the weight of " 32 wheat corns in the midst of the ear " ; again about 1600, as " 24 barley corns, dry and taken out of the middle of the ear." Still later the artificial grain (^\f oz. Troy) is defined as "one grain and a half of round dry wheat." The Greeks made four grains of barley equivalent to the keration or carob seed. From this is derived the carat, the measure by which diamonds and pearls are weighed. The grain of barley and the carat have been used by all European countries as the basis of existing weights. 103. Great inconvenience was long experienced from this lack of uniformity, so that Parliament in 1821 passed an act adopting the Imperial Pound Troy as the standard of weight. It was also enacted that of the 5760 grains contained in the pound Troy, the pound avoir- dupois should contain 7000. The unit pound is defined by a piece of metal kept in the standard office. The ounce, grain, etc., are subdivi- sions of the pound. 104. Avoirdupois Weight 16 drams (dr.) = 1 ounce (oz.) 16 ounces = 1 pound (lb.) 100 pounds = 1 hundredweight (cwt.) 2000 pounds := 1 ton (T.) 112 lb. = 1 long cwt. and 2240 lb. = 1 long ton are used in the customhouse and in weighing coal and iron at the mines. The c in cwt. stands for the Latin word centum, a hundred. Lb. is a contraction of the Latin word libra, pound. Pound is from the Latin word pondus, a weight. Ounce is from the Latin word uncin, a twelfth part. Dram is from the Latin word drachma, a handful. 62 ME As CUES G3 105. Tkoy Wkkjiit 24 grains (gr.) = 1 peiinyweiglit (pvvt.) 20 pennyweights = 1 ounce Troy 12 ounces Troy = 1 ])oun(l Troy This weight is used for the precious metals and jewels. The ounce Troy and pound Troy must be carefully distinguished from the ounce and pound avoirdupois. The grain, however, is the same throughout. 437.5 grains = 1 ounce avoirdupois 480 grains = 1 ounce Troy 7000 grains = 1 pound avoirdupois 5760 grains = 1 pound Troy 106. Apothecaries' Weight 20 grains = 1 scruple (sc. or 9) 3 scruples = 1 dram (di-. or 3 ) 8 drams = 1 ounce (oz. or ^ ) 12 ounces = 1 pound 5700 grains = 1 pound This table is used in compounding drugs and medicines. Scruple is from the Latin word scrupulum, a small weight. Of the above measures of weight, avoh'dupois is the most generally used. 107. Measures of Length. The ancients usually derived their units of length from some part of the human body. Thus, we find the fathom (the distance of the outstretched hands), the cubit (the length of the forearm), and later the ell (the distance from the elbow to the end of the finger), the foot (the length of the human foot), the span (the distance between the ends of the thumb and little finger when outstretched), the jJalm (the width of the hand), the dir/if (the breadth of the finger). The Roman foot was subdivided into four palms, and the palm into four digits. The division into inches or uncice (a twelfth part) applied not only to the foot but to anything. 108. For longer measures there was still less uniformity. We find the Hebrew's half-da fs journey ; the Chinese /t'A, the distance a man's voice can be heard upon a clear plain ; the Greek stadium, prob- 64 MEASURES ably derived from the length of the race course ; the Roman pace of five feet ; the furlong, the length of a furrow. The mille passus, a thousand paces, is the origin of the modern 7nile. 109. In 1374 the inch is defined in English law as the length of " three barley corns, round and dry." Later, other arbitrary measures of length were adopted by the government. The standard unit in England and the United States is the yard. The standard yard is the length of a metal bar preserved in the office of Standard Weights and Measures. The standard foot and inch are subdivisions of this standard yard. 110. Common Measures of Length 12 inches (in.) = 1 foot (ft.) 3 feet = 1 yard (yd.) 5i yards or 16i feet = 1 rod (rd.) 320 rods or 5280 feet = 1 mile (mi.) The furlong, equal to 40 rods, is seldom used. The fathom, equal to 6 feet, and the knot or geographical mile, equal to one minute of the equatorial circumference of the earth (6080 feet), are sometimes used. 111. Surveyors' Measures of Length 7.92 inches = 1 link (li.) 100 links = 1 chain (ch.) = (4 rd.) 80 chaifts = 1 mile 112. ^Measures of Surfaces 144 square inches (sq. in.) = 1 square foot (sq. ft.) 9 square feet =: 1 square yard (sq. yd.) 30^ square yards = 1 square rod (sq. rd.) 160 square rods = 1 acre (A.) 640 acres = 1 square mile (sq. mi.) 1 square mile = 1 section. 36 sections = 1 township (twp.) MEASURES 65 113. Measures of Solids 1728 cubic inclies (cu. iu.) — 1 cubic foot (cu. ft.) 27 cubic feet = 1 cubic yard (cu. yd.) The cord, equal to 128 cubic feet, is a rectangular solid 8 feet long, 4 feet wide, and -1 feet high. The common use of the word is, how- ever, a pile of wood 8 feet long and 4 feet high, the widtli of the pile varying with the length of the stick. 1 cubic yard = 1 load 24| cubic feet = 1 perch 114. Measures of Money. Originally, among primitive people, buying and selling was carried on by barter, or the actual exchange of commodities. The inconveniences arising from transactions of this kind brought about the adoption of a medium of exchange, or money. Money, usually consisting of gold and silver, was used at a very early period in the world's history. Gold and silver seem at first to have been exchanged for commodities by weight. Business trans- actions were then still further simplified by the introduction of coins and paper money. Finally, as in the case of weights and measures, governments adopted definite standards of money value. 115. United States Money 10 mills = 1 cent (ct.) 10 cents = 1 dime (d.) 10 dimes = 1 dollar (%) 10 dollars = 1 eagle (E.) 116. English Money 12 pence (rl.)= 1 shilling (.s.) = 10.2433 20 shillings = 1 pound (£) = ^4.8665 117. French Money 10 centimes = 1 decline 10 decimes = 1 franc = .| 0.193 LTMAX'S ADV. AR. 5 66 MEASURES 118. German Money 100 pfennigs = 1 mark (M.) = 80.238 119. jNIeasures of Number 12 units = 1 dozen (doz.) 12 dozen = 1 gross (gro.) 12 gross = 1 great gross (gt. gro.) Also 24 sheets of paper = 1 quire 20 quires = 1 ream 120. Liquid Measure 4 gills (gi.)=l pint (pt.) 2 pints = 1 quart (qt.) 4 quarts = 1 gallon (gal.) = 2-31 cu. in. 31i gallons = 1 barrel (bbl.) 121. Dry Measure 2 pints = 1 quart 8 quarts = 1 peck (pk.) 4 pecks = 1 bushel (bu.)= 21.10.42 cu. in. The Winchester bushel is the standard nieusure for dry substances. It is a cylindrical vessel 18| in. in diameter and 8 in. deep, containing 2150.42 cu. in. Before the adoption of this and other standards by the English government there was even a greater variety of measures of capacity than of length and weight. 122. Reduction of Compound Numbers. Quantities like 5 mi. 10 1(1. 7 yd. 2 ft. iiiid 3 lb. 5 oz. are called com- pound numbers, because they are expressed in several denominations. MEASURES 67 Ux. 1. Reduce 25 yd. 2 ft. 11 in. to inches. Solution. 1 yd. = 3 ft. .-. 25 yd. = 25 x 3 ft. = 75 ft. 75 ft. + 2*^ ft. = 77 ft. 1 ft. = 12 in. .-. 77 ft. = 77 X 12 in. = 924 in. 924 in. +11 in. = 935 in. Note. The explanation shows that 25 and 77 are the multipliers and 3 ft. and 12 in. the multiplicands; but to shorten the operation, 3 and 12, regarded as abstract numbers, may be used as multipliers, since the product of 25 X 3 = the product of 3 x 25. Ex. 2. Reduce 1436 pt. to bushels, pecks, etc. 1436 no. of pt. 25 or 25 yd. = 900 in. _3 2 ft. = 24 in. 75 11 in. = 11 in. 2 .-. 77 25 yd. 2 ft. 11 in. = 935 in. 12 924 11 935 718 no. of qt. 89 no. of pk. + 6 qt. 22 no. of bu. + 1 pk. Solution. Since there are 2 pt. in 1 qt., in 1436 pt. there are as many quarts as 2 pt. are contained times in 1436 pt., or 718 qt. Since there are 8 qt. in 1 pk., in 718 qt. there are as many pecks as 8 qt. are contained times in 718 qt., or 89 pk. 6 qt. Since there are 4 pk. in 1 bu., in 89 pk. there are as many bushels as 4 pk. are contained times in 89 pk., or 22 bu. 1 pk. .-. 1436pt.= 22bu. Ipk. 6qt. EXERCISE 19 1. Reduce 3 A. 5 sq. rd. 12 sq. yd. to square yards. 2. Reduce 11000 sq. rd. to acres. 3. Reduce 2 gt. gro. 5 gro. to dozens. 4. Reduce 972 sheets to reams. 5. Reduce 20 cu. yd. to cubic inches. 6. Reduce 1000 oz. to pounds and ounces (avoirdupois). 68 MEASURES 7. Reduce 12 lb. 5 oz. 11 pwt. 20 gr. to grains. 8. Reduce 113 T. 7 cwt. 11 lb. to pounds. 9. Reduce 14763051 lb. to tons. 10. Reduce 5 sq. yd. 3 sq. ft. 91 sq. in. to square inches. 11. Reduce 46218385 sq. in. to acres. 12. Reduce 5 bu. 7 pk. 3 qt. to Cjuarts. 13. Reduce 34372 pt. to pecks. 14. Reduce 21 yd. to a decimal of a mile. 15. Reduce 2 pk. 3 qt. 1 pt. to a decimal of a bushel. 16. Reduce 0.0125 A. + 0.25 sq. rd. to square feet. 17. Reduce 0.01 of a cubic yard to cubic inches. 18. Reduce 43629145 in. to miles. 19. Reduce | of a peck to pints. 20. Reduce 2 qt. 1 pt. to a fraction of a peck. 21. Reduce 1 mi. 11 ch. to feet. 123. Addition and Subtraction of Compound Numbers. Compound addition and subtraction is tlie addition and subtraction of compound numbers of the same kind. The processes differ very little from the corresponding pro- cesses in the addition and subtraction of abstract numbers. Ex. 1. Add 4 lb. 7 oz. (Av.), 3 lb. 4 oz., 12 11). 10 oz., 9 lb. 5 oz. Solution. The work is as follows : 5, 1.5, 19, 26 oz. = 1 lb. 10 oz. 1, 10, 22, 25, 20 lb. 4 lb / oz 3 4 12 10 9 5 291b. 10 oz. MEASURES 69 Ux. 2. From 41 lb. 4 oz. (Av.) subtract 29 lb. 8 oz. The work is as follows : 41 lb 4 oz 1 lb., or 16 oz. + 4 oz. = 20 oz. 29 8 8 and 12 are 20 TTVi To ,7:;: . .. .-. 11 lb. 12 oz. 1 and 29 and 11 are 41. 124. Multiplication of Compound Numbers. Ux. Multiply 5 yd. 2 ft. by 7. 7 X 2 ft. = 14 ft. = 4 yd. 2 ft. 5 yd. 2 ft. 7x5 yd. = 35 yd. 7 35 yd.'+ 4 yd. 2 ft. = 39 yd. 2 ft. 39 yd. 2 ft. 125. Division of Compound Numbers. Compound divi- sion is of two kinds. The first is the converse of multi- plication. In this case the quotient is a compound number of the same kind as the dividend. In the second case the dividend and divisor are both compound numbers of the same kind, and the quotient is an abstract number. 126. The two cases arise from the fact that division may be regarded as the operation of finding one of two factors when the other factor and the product are given. Thus, 39 yd. 2 ft. is the product of 7 and 5 yd. 2 ft. ... 3^^^-^^^- = 5 yd. 2 ft., or '^ll^^AJ^ = ^1^=7, the divi- 7 ^ ' 5 yd. 2 ft. 17 ft. dend and divisor being reduced to the same denomination before dividing. Ux. 1. Divide 29 mi. 2 yd. 2 ft. by 8. Solution. 29 mi. ^8 = 3 mi. + a remainder of 5 mi. 5 mi. = 8800 yd. and 8800 yd. + 2 yd. = 8802 yd. 8802 yd. -f- 8 = 1100 yd. + a remainder of 2 yd. 2 yd. = 6 ft., and 6 ft. + 2 ft. = 8 ft. 8 ft. - 8 = 1 ft. .♦. 29 mi. 2 yd. 2 ft. - 8 = 3 mi. 1100 yd. 1 ft. 70 ME A SUB ES Ux. 2. Divide 139 lb. 8 oz. (Av.) by 4 lb. 8 oz. Solution. 139 lb. 8 oz. = 2232 oz. 4 lb. 8 oz. = 72 oz. 2232 0Z. -^72oz. = 31. 127. Check. Compound addition and subtraction may- be checked in the same way as addition and subtraction of simple numbers. Multiplication may be checked by division, and division by multiplication. EXERCISE 20 1. How many inches are there in 1 mi. 3 ch. ? 2. Add 14 lb. 3 oz., 5 lb. 7 oz., 31 lb. 11 oz. 3. From 17 cu. yd. 11 cu. in. subtract 5 cu. yd. 5 cu. ft. 4. From 11 bu. 1 pk. subtract 4 bu. 5 qt. 5. Multiply 30 A. 11 sq. rd. by 10. 6. Divide 159 A. 29.5 sq. rd. by 2. 7. How many bags containing 2 bu. 1 pk. each can be filled from a bin of wheat containing 256 bu. 2 pk. ? 8. How many revolutions will a bicycle wheel 7 ft. 4 in. in circumference make in traveling 25 mi. ? 9. How many times can a bushel measure be filled from a bin 8 ft. square and 6 ft. deep ? Will there be a remainder ? 10. How many gallons of water will a tank 4 ft. 7 in. by 2 ft. 11 in. by 1 ft. 3 in. contain ? 11. How many times is 7 ft. 6 in. contained in 195 mi. 280 rd. ? METRIC SYSTEM OF WEIGHTS AND MEASURES 71 12. How much coal is there in three carloads of 38 T. 3 cwt. 41 lb., 29 T. 7 cwt. 5 lb., 32 T. 17 cwt. 70 lb. ? 13. Wliat must be the length of a shed 7 ft. liigh and 9 ft. wide to contain 50 cd. of 16 in. wood ? 14. If a ton of coal occupies 36 cu. ft., what must Ije tlie depth of a bin 6 ft. wide by 7 J ft. long in order that it may contain 10 T. ? 15. Divide 320 rd. 4 yd. by 10 rd. 2 yd. 16. How many feet are there in | of a mile ? 17. Reduce 17 pt. to a decimal of a gallon. 18. How many steps does a man take in walking a mile if he advances 2 ft. 10 in. each step ? 19. The pound avoirdupois contains 7000 gr. Find the greatest weight that will measure both a pound Troy and a pound avoirdupois. Find the least weiglit that can be expressed without fractions in both pounds Troy and pounds avoirdupois. 20. A cubic foot of water weighs 1000 oz. avoirdupois. Find the number of grains Troy in a cubic inch of water. METRIC SYSTEM OF WEIGHTS AND MEASURES 128. Late in the eighteenth century France invented the metric system of weights and measures, but it was not made obligatory until 1837. Previous to this time there existed in France the same lack of uniformity in forming multiples and subniultiples of the units of measure as exists in our system at the present time. The metric system is now in use in most civilized countries except the United States and England. It was legalized by CongTess in the United States in 1866, but has not been generally adopted. In scientific work the system is quite generally used in all countries, MEASUBES 129. The unit of length is the meter. This is the fundamental unit, because from it everv other unit of measure or weisrht Is deriyed ; hence the /name metric system. The meter is theo- ' "^ retically one ten-millionth part of the /^^ distance of the pole from the equator. Though an error has since been dis- covered in the measurement of the dis- tance, the meter has not been changed, and a rod of platinum 39.37079 inches in length, deposited in the archives at Paris, is called the standard meter. c^ CO ^ — D^ -LO to C\J used to denote means 0.1 of a gram. 130. The unit of capacity is called the liter. It is a cube whose edge is 0.1 of a meter. 131. The unit of weight is the gram. The gram is the weight of a cube of distilled water at maximum density, whose edge is 0.01 of a meter. 132. The above units of measure. Together with the following prefixes, should be carefully memorized, because from them the whole metric system can be built up. 133. The Latin prefixes, deci. centi. milli, denote respectively 0.1, O.Ml. O.MOl of the unit. The Greek prefix micro is 0.000001 of a unit. Thus, deciuieter meter, and centicrram means 0.01 of a METRIC SYSTEM OF WEKIHTS AND MEASURES 73 134. The Greek prefixes, deea, hecto, kilo, myria, denote respectively 10, 100, 1000, 10000 times the unit. 'V\ms, Ivilometer means 1000 meters, and hectoliter means 100 liters. 135. In general, nothing beyond practice in arithmetical operations would be gained in reducing from tlie metric system to our system. Occasionally, however, such reduc- tions are necessary, hence, a few of the common equivalents are given in the tables. 136. Mkasures of Length 10 uiillinieters ("'"^) = 1 centimeter 10 centimeters (*^'") = 1 decimeter 10 decimeters (^"'> = 1 meter = 80.37 in. 10 meters ('") = 1 decameter 10 decameters '^'"^ = 1 hectometer 10 hectometers ("'">= 1 kilometer 10 kilometers (^"'' = 1 myriameter (^^""^ 137. Square Measure 100 square millimeters (""-^ = 1 square centimeter 100 square centimeters ('^^-^ = 1 square decimeter 100 square decimeters ^^'"-^ = 1 square meter 100 square meters ('"-^ = 1 square decameter 100 square hectometers *"'"-^= 1 square kilometer ^•^'"-^ This table may be extended by squaring each unit of length for the corresponding unit of square measure. The denominations given in tlie table are the only ones in common use. In measuring land, the square decameter is called the are. the square hectometer, the hectare = 2.47 acres, and the square meter, the pentare. 74 MEASURES 138. Cubic Measure 1000 cubic millimeters ('"'"^> = 1 cubic centimeter 1000 cubic centimeters ('^"•^^ = 1 cubic decimeter 1000 cubic decimeters ^<^'"^) = 1 cubic meter ('"^) This table may be extended by cubing each unit of length for the corresponding unit of cubic measure. The denominations given in the table are the only ones in common use. The cubic meter is used in measuring wood, and is called the stere. 139. Measures of Weight 10 milligrams ('"^^ = 1 centigram 10 centigrams ("^s) = 1 decigram 10 decigrams ^^s) — 1 gram 10 grams (e) = 1 decagram 10 decagrams (^^^ = 1 hectogram 10 hectograms (Hg) == 1 kilogram = 2.2 lb. 10 kilograms (*^s) = l myriagram 10 myriagrams '■^s) = 1 quintal 10 quintals (Q) = 1 tonneau (T) The metric ton or tonneau is the weight of one cubic meter of distilled water = 2201.62 pounds. 140. Measures of Capacity 10 milliliters C"!) = 1 centiliter 10 centiliters ('^i) = 1 deciliter 10 deciliters <«^) = 1 liter = 1 qt. nearly 10 liters (^) = 1 decaliter 10 decaliterse of a sphere wliose radius is 3 in. is | x 3.1416 X 3^^ cu. in. = 113.1 cu. in. MENS URA TION 111 206. Board Measure. In measuiing lmnl)er the board foot is used. It is a board 1 ft. long, 1 ft. wide and 1 in. or less thick. Lumber more than 1 in. tliick is m(;asured by the number of square feet of l)oards 1 in. thick to which it is equal. Thus, a board 10 ft. long, 1 ft. wide and 1}, in. thick, contaius 15 board feet. Lumber is usually sold by the 1000 board feet. A quo- tation of !^17 per jNI, means flT per 1000 board feet. EXERCISE 35 1. Find the cost of 12 boards 16 ft. long, 6 in. wide, and 1 inch thick at -f 18 per M. 2. How many board feet are there in a stick of timber 16 ft. by 16 in. by 10 in. ? 3. How much is a stick of timber 15 ft. by 2 ft. by 1 ft. 4 in. worth at #22 per M ? 4. How many board feet are used in laying the flooring of a two-story house 32 ft. by 20 ft., allowing 40 ft. waste ? 5. What is the cost of 25 21 -in. planks 16 ft. long by 1 ft. wide at .$22.50 per M? 6. What is the cost of 15 joists 12 ft. by 10 in. by 4 in. at $23 per M ? 207. Wood Measure. The unit of wood measure is the cord. The cord is a pile of wood 8 ft. by 4 ft. by 4 ft. A pile of wood 1 ft. b}^ 4 ft. by 4 ft. is called a cord foot. A cord of stove wood is 8 ft. long by 4 ft. high. The length of stove wood is usually 16 in. 112 MENS URA TtON EXERCISE 36 1. Find the number of cords of wood in a pile 32 ft. by 4 ft. by 4 ft. 2. At $5.75 per cord, how much will a pile of wood 52 ft. by 4 ft. by 4 ft. cost ? 3. How much will a pile of stove wood 94 ft. long 4 ft. high be worth at $2.75 per cord ? 208. Carpeting. A yard of carpet refers to the running measurement, regardless of the width. The cheaper grades of carpet are usually 1 yd. wide, and the more expensive, such as Brussels, Wilton, etc., are J of a yard wide. In carpeting, it is usually necessary to allow for some waste in matching the figures in patterns. Dealers count this waste in their charges. In computing the cost of carpets, dealers charge the same for a fractional width as for a whole one. Carpets may often be laid with less waste one way of the room than the other; hence, it is sometimes best to compute the cost with the strips running both ways, and by comparison determine which involves the smaller waste. EXERCISE 37 1. How many yards of Brussels carpet | of a yard wide will be required to cover the floor of a room 15 ft. by 13 ft. 6 in., tlie waste in matching being 4 in. to each strip ? Which will be the more economical way to lay the carpet ? 2. How much will it cost to cover the same room with Brussels carpet if a border | of a yard wide is used, the carpet and border being SI. 25 per yard, and the waste being 4 in. to each strip of carpet and -| of a yard of border at each corner ? MENSURATION 113 3. How much will it cost to cover the same room with ingi'ain carpet 1 yd. wide, at (-)7| ct. per yard, the waste being 6 in. to eacli strip ? 4. At $lA2}y per yard, how mucli will it cost to carpet a flight of stairs of 14 steps, each step being 8 in. high and 11 in. wide ? 5. A room is 16 ft. 10 in. by 14 ft. 9 in. How long must the strips of carpet used in covering the floor be cut, if tlie pattern is 14 in. ? (If laid lengthwise of the room, the length of 15 patterns must be used.) Will it be clieaper to run the strips lengthwise or across the room ? If the room is covered with carpet | of a yard wide at 11.35 per yard, how much will it cost? 209. Papering. Wall paper is sold in single rolls 8 yd. long, or in double rolls IG yd. long. It is usually 18 in. wide. There is considerable waste in cutting and matching paper. AVhole rolls may be returned to the dealer, but part of a roll will not usually be taken back. EXERCISE 38 1. How many rolls of paper are used in papering a room 14 ft. by 12 ft. 6 in., and 8 ft. high above the base- board, if the room contains 2 windows 6 ft. by 3 ft. 6 in. and 2 doors 7 ft. by 4 ft., the width of the border being 16 in., and 6 in. waste being allowed to each strip for matching ? 2. How much will it cost to paper the above room if the paper is 11 ct. per roll, the border being 18 in. wide, and the paper hanger working 8 hr. at 30 ct. per hour ? ltman's adv. ar. — 8 114 MENS URA TION 3. At 25 ct. per roll, how much will it cost to paper a room 18 ft. square and 9 ft. high above the baseboard, allowing for 2 doors, each 7 ft. by 3 ft. 9 in., and 3 win- dows, each 6 ft. by 3 ft. 4 in., the border being 18 in. wide ? 210. Painting and Plastering. The square yard is the unit of painting and plastering. There is no uniform j^i'^ctice as to allowances to be made for openings made by windows, doors, etc., and the baseboard. To avoid complications, a definite written contract should always be drawn up. EXERCISE 39 1. How much will it cost to plaster the walls and ceiling of a room 15 ft. by 13 ft. 6 in., and 9 ft. high, at 27J ct. per square yard, deducting half of the area of 2 doors, each 7 ft. by 31 ft., and 2 windows, each 6 ft. by 31 ft. ? 2. How much will it cost to paint the walls and ceiling of the same room at 121 ct. per square yard, the same allowance being made for openings ? 3. At 20 ct. per square yard, how much will it cost to paint a floor 18 ft. by 16 ft. 6 in. ? 4. Allowing 1 of the surface of the sides for doors, windows and baseboard, how much will it cost to plaster the sides and ceiling of a room 22 ft. by 18 ft. and 9^ ft. high, at 221 ct. per square yard ? 