QA foz Ce Consta6U*s lEtrucaticnal .Series. EIRST STEPS n AEITHIETIO. BY JAMES CUEEIE, A.M., Author of " A Practical Arithmetic for Elementary Schools,' *' Early and Infant School-Education," etc. THOMAS LAURIE, EDINBURGH. Price Sixpence. LIBRARY OF THK University of California. Received ^/^.^Ci , iSgy . Accession No, G 7 i/ 7^ . Class No. ■^/^7f VOPZQ VA Digitized by the Internet Archive in 2007*with funding from IVIicrosoft Corporation http://www.archive.org/details/firststepsinaritOOcurrrich C0nstaIjle's (^butatinnal 5tvics. FIRST STEPS IN ARITHMETIC BY JAMES CURRIE, A.M. ^ \ PRINCIPAL OF THE CHURCH OF SCOTLAND TRAINING-COLLEGE, EDINBURGH ; AUTHOR OF " EARLY AND INFANT SCHOOL-EDUCATION," ** COMMON SCHOOL EDUCATION," ETC. EDINBURGH : THOMAS LAURIE, COCKBURN STREET LONDON : SIMPKIN, MAKSHALL, AND CO. ; and HAMILTON, ADAMS, AND CO. '-7^7^ 1!^ '^f m f y PREFACE. This treatise of Arithmetic is designed to comprise all that is needed by the pupils of common schools, and by those of higher schools till they have completed their elementary education. It is not one of theory, since the instruction of pupils of their standing must be, in the main, practical ; nor, on the other hand, is it a mere collection of examples, since the only practical instruction worthy of the name is that which sets the processes before them in a rational way. It aims throughout at that just combination of theory with practice which is the greatest merit of an elementary text-book. The explanations are given concisely, and in the form in which they are likely to be soonest apprehended by the pupil ; whilst the exer- cises for practice will be found to be very numerous and carefully graduated. In particular. Notation and the four elementary operations, on a satisfactory knowledge of which the pupil's subsequent progress depends, are treated with great fulness. An introductory text-book of Arithmetic should not be a mere condensation of a higher one ; it should devote the space which it gains from the omission of certain of the more advanced rules to the ampler treatment of those which are fundamental. Where the arithmetic of a school is weak at all, it is in these rules that the weakness almost invariably lies : and it is in these rules, according to the testimony of all competent authorities, that the most material improvement in the teaching of the subject is to be looked for. In the arrangement of the treatise the author has kept in view the requirements of the Privy-Council for Elementary Schools and Pupil- Teachers, although he has not limited himself by them. The Miscellaneous Exercises at the end have been taken chiefly from the papers of the Privy-Council and Dick Bequest Examinations. For the convenience of junior classes the early chapters, treating of the elementary operations with simple numbers and with money, and forming pp. 1-64 of the present work, are published separately under the title of '' First Steps in Arithmetic." CONTENTS. PAGE Tables of Value, 5 Notation, 9 Addition, . 17 Subtraction, • . . 25 Multiplication, 32 Division, 39 Miscellaneous Exercises, 48 Compound Addition— Money, 51 Compound Subtraction— Money, 54 Compound Multiplication— Money, . . . . .55 Compound Division— Money, 58 Reduction— Money, 62 l^IlSCELLANEOUS EXERCISES, 66 Compound Rules— Weights and Measures, .... 70 Miscellaneous Exercises, 79 Bills op Parcels, 83 Practice, 84 Rule of Three, 92 Compound Rule of Three, 101 Measures and Multiples, 103 Vulgar Fractions, 106 Decimal Fractions, . . . 114 Simple Interest, 121 Compound Interest, 124 Discount, 125 Stocks, 127 Brokerags, 128 Insurance, 129 Profit and Loss, . . .130 Square Root, 131 Mensuration, 132 Miscellaneous Exercises, 137 TABLES OF MONEY, WEIGHT, AND MEASURES. MONEY. I. Money of Account. 4 farthings, /. = 1 penny, d. 12 pence = 1 shilling, s. 20 shillings = 1 pound, £ II. Coins in Circulation. Bronze. 2 farthings = 1 halfpenny, ^d. 2 halfpence = 1 penny. Silver. 4 threepenny pieces = 1 shilling. 8 groats = 1 shilling. 2 sixpences = 1 shilling. 2 shillings = 1 florin, Jl. 2 shillings and sixp. = 1 half-crown. 5 shillings = 1 crown, cr. Gold. 10 shillings 4 half-crowns 6 florins 2 crowns 20 shillings 8 half-crowns , _ , govereic-n 10 florins r- iso\ereian. 4 crowns Paper money is also in use. One pound-note represents the value of 20s. , or one sovereign ; and there are also five-pound notes, ten-pound notes, twenty-pound notes, fifty-pound notes, and one-hundred-pound notes. The guinea, formerly a gold coin = £1, Is., is still recognised as a standard value, though the coin itself is not in use : so the half-guinea, or 10s. 6d. 1 half-sovereign. WEIGHT. III. Avoirdupois Weight is used for all common goods. 16 drams, dr. = 1 ounce, oz. 16 oz. = 1 pound, lb. 28 lb =1 quarter, qr. 4 qrs. or 112 a =1 hundredwt. cwi. 20 cwt. = 1 ton, T. Also, 14 lb =1 stone, st. IV. Troy Weiglit is used for weighing the precious metals and jewellery. 24 grains, gr. = 1 pennyweight, dwt. 20 dwt. = 1 ounce, oz. 12 oz. = 1 pound, lb Note.— The lb Troy = 5760 gr. The lb Avoir. = 7000 gr. LENGTH. V. Lineal Measure is used for measuring length , and is hence often called long Tueasure. 12 inches, in. = 1 foot, ft. 3 feet = 1 yard, yd. 5^ yards = 1 pole, po. 40 poles = 1 furlong, fur. 8 fuiiongs = 1 mile, ml. Tradesmen use what is called a foot- rule of three feet long for measuring with, on which the feet are divided into inches, and the inches into eighth parts, tenths, or sixteenths. For longer measurements, a tape or line of 22 yards, similarly divided, is com- monly used. Obsolete measures, hut still used for special purposes, are the following :— 1 line = Ath inch. 1 palm = 3 inches. 1 span = 9 inches. 1 cubit =18 inches. 1 hand (for mea-^ suring height of >• = 4 inches. horses) ) 1 fathom (for inea-) ^ g f^et. surmg depth) ) 1 geographical mile = 1 mile 266 yds. [nearly. 1 league = 3 geog. miles. 1 degree = 60 geog. miles. VI. Cloth. Measure is used for measuring cloth. 2i inches = 1 nail, nl 4 nails = quarter, qr. 4 quarters = 