211. Roofing and Flooring. A square 10 ft. on a side, or 100 sq. ft., is the unit of roofing and flooring. The average shingle is taken to be 16 in. long and 4 in. wide. Shingles are usually laid about 4 in. to the weather. ME^ S URA TION 115 Allowing for waste, Jibout 1000 sliinglt's are estimated as needed for each square, but if the shingles are good, 850 to 900 are suflieient. Tliere are 250 sliingles in a bundle. EXERCISE 40 1. At #8.60 per square, how mucli will it cost to sliingle a roof 50 ft. by 22^ ft. on each side ? 2. How much wdll it cost to lay a hard-wood floor in a room 30 ft. by 28 ft., if the labor, nails, etc. cost i$22.50, lumber being i28 per M, and allowing 57 sq. ft. for waste ? 3. Allowing 900 shingles to the square, how many bundles will be required to shingle a roof 70 ft. by 28 ft. on each side? How much will the shingles cost at $^3.75 per M ? 4. At i? 12.50 per square, how much will the slate for a roof 40 ft. by 21: ft. on each side cost ? 212. Stonework and Masonry. The cubic yard or the perch is the unit of stonework. A perch of stone is a rectangular solid 16J ft. by 1^ ft. by 1 ft., and therefore contains 24| cu. ft. A common brick is 8 in. by 4 in. by 2 in. Bricks are usually estimated by the thousand, sometimes by the cubic f(5ot, 22 bricks laid in mortar being taken as a cubic foot. There is no uniformity of practice in making allowances for windows and other openings. There should be a defi- nite written contract with the builder covering this point. The corners, however, are counted twice on account of the extra work involved in building them. It is also gener- ally considered that the work around openings is more difficult, so that allowance is frequently made here. 116 MENS URA TION EXERCISE 14 1. If 60 ct. per cubic yard was paid for excavating a cellar 30 ft. by 20 ft. by 7 ft., and $4.75 a perch was paid for building the four stone walls, 18 in. thick and extend- ing 2 ft. above the level of the ground, what was the total cost? 2. How many bricks will be used in building the walls of a flat-roofed building 90 ft. by 60 ft. and 20 ft. high, if the walls are 18 in. thick and 500 cu. ft. are allowed for openings ? 3. How much will it cost to build the walls described in Ex. 2, if the bricks are $8.50 per ]M, and the mortar and brick-laying cost $3.50 per M ? 4. How many perch of stone will be needed for the walls of a cellar 30 ft. by 22| ft. and 9 ft. deep from the top of the wall, the wall being 18 in. thick ? How many perch will be needed for a cross Avail of the same thick- ness, allowing for half of a door 7 ft. by 4 ft. ? How much will the stone cost at -^4.50 a perch ? 213. Contents of Cisterns, Tanks, etc. The gallon or the barrel is the unit of measure for cisterns, tanks, etc. The liquid gallon contains 231 cu. in. and the barrel 311 gal. EXERCISE 42 1. How many gallons of water will a tank 10 ft. long, 3 ft. wide and 3 ft. deep contain ? How many barrels ? 2. How many gallons of Avater will a cistern 10 ft. deep and 10 ft. in diameter contain ? How many barrels ? 3. How many barrels will a cylindrical tank 5 ft. higli and 3 ft. in diameter contain ? MENSUBATION 117 4. How many barrels of oil will a tank 40 ft. long and 6 ft. in diameter contain ? 5. Show tliat to find the approximate num])er of gaHons in a cistern it is necessary only to multiply the number of cubic feet by 7^ and subtract from tlie product ^-J-^- of the product. Apply this method to eacli of tlie al)Ove exercises. 6. How many gallons will a cask contain, the bnng diameter being 24 in., the head diameter 20 in. and the lengtli 34 in. ? Suggestion . The average or mean diameter is '—^^ -* = 22 in. 214. Measuring Grain in the Bin, Corn in the Crib, etc. There are 2150.42 cu. in. in every bushel, stricken measure, and 2747.71 cu. in. in every bushel, heaped measure. EXERCISE 43 1. How many bushels of wheat does a bin 8 ft. by 7 ft. by 6 ft. contain ? 2. Show that multiplying by 0.8 will give the approxi- mate number of stricken bushels in any number of cubic feet, and dividing by 0.8 will give the approximate number of cubic feet in any number of stricken bushels. 3. Show that multiplying by 0.63 Avill give the approxi- mate number of heaped bushels in any number of cubic feet, and dividing by 0.63 will give the approximate num- ber of cubic feet in any number of heaped bushels. 4. How deep must a bin 10 ft. by 8 ft. be to hold 500 bushels of wheat ? 118 MEXS URA TION 5. A farmer builds a corncrib 20 ft. long, 10 ft. high, 8 ft. wide at the bottom and 12 ft. wide at the top. How many heaped bushels of corn in the ear will the crib hold when level full ? If the ridge of the roof is 3 ft. above the top level, how many bushels will the crib hold when filled to the ridge ? Suggestion. The average width of the crib is -^ — '^ '- — 10 ft. 6. How many stricken bushels of shelled corn are there in the above crib if 3 half bushels of ears make one bushel of shelled corn ? 215. Measuring Hay in the Mow or Stack. The only correct way to measure hay is to weigh it. Hovv^ever, it is sometimes convenient to be able to estimate the number of tons in a mow or stack. The results of such estima- tions can be only approximately correct, as different kinds of hay vary in weight. In well-settled mows or stacks, as nearly as can be estimated, 15 cu. yd. make one ton. When hay is baled, 10 cu. yd. make a ton. EXERCISE 44 1. Approximately how many tons of hay are there in a mow 40 ft. by 22 ft. and 15 ft. deep ? 2. Approximately how many tons of hay are there in a circular stack 21 ft. high and averaging 80 ft. in circum- ference ? 3. Approximately how many tons of hay are there in a rick averaging 35 ft. long, 15 ft. wide and 20 ft. high ? 216. Land Measure. The unit of land measure is the acre. In the Eastern states, the land was divided, as convenient, when settled, and the description of tracts of land refer to MEN SUE A TION 119 such natural objects as iiear-l)y bowlders, ponds, estal)lished roads, etc. l>ut all states whose lands have been surveyed since 1802 are divided by a system of meridians and paral- lels into townships G miles square. Each township con- tains 36 scpiare miles or sections. Each section contains 2 half sections and 4 quarter sections. Public lands arc located with i-cfcrence to a nortli and south line called the principal meridian and an east and west line called the base line. The nortli and south rows of townships are called ranges ^ and these rows are numbered from the principal meridian. ~ The townships are numbered — from the base line. A township is therefore designated by its ^ number and the number of its — Thus, A is township 4 N"., Range 3, W. What is 5? J/? S'i ' The 36 sections of a township are numbered as in the following diagram. The corners of all sections are per- manently marked by stones, or otherwise. A SECTION A TOWNSHIP 6 5 4 3 2 1 7 8 9 10 11 12 18 17 10 15 14 13 19 20 21 ■^2 23 21 30 29 28 27 20 25 31 32 33 31 35 36 N. i Section (320 A.) S.W. 1 (160 A.) W.i of S.E.i 'SO A S.E.i The divisions of sections into half sections, quarter sec- tions, etc., are shown in the diagram. Thus, the X. E. \ of N. E. \, section 6, means the northeast quarter of the northeast quarter of section 6. 120 MENSURATION EXERCISE 45 1. How many acres are there in a section ? In the S. W. \ of S. W. 1, section 16 ? In S. | of N. E. \, section 36 ? Locate these sections. 2. What will be the cost of a quarter section of land at $ bb an acre ? 3. How many rods of fence are necessary to inclose a quarter section ? 4. How many acres are there in a township ? 5. The sections of a township are separated and the township is separated from adjacent townships by a road 45 ft. wide, the section lines being in the middle of the road. How many acres are there in the roads of the township ? EXERCISE 46 1. The side of a square is 100 ft. Find the length of a diagonal. 2. One side of a right-angled triangle is 16 yd. and the other side is J of the hypotenuse ; what is the length of the hypotenuse ? 3. Find the volume of a pyramid Avhose base and faces are all equilateral triangles with sides 10 in. long. 4. The largest pyramid in the world has a square base with sides 764 ft. Its four faces are equilateral triangles. Find the number of acres covered by its base, the number of square yards in its four faces, and the height of the pyramid. 5. A cistern 22 ft. long, 10 ft. wide and 8 ft. deep is to be filled with water from a well 8 ft. in diameter and 40 ft. deep. If no water flows into the well while filling the cistern, find how far the Avater in the well is lowered. MEN SURA TION 121 6. Two persons start from the same place at the same time. One walks clue east at the rate of 3 mi. an hour, and the other due soutli at the rate of 3i mi. an hour. In how many hours Avill they be 30 mi. apart ? 7. What is the circumference of the earth if its diameter is 7916 mi. ? 8. Air being 0.00129206 as heavy as water, find in kilograms the weight of tlie air in a room 23"" long, 16"' wide and 10'" high. 9. A rectangular sheet of tin of uniform tliickness is SS'^'" wide and 2.7'" long, and weighs 356^. Find its thick- ness if tin is 7.3 times as heavy as water. 10. A plate of iron weighs 277. 54^^, and is 137*^'" long, 643""" wide, 43.1'"'" thick. How much heavier than watei' is iron ? 11. A tank is 2'" long, 5^^'" wide and 8*'" deep. How many liters of water will it contain, and how much will the water weigh ? 12. Sulphuric acid is 1.84 times as heavy as water. How many kilograms will a tank hold that is 2'" long, 75^"' wide and 50*^'" deep ? 13. A block of marble is 2 ft. long, 10 in. wide and 8 in. thick. What is the edge of a cubical block of equal volume ? 14. If 1 T. of hard coal occupies a space of 36 cu. ft., how many tons will a bin 10 ft. long, 7| ft. wide and 9 ft. deep liold ? 15. How much space will a car load of hard coal con- sistiug of 38 T. 14 cwt. 75 lb. occupy, if one ton occupies 36 cu. ft. ? 16. How long must a bin 20 ft. wide and 20 ft. deep be to hold the above car load of coal ? 122 MENS URA TION 17. Find correct to 0.001 the diagonal of a square whose side is 10 in., and the diagonal of a cube whose edge is 10 in. 18. What will be the expense of painting the walls and ceiling of a room whose height is 10 ft. 4 in., length 16 ft. 6 in. and width 12 ft. 3 in., at 15 ct. per square yard ? 19. At 11 ct. per square foot, how much will it cost to make a cement walk 5 ft. wide around a school yard in the shape of a rectangle, 18 rd. by 26 rd. ? 20. Two corridors of a public building intersect at right angles near the center of the building. If the corridors are 160 ft. and 140 ft. long respectively, and 20 ft. wide, how much will it cost to cover them with a hard-wood floor at $ 24 per thousand feet ? . 21. At $18 per M, how much will it cost to cover the floor of a barn 30 ft. long and 20 ft. wide with 2-inch planks ? 22. How much will it cost to fence the school yard mentioned in Ex. 19, with 1-inch boards, 6 in. wide, at $11. bO per M; the fence to be 4 boards high and built 2 ft. inside the walk ? 23. How many board feet are there in 150 rafters, 14 ft. long, 4 in. wide and 2 in. thick ? 24. How many bunches of shingles will be required to shingle a barn with a roof 60 ft. long and rafters 18 ft. long, the shingles being laid 4 in. to the weatlier with a double row at the bottom ? 25. What is the value of a log that will cut 36 1-inch boards, each 16 ft. long and 12 in. wide at 1| ct. per square foot ? MENS I 'JiA TION 123 26. How many board feet are there in a stick of timber 18J ft. long, lb in. wide and 12 in. thick ? 27. How many bricks will be used in l)uilding the walls of a building 120 ft. long, 60 ft. wide and 45 ft. higli, outside measurement, if tlie walls are 18 in. thick and no allowance is made for doors and windows ? 28. How many centimeters of lead are there in a piece of lead pipe 1'" long, the outer diameter being 5*"', and the thickness of the lead being 10""" ? 29. A race track 30 ft. wide with semicircular ends is constructed in a field 1050 ft. by 400 ft. Find the inside and outside lenofths of the track. Also find the area of the track and the area of the field inside the track. 30. Find the volume and convex surface of a cone, the diameter of the base being 16 in. and the altitude 18 in. 31. Find the volume and surface of a sphere whose diameter is 6 in. 32. Find the least possible loss of material in cutting a cube out of a sphere of wood 9 in. in diameter. 33. Find the least possible loss of material in cutting a spliere out of a cubical block of wood with edges 9 in. long. 34. Find the cost of making a road 200 yd. in length and 24 ft. wide ; the soil being first excavated to the depth of 14 in., at a cost of 20 ct. per cubic yard; crushed stone being then put in 8 in. deep at a cost of 40 ct. per cubic yard, and gravel placed on top 6 in. thick at a cost of 45 ct. per cubic yard. 35. A map of Kansas is made on a scale of 1 in. to 100 mi. The map measures 4 in. by 2 in. Find the area of the state. GRAPHICAL REPRESENTATIONS 217. Graphical methods of representing relations be- tween different measurements are so extensively used in many lines of work ^hat it seems best to give a brief treat- ment of the subject here. Such graphical representations as are given in the following exercises show relations pic- torially in a much clearer manner than can be shown by a mere statement of figures. Ex. 1. Explain graphically the relation between an inch and a centimeter. The two lines drawn accurately to scale represent graphically LlLL the relation between the inch and the i cm. centimeter. Ex. 2. Draw a line 1.5 in. long and find the number of centimeters in it. Ex. 3. Explain graphically the relation 1 lb- between the pound and the kilogram, given l«^s=2.2 lb. Ex. 4. Explain graphically the relation between a pint and a liter, given 1^ = 1.76 pt. Ex. 5. Erom a diagram find (a) the number of centi- meters in 4 in., (h) the number of liters in a gallon, (^c) the number of pounds in 5^^. Ex. 6. The values of manufactures produced in the United States, Germany, France and Great Britain in 18G0 were i^ 1907000000, $1995000000, $2092000000, 124 IT^'g = 2.2 lb GRAPHICAL liEPEESENTA TIONS 125 $2808000000 I'L'spectively, and in 181)4 they were 19498000000, 13357000000, ^2900000000, 14263000000 respectively. These facts may be represented grapliically as follows: /U.S. Q\ Germ. France Gt.Brit.l MILLIONS OF DOLLARS 1907 1995 2092 2808 oil Germ. "] France Gt.Brit. 19498 3357 2900 4263 These measurements, drawn accurately to a scale, show at a glance the comparative grow^th in manufactures produced in the different countries mentioned from 1860 to 1894. Fx. 7. The areas of England and Michigan are 50839 and 58915 square miles respectively. The populations are approximately 31000000 and 2421000. Represent graph- ically the comparative sizes and the comparative density in population of the two. The square roots of the numbers representing the areas correct to units' place are 225 and 243 respectively. The ratio between these England Michigan two numbers reduces to 5 to 5.4. If some convenient unit of measure be taken, and squares be constructed witli sides equal to 5 126 GRAPHICAL REPRESENTATIONS to 5.4 of these units, these squares will represent graphically the comparative areas. The comparative density in population will be represented by the number of dots that appear in each square, it being assumed that a dot represents 100000 in population. There will then be 310 dots in the square representing England and 24 in the square representing Michigan. ^x. 8. On a certain day between 6 A.M. and 7 P.M. the thermometer registers as follows : 6 A.M., 20° ; 7 A.M., 22.5°; 8 a.m., 27°; 9 a.m., 35°; 10 a.m., 42.5° ; 11 a.m., 48°; 12 M., 52°; 1 p.m., 5b° ; 2 p.m., 60°; 3 p.m., 62°; 4 p.m., 60°; 5 p.m., 50°; 6 p.m., 42°; 7 p.m., 35°. Illus- trate graphically this variation in temperature. Draw two straight lines perpendicular to each other. Measure off on the horizontal line OX equal spaces, each representing 1 hr., and on the perpendicular line OF equal spaces, each one representing 10°. The temperature at a.m. is shown at 0; at 7 a.m. at A, a distance TEMPERATURE 70°Ly 60" 50" 40' 30' A. M. M. P. M. I \ \ I \ I \ 06 7 8 910 1112 12 3^5 67 X of 1 unit along OX and | of a unit above OX parallel to OY; at 8 a.m. at B, a distance of 2 units along OX and .7 of a unit above OX parallel to OY. In the same way points may be located showing the tempera- ture at each hour. A continuous curve drawn through these points is the temperature curve for the day from a.m. to 7 p.m. This curve shows at a glance the variation in temperature between the hours given. GRAPHICAL REPRESENTATIONS 127 6 hr. Y Ux. 9. Two trains leave a certain place traveling in tlie same direction, one at the rate of 20 mi. an hour, and the other at the rate of 40 mi. an hour. If the second train leaves 3 hr. after the first, when and where will it pass the first? Let each space along OX represent 20 mi., and each space along OY represent 1 hr. At the end of the first hour the first train is at A ; at the end of the second hour at B,; and at the ^,,„ v P^ end of the sixth hour at P. At the end of the fourth hour the sec- ond train, which starts from 0', 3 spaces above O, since it starts 3 hr. later, is at .4'; at the end of the fifth hour at B' ; and at the end of the sixth hour at P. The point P, where the line OP and O'P cross, is the place where the second train overtakes the first. If from P perpendiculars PX and PF are dropped upon OX and OY, then the distances OX and OY will represent the space traveled and the time that has elapsed since the starting of the first train till the second one overtakes it. OX contains 6 distance spaces, and represents 120 mi., while OF contains 6 time spaces, and represents 6 hr. EXERCISE 47 For convenience in constructing the graphical repre- sentations required in the following exercises, the student should provide himself with paper ruled in small squares. 1. Illustrate graphically the comparative areas and the comparative density in population in the following cases : 128 GBAPIIICAL BEPEESENTA TIONS Area POFL LATIoN Q,) Alaska Greenland (J>) Mexico Texas (c) United States (including foreign possessions) British Empire 590884 8;:I7837 767258 265780 3806279 11391036 63592 12000 13606000 3048710 84907156 383165494 2. On Jan. 1, 1904, the thermometer registered the temperature at 1 a.m., and at each succeeding hour till midnight, at Ypsilanti, Michigan and Havana, Cuba, respectively as follows : 1 2 3 4 5 6 7 8 9 10 11 N. 23° 24° 24° 24° 22° 22° 21° 19° 18° 20° 21° 22° 64° 64° 63° 63° 63° 63° 62° 64° 67° 68° 70° 72° 1 2 3 4 5 6 7 8 9 10 11 Mt. 22° 22° 22° 21° 17° 16° 16° 15° 13° 12° 12° 12^^ 73° 73° 73° 73° 72° 70° 69° 68° 67° 66° 65° Qb" Illustrate each graphically. 3. The mean temperature for January (average for the 31 da. of the month) for the same hours and places as in Ex. 2 was as follows : 1234567 89 10 UN. 14.2° 14.3° 14.2° 14.2° 14.1° 14.3° 14.2° 14.5° 15.3° 17.1° 18.1° 19.7° 67.2° 67.0° 66.7° 66.4° 66.0° 65.6° 65.4° 66.6° 68.9° 71.0° 73.1° 74.0° 12 3 4 5 6 7 8 9 10 11 Mt. 20.4° 20.7° 20.6° 19.9° 18.8° 18.1° 17.5° 16.5° 16.0° 15.4° 14.5° 16.6° 74.7° 74.9° 75.0° 74.7° 74.2° 72.9° 71.6° 70.6° 69.8° 69.0° 68.4° 67.9'^ Illustrate graphically. GRAPHICAL n EPRESENTATIONS 129 4. Illustrate graphically, as in Ex. 9, the point where and time at which the two trains given in the annexed time-table pass each other. GoixG East Going West A.M. Miles Miles A.M. 10.00 284 Detroit 12.35 8.54 247 Ann Arbor 37 1.25 8.00 7.50 214 ( Lv. -, , Ar. ) i . Jackson _ r I Ar. Lv. ) 76 2.20 2.25 6.10 164 Battle Creek 121 3.30 4.55 141 Kalamazoo 144 4.10 3.25 3.15 93 (Lv. ^,., Ar. ) ] . Niles ^ >■ I Ar. Lv. ) 192 5.28 5.33 1.55 56 Michigan City 228 6.32 12.40 13 Kensington 271 7.30 12.00 Chicago 284 8.00 night A.M. 5. A cyclist starts at 7 a.m. from a town and rides 2 hr. at the rate of 10 mi. an hour. He rests 1 hr. and then returns at the rate of 9 mi. an hour. A second cyclist leaves the same place at 8 A.M. and rides at the rate of 6 mi. an hour. When and where will they meet ? 6. Two cyclists start from the same place at the same time. The first rides for 2 hr. at the rate of 9 mi. an hour, rests 15 min., and then continues at 6 mi. an hour. The second one rides without stopping at the rate of 7 mi. an hour. Where will the sefcond cyclist overtake the first ? 7. The average yield of wheat per acre in the United States for the years from 1893 to 1903 in bushels was as follows: 11.4, 13.2, 13.7, 12.4, 13.4, 15.3, 12.3, 12.3,15.0, 14.5, 12.9. The highest Chicago cash price per bushel for the same years given in cents was: 64.5, 63f, 64|, 93i, 109, 70, 691 755^ 791 773^ 87. Illustrate grapliically, putting the tw^o curves in one figure. lyman's adv. ar. — 9 130 GRAPHICAL REPRESENTATIONS 8. The average yield of corn per acre in the United States for the years from 1893 to 1903 in bushels was as follows: 22.5, 19.4, 26.2, 28.2, 23.8, 21.8, 25.3, 25.3, 16.7, 26.8, 26.5. The highest Chicago cash price per bushel for the same years given in cents was : 36 J, 17^, 26|, 23|, 271 38, 311 401, 671 57^, 433. Illustrate graphically, putting the two curves in one figure. 9. The average summer daily temperature in Paris at the foot and top of the Eiffel tower in 1900 was as follows : 2 4 6 8 10 N. 2 4 6 8 10 Mt. 57.2° 55.4° 58.1° 63.5° 67.8° 69.8° 70.1° 69.8° 68° 62.1° 60.7° 58.9° 57.4° 55.7° 57.2° 58.1° 60.8° 63.5° 63.9° 64° 64.4° 61.2° 60.7° 59.1'' Illustrate graphically, putting the two curves in one figure. RATIO AND PROPORTION 218. The ratio of one number to another of the same kind is tlieir quotient. The former number is called the antecedent, and the hitter the consequent. The terms of the ratio therefore bear the same rehilion to each other as the terms of a fraction. Thus, the ratio of a to h may be written a : h (read the ratio of a to i), - or a^h. The forms a : 6, and -, are generally used. The ratio of 3 ft. to 5 ft. is 3 : 5. This may also be expressed by | or 0.6. 219. The ratio is always an abstract number, since it is the relation of one number to another of the same kind. There can be no ratio between 5 hr. and % 10, nor between 7 lb. and 6 ft. But there can be a ratio between 3 ft. and 6 in., since the quantities are of the same kind. Both terms must, however, be reduced to the same unit. Thus, 3 ft. = 36 in., and 36 in. : 6 in. = ^{- = 6. The ratio - is called the inverse or reciprocal of the . a ^ ratio -• EXERCISE 48 1. How is the value of a ratio affected by multiplying or dividing both terms by the same number ? 2. How is the value affected by multiplying or dividing the antecedent ? by multiplying or dividing the consequent? 131 132 BATIO AND PROPORTION Express the ratio of : 3. 100 to 25. 7. 115 to 50 cents. 4. 16| to 100. 8. 71 to 371. 5. 331 to 100. 9. I to 16|. 6. 2 "™ 4 «^" to 50 ^'". 10. 121 to 100. 11. 14 lir. 30 mill. 3 sec. to a day. 12. 2 mo. 10 da. to a year. 13. What number has to 10 the ratio 2 ? to 5 the ratio 0.3? 14. li X : S = 5, find x. 15. li X : 1 = 2, find x. 16. Which ratio is the greater, -f^ or -f^ ^ if ^^^ if ^ 17. The ratio of the circumference of a circle to its diameter being 3.1416, find the diameter of a circle whose circumference is 125 ft. correct to inches. 18. A map is drawn on the scale of 1 in. to 75 mi. In what ratio are the lengths diminished ? In what ratio is the area diminished ? 19. Two rooms are 14 ft. long, 12 ft. wide, and 12 ft. long, 10 ft. wide respectively. What is the ratio of the cost of carpeting them ? 20. What is the ratio of a square field 20 rd. on a side to one 25 rd. on a side ? 21. What is the ratio of the circumferences of two cir- cles whose diameters are 2 in. and 4 in. ? of two circum- ferences whose diameters are 5 in. and 7 in. ? of two circumferences whose diameters are d and c?'? Hence in general the ratio of two circumferences is equal to wliat ? RATIO AND PROPORTION 133 22. What is the ratio of the areas of two circles whose radii are 3 in. and. 5 in. ? of the areas of two circles whose radii are 4 in. and 6 in.? of the areas of two cir- cles whose radii are r and r' ? Hence in general the ratio of the areas of two circles is equal to what ? 23. What is the ratio of the volumes of two spheres whose radii are 2 in. and 3 in.? of the volumes of two spheres whose radii are 5 in. and 6 in.? of the volumes of two spheres whose radii are r and r' ? Hence in general the ratio of the volumes of two spheres is equal to what ? 220. Specific Gravity. The specific gravity of a sub- stance is tlie ratio of its weight to the weight of an equal volume of some other substance taken as a standard. 221. Distilled water at its maximum density, 4° C, is the standard of specific gravity for solids and liquids. 222. Since I""'' of water weighs 1 gram, the same num- ber that expresses the weight of any substance in grams will also express its specific gravity. Thus, 1^"'^ of water weighs 1^; hence, 1 is the specific gravity of water. 1"^^ of lead weighs 11.35^; hence, this being 11.35 heavier than an equal volume of water, the specific gravity of lead is 11.35. Specific Gravities of Substances Copper . . 8.92 Tin 7.29 Sea Water . . 1.026 Iron (cast) 7.21 Anthracite Coal 1.30 Sulphuric Acid 1.811 Gold. . .19.26 Cork . . . .0.24 Milk . . . .1.032 Lead . . 11.30 Pine .... 0.65 Alcohol . . . 0.84 Platinum . 21.50 Oak 0.845 Ice 0.92 Mercury . 13.596 Beech .... 0.852 Rock Salt . . 2.257 1 cu. ft. of water weighs about 1000 oz., or 62.5 lb. 134 RATIO AND PROPORTION Ex. 1. A mass of cast iron weighs 3500 lb. How many cubic feet does it contain ? Since 1 cu. ft. of water weighs 62.5 lb., 1 cu. ft. of iron weiglis 7.21 X 62..5 lb. 3500 '.21 X 62.5 7.77, the number of cubic feet. Ex. 2. In France wood is sold by weight. How much does 1 stere of beech wood weigh, allowing ^ for space not filled ? Since 1 ""^ of water weighs 1000 ^s, 1 stere of beech wood weighs 0.852 X 1000 Kg _ i of 0.852 x 1000 ^g = 568 Kg. EXERCISE 49 1. What is the ratio of the weight of 1 stere to 1 cord of oak wood, allowing J for waste space ? 2. Allowing ^ for waste space, how many tons of coal will a bin 9 ft. long, 8 ft. wide and 8 ft. deep hold ? 3. What is the weight of a cubic decimeter of each of the substances in the above table ? of a cubic foot ? 4. A flask will hold 6 oz. of w^ater. How much alcohol will it hold ? how much mercury ? 5. To what depth will a cubic foot of cork sink in sea water ? in alcohol ? 6. How much does a piece of copper 20'"'" long, IS'^'" wide and 5"^™ thick weigh ? 7. If 1 lb. of rock salt is dissolved in 1 cu. ft. of water without increasing its volume, what will be the specific gravity of the solution ? 8. How much does a l)oat weigh that displaces 7000 cu. ft. of water? 9. If a boat is capable of displacing 3000 cu. ft., what weight will be required to sink it ? RATIO AND PROPORTION 135 223. Proportion. A proportion is an equality of ratios and is expressed in the following way : a e a : h: = a : d, a : b : : c : d. 224. The method of solving problems by proportion is often called the Rule of Three, since problems which give three quantities so related that two of them sustain the same ratio to each other as the third to the quantity required, can readily be solved by proportion. 225. Thus, if any three of the four terms of a proportion are known, the other one can be found. If 1=^, then, :r=3 X f = 2f 01 C 1 ^ c Check by putting 2\ for x, then =1 = -, or — = -. J i ^7 3 7 21 7 226. The first and last terms of a proportion are called the extremes, and the second and third terms the means. 227. In any proportion the product of the mean% is equal to the product of the extremes. a c If 7 = -, then by clearing of fractions ad = he. This d •proves the proposition, since a and d are the extremes, and h and c the means. 228. If 1 lb. of sugar costs 4 ct., 2 lb. Avill cost 8 ct. and 4 ct. : 8 ct. = 1 lb : 2 lb. At the same rate 3 lb. would cost 12 ct., etc. The ratio of costs in each case is equal to the ratio of the weights. The cost of sugar is said to be directly proportional to its weight. 136 BATIO AND PROPORTION Ex. The Washington monument is 555 ft. high. What is the lieight of a post that casts a shadow 1 ft. 9 in. when the monument casts a shadow 192 ft. 6 in. ? Solution by proportion. Let X = the height of the post. Then j^^V7d_ 555 192.5 111 0.05 .-. X = W^ X l.n = 5.04 ft. 1.1 Solution by unitary analysis. A shadow 192 ft. 6 in. long is cast by a monument 555 ft. high. .-. a shadow 1 ft. long will be cast by a post — — — ft. hioh. ^ ^ ^ 192.5 , ^- --- .'. a shadow 1 ft. 9 in. long will be cast by a post -^-^ ^^r— ft- high, or 5.04 ft. high. ^^"-^ 229. If 1 man can do a certain piece of work in 6 days, 2 men working at the same rate will do the work in 3 days, 3 men will do it in 2 days, etc. 2 men do the work in ^ the time that 1 man will do it ; 3 men in 1 the time, etc. Hence, as the number of men increases, the time diminishes in the same ratio. If 2 men do the work in 3 days, 3 men will do it in f of 3 days, or 2 days. There- fore the ratio of the number of men, |, is equal to the corresponding ratio of time inverted. Hence, the number of men is said to be inversely proportional to the time. Ex. The crew and passengers of a steamship consisted of 1500 persons. The ship had sufficient provisions to last 12 weeks when the survivors of a wreck were taken on board. The provisions were then consumed in 10 weeks ; how many were taken on board ? RATIO AND PROPORTION 137 Solution hy jiroportion. Let X equal the total number on board. Then _^ = ir, 1500 10 or X — = 1800, and 1800 - 1500 = 300, the number taken on board. Solution hy unitary analysis. There are provisions for 12 weeks for 1500 persons. .-. there are provisions for 1 week for 12 x 1500 persons. 1500 X 1*^ .-. there are provisions for 10 weeks for — -^ persons or 1800 persons. .-. 1800 — 1500 = 300, the number taken on board. EXERCISE 50 State which of the following are directly proportional and which are inversely proportional : 1. The price of bread, the price of flour. 2. The number of workmen, the amount of work done in a given time. 3. The number of workmen, the time required to do a given amount of work. 4. The height of the thermometer, the temperature. 5. The velocity of a train, the time required to go a given distance. 6. The number of horses bought for a given sum, the price per horse. 7. The price of freight, the distance carried. 8. The area of a circle, the length of its diameter. 138 BATIO AND PROPORTION 9. In how many ways can the terms of the proportion 2 : 3 = 8 : 12 be arranged without destroying the proportion? 10. Tlie assessed value of a certain town is $7500000, and bonds for $6000 are issued. What part of tliis does a person worth 1 10000 pay? 11. A shadow cast by a post 6 ft. higli is 9 ft. 3 in. How long is the shadow cast by a church steeple 150 ft. high ? 12. A merchant fails for $12,300 and his property is worth $5720. How much will he pay a creditor Avhom he owes $2500? 13. A clock is set at noon on Monday; at 6 p.m. on Wednesday it is 2 minutes and ' 20 seconds too slow. Supposing the loss of time to be constant, what is the cor- rect time when the clock strikes 12 on Sunday noon? 14. There are two kinds of thermometers used in this country, Fahrenheit, used to register temperature, and Centigrade, used largel}^ in scientific work. The freezing point of water is 32° and 0° resj^ectively, while the boiling point is 212° and 100° respectively. 68° Fahrenheit cor- responds to what temperature Centigrade and 5-1° Centi- grade to what temperature Fahrenheit ? 15. There is another kind of thermometer known as Reaumur, the freezing and boiling points being 0° and 80° respectively. Express in Reaumur scale 70° on each of the other two. 16. The boiling point of alcohol is 78° Centigrade ; what is the boiling point of alcohol on each of the other two ? 17. A grain of gold can be beaten into a leaf of 56 sq. in. How many of these leaves will make an inch in height if 1 cu. ft. of gold weighs 1215 lb.? RATIO AND PROPORTION 139 18. Divide 60 into two parts proportional to 2 and 8. 19. Divide 90 into parts proportional to 2, 3 and 4. 20. Two men start in business with a capital of fTSOO. One of them furnishes ^4000 and the other 18500. At tlie end of a year the profits are 'i^3250. How much is each man's share? 21. A man starts in business with a capital of 85000 and in 3 months admits a partner with a capital of $>4500. At the end of the year the profits amount to 88750. How much is each man's share ? 22. A piece of work was to have been done by 10 men in 20 days, but at the end of two days 8 men left. How long did it take the remaining 7 men to complete the work ? 23. If the interest on 8325 is 872.50 in a given time, how much is the interest on 8850 for the same time? 24. Two cog wheels work together ; one has 36 cogs and the other 14. How many revolutions does the smaller one make while the larger one makes 28 revolutions? METHOD OF ATTACK 230. Ill solving any aritlimetical problem tlie student will find the following suggestions useful : (1) The first essential is a thorough understanding of the proper relations between the conditions given. This requires some form of analysis leading to a complete state- ment of the conditions. (2) The solution should involve no unnecessary work. Cancellation and other convenient short methods should be used if possible. (3) All arithmetical work should be carefully checked. The student must realize tliat accuracy is of the highest importance and that to secure accuracy his work must always be checked. Any arithmetical work that has an error in it is valueless. The check also gives the student a means of knowing for himself whether he has a correct result or not. He has no need of answers to his problems. Ex. 1. If the time of the beat of a pendulum varies as the square root of its length, and the length of a pendu- lum that beats seconds is 39.2 in., find the length of a pendulum that beats 50 times a minute. Solution. The given pendulum beats GO times per minute, the required pendulum beats 50 times per minute. Since the longer the pendulum the more slowly it beats, the re- quired pendulum is longer than the given one. Therefore, the square root of the lengths of the pendulums are in the ratio f§, or ^. 140 METHOD OF ATTACK 141 Let I = the length of the required pendidum. -I hen, — ; = -, V;39.2 5 I 62 or = — , 39.2 52' ; 6 X 6 X 39.2 . 6 X 6 X 39.2 x 4 . -« 1,0 • or I = in. = in. = 00.448 in. 5x5 100 Check either by changing the order of the factors and performing the multiplication again, or by casting out the nines. Ux. 2. The greatest possible sphere is cut from a cube, one of whose edges is 3 ft. Find the portion of the cube cut away. Solution. The volume of the cube is 3^ cu. ft. The volume of the sphere is f tt x (|)^ cu. ft. Therefore the portion cut away is 3^ cu. f t. — | tt x (f )^ cu. ft. Without performing the operations indicated the student can by cancellation and combination of terms v^rite the result thus, 3-2^3 _?^cu.ft. = 32(?^I^^^^cu.ft. = 9x 1.4292 cu.ft. = 12.8628 cu. ft. Check as before. Ex. 3. Find the area of a square field whose diagonal is 50 rods. Solution. Let x = one side of the square field. Then x^ + x^ = bO% or 2 x2 = 502. 502 .-. x^, or the area of the field in square rods, = '-— sq. rd. = 7|f acres. Check each step in the work. Ux. 4. Find the area of the circle which is equal in area to two circles whose radii are 5 in. and 7 in. 142 METHOD OF ATTACK Solution. Let r = the radius of the required circle. Then its area in square inches = irr'^ = tt x 5- + tt x 7^ = 7r(52 + 7^) = TT X 74, or 232.48 sq. in. Check each step in the work. Here, instead of multiplying tt by 25 and then by 49 and adding the results, time is saved by adding 25 and 49 and multiplying tt by the sum, 74. 231. The foot pound is used as a unit of work. This unit is defined as tlie amount of work required to over- come the resistance of one pound through a space of one foot. The rate of work is generally defined by using the term horse poiver. An engine of one horse power can do 33000 foot pounds of work in one minute, i.e. can over- come a resistance of 33000 pounds through a space of one foot in one minute. Ex.''5. AVhat horse power is an engine exerting that draws a train with a uniform speed of 40 miles an hour against a resistance of 1000 pounds ? Soluiion. The amount of work done in one hour is 1000 x 40 X 5280 foot pounds. rrx. ^ f 1 J • • ^ ' 1000 X 40 X 5280 . ^ The amount of work done in one minute is foot 1 60 pounds. 2 ig rr. i ,, -PI- 1 • 1000 X ^0 X ^2^0 , Therefore, the rate of doing work is — - horse power ^p X nm 3 = x_j-2< horse power = 106| horse power. Check each step. 232. The student will notice that in each of the above exercises, first, the relations between the given conditions are carefully established ; and second, a complete statement of these conditions is frriften out and the work shortened as mucli as possible by cancellation or otherwise^ before the processes METHOD OF ATTACK 143 of multiplication and dividon are used. Frequently stu- dents in solving such problems will perform tlie operations indicated at each step, thus doing a hxrge amount of un- necessary work. By carefully studying these model solu- tions the student will see where the unnecessary work can be avoided. As indicated in Art. 41, it is a good plan, whenever pos- sible, to estimate the result mentally and to compare this rough estimate with the result found by solving the prob- lem. This will prevent large errors and such errors as arise from misplacing the decimal point. EXERCISE 51 1. Find the area bounded by 6 eqaal coins whose centers are at the vertices of a regular hexagon, the diam- eter of each coin being 2.38'^'". 2. A crescent is bounded by a semi-circumference of a circle whose radius is 15 inches, and by the arc of another circumference whose center is on the first arc produced. Find the area and perimeter of the crescent. 3. A horse is tied with a 50 ft. rope to one corner of a barn 30 ft. by 40 ft. Find the area he can graze over. • 4. A well 30 ft. deep and 4 ft. in diameter is to be dug. If a cubic foot of earth weighs 12 lb., how much work is to be done? 5. A horse drawing a wagon along a level road at tlie rate of 2 mi. an hour does 29216 foot pounds of work in 3 min. What pull in pounds does he exert in drawing the wagon? 144 METHOD OF ATTACK 6. A uniform heavy bar, 12 ft. long and weighing 80 lb., rests on 2 props in the same horizontal plane, so that 2 ft. projects over one of the props ; find the distance be- tween the props so that the pressure on one may be double that on the other ; also find the pressures. 7. It is proved in geometry that similar volumes are to each other as the cubes of their like dimensions. If a cubical bin whose edge is 4 ft. holds 52 bu. of wheat, how many bushels will a bin 6 ft. on an edge hold? 8. The temperature remaining the same, the space oc- cupied by a gas varies inversely as the pressure. At a con- stant temperature a mass of air occupies 25 cu. ft. under a pressure of 10 lb. to the square inch ; what space will it occupy under a pressure of 26 lb. to the square inch? 9. A cubic foot of water weighs 1000 oz., and the pressure of the air is 336 oz. per square inch ; find the pressure on a square foot at a depth of 10 ft. below the surface of a pond. 10. If the specific gravity of mercury is 13.598 and the weight of a cubic inch of water is 252.6 grains, find the pressure of air per square inch in pounds when the mercury in the barometer stands at 30.5 in. 11. An iceberg (specific gravity 0.925) floats in sea water (specific gravity 1.025). Find the ratio of the part out of water to the part immersed. 12. A piece of lead placed in a cylindrical vessel, the radius of whose base is 1.2^"^, causes the liquid in the vessel to rise 3^"™. What is the volume of the piece of lead, and how much does it weigh if lead is 11.2 times as heavy as water ? MISCELLANEOUS EXERCISE 145 MISCELLANEOUS EXERCISE 52 Express the ratio of : 1. A cubic decimeter to a liter. 2. A cubic centimeter to a cubic uiillimeter. 3. A cubic decimeter to a cubic meter. 4. A kilogram to a centigram. 5. A meter to a yard. 6. A quart to a liter. 7. A kilogram to a pound. 8. A milligram to a kilogram. 9. A kilogram to 40 grams. 10. A kilometer to 200 centimeters. Find the value of : 11. (60--i/)x3. 13. (W-5)x6. 12. 120 ^ 12 x 50 14. (3|9 + 2) x4. 15. /522 2727 144 8x9^, 12 V 6 22 180 4x57'^ 16. (26-13)x7 2+15-3 18. 17. 5 X 8-17 X 2 17-14 19. Ti:+3i ^j 31 If lixf 20. What is a decimal fraction ? 21. How is the units' place distinguished ? 22. What is the place value of a digit one place to the right of units ? three places to the right ? lyman's adv. ar. — 10 146 MISCELLANEOUS EXERCISE 23. What is the importance of the symbol in the deci- mal scale of notation ? 24. If a decimal fraction is multiplied by a digit in units' place, do the place values of the digits in the product dif- fer from the place value of the digits in the multiplicand ? If the decimal fraction is multiplied by the same digit two orders lower, is there a difference in the place value of the digits in the product ? 25. If a decimal fraction is divided by a digit in units' place, do the place values of the digits in the quotient dif- fer from the place values of the digits in the dividend? If the decimal fraction is divided by the same digit three orders higher, what is the difference in the place values of the digits in the quotient ? 26. What is a divisor of a number ? a common divisor of two or more numbers ? the greatest common divisor of two or more numbers ? 27. What is a multiple of a number ? a common multi- ple of two or more numbers ? the greatest common multiple of two or more numbers ? 28. What is a prime number ? What is a prime factor of a number ? When are two numbers prime to each other ? 29. What is the shortest piece of rope that can be cut exactly into pieces 12, 15 and 20 ft. long ? 30. Find the 1. c. m. of the first five odd numbers, also of the first six even numbers. 31. Find the g. c. d. of 125, 340 and 735. 32. Evaluate 3| + 5^ 4- 7^ + 9^. 33. Evaluate | + ^^ + iV + 2V " I - I'o " A- 34. Evaluate 4^^ + 2 x 5f - 3 x | - i. MISCELLANEOUS EXERCISE 147 35. A cubic foot of water weighs 1000 oz. How many tons, etc., of water are there in a canal 80 ft. wide, 8 ft. deep and 10 mi. long ? 36. How niany feet per second are equal to 40 mi. an hour ? 37. Find the square root of 0.4 ; the cube root of 0.27. 38. If I walk 7.2^'" in 1 lir., how far shall I go in 6 hr. and 20 min. at the same rate ? 39. How many cubic centimeters of air are there in a room 9^'" long, 6 J'" wide and 3.15™ high ? 40. What is the area of a cube that has the same volume as a box 2 ft. 6 in. by 2 ft. 3 in. by 2 ft. ? 41. How many cubic meters of water pass under a bridge in one minute when the river is 20™ wide, 4™ deep and is running 3^™ per hour? 42. Write three numbers of four figures each that are divisible by both 8 and 3. 43. Write three numbers of six figures each that are divisible by both 9 and 11. 44. Replace the zeros in 205006 so that the number may be divisible by both 9 and 11. 45. What is the cost per hour of lighting a room with 40 burners, each consuming 2 J cu. in. of gas per second, the price of gas being ^1.25 per thousand cubic feet? 46. A roller used in rolling a lawn is 6| ft. in circum- ference and 2|^ ft. wide. If the roller makes 10 revolutions in crossing the lawn once and must pass back and forth 12 times to cover the whole lawn, find the area of the lawn. 47. Find the sum of J + J + 2% + 3% correct to four deci- mal places. 148 MISCELLANEOUS EXERCISE 48. Find each of the following products correct to five significant figures : (a) 20.361 X 40.482. (h) 1.5674 x 75.429. (0 824.763 X 45. {d) 103.64 x 0.033. (e) 0.423x0.00765. 49. YmA each of the following quotients correct to 0.01 : (a) 22-3.1416; (h) 42.567 h- 21.268 ; «) 0.4-0.75; (cZ) 237.64-2.1473.; (e) 2-9.97. 50. Find the cost of carpeting a room 12 ft. 3 in. long and 10 ft. 9 in. wide with carpet 27 in. wide at $1.12 a yard. 51. Find the cost of 8 T. 1450 lb. of coal at 17.25 a ton. 52. Multiply 7644 by 331 and divide the result by 16f. 53. Divide 8350 by 25 and multiply the result by 12i. Find the value of : 54. 0.0001x0.0001; 6.74x21.023. 55. 1.1 X 0.011; 7.6 xO.76. 56. 2.5 X 25 X 250 , 2.5 x 0.25 x 0.025. 57. 0.002 X 3.01 ; 0.0005 x 0.01 x 5000000. 58. 15.625-25; 0.15625^2.5. 59. 8-0.002; 50-0.25. 60. 9.065-0.049; 0.005-0.01. 61. 0.00128-8.192; 1708.4592-0.00024. Find correct to 4 decimal places : 62. 0.138138 + 0.1425876 + 2.060606 + 0.008964. 63. 7.427525-2.347596. 65. 0.33i-0.37|. 64. 0.331 X0.37J. 66. 0.0404^7692. MISCELLANEOUS EXERCISE 149 67. If the length of Jupiter's day is 9 hr. 56 min., how many more days has Jupiter than the earth in one year? 68. If i500 can be counted in one minute, how long will it take to count -$1000000 ? 69. What is the difference between the daily income of a man whose salary is $1200 a year and of one wJiose salary is $1600? 70. Counting 12 hr. a day, liow long would it take to count a billion at the rate of 750 a minute ? 71. How many days old was a person Oct. 5, 1904, who was born July 27, 1861 ? 72. The ancient Roman mile is 0.917 of the English mile. Express the diameter of the earth (7926 English miles) in Roman miles. 73. The diameter of a fly wheel is found by measure- ment to be 20.12 in. Find its circumference. 74. The specific gravity of copper is 8.97; of gold, 19.36 ; of lead, 11.36. Find the weight of a lump of each equal in bulk to a liter of water. 75. The diameter of the earth is 7926 mi. The sun's diameter is 111.454 times the earth's diameter. Find the sun's diameter correct to miles. 76. A lump of iron containing 12 cu. ft. is drawn out into a rod 50 ft. long. What is the diameter of the rod ? 77. The true length of the year is 365.2426 da. What error is made by calculating the year as 365 da., and add- ing a day every leap year, omitting three leap years in four centuries ? 78. The edge of a cube is 12 in. What is the edge of a cube three times its volume ? 150 MISCELLANEOUS EXERCISE 79. How many miles an hour does a person walk who takes two steps a second and 1900 steps to the mile ? 80. Express in words 0.12071 and 12000.00071. 81. How many steps 0.8 of a meter long Avill a person take in walking 10^"* ? 82. A clock which gains one minute in 10 hr. is correct on Monday noon. What is the correct time when it indi- cates Monday noon of the next week ? In scientific work, when numbers depend upon measurements and therefore cannot be expressed with absolute accuracy the index nota- tion is frequently used. Thus, the wave length of blue light, deter- mined by the physicist to be 0.000431'"'" would usually be written 4.31 X lO""*'"*^. The distance from the sun to the earth is determined by the astronomer to be approximately 93000000 mi. In index nota- tion it would be written 9.3 x 10^ mi. 83. Express the following in the index notation : 0.0000025; 36500000000; 2000V000; 41100000. 84. Express in the common notation 1.1 x 10~^; 3.6 x 10^; 4.321x10-8; 5x10-4; 5x106. 85. From 3542g subtract 2131g. 86. Find the sum of 34. 6^2, 121. 51^2, and 25.11i2 and express it in the decimal notation. 87. If brass weighs 525 lb. per cubic foot, find the weight of a circular brass plate 21 in. in diameter and ^ in. thick. 88. If a cubic foot of gold may be made to cover uni- formly 432000000 sq. in., tind the thickness of the gold. 89. If a gallon of water contains 277.274 cu. in., and a cubic foot of water weighs 1000 oz., liow much does a pint of water weigh ? How many gallons will weigh a ton ? MISCELLANEOUS EXERCISE 151 90. Four circles each 1 ft. in diameter are so placed tliat two of them touch two of the others, and the remaininof two both touch three of the others; find the area of the figure whose angles are at the four centers. 91. What (standard) time is it in Boston when it is 4.30 P.M. in San Francisco? 92. A ship's clock is corrected at 1 o'clock each day. If the ship passes over 10° 30' each day, wliat change must be made in the clock (a) if the ship is sailing from W. to E. ; (/)) from E. to W. ? 93. Find the remainders (without dividing) after 471321 has been divided by all of the numbers (except 7) from 2 to 12 inclusive. 94. Show without dividing that 133056 is divisible by 792. 95. A ship's clock is corrected every day at 1 p.m.; how much must it be put back or forward at 1, if the ship has passed over 11° of longitude from east to west ? 96. When it is noon (standard time) Wednesday, Dec. 7, at Chicago, what time and date is it at Rome ? at Tokyo ? 97. A meter is defined as 1 x lO"*" of the distance from the pole to the equator. Find the circumference of the earth in kilometers. * 98. Find the circumference of the earth in miles if the meter is equal to 39.37079 in. 99. If 1 cu. ft. of water weighs 1000 oz., and platinum is 20.337 times as heavy as water, how many feet of platinum wire g^-o^Q-Q of an inch in diameter would weigh a grain ? PEKCENTAGE 233. I = j% = 0.50 = 50% = 50 per cent, 33 and J = ^ = 0.331 = 331 % = 331 per cent. These are different ways of denoting the same fractional part. In business operations it is customary to express fractions in hundredths, but in stating problems the denominator 100 is omitted and the per cent symbol, %, or the expression per cent is used. Percentage is there- fore only an apj)lication of the decimal fraction and not a separate department of arithmetic. 234. The word percentage is derived from the Latin 2)er centum, meaning bi/ the htmdredths. 235. The number denoting how many hundredths are taken is called the rate per cent. Thus, if 5% of a number is to be taken, 5 is called the rate per cent, and 5% the rate. 236. The following examples illustrate several closely related operations frequently used in business transactions. Ux. 1. What is 8% of -f750? Solution. 8 % of $750 = 0.08 of |750 = |60. Ux. 2. 12 is what per cent of 240 ? Solution. Let x % = the rate. Then a: % of 240 = 12, 152 PERCENTAGE 158 Ex. 3. 20 is G% of wliiit nun-iber ? Solution. Let x = the n umber. Then 6 % of a; = 20, 0.06 ^ EXERCISE 53 1. Express the following fractions in per cent, also n« rlppimnU- 13 9 51211312. db aeoillldib . "2", p 10' 6' "8' 5^' T' 10^' ^~' 2. 3 is what per cent of 4 ? 8 is what per cent of 4 ? 18 is what per cent of 27 ? 25 is what per cent of 200 ? 7 is what per cent of 2 ? 3. The population of a town is 7250. What is the population live years later if it has increased 7% in that time ? 4. A town of 11750 inhabitants decreases 12% in ten years. What is its population after this loss ? 5. Express the following as decimals: |%, 331%, 0.5%, 125%. 6. What is -1% of 75? 1% of 100? 0.1% of -| ? f%ofi^? 7. Write as per cent l-J, 2|, i O.OOl 10, 2, 0.25, 2.5, 0.16|. 8. Tlie attendance in a certain school increased in one year from 318 to 425 ; find the rate per cent of increase. 9. In a certain school there are 291 boys and 315 girls. What percentage of the attendance is boys and what per- centage is girls ? 10. In a certain town the total school enrollment is 962 ; of this 156 are in the high school. What percentage of the whole enrollment is in the high school ? 154 PERCENTAGE 11. If 0.8% of those living at the age of 24 die within a year, how many out of 6625 persons of this age die during that period ? 12. At the age of 15, 735 out of 96285 die within a year. Wliat is the rate per cent of deaths ? 13. At the age of 25, 718 out of 89032 die within one year. Is the death rate higher or lower than at the age of 15? 14. A man owns a farm worth -^7500. His annual taxes are -$68.50. How much must he make in order to clear 6 % from his farm each year ? 15. A house depreciates in value each year at the rate of 5% of its value at the beginning of the year, and its value at the end of three years is §4225; find the original value. 16. A man sold two horses for f 200 each; on the pur- chase price of one he made 20%, and on the other he lost 25%. Did he gain or lose and how much ? 17. The wholesale grocer buys coffee at 25 ct. per pound and sells it at 30 ct. The retail grocer bu^-s it at 30 ct. and sells it at Sl^ ct. What per cent does each make ? 18. If a person spends 60% of his income and saves i 1000, what is his income ? 19. Which investment returns the larger per cent, flour costing i 1.98 per hundred pounds and sold for $2.10, or sugar costing 3J ct. a pound and sold for 4-|- ct. ? 20. A man owning a f interest in a store sold ^ of his interest. What per cent of his share did he sell, and what per cent of the store did he still own ? PJtRCENTAGE 155 21. A mercliant sold out his stock of goods at a discount of 10% of the cost and realized 't>14T56.84. How much did his goods cost him ? 22. A house rents for #300 a year, which represents 6% of its value. How much is it worth ? 23. In 1880 the population of the United States was 50152866, in 1890 it was 63069756, and in 1900 it was 75994575. During which decade was the per cent of in- crease greater and hoAV much ? 24. What is the difference, in square yards, between | of an acre and | % of an acre ? 25. The population of a city is 14560, and is 35% more tlmn it was 10 yr. ago. then ? What was the population 26. On Nov. 1, 1897, the amount of money in circulation in the United States was : gold (including gold certifi- cates), -1^576000000; silver (including silver certificates), 1496000000; paper, .15634000000. Nov. 1, 1902, the cor- responding amounts were 1967000000, $623000000 and 1736000000. What was the per cent of increase in each case during the 5 ji\^ and what was the total per cent of increase ? 27. The following tables show the total receipts and disbursements of three of the largest life insurance com- panies in the United States for the year 1902 : Total Income Expenses and Taxes Death Claims Other Disburse- ments 1073636984 782424835 330651136 183485217 156329328 54403289 252617938 163663466 60459793 316541543 185702274 69056722 156 PERCENTAGE Find the per cent of the total income remaining in the hands of each compan}^ at the end of the year. Find the per cent of expense to income and of death claims to income in each case. 28. In 1890 the total foreign population in the United States was 9249547, of whom 2784894 were born in Ger- many and 1871509 in Ireland. The population of the United States in 1890 being 63069756, what per cent of the population was born in Germany, and what per cent in Ireland ? 29. In 1890 the total number of negroes in the United States was 7470000, which was 11.8% of the total popula- tion at that time. Determine the population correct to thousands. 30. In 1898 the total value of the exports from the United States was sfi^ 1231482330, the total value of im- ports was '1616049654. By what per cent did the value of the exports exceed the value of the imports ? COMMERCIAL DISCOUNTS 237. Manufacturers, publishers and wholesale dealers have a fixed price list for their products. Their customers are allowed certain discounts from their list price, deter- mined by the current market value. Thus, a book may be published at $1.50 with a discount of 20^ to dealers. The $1.50 is the list price and 20 fo is the discount. Tlie list price less the discount ($1.50 — 20^ of $1.50 = $1.20) is the net price, or cost. 238. To avoid the inconvenience and expense of issuing a new- catalogue whenever the market values change, business houses gen- erally print a new trade price list giving new discounts, without issuing a new catalogue. The discount is changed either by increas- ing or diminishing the single rate of discount already allowed, accord- ing as the cost of production is diminished or increased. If the discount is to be increased, the change is generally made by quoting a further discount. Thus, in a catalogue of electrical goods a 32 candle power lamp is quoted at ^1.20. In trade price list j\., accom- panying the catalogue, a discount of .50 % is allowed on small orders. In trade price list B, issued later on account of a change in the cost of production, a discount of 50% and 15% is allowed. A dealer buy- ing 10 lamps according to trade price list A would pay 10 x f 1.20 - 50% of 10 X ^1.20 = 16, while according to trade price list B he would pay $6 - 15% oi $Q = $5.10. The discount is frequently increased in case of large orders. Thus, in the above trade price list, a discount of 50 % is allowed on all orders for less than 25 lamjxs, a discount of 50 % and 20 % is allowed on all orders for 25 to 100 lamps, and a discount of 50%, 20% and 10% on orders for 100 or over. 157 158 COMMER CIA L DISCO UN IS 239. Bills are generally made out payable in 30, 60 or 90 days, subject to a certain discount for cash, or if paid before due. Business houses usually print on their bill heads their terms of discount for cash, e.g. " Terms : 60 days, or 2% discount for cash." "Terms: net 90 days, or 3% in 10 days." Ux. 1. On March 12, 1903, E. C. Horner & Co. bought of James Bros., Chicago, 50 plows, listed at §6.50, less 25% and 10%. Terms: 90 days, 3% in 10 days. E. C. HoRXER & Co. Terms: OOdaj^s; 3% 10 days. Bill Rendered Chicago, III., March 12, 1903. Bought of James Bros. 50 Plows @ $6.50 Discount, 25% Discount, 10% $325 81 00 25 243 24 75 38 219 37 If Horner & Co. avail themselves of cash payment, they will deduct 3% of $219..37 = ^6.58, and send the remainder, $212.79, to James Bros. If the bill is not j)aid till the 90 days expire, they will send $219.37. Ux. 2. Find the cost of a bill of goods amounting to i75 less 20%, 5% and 2% for cash. Solution. Let x = the cost. Then x = 0.98 x 0.95 x 0.80 of $ 75 = $ 55.86. Atrnli/fils. $75 is the list price. Then $75 - 20% of $75 = 0.80 of $75 is the amount left after the first discount. And 0.80 of $75 - 5% of 0.80 of $75 = 0.95 x 0.80 of $75 is the amount left after the second discount. And 0.95 x 0.80 of COMMERCIAL DISCOUNTS 159 175-2% of 0.95x0.80 of .| 75 = 0.98 x 0.95 x 0.80 of i$75 is the amount left after the third discount. .-. 0.98 X 0.95 X 0.80 of |75 = |55.8G is the net price or cost. Second Sohition. 5 )$ 75 = list price. 1 15 = 20% discount. 2())p0 ^■] = 5% discount 50 )1 57 ^ 1.14 = 2% discount for cash. $ 55.86 = cost of the goods. I^x. 3. What must be the list price of goods in order to reahze $243 after deducting discounts of 25j/o, 10 fo and 10 fo'? Solution. Let x = the list price. Then 0.90 x 0.90 x 0.75 of a: = |243. 3 400 . ^ _ !| 243 _ ^ m X 190^0 _ 1 400. 0.90 x 0.90 X 0.75 ^19 x /7^ EXERCISE 54 1. Find the net amount of the bill to render in each of the following cases : (a) .|T50 less 33i/o. lb) -$1250 less 25/g and 15fo. (c) 1525 less 20^6, 10 fo and 5fo. Id) 1525 less 3% 10 fo and 20 fo. (e) $5050.75 less oO fo and 10 fo. 2. March 1, 1903, tlie iManhattan Electrical Supply Co. sold George J. Fiske & Co. the following bill of goods, 60 da., 2fo 10 da. : 2 electrical gongs at $17.22 each, less 40^0 and- lO fo; 2 hotel annunciators at $ 15 each, less 60 fo : 2 spools of wire at 75 ct. each, less 50 fo and 10^. Find the amount to be remitted if paid March 11, and write the bill rendered. 160 COMMERCIAL DISCOUNTS 3. A piano listed at $ 750 was sold at a discount of 40 fo and 10 fo. If the freight was 84.87 and drayage $3, what was the net cost of the piano ? 4. Find the net cost of a piece of Rogers's statuary listed at $65 and discounted at 35^, 20 fo, 10 fc and 5fo. 5. A merchant buys f 1750 worth of goods at a discount of 331/0 and 10 fo. If he sells the goods at the list prices, what is the rate of gain on the cost ? 6. A car load of flour weighing 195 hundredweight cost a grocer $ 1.85 a hundredweight. If he is allowed a discount of 1^ for cash and sells the flour for $2.10 a hundredweight, how much does he make ? 7. Which is the greater, a discount of 10^, 10% and 10/0, or a discount of 20 fo, bfo and 5fo? 8. A merchant buys goods at a discount of -10 fo and 10 fo and sells at a discount of 30^ and 5fo. What is his gain per cent ? 9. A certain publishing house allows a discount of l(J|/o on all orders under $100, 16|/o and 10 fo on all orders between $100 and $500, and 16ffo, 10 fo and bfo on all orders above $ 500. If three dealers wish to send in orders amounting to $60, $175 and $350 respectively, how much will each one gain if they combine their orders ? 10. Which is the better discount for a buyer to take : (a) 331/0, 10/0 and bfo, or 40/. ? (5) 10/0, 10/0 and Bfo, or 25 fo ? (c) 40/0 and 15/0, or 40/), lO/o and 5fo? Id) 60 fo and 15 fo, or 60 fo? 11. How much above the cost must a book marked $2 be sold, if 10 fo is taken from the marked price and a profit of 10 fo on the cost is still made ? COMMERCIAL DISCOUNTS IGl 12. One firm offers to sell -1500 wortli of galvanized pipe at a clisconnt of 40 J^, 10^ and b'/o, and another firm offers a discount of U\f, 20/o and lO/o. Wliicli is the better rate of discount and what is tlie difference in dollars ? 13. Office furniture amounting to $ 750 was inventoried at the end of the first year at 25 ^^ below cost and at the end of tlie second year at 15 ^o below inventory. What was the loss in value ? 14. If a grocer buys sugar at 3.42 ct.'per pound and sells it at 4 ct., what is his gain per cent? 15. A dealer marked his goods at 33^ J^ above cost, but sold at a certain per cent discount and still made 15^ on the cost. What was the rate per cent of discount ? 16. What three equal rates of discount are equivalent to a single rate of 27.1^? lyman's adv. ar. 11 MARKING GOODS 240. ]Most merchants use a private mark to indicate the cost and selling price of goods. They usually select some word or phrase containing 10 different letters and use it as a key. These letters are used to represent the 9 digits and 0. In this way the cost and selling price Avill be understood only by those who know the key. Two different keys are generally selected, one to mark the cost and the other to mark the selling price. One or more extra letters, called repeaters, are used to avoid the repetition of a figure and to prevent giving any clew to the private mark used. The cost is usually written above and the selling price below a line. 241. The words equinoctial (omitting the last i) and importance are adapted for use as keys, since they both contain 10 different letters. These words give the fol- lowing keys : 1234567890 e q u i n c t a I i m 2? r t a 71 c e Repeaters x and ?/. Thus, if a merchant pays $29.98 per dozen for hats, and sells them for !$3.50 each, he would mark them ^ • EXERCISE 55 1. Explain why, if the cost of a dozen articles is divided by 10, the result will give the retail price of one article with a profit of 20% added. 162 MARKING GOODS lf>3 2. Explain wliy, to make a profit of 33^%, the cost of a dozen articles may be divided by 10 and -1 of the result added. 3. Determine sliort methods of finding the retail price of one article when the cost per dozen is given and the dealer wishes to make a profit of 35% ; 37| % ; 40% ; 50% ; 60%. 4. A merchant buys shirts for -$12.50 per dozen. For what price must lie sell them to make 50%? 40%? 5. A merchant retails neckties at 50 ct. and makes 50% . How much did they cost him per dozen ? Using equinoctal and importance as keys, mark the cost and selling price of the following articles : 6. Gloves costing -$5 per dozen and selling for $6.50. 7. Hats costing $22.50 per dozen and selling at 20% gain. 8. Caps costing $7.50 per dozen and selling at 33^% gain. 9. Shoes costing $1.98 and selling at 25% gain. 10. Rubber boots costing $2.68 and selling at $3.75. 11. Make a key of the letters contained in the words Cumberland and Charleston spelled backward, and mark the articles given in Ex. 6 to 10. 12. A merchant sold a bill of goods that cost $125 ; the asking price was 30% in advance of the cost, from wliich a wholesale discount of 15% was allowed. What was the per cent gain ? 13. An invoice of hats costing $112 is marked so as to sell at 40% profit. Does the merchant gain or lose if the hats are sold at 30% discount from the marked price ? COMMISSION AND BROKERAGE 242. Farmers, produce dealers, manufacturers and others frequently find it more convenient to employ a third person to dispose of their goods, instead of selling direct to consumers. The person who sells the goods is called a commission merchant, an agent or a broker. The pay received for such services is called commission or brokerage. 243. Produce is usually shipped to a commission mer- chant, and sold by him in his own name. The proceeds less the commission, or the net proceeds, are sent to the shipper or consignor. If a commission merchant is buying goods for a customer, he charges the cost plus the com- mission. The amount of commission varies in different lines of business. 244. A broker buys and sells without having possession of the goods, and generally does not make contracts in his own name. 245. Commission, or brokerage, is usually computed at a certain per cent of the amount realized on sales, or in- vested for the customer. In buying and selling certain kinds of merchandise, it is customary to pay a certain price per unit of measurement or weiglit; as grain per bushel, hay per ton, etc. 164 COMMISSION AND BROKERAGE 165 EXERCISE 56 What is the conimissiuu uii : 1. 1750.50 at 2% ? 4. 1350.45 at 10%^ 2. il2368ati%? 5. i|3764ati%? 3. 875429.75 at i%? 6. i5250at7|%? 7. The sale of 1000 bii. of grain at |- ct. a bushel ? 8. The sale of 25 T. of hay at 50 ct. a ton ? 9. The sale of 40 liead of cattle at 50 ct. a head ? 10. The sale of 1500 bales of cotton at 25 ct. a bale ? 11. The sale of 22 horses @ .$125 a head at 2% ? Find the amount to invest and the commission when the following remittances and rates of commission are given : 12. $1030 at 3%. 15. 86300 at 5%. 13. 15025 at 1%. 16. f 1100 at 10%. 14. 88020 at i%. 17. 82562 at 21%. Find the net proceeds and commission on each of the following sales : 18. 200 bbl. of apples @ 83, less freight 862.50, com- mission 5%. 19. 5000 bu. of wheat @ 72 ct., less 8102.50 freight, 825 storage, \% insurance and 2% commission. * 20. 500 bbl. of beef @ 819.50, less 48 ct. a barrel freight, 87.50 storage and 2|% commission. 21. 1500 doz. eggs @ 22 ct., less 89.50 express and 10% commission. 22. 12 bales of cotton averaging 475 lb. @ 9^ ct. a pound, less 842.50 freight, 81.25 a bale storage and 2^% commission. 166 COMMISSION AND BROKERAGE 23. An agent charges $20 for advertising the sale of a farm, and 3% commission. He sells the farm for $7500. What are the net proceeds and the agent's commission ? 24. A collector is given a bill of $1350 to collect at 5% commission. He succeeds in collecting 85 ct. on the dollar. How much is due his employer, and what is his commission ? 25. A miller orders his agent to buy him 2500 bu. of Avheat @ 80 ct. If the agent charges 3% commission, and freight and drayage charges are $95.75, what is the total cost of the wheat ? 26. A merchant sends his agent $1836 to buy an equal number of yards of each of three grades of muslin at 3, 4 and 5 ct. a yard respectively, after deducting 2% commission. How many yards of each kind does he get, and what is the agent's commission ? 27. A manufacturer sold $20000 worth of goods through his agent at 2% commission, and instructed him to purchase raw material with the proceeds at 1% commission. Find the net proceeds of the sale, the amount invested in raw material, and the agent's entire commission. 28. A dealer sent two car loads of hay weighing 27 T. to his broker in New York, who sold it for $16 a ton, and remitted $418.50. If the dealer paid $8.50 a ton, the freight cost 16| ct. a hundredweight, and storage was $12.50, how much did he make, and what was the broker's commission per ton ? 29. A book agent sells, during July and August, 77 copies of a certain book at 40% commission. If he sells 20 copies in full leather binding @$6.50, 25 copies in half leather @ $5.25, and 32 copies in cloth @ $4, how much does he make if his expenses average $1.25 a day? INTEREST 246. Interest is money paid for the use of money. 247. Tlie sum loaned is called the principal. 248. Tlie rate of interest is the rate per cent per annum of the principal paid for tlie use of money. In the absence of a specific contract the rate of interest is fixed by law in most states. The rate thus determined is called the legal rate. By special contract interest may be received at a higher rate than the legal rate. The maximum contract rate is fixed by law in most states. Interest in excess of the maximum contract rate is called usury. The penalty for usury is fixed by law in states where it is forbidden. 249. The principal plus the interest is called the amount. 250. The practical business problem of most frequent occurrence in interest is to find the interest when the prin- cipal^ rate^ and time are given. 251. In computing interest without tables it is usually tlie custom to reckon the year as 360 da., the month as .^^2 of ^ year and the day as -^-^ of a month or ^i^ of a year. (See, however, § 254.) Ex. Find the interest and amount of $720 for 2 yr. 6 mo. 15 da. at 6^. Solution. 2 yr. 6 mo. 15 da. = f|f yr. The interest for one year = 0.06 of ^720. .-. the interest for f|A yr. = fi§ x ^^^ x $720 = $109.80, The amount is |720 -f- 1109.80 = $829.80. 167 168 INTEREST EXERCISE 57 Find the interest and amount of : 1. $100 for 1 yr. 4 mo. at Q% 2. $125 for 6 yr. 1 mo. 20 da. at If. 3. $150 for 5 yr. 9 mo. 11 da. at If. 4. $50 for 4 yr. 11 mo. 10 da. at {jf. 5. $1000 for 5 mo. 10 da. at If. 6. $350 for 3 yr. 9 mo. at If. 7. $1500 for 2 mo. at 6/o. 8. $25 for 1 yr. at %f. 9. $1200 for 2 yr. 6 mo. at bf. 10. $500 for 2 yr. 3 mo. 15 da. at -if. 252. A short ynethod of computing interest at Qf is based on the year of 12 mo. of 30 da. each. This method is sometimes called the ^f method. The interest on ,|1 for 1 yr. at 6% = $0.06. The interest on %l lor 1 mo. at 6% = J^ of ^^-^^ = 1 0.005. The interest on $1 for 1 da. at 6% = gV of $0,005 = $0.0001. Ex. Find the interest on $250 for 2 yr. 4 mo. 12 da. at Qf. The interest on $ 1 for 2 yr. at 6%= 2 x $0.06 = $0.12. The interest on $1 for 4 nio. at 6% = 4 x $0,005 = $0.02. The interest on $1 for 12 da. at 6% = 12 x $0.000i = $0.002. .-. the interest on $1 for 2 yr. 4 mo. 12 da. at 6% = $0,142. .-. the interest on $250 for 2 yr. 4 mo. 12 da. = 250 x $0,142 = $35.50. 253. To find the interest at bf, subtract J- of the interest at 6 % ; at 7 %, add 1 of the interest at 6 %, etc. SIMPLE INTEREST 109 EXERCISE 58 1. By the 6% nictliod find llie interest on ^r^lOO for 2 yr. 3 mo. 10 da. at 4 % ; at 4| % ; at «Ji % ; at 7 ^ % ; at 3% ; at 31%. Find the interest on : 2. $325 for 1 yr. 2 mo. at 6%. 3. $450 for 2 yr. 3 mo. 14 da. at 5,] %. 4. #315.T5 for 2 mo. 15 da. at 8%. 5. 12000 for 30 da. at 6%. 6. $115.50 for 3 mo. 10 da. at 1},%. 7. $387.50 for 6 mo. at 5%. 8. $524.70 for 60 da. at (5%. 9. $97.30 for 3 mo. 10 da. at 7 %. 10. $80.60 for 1 yr. 6 mo. 15 da. at 3%. 254. Exact Interest. To find the exact interest we must take the exact number of days between dates and reckon 365 da. to a year. Exact interest is used by the United States government, by some banks, and to some extent in other business transactions. Ux. Find the exact interest on $2500 from April 10 to Sept. 5 at 5%. 148 = the number of days from April 10 to Sept. 5. .-. the interest on |2500 for 118 da. at 50/ - iliiii§2ifM^=: i$50.68. ^' 100 X 305 170 INTEREST EXERCISE 59 Find the exact interest on : 1. 1575 from July 5 to Sept. 5 at 7 %. 2. 1125 from Jan. 1 till Nov. 1 at 6 %. 3. 110000 from March 10 till June 1 at 5%, 4. 1375.30 from April 25 till Aug. 1 at 6%. 5. Find the amount of ^55 375 at G% exact interest from Nov. 11, 1903, till July 27, 1905. 6. May 10, 1903, -$500 is loaned at 6%. Find the amount due Sept. 1, 1905, exact interest. 7. If 1500 is loaned on July 28, 1905, when will it amount to 8720 ? 8. What is the difference between the exact interest and the common interest on 81000 from July 1 till Nov. 1 at 6%? If the exact number of days between dates and 360 days to the year are taken, how much does the com- mon interest differ from the exact interest ? 9. Show that the difference between the common inter- est and the exact interest is ^^ of the former and ^^ of the latter. 10. Hence, show that exact interest may be obtained by subtracting y^g part from the conuiion interest, and the common interest may be obtained from the exact interest by adding ^^ part of itself. Exact interest is the fairest, but on account of its incon- venience without tables is not generally used. 255. The following is a section of an interest table for the year of 365 da. at 6 % : SIMPLE INTEREST 171 l).v\> iii..,i ■_'(M)0 .SODI) 4(iUit .".(((III i;(i III Tniin >iiiiii j 60 61 62 63 64 65 m 67 68 69 70 71 72 73 74 75 76 77 78 79 80 9.863 10.027 10.192 10.356 10.521 10.685 10.849 11.014 11.178 11.342 11.507 11.671 11.836 12.000 12.164 12.329 12.493 12.658 12.822 12.986 13.151 19.726 20.055 20.384 20.712 21.041 21.370 21.699 22.027 22.356 22.685 23.014 23.342 23.671 24.000 24.329 24.658 24.986 25.315 25.644 25.973 26.301 29.589 30.082 30.575 31.068 31.562 32.055 32.548 33.041 33.534 34.027 34.521 35.014 35. 507 36.000 36.493 36.986 37.479 37.973 38.466 39.959 39.452 39.452 40.110 40.767 41.425 42.082 42.740 43.397 44.055 44.712 45.370 46.027 46.685 47.342 48.000 48.658 49.315 49.973 50.630 51.288 51.945 52.603 49.315 50.137 50.959 51.781 52.603 53.415 54.247 55.068 55.890 56.712 57.534 58.356 59.178 60.000 60.822 61.644 62.466 63.288 64.110 64.932 65.753 59.178 60.164 61.151 62.137 63.123 64.110 65.09(5 66.082 67.068 68.055 69.041 70.027 71.014 72.000 72.986 73.973 74.959 75.945 76.932 77.918 78.904 69.041 70.192 71.342 72.493 73.644 74.795 75.945 77.096 78.947 79.397 80.548 81.699 82.849 84.000 85.151 86.301 87.452 88.603 89.753 90.904 92.055 78.904 80.219 81.534 82.849 84.1(J4 85.479 86.795 88.110 89.425 90.740 92.055 93.370 94.685 96.000 97.315 98.630 99.945 101.260 102.575 103.890 105.205 88.7(J7 90.247 91.726 93.205 94.685 96.164 97.644 99.123 100.603 103.562 103.562 105.041 106.521 108.000 109.479 110.959 112.438 113.918 115.397 116.877 118.356 Years 1000 2000 3000 4000 5000 6000 7000 8000 9000 1 2 3 4 5 ■ 6 60 120 180 240 300 360 120 240 360 480 600 720 180 360 540 720 900 1080 240 480 720 960 1200 1440 300 600 900 1200 1500 1800 360 720 1080 1440 1800 2160 420 840 1260 1680 2100 2520 480 960 1440 1920 2400 2880 510 1080 1620 2160 2700 3240 Ux. By the use of the table find the interest on -^4650 for2yr. 67 da. at 6 /o. for 2 yr. for 67 da. The interest on .'$4000 = $480 + $44.06 The interest on 600 = 72 + 6.61 The interest on 50 = 6 + 0.55 The interest on $4650 = $558 + $51.22 = $609.22. 172 INTEREST EXERCISE 60 By the use of the table find the interest on : 1. 1500 for {^^ da. 2. 11000 for 60 da. 3. 15225 for 73 da. 4. 110575 for 1 yr. 60 da. 5. $1846 for 2 yr. 80 da. 6. 11710 for 75 da. 7. il250 for 63 da. 8. 12120 from July 2 till Sept. 5. 9. 1648.60 from Jan. 10 till March 15. 10. -11410 from May 1 till July 10. 256. In any problem in interest there are four elements involved, tlie principal, the rate, the time and the interest. When any three of tliese are given, the other can be found. As indicated above, the practical business problem is to find the interest when the principal, rate and the time are given. However, the principles involved in the following illustrative examples are frequently met with in business : Ex. 1. What principal will produce f 72 interest in 1 yr. 6mo. at6fo? Solution. Let x = the principal. Then « x (j%oi x = ^ 72. .-. x = ~^^'^^ =$800. 3 X 0.06 SIMPLE IXTEHEST 173 E.r. 2. At what rate will 6^800 produce 872 interest in 2 yrs. ? Solution. Let a: %= the rate. Then .r % x 2 of ^ 800 = 8 72. • 7- 0/ — ^ ' - — 9 _ 4. 1 0/ E.V. 3. In what time will -S 1000 produce >=70 interest at Solution. Let .r = the time. Then x x 4% of 8 1000 = 870. S70 0.04x81000 '■ .-. .r = 1 yr., or 1 yr. 9 mo. Ex. 4. What prinei})al will amount to 81238 in 6 mo. 10 da. at 6 ^ ? Solution. Let .r = the principal. Then x + 6 % x J§§ of x = 8 12:]8 = the amount. .-. x = ^^-^^^ = 81200. 1 + 0.0(5 X ie EXERCISE 61 Find the rate at which : 1. 8750 will produce 867.50 interest in 1 yr. 6 mo. 2. 82000 will produce 8105 interest in 9 mo. Find the time in which : 3. 8250 will produce 825 interest at 5^. 4. 81200 will produce 890 interest at 6f). 9 9 174 INTEREST 5. 1850 will produce -f 106.25 at 5/o. 6. $2000 will produce $105 at If. What principal will produce : 7. il08 interest in 1 yr. 6 mo. at 6^ ^ 8. $61.25 interest in 2 yr. 6 mo. at 7^ 9. $262.50 interest in 1 yr. 6 mo. at 5% ? What principal will amount to : 10. $575 in 2 yr. 6 mo. at Qf ? 11. $1050 in 1 yr. at 5/o? 12. $1570 in 1 yr. 2 mo. at 4fo ? 13. A man with $25000 invested in his business makes 121^ profit annually. He sells out and invests the $25000 at 6fo and works on a salary of $2000 per annum. Does he make or lose by the change and how much ? 14. A man invests $20000 in business and makes $6000 in one j^ear on his sales. If the total expenses of running the business are $ 3500, what rate does he make on his money ? 15. A house and lot costs $1800 and rents for $16 a month. If taxes, insurance and repairs cost $72 a year, what rate is earned on the investment ? 16. Jan. 1, 1900, $450 are deposited in a savings bank at 3/o. Find the amount due July 3, 1900. 257. Compound Interest. In compound interest the interest is added to the principal at the end of each COMPOUND INTEREST 11 interest period. Then the amount becomes the new prin- cipal for the next interest period. Unless otherwise stated, interest is compounded annually, though it may be compounded semiannually, (juarteriy, etc., l)y agreement. In most states compound interest cannot be collected by law, but pay- ment of it does not constitute usury. Ux. Find the compound interest on 8500 for 3 yr. 4 mo. 15 da. at 4%. Solution. $500 = principal first year. 0.04 20.00 = interest first year 500 $520.00 = amount first year = principal second year. 0.04 20.80 = interest second year. 520 $540.80 = amount second year = principal third year. 0.04 21.63 = interest third year. 540.80 $562.43 = amount third year = principal fourth year. Interest on $562.43 for 4 mo. 15 da. at 4% = 88.44. $562.43 + $8.44 = $570.87 = amount for 3 yr. 4 mo. 15 da. 500 $70.87 = compound interest for 3 yr. 4 mo. 15 da. 258. The chief use of compound interest is among large investors, such as life insurance companies, building and loan associations, private banking establishments, etc., who wish to compute the income from reinvestment of interest when due. For such work compound interest tables are used. 176 INTEREST Tlie following is a section of such a table : Periods 1 Per Cent 1^ Per Cent 2 Per Cent 3 Per Cent 4 Per Cent 1 1.0100000 1.015000 1.020000 1.030000 1.040000 2 1.0201000 1.030225 1.040400 1.060900 1.081600 3 1.0303010 1.045678 1.061208 1.092727 1.124864 4 1. 0406040 1.061364 1.082432 1.125509 1.169859 5 1.0510100 1.077284 1.104081 1.159274 1.216653 6 1.^615201 1.093443 1.126162 1.194052 1.265319 7 1.0721353 1.109845 1.148686 1.229874 1.315932 8 1.0828567 1.126493 1.171660 1.266770 1.368569 9 1.0936852 1.143390 1.195093 1.304773 1.42.3312 10 1.1046221 1.160541 1.218994 1.343916 1.480244 11 1.1156683 1.177949 1.243374 1.384234 1.539454 12 1.1268250 1.195618 1.268242 1.425761 1.601032 13 1.1380'.)32 1.21.3552 1.293607 1.468534 1.665074 14 1.1494742 1.231756 1.319479 1.512590 1.731676 15 1.1609689 1.250232 1.345868 1.557967 1.800944 16 1.1725786 1.268985 1.372786 1.604706 1.872981 17 1.1843044 1.288020 1.400241 1.652848 1.947901 18 1.1961474 1.307341 1.428246 1.702433 2.025817 19 1.2081089 1.326951 1.456811 1.753506 2.106849 20 1.2201900 1.346855 1.485947 1.806111 2.191123 Solution of the above example by means of the tables. The amount of -|1 for 3 yr. at 4% is S^ 1.12486. The amount of -|500 will be 500 x -11.12486 = $562.43. The example may now be completed by using the tables for simple interest for 4 mo. 15 da., or as on p. 175. EXERCISE 62 1. Find the compound amount and the compound interest of 12000 for 3 yr. 6 mo. at 4% payable semi- annually. Note. It is evident that if interest is 4 % compounded semiannnally for 3 yr. 6 mo., the amount is the same as if the rate is 2% com- pounded annually for 7 yr. ANNUAL INTEREST 177 2. What is tlie difference between the simple iind com- pound interest on $750 for 2 yr. 7 mo. iit 5%? 3. Find tlie amount of ifSOOO compounded annually for 4 yr. at 4%. 4. Find tlie amount of 88500 compounded semiannually for 5 yr. at 8% ; at 4% ; at (>%. 259. Annual Interest. If a note or other written agree- ment contains the expression "with annual interest"' or '^ with interest payable annually," the interest is due at the end of each year, and if not then paid, will draw simple interest until paid. Such a note or agreement is said to bear annual interest. As in the case of compound interest, in most states annual interest cannot be collected by law, but does not constitute usury. Ux. George Reed borrowed $1500 at 7%, and agreed to pay interest annually. Having paid no interest, he wishes to settle at the end of 3 yr. 3 mo. 20 da. What is the amount due ? The simple interest on $ 1500 for 3 yr. 3 mo. 20 da. at 7% = 8347.08. Then, in addition to this, the simple interest on 1105 at 7% for 2 yr. 3 mo. 20 da. $105 at 7% for 1 yr. 3 mo. 20 da. $105 at 7% for 3 mo. 20 da. or on $105 at 7 % for 3 yr. 10 mo. = |28.18. Hence, principal borrowed = $1500 Simple interest = 3-47.08 Simple interest on interest not paid when due = 28.18 .-. total amount due at annual interest = $1875.26 lyman's adv. ar. — 12 178 INTEREST EXERCISE 63 1. What is the difference between the simple interest and the annual interest in the preceding example ? How long is it after the date on which the money is borrowed before the annual interest begins to differ from the simple interest ? 2. What is the difference between the compound in- terest and the annual interest in the preceding example ? How long is it before the compound interest begins to differ from tlie simple interest? from the annual interest? 3. Find what f 2500 will amount to in 4 yr. 10 mo. 18 da. at 5% simple interest and at 5% compound interest. 4. Sept. 1, 1896, a man borrows $500 at 6% interest, payable annually. If nothing is paid until Dec. 1, 1901, how much is due ? 5. Notes are sometimes given with interest coupons attached. These coupons draw interest, frequently at a higher rate than the note itself, if not paid when due. A coupon note for 12200 is issued July 1, 1896, at 6% inter- est. Nothing is paid until July 1, 1902. Find the amount at that date, the coupons bearing 7% if not paid when due. 260. Promissory Notes. A promissory note is a written promise to pay to a certain person named in the note a specified sum of money on demand, or at a specified time. 261. A promissory note is negotiable, i.e. can be trans- ferred from one owner to another by indorsement when it is made payable to the order of a definite person, or to hearer. PEOMISSORY NOTES 179 The following is a coininon form of a negotiable promissory note : //^^^. ^etvait, muk., nux^k , 27, f^O^. c/t^tl/ daya^ after date I promise to ])ay to the order of /f&tcvif joi^fi&Qy an& tko-^l^(^'}^cL dncL ^ dotU^a., lUO ' value received, ^v~itk vnt&v&Qyt at 6%. Mo. ^3. Due CCnclv&ii^ ^aknoyayi. Andrew Johnson is the maker of this note, Henry James is the payee, and $1000 is the face. 262. The above note would be non negotiable if the words ''the order of" were omitted. In that case the note would be payable to Henry James only. 263. If a note is sold by the payee, he must indorse it by w^riting his name across the back. 264. There are three common forms of indorsement : (1) In blank, the indorser simply writing his name across the back, thus making the note payable to the bearer. Indorsement in Blank (2) In full, the indorser di- recting the payment to the order of a definite person. Indorsement in Full c^a-2^ ta tk& avd&v aj^ ^viy-l&u CLnct f'foJjiyk'. 180 INTEREST (3) Qualified, the indorser avoiding responsibility by writ- ing the words '' without re- course " over his name. Qualified Indorsement S^ciu ta tk& avcC&v o-i ofvlf-l&i^ cincL ffat^k. lAMVio-iit V£.(S^auv^&. OR SIMPLY 265. By indorsing a note either in blank or in full, the payee becomes responsible for its payment if the maker fails to pay it. The indorsement in full will insure greater safety since in this case the note is made payable to a definite person. 266. A note made payable to Henry James, or bearer, is also negotiable, but does not need indorsement. 267. The custom of allowing three days of grace in the payment of a note has been abolished in many states and is rarely used in others. 268. The note ou pag'e 179 will mature March 27 + 60 days, or May 26, if no grace is allowed. It will mature March 27 + 63 days, or May 29, if grace is allowed. In states where grace is allowed this is indicated by writing in the note, " Due May 26/29." 269. In most states a note falling due on Sunday or a legal holiday must be paid the preceding business day. EXERCISE 64 1. Write a 30-day promivssory note for i500, payable to the order of James Black, bearing the legal rate of interest in your state. By indorsement make the note payable to Henry Wood. PARTIAL PAYMENTS 181 2. What is the maximnni contract rate of interest in your state ? What is the penalty for usury ? 3. Write a 60-day promissory note for f 100, with your- self as maker and Charles Jennings as payee, the note bearing the date May 5, 1903. If the note is payable to Charles Jennings, or bearer, in what way may it be trans- ferred ? If made payable to Charles Jennings, or order, in what way may it be transferred ? 4. Find the date of maturity of the note required in Ex. 3. Add days of grace if they are used in your state. 5. Find the interest on the above note. 6. 1250. Ypsilanti, Mich., April 11, 1905. Ninety days after date I promise to pay to the order of William Jordan tw^o hundred fifty and y^U^Q dollars, value received, with interest at 5%. L. M. Davis. When is the above note due ? Wliat is the amount at maturity ? Who pays the note ? Who receives the money? Who receives the note when paid? What indorsement is necessary if the note is sold to John Brown ? 270. Partial Payments. Frequently the maker of a note, not being able to pay the whole amount at one time, makes several partial payments, which are indorsed on the back of the note with the date of payment. Ux. April 6, 1902, a man buys a farm for $7500, pay- ing 15000 in cash, and giving a note for the remainder at 6%, with the privilege of paying all or part of it any time 182 INTEREST within 3 yr. The following payments are made and indorsed on the note by the payee : Oct. 1, 1903, $ 500 April 1, 1904, I 50 Oct. 1, 1901, -^1000 What amount is due April 6, 1905 ? Face of note = |2500.00 yr. mo. da. Oct. 1,1903 = 1903 10 1 April 6, 1902 = 1902 4 6 The interest on |2500 for 1 yr. 5 mo. 25 da. at 6% = | 222.92 Amount due Oct. 1, 1903 . . = 2722.92 Payment . = 500.00 Balance due = new principal . • = 12222.92 yv. mo. da. Oct. 1, 1904 = 1904 10 1 Oct. 1, 1903 = 1903 10 1 1 The interest on $2222.92 for 1 yr. at G% . . = 8 133.38 Amount due Oct. 1, 1904 = 2356.30 Payment April 1, 1904 (less than interest due April 1, viz. $60.69) = 50.00 Payment Oct. 1, 1904 = 1000.00 Balance due = new principal . . . . = $1306.30 yr. mo. da. April 6, 1905 = 1905 4 6 Oct. 1, 1904 = 1904 10 1 6 5 The interest on $1306.30 for 6 mo. 5 da. . . = 40.28 .-. the amount due April 6, 1905, is . . . =$1346.58 PARTIAL PAYMENTS 183 Check Difference between Dates Partial I>ikfekence8 yr. mo. da. yr. mo. da. Date of settlement 1905 4 6 1 5 25 Date of note 1902 4 6 1 Difference in time =3 6 2 11 80 = ::;yr. 271. Tlie above exami)le is solved by the United States Rule of Partial Payments, which is the legal method in most states. i>y this method the amount of the note is found to the time Avhen the payment, or the sum of the payments, equals or exceeds tlie interest due. From this amount the payment, or sum of the payments, is sub- tracted. This operation is repeated to the time of the next payment and so on. 272. It will be seen that three cases may arise under this rule : (1) The payment may be exactly equal to the interest due. In this case the payment simply cancels the interest, and the balance due remains the same as the original principal. (2) The payment maiy be greater than the interest due. In this case the balance due is diminished by the amount the payment exceeds the interest due. (3) The payment may be less than the interest due. In this case if^the unpaid balance of the interest were added, the principal would be increased, and the debtor would be paying more interest than if he had made no payment at all. Therefore, when the payment is less than the interest due, no change is made at that time in the principal ; but the interest is reckoned to the date when the sum of the payments does exceed the interest due, and then the sum of these payments is subtracted. 273. The following metliod of solving problems in partial payments, called The Merchants* Rule, is used 184 INTEREST among many business men when the note runs for one year or less : Ex. Sept. 1, 1904, a merchant takes a note for $827.50 from a customer in payment for some goods. The note is to run for 1 yr. at 6%. During the year the following payments are made: Nov. 1, 1904, !^75; April 1, 1905, 1 100; Aug. 1, 1905, 850. Find tlie amount due Sept. 1, 1905. The amount of 1327.50 for 1 yr. at 6% = $347.15 The amount of -"^75 for 10 mo. at 6% = ^ 78.75 The amount of 1100 for 5 mo. at (')% = 102.50 The amount of |50 for 1 mo. at 0% = .50.25 231..50 Balance due Sept. 1, 1905 i|115.65 274. By this method the amount of the note is found from the date of the note to the time of settlement. The amount of each payment is also found from its date to the time of settlement. The sum of the amounts of the payments is then subtracted from the amount of the principal. 275. Some states, e.g. New Hampshire and Vermont, have rules of their own for solving problems in partial payments. In such states it is left for the teacher to present the rule. EXERCISE 65 1. Which of the above methods is better for the debtor? Which is better for the creditor ? 2. $325 Cleveland, Ohio, May 15, 1897. Three years after date I promise to pay W. W. Johnson, or order, three hundred twenty-five dollars, value received, with interest at 6%. Henry George. Indorsements: May 15, 1897, 122.75; May 15, 1898, 822.75; June 29, 1900, $100 ; Jnne 12, 1902, f 50. Find the amount due June 12, 1904. PARTIAL PAYMENTS 185 3. Jan. 30, 1897, Arthur Ross borrowed 1 150 ; May 10, 1897, 1125; and Dec. 10, 1902, flOO, all at 7% interest. He paid Oct. 1, 1901, -flOO; and Dec. 10, 1902, .$100. Find tlie amount due Marcli 10, 190^3. 4. March 1, 1897, a man buys a farm for $5 6000. He pays $3000 in cash and gives a note for the remainder. He makes a payment of $500 on each of the following- dates : March 1, 1898 ; March 1, 1900 ; Sept. 1, 1900 ; and March 1, 1901. Find the amount due March 1, 1902. 5. 13500. Ypsilanti, Mich., Aug. 15, 1899. Five years after date T promise to pay John Robinson, or order, three thousand five hundred dollars, value received, with interest at G%. James Rowe. Indorsements: Nov. 5, 1902, $300; March 14, 1904, $200; May 14, 1905, $2000. Find the amount due Aug. 14, 1905. 6. A note for $5000, dated March 1, 1903, and payable two years from date, with interest at 5%, has on it the following indorsements : April 1, 1903, $500 ; June 1, 1903, $500; Sept. 1, 1903, $200; and May 1, 1904, $500. Find the amount due March 1, 1905. 7. Dec. 10, 1903, a merchant takes a note for $260 to run 1 yr. at 7%. During tlie year the following payments are made: Feb. 1, 1904, $50; June 21, 1904, $25; Oct. 10, 1904, $100. Find the amount due Dec. 10, 1904. BANKS AND BANKING 276. A bank is an institution that deals in money and credit. Credit is a promise to pay money in tlie future. Tlie chief instruments of credit are checks, drafts, and notes. It is a mistake to say that banks deal only in money. Their most profitable business is in credit transactions. Banks, however, have a capital of their own w-hich serves as a guarantee fund. Neither do all bank deposits represent money intrusted to banks by individuals. Most of them represent credit loaned to individuals by banks. Thus, if a bank accepts a promissory note for $5000, it may give in return a deposit credit for |5000 less the discount and will thereby add that sum to its deposits. 277. There are several kinds of banks, among which are national banks, organized under the National Banking Act of 186-3 and the amendments that have been made thereto ; state banks, organized under the laws of the state in Avhich they are situated ; savings banks ; and private banks. 278. National banks are subject to rigorous supervision by federal authorities. All banks organized under state laws are subject to similar supervision by state authorities. 279. The chief functions of banks are to receive deposits, to lend money on promissory notes, bonds, and mortgages, to discount merchants' notes befoi-e they are due, and to buy, sell, and collect drafts or bills of exchange. National 186 BANKS AND BANKING 187 banks also issue bank notes wliicb circulate as a medium of exchange. Savings banks and a few others allow interest on deposits. 280. On opening an account with a bank, the customer is usually given a pass book in which the dates and amounts of all deposits are entered on the credit side. If the customer wislies to draw money from the bank, or to pay a debt, he fills out a check similar to the following, and the dates and amounts of all such checks are entered on the debit side of his pass book : ^^o Kew Yovh, 190. __ jFourdj National Bank of Neto godt Paij to Ruii'Pv 171. fSvaaiyyi ^..or or del $600.00._. S't'V-& fuvnclv&cl.^ no luo Dollars. ^&cyvcj.& €. .%x. George E. Fox is the drawee or maker of this check, and Ralph M. Brown is the payee. 281. As in the case of promissory notes, checks may be made payable to payee or order, or to payee or bearer. The same rules of indorsement apply. When the deposi- tor wishes to draw money at the bank, the check is made payable to self. 188 BANKS AND BANKING 282. Banks also issue certificates of deposit Certificate of Deposit C ^ = rf g J; ox) « ^o Ypsilanti, Mich 190 -. ha. ^.deposited in the jFirst National Uaiilt of gfpsilanti, Dollars, payable to the order of. subject to the rxdes of this Banh, on tlie return of this Certificate, properly endorsed. Cashier. Teller. The money deposited on such a certificate is not subject to check and can be withdrawn only upon the presentation of the certificate properly indorsed. EXERCISE 66 1. Write a check for $35.75 in each of the forms indicated above, with yourself as drawer and Robert Lyons as payee. If necessar}^, indorse the check as when presented for payment. 2. July 22, 1903, a man deposits fl25 in a savings bank. The bank pays 3% on money left on deposit 3 mo., and 3|% if left 6 mo. or longer. Money must be deposited the first of the month to draw interest for that calendar month. If the money is drawn out Nov. 1 and interest paid for full months, how much does the man receive ? if drawn out Dec. 22 ? if drawn out March 1, 1904? BANKS AND BANKING 189 3. A man owns a certificate of deposit for f 500 dated Aug. 1, 1908. Feb. 1, 1904, he presents the certificate and draws #200. What is the face of the new certificate issued ? 4. Show tliat the following statement of the resources and liabilities of a savincrs bank will balance. Resources Loans and discounts Bonds, mortgages and securities Premiums paid on bonds Overdrafts Furniture and fixtures Other real estate . Items in transit Due from banks in reserve cities Exchanges for clearing house U. S. and National bank currency Gold coin .... Silver coin .... Nickels and cents . ^214,193.14 20,742.4 70,470.00 68,200.00 2,365.00 83.28 $946,542.72 504,171.35 1,218.75 1,471.70 13,801.50 85,337.01 13,409.02 Liabilities Capital stock paid in Surplus fund Undivided profits, net . Commercial deposits Certificates of deposit . Due to banks and bankers Certified checks Cashier's checks . Savings deposits . Savings certificates 1600,991.33 94,897.63 192,674.51 12,124.52 10.455.21 696,425.79 69,655.31 1 200,000.00 36,000.00 19,781.58 190 BANKS AND BANKING 283. If his financial standing is high, a person may borrow money from a bank by giving his individual note. The bank may, however, ask for security. In this case the borrower must have some responsible person indorse the note, or he must deposit collateral security in the form of stocks, bonds, etc. 284. The following is a common form of a bank note : ~ Detroit, Midi., ^W/sl. 6, 19 0¥- 3^kv&& viantkoy after date, J--- promise to pay to the order of_ cJ^k& (k-a^yvyyidyV^ioyt c/tatiancit Bank 3^^v-a ku.ncLi&cL and y^ Dollars at Tlie Coimnercial J^atl. Bank of Detroit, Mich., with interest at 6 per cent per annum itntil due, and seven per cent per annum thereafter until paid. Value received. If Mr. Rowe wishes to borrow $200 he takes the above note to the bank and, if necessary, either furnishes an indorser or collateral security, such as bonds, etc., which he assigns to the bank. If the bank authorities are satisfied, he receives $200 — |3 = 8197, the interest for 3 mo. being deducted. This interest is called bank discount. Days of grace are now rarely used by bankers. 285. In discounting notes, banks count forward by days or months as stated in the note and usually reckon 360 days in the year. Thus, a note dated July 22 at 60 days will mature July 22 + 60 days, or Sept. 20, A note dated July 22 at 2 mo. will mature Sept. 22. BANKS AND BANKING 191 EXERCISE 67 Each of the following notes is ; discounted on the date of issue. Find the ( late of matui •ity and the discount. Date of Note Time Fa(E Katk 1. Jan. 2, 1905 60 da. 11000 6% 2. Aug. 14, 1905 3 mo. $525 5% 3. Aug. 1, 1905 90 da. $387.50 6% 4. April 20, 1905 30 da. $500 6% 5. June 27, 1905 2 mo. $325 T% 286. Business men frequently take notes due at some future date from their customers, and in case money is needed before the notes are due, sell them to a bank. (Such a note is shown on page 179.) The seller must give satisfactory security. The bank pays the sum due at maturity less the discount from the date of discount to the date of maturity. The sum paid by the bank is called the proceeds. These notes may or may not bear interest. The following examples will illustrate both cases : Ux. 1. A note for $527.30, dated Aug. 31, 1904, due in 90 da., without interest, was discounted at the bank Oct. 10, at 6%. Find the proceeds. Solution. Face of note =$527.30 Discount for 50 da. = 4.39 Proceeds = $522.91 Ux. 2. A note for $378.50, dated Aug. 1, 1904, due in 4 mo. at 6%, was discounted at the bank Oct. 1, 1904, at 6%. Find the proceeds. Solution. Face of note = $378.50 Interest for 4 mo. = 7.57 Amount discounted = $386.07 Discount for 2 mo. = 3.86 Proceeds = $382.21 192 BANKS AND BANKING EXERCISE 68 Find the discount and proceeds of each of the following non-interest bearing notes : Time 30 da. 2 mo. 90 da. 60 da. 3 mo. Find the discount and proceeds of each of tlie following interest-bearing notes : „ T^ m Rate of Date of Rate of Face Date Time t t^ t^ Interest Discount Discount Face Date 1. $500 July 1 2. •1225.75 April 10 3. $253.30 Dec. 14 4. 1150.40 Aug. 12 5. 11250 Nov. 1 Date of Discount Rate of Discount July 10 7% May 1 6% Jan. 2 6% Sept. 5 5% Dec. 1 6% 6. 11500 Aug. 10 90 da. 6% Sept. 1 7% 7. $97.30 Oct. 2 60 da. 7% Nov. 1 6% 8. 1152.20 Sept. 4 4 mo. 5% Oct. 20 6% 9. $750.50 Jan. 4 30 da. 4% Feb. 1 7% 10. $431.40 June 17 2 mo. 6% July 10 6% 11. A merchant's bank account is overdrawn $2150.75, and he presents to the bank the following notes, which are discounted Dec. 5 at 6% and placed to his credit. What is his balance ? . Face Date Time f'^™ ^^ Interest $500 Nov. 12 60 da. 5% $1250.25 Sept. 30 90 da. 4% $727.40 Oct. 25 3 mo. no interest 12. For how much must I give my note at the bank, discounted at 6% and due in 60 da., to realize $1500? Sur/fjestum. Find the proceeds of te.vUa.^v Baa/o (Eo-. 196 EXCHANGE This draft is called a time draft and would be taken to Hawkes & Co. as before, who, if they intended to pay it, would write across the face in red ink : Accepted Aug. 21, 1903. Hawkes Sf Co. After writing these w^ords across the draft, Hawkes & Co. have agreed to pay $1000, and the draft becomes the same as a promissory note. 291. Fluctuations of Exchange. If the banks of San Francisco have sokl drafts for a larger sum on the New York banks than they have on deposit in New York, it will be necessary to send money enough to New York to balance the account. The money is usually sent by ex- press at some expense, which must be borne by the pur- chaser of the drafts. In this case a draft on New York would cost slightly in excess of its face. This excess is called a premium. A draft sold at less than its face is said to be sold at a discount. Premiums and discounts are usually quoted as a per cent of the face of the draft. Thus, a quotation of ^^ % premium means that a draft for .f 100 may be purchased for -1100.10. Sometimes the quota- tion is a certain amount per $1000. Thus, if New York exchange is quoted at $1.50 premium at New Orleans, a draft for $1000 will cost $1001.50. The above quotations refer to sight drafts. Time drafts are dis- counted by banks in the same manner as promissory notes. New York City is the greatest financial center of the United States, and so much business is transacted through the New York banks that New York exchange is generally at a premium. Consequently banks are always willing to cash such checks at par value. People in New York usually pay their indebtedness outside of the city by checks or drafts on New York banks, which find a ready sale at any bank. 292. The Clearing House is an institution organized by the banks of every large city to facilitate settlement of claims against one another. EXCHANGE 107 Clerks from each bank bring daily to tlie clearing house the checks, etc., due them from all other associated banks, each bank being represented by a separate package. Balances are struck between the credits and debits of each bank against all others, and the manager certifies the amount which each bank owes to the associated V)anks or is entitled to receive from them. The banks whose debits exceed their credits pay in the balance to the clearing honse, which issues clearing house certificates to the banks whose credits exceed their debits. In the New York Clearing House, which is the largest in America, nearly sixty billion dollars of clearings were made in 1904, with only three billions of dollars of balances paid in money. EXERCISE 69 1. A quotation of # 2.50 preniiuni is equivalent to what per cent ? 2. What is the cost in Kansas City of a draft on New York for $ 67.50 at \ f premium ? 3. What is the cost in Galveston of a draft on New York for ^4380.50 at 12.50 premium ? 4. In Ex. 2 and 3 which city is owing the other money? 5. Find the cost of a draft for f 500 payable in 60 days after sight, exchange being \ Jo premium, interest 6 Jo. 6. Find the cost of sending #67.50 by telegraphic money order if a 10-word message costs 40 ct. and each word in excess of 10 costs 2 ct. * 7. How much would it cost to send the same amount by postal money order ? by express money order ? 8. A draft on New York for 810000 costs 89980 in Chicago. Is exchange at a premium or a discount? What is the rate of exchange ? The balance of trade is in favor of which city ? 9. A merchant has a 60-day note for 81200 discounted at the bank at 6 Jo and purchases a draft with the proceeds, 198 EXCHANGE exchange -fl.OO. He sends the draft to a creditor to apply on account. How much is phiced to his credit ? 10. A Chicago banker discounts a draft for -^2500, pay- able at St. Louis 60 days after siglito What are the pro- ceeds, exchange at |^ % discount, interest 6 % ? FOREIGN EXCHANGE 293. Foreign exchange is the same in all essential fea- tures as domestic exchange, the difference being that exchange takes place between cities in different countries. 294. A draft on a foreign country, usually called a bill of exchange, is payable in the currency of the country on which it is drawn. 295. Foreign bills of exchange are generally written in duplicate, called a set of exchanges, and are of the follow- ing form : JYew York, CUicf, /^, 1903. Exchange for £100. ^&7^ clciyQ. after siglit of this first of exchange (second of same date and tenor unpaid) pay to the order of €. /if. Tfl&n^d cy}i,& kit ncli &ct jilaa^.')^cL^ at&i tvytcf, and charge the same, ivithout further advice, to To fSavi^cf JSvcytkeAA., ^t^cyuje. €. S'cyx.. jCo-nclcy/b. Jfo. f 736^6. FOREIGN EXCHANGE 199 The duplicate sLil)stitiites "second of exchange" for *' first of ex- change " and " first of same date" for " second of same date," in the original. Eillier one being paid, the other becomes void. 296. 1 he par of exchange between two countries is the value of the monetary unit of one expressed in Unit of tlie other. Thus, the gokl in the English pound is worth if) 4. 8665. Exciiange on Paris and other countries using the French monetary system is usually quoted at so many francs to the dollar, sometimes at so many cents to the franc. The par of exchange is about 5.18^ francs to the dollar, or 19.3 cents to the franc. Exchange on Germany is quoted at so many cents on 4 marks. The par of ex- change is 95.2 ; quoted in cents per mark it is 23.8. EXERCISE 70 1. What is the cost of a draft on London for £ 150, exchange §4.925? 2. What is the cost of a draft on Paris for 1200 francs, exchange 5.20? 3. In either Ex. 1 or 2 is the balance of trade in favor of the United States? 4. A tourist purchased a letter of credit and drew £ 80 at London, 1500 francs at Paris, 750 marks at Berlin. How much did the letter cost him if exchange is | % premium at London, | % premium at Paris, and ^ % pre- mium at Berlin? 5. What is the cost of a draft on Leipsic for 525 marks, exchange 96? exchange 24? 6. What is the cost of a draft on London for £ 75, exchange rif4.857? STOCKS AND BONDS . 297. A corporation or stock company is an association of individuals under the laws of a state for the purpose of transacting business as one person. Large-scale produc- tion is now usually conducted by corporations. A corporation is managed by officers elected by aboard of directors who are chosen by the stockholders, each stockholder having as many votes as he owns shares of stock. The capital stock is divided into a certain number of shares, the par value of which is determined by the number of shares into which the stock is divided. Thus, a capital stock of -150000 may be divided into 500 shares of $100 each, or 2000 shares of -125 each, etc. Stockholders may own any number of shares and participate in the profits according to the number of shares they own. 298. If a company is prosperous and makes more than its expenses, a dividend is paid to the stockholders. The dividend is usually a certain per cent of the par value of the stock, or sometimes so many dollars per share. If the rate of dividend is higher than the current rate of interest, there usually will be a demand for the stock and it will sell at a premium. If the rate of dividend is lower, the demand will be sliofht and the stock will sell at a discount. & 299. Companies frequently issue two kinds of stock, preferred and commoji. The holders of preferred stock are entitled to first share in the net earnings of the corporation up to a certain amount, usually from 5% to 7% of the par value. The holders of common stock are entitled to a share, or all of what is left after tjie dividend on the preferred stock is paid. 200 STOCKS AND BOXDS 201 Stock is sometimes issued to the stockholders of a corporation without a corresponding increase in the vahie of the property. Such stock is called watered stock. A corporation may be prohibited by its charter, or by law, from paying dividends in excess of a certain amount. ThuSj if a corporation with a capital stock of «| 100000 makes $16000 and wishes to pay this amount in dividends, but is prohibited from paying more than 8 %, watered stock, equal in amount to the capitalization of the corporation, may be issued to the stock- holders and an 8% dividend (= | IGOOO) may be declared upon this new basis. 300. Since it is difficult for individuals to buy and sell stock personally, the business is usually transacted through a stock broker, who charges a small per cent, called broker- age (usually J% ), of the par value of the stock bought or sold. The stock broker generally belongs to an organi- zation called a stock exchange. The New York Stock Exchange is the most important in America. 301. Generally each stockholder is responsible only to the extent of the par value of the stock he owns. In the case of national banks, however, a stockholder is liable to the amount of the par value of his stock in addition to the amount paid for the purchase of the stock. 302. Investors often buy stocks and hold them for the dividend they yield. Speculators buy them to sell at a profit. Speculators usually buy on a margin, that is, they pay only a part of the purchase price and borrow the rest by depositing the certificate of stock as cojlateral. A man who buys stock on a 20-point margin pays down 20% of the par value and borrows the rest. A "bull" is a buyer of stocks which he hopes to sell at a profit. He acts on the belief that prices will go up. A "bear" is a seller of stocks which he does not possess, but borrows on the belief that prices will go down. Tlius, if a stock is quoted at 50, a " bear," thinking it will go down to 45, may sell at 50, and deliver borrowed stock to the purchaser. If the stock goes down to 45, he will purchase it and return it to the owner, thus realizing a profit of 5 points. This is called selling stocks " short." Bears are said to be " short " of stock and bulls " long." 202 STOCKS AND BOXDS 303. When for any reason a stock company finds the amount of money paid in by stockholders insufficient, it generally borrows money and issues bonds, secured by a mortgage on the property of the company. These bonds are written agreements to pay a certain sum of money within a stated time and at a fixed rate of interest. Bonds have a prior claim over any kind of stock. 304. National governments, states, counties, and cities often issue bonds, but without mortgages, the credit of such organization being considered good. Registered bonds are issued in the name of the owner, and are made payable to him or his assignee. Interest, when due, is sent direct to the owner. Coupon bonds are payable to bearer, and have small interest coupons attached, which are cut off when due, and the amount of interest is collected either personally, or through a bank. There is a coupon for each interest period. Bonds are usually named from the rate of interest they bear, or from the date at which they are payable. Thus, Union Pacific 4's means Union Pacific bonds bearing 4% interest. U.S. 4's reg. 1907 means United States registered bonds bearing 4% interest and due in 1907. Western Union 7's coup. 1900 means Western Union coupon bonds bearing 7% interest and due in 1900. 305. The following quotations show the prices paid for stocks and bonds on a certain day. The daily newspaper will furnish the best source for quotations.. Stocks Bonds Amalgamated Copper . 72| U.S. New 4's reg. . . . 135i A. T. and S. F. . . . 88| U.S. New 4's coup. . 136i A. T. and S. F. preferi ■ed . 98| U.S. 3's reg . 107i Canadian Pacific . . . 131 U.S. 3's coup. . . . . 108 National Biscuit Co. . . 98| Atchison 4's . . . . • 10-21 National Biscuit Co. pre- N.Y. Central 3j's . . . 103| ferred . 105 J C. B. and Q. 4's . . . 93^- N.Y. Central . . . . 143 C. R. I. and P. 4's . . . 105 STOCKS AND BONDS 203 Stocks Railway Steel Spring. . . Railway Steel Spring pre- ferred U.S. Steel ........ U.S. Steel preferred . . . \Vabash Wabash prefei-red . . . . AVesteru Union preferred . 33] 872^ 37i 37^ Bonds Southern Ry. 5's . . Detroit Gas Co, .5's Chicago and Alton 3]'s Hocking Valley 4^'s . B. and O. 4's . . . . U.S. Steel 5's . . . 116 lOOi 70J 100^ mi 306. Quotations are usually made at a certain per cent of the par value of the stock or bond. Thus, the quota- tion of 72| for Amalgamated Copper means 72|% of the par value of one share. The purchaser must pay his broker 72^ T2| + 1^ = 72|, and the seller will receive from his broker 95 -8- 307. In the following examples the par value of a share will be taken as -$100 unless otherwise stated. Brokerage at 1^% must be taken into account in each case where not otherwise stated. Ux. 1. A person buys 100 shares of A. T. and S. F. as quoted above, and sells 6 mo. later for 85|^, having received a dividend of 2%. Does he gain or lose, and how much, money being w^orth 6% per annum? Solution. Each share costs and is sold for .-. the gain on each share is .-. the gain on 100 shares is The dividend received = .-. the total gain is The amount invested is 83f + i = 83J, 851 _ 1 =. 85. 85-83^ = 1^, or $1.50. 100 X 11.50 = ^150. 2% of $10000 = 1200. $150 + $200 = $350. 100 X $83^ = $8350. The interest for 6 mo. is h of 6% of $8350 = $250.50. .-. $350 - $250.50 = $99.50 = the net gain. 204 STOCKS AND BONDS Ex. 2. A man sells short 100 shares of Canadian Pacific at 131 and three days later *' covers " (that is, buys the stock) at 128|. What is his net profit ? Solution. Each share sold yields 131 ~\ = $130| Each share is bought for I28f + i = 128| Therefore the gain on each share is f 2 Therefore the net gain on 100 shares is 100 x ^2 = |200. EXERCISE 71 1. The capital stock of a company is f 1000000, ^ of which is preferred, entitled to a 7 % dividend, and the rest common. If 147500 is distributed in dividends, what rate of dividend is paid on the common stock ? 2. A person buys 302 shares of stock, par value 810, for 17 a share, paying 5 ct. a share brokerage. 6 mo. later, after having received a 5% dividend, he sells for 89.75 a share. How much does he make, money being worth 6% ? 3. If, in Ex. 1, 1 77500 is distributed in dividends, which is the better stock to own, common or preferred ? 4. Which is the better property to own, 8 1000 stock in a company at 6 %, or one of its 8 1000 bonds at 4 % ? 5. Which is the safer against loss by theft, a coupon bond, or a registered bond ? Which is the more readily transferred ? 6. Why are U.S. 4's registered quoted at 135], while U.S. 4's coupon are quoted at 136^^? 7. How much will 50 shares of Amalgamated Copper cost ? 8. How much will 75 Atchinson 4's cost ? 9. How much will 100 shares of New York Central cost? STOCKS AND BONDS 205 10. How miicli will 100 New York Central 3.] % bonds cost ? 11. Which should you prefer to own, the 100 shares of stocks or the 100 bonds mentioned in Ex. 9 and 10 ? 12. What is the greatest number of Canadian Pacitic shares that can be purchased for ^ 5000 ? 13. Which is the better investment, a 4% mortgage or Southern 5's as quoted ? 14. Which is the better investment, B and O. 4's or U.S. 5's as quoted? 15. What sum must be invested in Atchison 4's at 102| to secure an annual income of 84120 ? 16. What rate of income will U.S. 3's registered yield ? 17. If I pay $3762.50 for U.S. Steel preferred, how many shares do I buy ? 18. How much must I pay for B. and O. 4's to yield an income of 5% on my investment? of 6% ? 19. What income will a man receive from an invest- ment of $21625 in U.S. 3's coup. ? 20. What dividend can a company declare on a capital stock of $50000 whose net earnings are $7500? • 21. A certain bank pays a semiannual dividend of 3^% on its stock ; what is the annual dividend on 25 shares ? 22. How much must I pay for 5% bonds that the investment may yield 6% income? for 4% bonds? for 3% bonds? 23. A man owms 100 shares of Amalgamated Copper stock. If the company declares a dividend of 5% payable in stock, how much stock will he then own ? 206 STOCKS AND BONDS 24. My broker, after selling for me 200 shares of Wabasli preferred, remitted to me i^9987.50. At what price did he sell the stock ? 25. How much must be invested in U.S. 3"s coup, to bring an annual income of $ 500 ? 26. A bank with a capital stock of 1 150000, declares a semiannual dividend of 3%. What is the amount of the dividend, and how much will a person receive who owns 25 shares ? 27. A gas company declares a 6% dividend and dis- tributes $120000 among its stockholders. What is its capital stock ? 28. A cement company divides ^^ 80000 among its stock- holders. What is the rate of dividend, the capital stock being $ 1000000 ? How much is paid to a person who owns 902 shares of ^ 10 each ? 29. A broker bought for a customer 500 shares of cop- per stock, par value 125, at a total cost of 818015.63. Find the market quotation and brokerage. 30. A man bought 200 shares of New York Central at 143. The market price declined till it reached 139 and then rallied to 141^. Believing that another decline was coming, he sold 500 shares (300 of them short) at 1411. The price continued to rally, however, and he covered by buying 300 shares at 1421. What was the net loss on the whole transaction, making no allowance for interest, but allowing |% brokerage for each sale and purchase? INSURANCE 308. There are two general classes of insurance : in- surance on the person in the form of life, endowment, accident, and health insurance, and insurance on property in the form of fire, marine, live stock, tornado, plate glass, boiler insurance, insurance against bad debts, etc. PROPERTY INSURANCE 309. Tlie principal kinds of property insurance are fire insurance, or insurance against loss by fire ; marine insur- ance, or insurance against loss of vessels at sea, or property on board of vessels at sea ; tornado insurance, or insurance against loss by storms, etc. 310. The written agreement between the company and the person insured is called the policy, and the sum to be paid by the company in case of loss, the face of the policy. The person insured is called the insured, and the amount paid by the insured to the company for insurance, the premium. 311. The premium is usually computed as a certain per cent of the face of the policy, or as a certain sum on each 8100 of insurance. In either case it is called the rate of insurance. Ex. A house valued at 85000 is insured for | of its value at 1.1% per annum. What is the annual premium ? 207 208 INSURANCE How much would the owner lose if the house were burned after seven premiums had been paid ? How much would the company lose ? Solution. Valuation of house is f of ^5000 = $4000. Premium = 1.1 % of $4000 = $44. Loss of owner = $5000 - $4000 + 7 x $44 = $l308o Loss of company = $4000 - 7 x $44 = $3692. The above rate of insurance might have been stated as 11.10 on each |100 insured. EXERCISE 72 1. A house valued at 16000 is insured for | of its value at 1% per annum. What is the annual premium ? How much does the owner lose if the house is burned after 10 premiums have been paid ? How much does the company lose? 2. How much would the owner lose in case the house were damaged by fire to the extent of il500 after 3 pre- miums had been paid ? 3. How much would the owner lose if the house were damaged by fire to the extent of ^^350 after 9 premiums had been paid ? 4. A residence valued at $3500 is insured for | of its full value at I % per annum. The company will insure the house for 3 yr. on the payment of 2^ times the annual premium in advance. How much will it cost to insure the house for 3 yr. ? They will insure for 5 yr. on the payment of 4 times the annual premium in advance. How much will it cost to insure the house for 5 3^-. ? PBOPEllTY INSURANCE 209 5. How much will it cost to insure a manufacturing plant valued at !i^ 65000 at |% and the machinery valued at . 130000 at ^9^%? 6. The insurance in Example 5 is placed in f(nir compa- nies, as follows : building, 125000, i 20000, 115000, 15000; machinery, 112000, ^8000, *(.;000, 14000. What is the annual premium paid each company ? 7. A farmer takes the following insurance on his prop- erty : house valued at $2500 at !{ % ; barn valued at % 1800 at 1 1 % ; live stock valued at ^2600 at -|% ; grain valued at $1800 at 1%; he also takes tornado insurance for 13000 and paj^s 40 ct. per -SlOO for 5 yr. He pays 4 times the annual premium for fire insurance for 5 yr. and 3 times for live stock insurance. What is his total premium for 5 yr. ? 8. A dealer in Buffalo ordered his Chicago agent to buy 4000 bu. of wheat at 72 ct., 2500 bu. of oats at 26 ct., 7200 bu. of corn at 37 J ct., paying 2% commis*sion for buying. The grain was shipped by boat, and a policy at 1J% taken to cover the cost of grain and all charges. Wliat Avas the amount of the policy and what was the premium ? 9. In a town where the regular police force consists of 20* or more patrolmen a company will insure a bank against burglary for 1 yr. for 50 ct. per $100 up to $3000, and 25 ct. per $100 above that amount. How much Avill it cost to insure a bank for $50000 against burglary in such a town ? 10. The annual premium for insuring a plate glass win- dow 6 ft. by 10 ft. is $3.30. How much will it cost a merchant to insure two such windows for 5 yr. ? LYMAX'S ADV. AR. 14 210 INSURANCE LIFE AND ACCIDENT INSURANCE 312. Life insurance is an agreement to pay to the heirs of a person a specified sum upon his death. 313. Endowment insurance is an agreement to pay a specified sum to the person insured if living at the end of a definite period of years, or to his heirs in case of death within that period. 314. Accident insurance is an agreement to pay the person insured a weekly indemnity for loss of time while incapacitated from accident, or a fixed amount in case of permanent injury, such as the loss of both hands, both feet, the entire sight of the eyes, etc., or a fixed amount to his heirs in case of death by accident. 315. Health insurance is an agreement to pay a weekly sum in case of sickness from specified diseases. In addi- tion to the weekly indemnity, health insurance sometimes guarantees the payment of all doctor's fees and special amounts to cover cost of surgical operations. 316. The following tables show the annual rates per $1000 charged by one of the leading life insurance com- panies doing business in the United States. These rates are for life and endowment policies. Insurance companies also issue rates payable semiannually or quarterly. Such rates are slightly in advance of the annual rate, due to the fact that interest is charged on the amounts not paid at the time when the whole premium is due. LIFE AND ACCIDENT IXSUBAXCE 111 WHOLE LIFE POLICIES PARTICIPATING Age Payments 20 15 10 5 Single FOR Life Payments Payments Payments Payments Payment 18 $28 05 $33 75 $45 37 $80 70 $364 89 19 28 47 34 24 46 03 8184 369 % 20 28 90 34 76 46 71 83 02 375 19 21 $19 50 29 35 35 29 47 41 84 24 380 58 22 19 93 29 82 35 84 48 13 85 50 386 15 23 20 38 30 30 3(3 41 48 88 86 80 391 89 24 20 86 30 81 37 00 49 65 88 14 397 82 25 2135 3133 37 61 50 45 89 52 403 93 26 2187 3187 38 24 5128 90 95 410 24 27 22 42 32 43 38 90 52 14 92 43 416 74 28 22 99 33 01 39 57 53 02 93 5K) 423 45 29 23 59 33 61 40 28 53 i^ 95 53 430 37 30 24 22 »4 24 4101 54 89 97 16 437 50 31 24 89 34 89 4177 55 87 98 84 444 86 32 25 59 35 58 42 55 5() 89 100 58 452 44 33 2(5 33 36 29 43 37 57 94 102 38 460 25 34 27 11 37 03 44 21 59 03 104 23 4(xS 30 35 27 93 37 80 45 10 60 16 106 14 476 58 36 28 80 38 61 46 01 6133 108 11 485 12 37 29 72 39 45 46 97 62 &1 110 15 493 91 38 30 69 40 34 47 96 63 80 112 26 502 95 39 3171 4126 48 99 65 10 114 43 512 24 40 32 80 42 24 50 07 66 45 116 67 52180 41 33 95 43 2(1 5120 67 85 118 98 531 ()2 42 35 17 44 34 52 38 69 30 12137 54171 43 36 47 45 48 53 62 70 82 123 83 552 07 44 37 84 46 68 54 92 72 40 126 38 562 70 45 39 31 47 95 56 28 74 04 129 01 573 59 46 40 86 49 30 57 72 75 75 13172 584 76 47 42 52 50 73 59 23 77 54 11:^4 52 59(5 18 48 44 29 52 25 60 82 79 40 137 42 607 85 -49 46 17 53 87 62 49 81 35 140 41 619 76 50 48 17 55 59 64 26 83 38 143 48 63189 51 50 31 57 43 6()13 85 50 146 (55 644 22 52 52 58 59 38 68 10 87 72 149 92 ma 74 53 55 00 6147 70 19 J)0 03 153 28 669 43 54 57 59 63 71 72 40 92 46 156 74 682 28 55 60 34 66 10 74 75 94 99 1()0 30 695 27 56 63 28 68 6() 77 24 97 (>6 l(i3 97 708 38 57 66 42 7141 79 13.57 YEAR Loan Cash Value Paid up Policy Extended Insurance En- dow- Years meni Days 1 3 $82 $92 $147 10 196 4 118 132 203 13 84vS 5 155 173 259 15 $41 6 193 215 314 14 121 7 233 259 368 13 198 8 274 305 421 12 272 9 316 352 474 11 343 10 360 401 525 10 411 11 405 451 576 9 477 12 453 504 626 8 541 13 502 558 675 7 602 14 553 615 724 (3 660 15 606 674 771 5 716 16 659 733 818 4 781 17 716 796 865 3 842 18 774 861 910 2 898 19 835 928 955 1 951 20 1000 EXERCISE 73 From the tables find the annual premium required for : 1. A life policy for % 2500, age 24. 2. A ten-payment life policy for $ 4000, age 29. 3. A ten-year endowment policy for %> 5000, age 40. 4. A twenty-payment life policy for $3000, age 87. 5. A twenty-year endowment policj^ for $6000, age 37. 6. At age 24 Mr. Robbins takes out a life policy for 15000 ; if he dies at the age of 41, how much does tlie face of the policy exceed the premiums paid ? 7. If money is worth 6^ per annum, what do the premiums paid in Ex. 6 amount to ? How much does tlie face of the policy exceed the amount ? LIFE AND ACCIDENT INSURANCE 215 8. At jige 35 Mr. Andrews takes out a -t 5000 twenty- payment life policy ; what is the face of the paid-up life policy that will he given to him if he stops ])aying premiums and surrenders his ])()licy at age 4(j ? What is the guar- anteed cash value of the policy at age 45 ? 9. At age 31 a man took out a ^ 2500 life policy and at age 36 a % 1500 twenty-five-year endowment policy and a $ 1000 twenty-year endowment policy. How much does the insurance exceed the premiums paid if he dies at the age of 43 ? 10. If the annual dividends on a twenty-payment life policy, age 35, average 21 fo of the premiums, liow mucli has a 8 1000 policy cost at the end of 20 years, money being worth 5 fo ? Sugr/estion. $ 37.80 - 21 % of $ 37.80 = $29.86 = the average yearly cost, and 20 x 129.86 + 10 x d% of 20 x i$ 29.86 = 1895.80 = the total cost. 11. If dividends are not paid annually, but are allowed to accumulate for a period of twenty years on the above twenty-payment life policy, the insured would be privi- leged to withdraw the accumulated surplus in cash and still retain a full-paid policy for i 1000 payable at deatli. Should the accumulated surplus amount to 8 391.78 at the end of twenty years, how much does the policy cost, money being worth 5^0 ? 12. Mr. Young takes out a $ 5000 fifteen-payment life policy Nov. 19, 1887, at age 40. In 1902, instead of con- tinuing the insurance, he surrenders for a cash value of §4036.75, which includes the accumulated dividends. Allowing il5 per annum per $1000 for protection af- forded, what rate of interest has his money earned in the 15 years ? TAXES AND DUTIES 319. The expenses of the United States government for pensions, army and navy, salaries of the President, con- gressmen, and other officials, etc., amount to something over 1 1000000 a day. The state must have money for the care of the insane, blind, deaf and dumb, criminals ; for educational purposes, salaries of state officials, etc. The county needs money for public buildings, bridges, salaries, educational purposes, etc. The city and village must have public improvements, fire protection, police, schools, etc. These expenses are met by taxes. TAXES 320. The expenses for the support of the state, county, city, etc., are paid by taxes on real estate and personal property. In addition to the property' tax most states collect a poll tax of from 11 to f 3 from each male citizen over 21 years of age and under 50. 321. The rate of taxation is usually expressed as a certain number of mills on each dollar, or as a certain number of cents on each $ 100 of valuation. EXERCISE 74 1. Tlie valuation of the property of a certain county is $ 7500000. If the general state tax and tlie general county tax are each 60 ct. on each $ 100 and in addition the 216 TAXES AND DUTIES 217 bridge tax is 40 ct. and the school tax 30 ct., what is tlie total tax of the county and what is the amount set aside for each of the above purposes ? 2. What are the taxes of a man who owns 160 acres of land in the above county worth ^ 60 an acre and assessed at I of its value, and personal property amounting to 11850? 3. The total assessed value of property in Michigan in 1901 was '^1578100000. What amount did the State University receive in 1903 from a | of a mill tax ? 4. How much of this tax did a farmer have to pay who owns 200 acres of land valued at ^ lb an acre and assessed at I of its value ? 5. A certain city is bonded for $ 6000 ; its taxable property is valued at 8 7500000. How much of the above bonded indebtedness does a man worth %> 10000 j)ay ? 6. Suppose the above city wishes to build a high school building valued at $ 50000, what will be the tax on each 1 100 ? 7. The taxable property of a certain county is $125000000. What will be the tax on each $>100 to build a courthouse worth ^ 90000 ? 8. The Michigan State Normal College received from the state, in 1903, -$103200. How much of this did a man pay who owns -f 7500 worth of taxable property, the state having property listed at $ 1578100000 ? 9. The assessed value of a town is -^S^ 250000 and tlie amount of tax to be raised is -$3500; Avhat is the rate of taxation ? 218 TAXES AND DUTIES DUTIES 322. The income for the support of the national gov- ernment is derived largely from custom revenue (tariff or duty on imports, collected at customhouses established by the government at ports of entry), and internal revenue (taxes on spirits, tobacco, etc.). 323. ^lerchandise brought into the country is subject to ad valorem duty (a certain per cent of the cost of the goods), specific duty (a certain amount of weight, number, or measure, without regard to value), or both ad valorem and specific duty. Some goods are admitted duty free. Illustrations. Cut glass and laces pay an ad valorem duty of 60%. Machinery pays 45%. Tin plates pay a specific dutj'- of li ct. per pound, horses valued at $150 or less pay $30, and wheat pays 25 ct. per bushel. Cigars pay a duty of $4.50 per pound and 25%, and lead pencils pay 45 ct. per gross and 25%. Books published in foreign lan- guages are admitted duty free. EXERCISE 75 1. What will be the duty on 1 T. 4 cwt. of tin plate ? 2. What will be the duty on 20 gross of lead pencils? 3. What is the cost per gross of lead pencils on Avhich the two rates of duty are equal ? 4. The duty on ready-made clothing is 50%. What is the duty on -16000 worth? 5. If the duty on linen collars and cuffs is 40 ct. per dozen and 20%, wliat is the duty on 10 doz. collars at 75 ct. a dozen and 10 pairs of cuffs at 25 ct. a pair ? 6. What is the duty at 50% on 500 doz. kid gloves at 75 francs a dozen ? 7. Find the duty on an importation of ^750 8s. 4:d. worth of English crockery at 40%. THE PROGRESSIONS 324. By a series is meant a succession of terms formed according to some common law. 325. An arithmetical progression (A. P.) is a series in which each term tlifl'ers from the preceding by a constant quantity called the common difference. Thus, 2, 5, 8, 11, •••, and 1.5, 10, 5, 0, - 5, - 10, •••, are arithmetical progressions. In the first, 3 is the common difference and is added to each term to form the next; in the second, —5 is the common difference and is added to each term to form the next. . 326. A geometrical progression (G. P.) is a series in which each term differs from the preceding by a constant multi- plier called the ratio. Thus, 2, 4, 8, 16, ••., and 18, - 6, 2, - |, |, ..., are geometrical pro- gressions, the ratios being respectively 2 and — -|. 327. Last Term. If a is the first term, I the last term, d tlie common difference, r the ratio, and n the number of terms, we have from the definitions, — 1st 2(1 3d «th A. P. n ('/ + '/) {a + 2(1) ... a+{n-\)d G. P. a or ar^ • n—i .'. the formulas for tlie last of n terms are: A. P. l = a + (n - l)d. G. P. 1 = a/"'-i. 219 220 THE PROGRESSIONS Mr. Find the last term in an xV. P. in which the first term is 10, the common difference 4, and the number of terms 12. Solution. I = a-\- (n- l)d = 10 + (12 - 1) x 4 = 54. Mx. Find the hist term in a G. P. in which the first term is 2, tlie common ratio 3, and the number of terms 5. Solution. I = a?-''-'^ =2x3^ = 162. 328. Sum of Series. A. P. Take the series o, 5, 7, 9, 11, in which « = 3, d = 2, I = 11, and the snni (S)= 35. Then ^^= 3+ (3 + 2)+ (3 + 4) + (3 + G) +(3 + 8), and in reverse order ^^ z. 11 + (11 - 2) + (11 - 4) + (11 - G) + (11 - 8). Adding and canceling tlie 2, 4. 6, and 8, 2 ^^ = (3 + 11) + (3 + 11) + (3 + 11) + (3 +11) + (3 + 11) 3=5(3 + 11). .-. 5: = |(3 + 11).=:35, or the sum of the series equals one half of the number of teryns times the sum of the first and last terms. Take the general series a, a + d, a + 2 d, •••, a -\- {n — l)d. In this series it will be noticed that each term is formed by adding to the first term the common difference mnltiplied by the number of the term less one. Then 5 = rt + (rt + j^2 ^^- ^^^^ ^'^^'^^ second, :] times as far the next second, 5 times as far the third second, and so on, how far will it fall in the twelfth second? How far will it fall in 12 sec. ? 5. Find the 8th term in the series 1, \, J, •••. 6. Kind the snm of 1 + }^ + i + •■• to infinity. 7. Plnd the 7th term in the series 4, —2, 1, •••. 8. Find the valne of 0.423. 9. Find the 5th and 9th terms of the series 3, 6, 12, •••. 10. Find the 9th term of the series g^^, 3^2, iV' *"• 11. Snm to 5 terms the series 9, —6, 4, •••. 12. Find the value of 0.2: 0.28; 0.24; 1.7145. 13. Find the sum of 3 + 0.3 + 0.03 H to infinity. 14. Find the sum of the first 25 odd numbers ; the first 25 even numbers. 15. What is the distance passed through by a ball before it comes to rest, if it falls from a height of 40 ft. and rebounds half the distance at each fall ? LOGARITHMS 331. Early in tlie seventeenth century, John Napier, a Scotchman, invented logarithms, by the use of which the arithmetical processes of multiplication, division, evolution and involution are greatly abridged. 1 2 1 4 2 8 3 16 4 32 5 64 6 128 7 256 8 512 9 1024 10 2048 11 4096 12 8192 13 16384 14 32768 15 65536 16 131072 17 262144 18 524288 19 1048576 20 332. Many simple arithmetical operations can be performed by the use of two columns of numbers, as given in the annexed table. The left-hand column is formed by writing unity at the top and doubling each number to get the next. The right-hand column is formed by writing opposite each power of 2, the index of the power. Thus 512 = 2^, the number opposite 512 indicating the power of 2 used to produce 512. Ex. 1. Multiply 4096 by 64. From the table 4096 = 21^ and 64 = 2«. .-. 4096 X 64 = 212 ^ 2^ = 212+6 ^ ois ^ 262144 (from table). The student should notice that the simple oj)eration of addition is substituted for multiplication, the product being found in the left-hand column opposite 18, the sum of 12 and 6. Ex. 2. Divide 1048576 by 2048. 1048576 -- 2048 = 2'^ - 2" = 220-11 = 2^ place of division) . 224 512 (subtraction takes the LOGARITHMS 225 Rr. 3. Find sy;J2768. \/;32708 = v^L>i5 = 2V = 23 = 8 (division takes the place of evolu- tion). In the preceding table tlie niind)ers in the right-hand column are called the logarithms of the corresponding numbers in the left-hand column. 2 is called the base of this system. Tlierefore, the logarithm of a number is the exponent by tvMch the base is affected to j^Toduce the number. 333. Any other base than 2 might have been used and columns similar to the above formed. In practice 10 is always taken as the base and the logarithms are called common logarithms in distinction from the natural loga- rithm, of which the base is 2.71828. Common logarithms are indices^ positive or 7iegative^ of the poiver o/ 10. From the definition of common logarithms, it follows that since 100=1, log 1 = 0. 101 ^ 10, log 10 = 1. 10-2 = 100, log 100 = 2. 103 = 1000, log 1000 = 3. etc. 334. Since most numbers are not exact powers of 10, logarithms will in general consist of an integral and deci- mal part. Thus, since log 100 = 2 and log 1000 = 3, the logarithms of numbers between 100 and 1000 will lie between 2 and 3, or will be 2+ a fraction. Also since log 0.01 = - 2 and log 0.001 = - 3, the logarithms of all numbers between 0.01 and 0.001 will lie between — 2 and — 3 or will be — 3 -f ^ fraction. The integral part of the logarithm is called the characteristic and the decimal part the mantissa. LYMAN 'S AT)V. AR. — 15 10-1 ^ 0.1, log 0.1 = - 1. 10-2 = 0.01, log 0.01 = - 2. 10-3 ^ 0.001, log 0.001 = - 3. 10-4 = 0.0001, , log 0.0001 = - 4. etc. 226 L0GARITH2IS 335. The characteristic of the logarithm of a number is independent of the digits composing the number, but depends on the position of the decimal point. Charac- teristics, therefore, are not given in the tables. Thus, since 246 lies between 100 and 1000, log 246 will lie between 2 and 3, or will be 2 + a fraction. Again since 0.0024 lies between 0.001 and 0.01, its logarithm lies between —3 and —2, or log 0.0024 = — 3 -f a fraction. 336. From the above illustrations it readily appears that the characteristic of the logarithm of a nuynber^ partly or ivholly integral^ is zero or positive and one less than the number of figures in the integral part. 337. The characteristic of the logarithm of a pure deciynal is negative and one more than the number of zeros preceding the first significant figure. EXERCISE 77 1. Determine the characteristic of the logarithm of 2 ; 526; 75.34; 0.0005; 300.002; 0.05743. 2. If log 787 = 2.8960, what are the logarithms of 7.87, 0.0787, 78700, 78.7? 338. The mantissa of the logarithm of a number is independent of the position of the decimal point, but depends on the digits composing the number. Mantissas are always positive and are found in the tables, for mov- ing the decimal point is equivalent to multiplying the number by some integral power of 10, and therefore adds to or subtracts from the logarithm an integer. LOGARITHMS 227 Thus, log 76.42 = log 76.42, log 764.2 = log 76.42 x 10 = log 76.42 + 1, log 7642 = log 76.42 x 10'^ = log 7(5.42 + 2, log 7.642 = log 76.42 x lO-^ = log 76.42 + ( - 1 ). So that the mantissas of all numbers composed of the digits 7642 in that order are the same, since moving the decimal point affects the characteristic alone. Log 0.0063 is never written - 3 + 7993, but 3.7993. The minus sign is written above to indicate that the characteristic alone is negative. To avoid negative characteristics 10 is added and subtracted. Thus, 3.7993 = 7.7993 - 10. 339. The principles used in working with logarithms ai-e as follows : I. The logarithm of a product equals the sum of the logarithms of the factors. II. The logarithm of a cpwtient equals the logarithm of the dividend minus the logarithjii of the divisor. III. The logarithm of a poiver equcds the index of the power times the logarithm of the number. IV. The logarithm of a root equcds the logarithm of the number divided by the index of the root. For let 10^ = n and 10^ = m, then log n = X and log m = y. Therefore, since mn = 10-^+J', log nni = X -\- y = log ;i + log ??i; and ■ 71 -^ m = 10'-^, then log— =x — y — log n — log m. m 228 LOGARITHMS^ Also rf = (100 " = 10'% then log n'" = rx = r log n. Finally y/n = 8 and in row N find 4. The mantissa 5843 will be found at the intersection of the row 38 and column 4. .-. log 384 = 2.5843. What is log 3.84 ? log 38.4 ? log 0.0384 ? (2) When the number consists of more than three figures. Find as above the mantissa of the logarithm of the number consisting of the first three figures. To correct for the remaining figures interpolate by assuming that^ for differences small as compared ivith the numbers., the differ- ences betiveen numbers are proportional to the differences between their logarithms. This statement is only approxi- mately true, but its .use leads to results accurate enough for ordinary computations. Ux. Find log 3847. iVIantissa of log 3850 = 5855. Mantissa of log 3840 = 5843. 10 0.0012. Mantissa of log 3847 = 5813 + j\ of 0.0012 = 5851. The difference between 3840 and 3850 is 10, the difference between the mantissas of their logarithms (5855 — 5843) is 0.0012. Assuming that each increase of 1 unit between 3840 and 3850 produces an in- crease of 1 tenth of the difference in the mantissas, the addition for 3847 will be 7 tenths of 0.0012 or 0.00081. 5813 + 0.00084 = 5851. Therefore, the mantissa of log 3847 = 5851. 230 LOGARITHMS N 1 2 3 4 5 6 7 8 9 10 0000 0043 0086 0128 0170 0212 0253 0294 0:334 0374 11 0414 0453 0492 0531 0569 0(507 0645 0682 0719 0755 12 0792 0828 0864 0899 09;34 0969 1004 1038 1072 110(5 13 1139 1173 1206 1239 1271 1303 1335 1367 1:399 1430 14 15 1461 1492 1523 1553 1584 1614 1644 1673 1703 1732 1761 1790 1818 1847 1875 1903 1931 1959 1987 2014 1() 2041 2068 2095 21 '>2 2148 2175 2201 2227 2253 2279 17 2304 2330 2355 2380 2405 24:30 2455 2480 2504 2529 18 2553 2577 2601 2625 2648 2672 2695 2718 2742 2765 19 20 2788 2810 2833 2856 2878 2900 2923 2945 2967 2989 3010 3032 3054 3075 3096 3118 3139 3160 3181 3201 21 3222 3243 3263 3284 3304 3324 3345 33(i5 3385 3404 22 3124 3444 3464 3483 3502 3522 3541 3560 3579 3598 23 3617 3636 3655 3674 3692 3711 3729 3747 3766 3784 24 25 3802 3820 3838 3856 3874 3892 3909 3927 3945 3962 3979 3997 4014 4031 4048 4065 4082 4099 4116 41:33 26 4150 4106 4183 4200 4216 4232 4249 4265 4281 4298 27 4314 4330 4:346 4362 4378 4393 4409 4425 4440 445(5 28 4472 4487 4502 4518 4533 4548 4564 4579 4594 4(509 29 30 4624 4639 4654 4669 4683 4698 4713 4728 4742 4757 4771 4786 4800 4814 4829 4843 4857 4871 4886 4{H)0 31 4914 4928 4;>42 4955 4969 4983 4997 5011 5024 50:58 32 5051 5065 5079 5092 5105 5119 5132 5145 5159 5172 33 5185 5198 5211 5224 5237 5250 5263 5276 5289 5302 34 35 5315 5328 5340 5353 5366 5378 5391 5403 5416 5428 5441 5453 5465 5478 5490 5502 5514 5527 5539 5551 36. 5563 5575 5587 5599 5011 5623 5635 5647 5658 5670 37 5682 5694 5705 5717 5729 5740 '5752 5763 5775 578(5 38 5798 5809 5821 5832 5843 5855 5866 5877 5888 5899 39 40 5911 5922 5933 5944 5955 5966 5977 5988 5999 (3010 6021 6031 6042 6053 6004 6075 6085 6096 6107 6117 41 6128 6138 6149 6160 6170 6180 6191 6201 6212 (5222 42 6232 (5243 6253 6263 6274 6284 6294 6304 6314 (5:i25 43 6335 6345 6355 6365 6375 6:385 6395 6405 6415 6425 44 45 6435 6444 6454 6464 6474 6484 6493 6503 6513 6.-22 6532 6542 6551 6561 6571 6580 6590 6599 6609 6618 46 6628 ()637 6(546 6656 6665 6675 6684 6693 6702 (5712 47 6721 6730 <5739 6749 6758 6767 6776 6785 6794 6803 48 6312 6821 6830 6839 6848 6857 6866 6875 6884 6893 49 50 6902 6911 6920 6928 6937 (5946 6955 6964 6972 6i)81 6990 6998 7007 7016 7024 7033 7042 7050 7059 70(57 51 7076 7084 7093 7101 7110 7118 7126 7135 7143 7152 52 7160 7168 7177 7185 7193 7202 7210 7218 7226 7235 53 7243 7251 7259 7267 7275 7284 72<»2 7300 7:308 731(5 54 7324 7332 7340 7348 7356 7364 7372 7380 7388 739(3 N 1 2 3 4 5 6 7 8 9 USE OF TABLES 231 N 1 2 3 4: 5 6 7 s {) 55 7404 7412 7419 7427 7435 7443 7451 7459 74(56 7474 ")(; 7482 741M) 7497 7505 7513 7520 7528 75;;() 754:5 7551 57 7559 75()(; 7574 7582 7589 7597 7(i()4 7(512 7619 7(527 58 Hi'M 7642 7()49 7657 7664 7672 7679 7686 7(594 7701 5!) 60 7709 7716 7723 7731 773.8 7745 7752 77(50 77(57 7774 7782 7789 771KJ 7803 7810 7818 7825 7832 7839 784(5 (il 7853 7860 78 8195 8202 8209 8215 8222 8228 8235 8241 8248 8254 9 9474 9479 9484 9489 81) 90 9494 9499 9504 9509 9513 9518 9523 9528 9533 9538 9542 9547 9552 9557 9562 95(56 9571 9576 9581 9586 1)1 9590 9595 9600 9605 9609 9614 9619 9(524 9628 9(533 92 9638 9()43 9647 9652 9657 9()61 9(56(5 9671 9675 9680 93 9(585 9689 9694 9699 9703 9708 9713 9717 9722 9727 94 95 9731 9736 9741 9745 9750 9754 9759 9763 9768 9773 9777 9782 9786 9791 9795 9800 9805 9809 9814 9818 96 982,^ 9827 9832 9836 9841 9845 9850 9854 9859 98(53 97 9868 9872 9877 9881 9886 9890 981U 9899 9903 9W8 98 9912 9917 9921 9926 9930 9934 9939 9943 9948 9952 99 9956 9961 9965 9969 9974 9978 9983 9987 9991 99<)6 X 1 2 3 4 5 6 7 8 9 232 LOGARITHMS EXERCISE 79 1. Find log 1845. 2. Find log 6.897. 3. Find log 0.04253. 342. To find the number corresponding to a given logarithm. The number corresponding to a logarithm is called the antilogarithm. The characteristic determines the position of the deci- mal point. (1) If the mantissa is found iyi the tables^ the number is found at once. Ex. 1. Find antilog 3.5877. The mantissa is found at the intersection of column 7 and row 38. .-. antilog 3.5877 = 3870. (2) If the exact mantissa is not found in the tables^ the first three figures of the corresponding number can be found and to them can be annexed figures found by interpolation. Ex, 2. Find antilog 3.5882. log 3880 = 3.5888 log required number = 3.5882 log 3870 = 3.5877 log 3870 = 3..5877 10 0.0011 log req. no. - log 3870 = 0.0005 3870 + f A of lo] = 3874.54+. The two mantissas in the table nearest to the given mantissa are 5888 and 5877 differing by 0.0011. Their corresponding numbers, since the characteristic is 3, are 3880 and 3870, differing by 10. The USE OF TABLES 233 diiference between the smaller mantissa 5877 and the required man- tissa 5882 is 0.0005. Since an increase of 11 ten thousandths in mantissas corresponds to an increase of 10 in the numbers, an increase of 5 ten thousandths in mantissas may be assumed to corres])oiid to an increase oi'{\ of 10 in the numbers. Therefore the number is 3870 +(r\ of 10) = 3874.54+. EXERCISE 80 1. Find antilog 2.9445; aiitilog 2.40G5. 2. Find antilog 1.6527 ; antilog 3.7779. 3. Find antilog 1.9994; antilog 0.7320, 343. Tlie cologarithm of a number is the logaritlnn of its reciprocal. The cologarithm of 100 equals the loga- rithm of Y^Q-, i.e. — 2. As the cologarithm of a number equals the logarithm with its sign changed, adding the cologarithm will give the same result as subtracting the logarithm. The former is sometimes more convenient. Since log 1 = 0, .-. log _ = log 1 — log n =0 — log n, n therefore colog n = — log n. To avoid negative results it is often more convenient to add and subtract 10. Then colog n = 10 — log n — 10. Ex. 1. Find colog 47.3. log 1 = 10.0000 - 10 log 47.3 = 1.6749 colog 47.3 = 8.3251 - 10 In subtracting 1.6740 or any other logarithm from 10. the result may be obtained mentally by subtracting the right hand figure from 10 and all the others from 9. 234 LOGARITHMS 452 X 23 Ex. 2. Find the value of 5871 X 29 log ^^- ^ "-^ = log 452 + log 23 - log 5371 - log 29 ^ 5371 X 29 ^ "^ "^ ^ = log 452 + log 23 + colog 5371 + colog 29 log 452 = 2.6551 log 23 = 1.3617 colog 5371 =6.2699 -10 colog 29 = 8.5376 - 10 log 0.066728+= 8.8243 - 10 Therefore '^''^" ^ "'^ = 0.06672S+. 5371 X 29 Ux. 3. Find log 50'. log 50? = I log 50 log 50 = 1.6990 f log 50 = I of 1.6990 = 1.2742 1.2742 = log 18.8 .♦. 50i- = 18.8. EXERCISE 81 Find the value of : 1. (5x4^7)*. 6. 0.0625 -^ 0.25. 1 ^ 31 X 47 X 53 2- 225' " 29x43 x50' 3/ 23 X 30 3 ; ) 4| % ; (/?) five sixty- fourths ; (cZ) three thousand one hujidred jB.fteen thou- sandths. 34. Write a number that shall be at the same time simple, composite, abstract and even. State why it fills each of these requirements, 35. Add together 15262986957 and 3879, and multiply the 19th part of the sum by 76. 36. In trying numbers for factors, why is it unnecessary to try one larger than the square root of the number? 37. Find the cost, at 25 ct. a rod, of building a fence round a square 10-acre field. 38. How many cords of wood can be stored in a shed 16 ft. long, 12 ft. wide and 6 ft. high? 39. Find the sum of 11, | x 1-|, 3, -^q. Express the result as a decimal. 248 EXERCISES FOR REVIEW 40. If I sell I of a larm for what -| of it cost, what is my per cent of gain ? 41. I sell goods at 15% below the market price and still make a profit of 10%. What per cent above cost was the market price ? 42. How was the principal unit of the metric S5^stem determined ? Explain the relation between this unit and the metric units of capacity and weights. 43. Find the cube root of 4.080659192. 44. Prove that the product of any three consecutive numbers is divisible by 6 or by 24. Determine when it is divisible by 6 ; when it is divisible by 24. 45. The diameters of four spheres are 3.75, 5, 6.25 and 7.5. Prove that the volume of one of them is equal to the volume of the remaining three. 46. A merchant buys goods to the amount of $4000 ; in order to pay for them he gets his note for 60 da. dis- counted at a bank. If the face of the note is $4033.61, what is the rate of discount ? 47. Prove that the exact interest of any sum for a given number of days is equal to the interest of the same sum for the same number of days (as usually computed) diminished by y^g- of itself. 48. A sells a certain amount of 5% stock at 86 and invests in 6% stock at 103; by so doing his income is changed $1. What amount of stock did he sell? Was his income increased or diminished ? 49. Divide | by |- and demonstrate the correctness of tlie work. NEW YOBK STATE UEGENTS' EXAMINATIONS 249 50. Multiply 42.35 by 3.14159, using tlie contracted method and finding the result correct to two decimal places. Prove the work by division, using the contracted method. 51. A man borrows >^4500, and agrees to pay princi[)al and interest in four equal annual installments. If the rate of interest is 6%, what will be the amount of each annual payment ? 52. AVhen it is Monday, 7 A.M., at San Francisco, longi- tude 122° 24' 15" W., what day and time of day is it at Berlin, longitude 13° 23' So" E. ? 53. When exchange is at 5.18, find the gain on 100"^ of silk bought in Paris at 2 francs a meter and sold in New York at 89 ct. a 3^ard, the duty being 6% ad valorem. 54. Find the face of a sisfht draft that can be bou"-ht o o for '1)585.80 when exchange is at a premium of |%. 55. Divide 0.8487432 by 0.075637 and multiply the quotient by 0.835642. Find the result correct to three decimal places, using the contracted methods of division and multiplication of decimals. 56. Express in Avords each of the following: 600.035, u.uo^, ouogQQQ, 5000, looo- 57. A body on the surface of the earth weighs 27 lb. Assuming that the radius of the earth is 4000 mi., find the weight of the same body 2000 mi. above the surface. (The weight of a body above the surface of the earth varies in- versely as the square of the distance from the center of the earth.) 58. Washington is 77° 3' W. longitude and Pekin 116° 29' E. longitude. When it is 9.30 p.m., Tuesday, Dec. 31, 1901, at Washington, what is the time of the day, the day of the week, and the date at Pekin ? 250 EXERCISES FOR REVIEW 59. Find the exact interest on 1590 from Sept. 18, 1893, to March 1, 1894, at 4|%. 60. Is the merchants' rule or tlie United States rule for computing partial payments more favorable to the debtor ? Give reasons. 61. A locomotive runs | of a mile in | of a minute. At what rate au hour does it run ? (Give analysis in full.) 62. The edges of a rectangular parallelopiped are in the proportion of 3, 4 and 6; its volume is 720 cu. in. Find its entire surface. 63. A note for 8250, due in 1 yr., with interest at 6%, is dated Jan. 1, 1892. What is the true value of this note Oct. 1, 1892 ? 64. At 10 A.M. Jan. 5 a watch is 5 min. too slow; at 2 P.M. of Jan. 9 it is 3 min. 20 sec. too fast. When did it mark correct time ? 65. A gallon contains 231 cu. in. ; a cubic foot of water weighs 62.5 lb.; mercury is 13.5 times as heavy as Avater. How many gallons of mercury will weigh a ton ? 66. Find the face of a note that will yield 1 861.44 pro- ceeds when discounted for 90 da. at 6%. 67. A merchant buys goods listed at 12500, getting successive trade discounts of 20, 10 and 5 ; he sells his goods at 20% above the cost price, taking in payment a note at 60 da. without interest ; he then gets the note discounted at 6% and pays his bill. Find his entire gain. 68. A person deposits $100 a year in a savings bank that pays 4% interest, compounded annually. How much money stands to his credit immediately after the fifth deposit ? NEW YORK STATE llEGENTS' EXAMINATIOXS -2.^)1 69. Cluin«;t' '2()^)'j-]'2 ill tliu (|uiiiai'y scale to an e<[uivalent number in the deeinial scale, and prove the work. 70. A New York merchant remitted to London throui^di his broker £12000 18s. [hi Find the cost of the draft if exchange is at 4.81>| and brokerage is {%. 71. In extracting the cube root state and explain the process of («) separating into periods, (b) forming the trial divisor, (^) completing the divisor. 72. A merchant buys goods at a list price of 8800, o-ettincT discounts of 10, 20 and 5 with 60 da. credit, or a further discount of 5% for cash. How much will he gain by borrowing at ()% to pay the bill? 73. At a certain election 510 votes were cast for two candidates; | of those cast for one equaled | of those cast for the other. How many votes were cast for each candidate ? 74. If the cost of an article had been S% less, the gain would have been 10% more. What was the per cent gain ? 75. Prove that the excess of 9's in the product of two numbers is equal to the excess in the product of the excesses in the two factors. 76. Derive a rule for marking goods so that a given reduction may be made from the marked price and a given profit still made on the cost. 77. The greatest common divisor and the least common multiple of two numbers between 100 and 200 are respec- tively 6 and 3150. Find the numbers. 78. How much will the product of two numbers be in- creased by increasing each of the numbers by 1 ? Give proof. 252 EXERCISES FOR REVIEW 79. The longer sides of an oblong rectangle are 15 ft. and the diagonal is 20 ft» Find its area. 80. Find the fourth term of the following proportion and demonstrate the principle on which the operation is based : 8 : 12 = 10 : rr. 81. Demonstrate the following: If the greater of two numbers is divided by the less, and the less is divided by the remainder, and this process is continued till there is no remainder, the last divisor will be the greatest common divisor. 82. Find in inches to two places of decimals the diagonal of a cube whose volume is 9 cu, ft. 83. Compare the standard units of money of the United States, England, France, and Germany as to relative value. Find the value of $100 in each of the other units. 84. A dealer sent a margin of 81500 to his broker, April 16, 1905, and ordered him to buy 100 shares of American Sugar stock. The broker filled the order at 131 1 and sold the stock May 1 at 1261 charging 1% brokerage each way and 6% interest. How much money should be returned to the dealer ? 85. A four months' note for 1 584, without interest, is discounted at a bank at 5% on the day of its issue. Find the proceeds of the note. 86. What is the difference between a discount of 10% and two successive discounts of 5% each on a bill of $832 ? 87. If I buy cloth at $1.20 a yard, how must I sell it so as to gain 25% ? 88. Find the cost of paving a walk 140"^™ wide and | of a kilometer long at $1.25 a square meter. NEW YORK STATE liEGENTS' EXAMINATIONS 253 89. Indicate tlie factors which, multiplied together, equal the square root of 441. 90. A newsboy buys 144 daily papers at 20 ct. a dozen, and sells them at 8 ct. each. At the end of da. lie has 18 papers on hand. How much has he made (lurin<^r the week ? 91. The diameters of two concentric circles are 20 ft. and 30 ft. Find the area of the ring. 92. What yearly income will S^ 2267.50 produce when invested in U. S. 4's at 113|, brokerage J% ? 93. Find the amount of 8486.50 for 1 yr. 5 mo. and 17 da. at 5|% simple interest. 94. I buy stocks at 4% discount and sell at 4% pre- mium ; what per cent proht do I make on the investment ? 95. A merchant buys goods to the amount of f 1575 on 9 months' credit; he sells them for 81800 ca^h. Money being Avortli 6%, how much does he gain? 96. Find the cost, at 60 ct. a yard, of carpeting a room 16 ft. 4 in. wide and 21 ft. 6 in. long with carpet 27 in. wide, if the strips of carpet run lengthwise. 97. Find the cost at 45 ct. a roll of papering the walls of a room 16|- ft. long, 15 ft. wide, and 12 ft. high, mak- ing no allowance for openings. 98. Find the cost of plastering the four walls and the ceiling of a room 15 ft. long, 12 ft. wide and 9 ft. high at 15 ct. a sq. yd., allowing 6 sq. yd. for openings. Approved Text-Books in Algebra By WILLIAM J. MILNE, Ph.D.. LL.D. 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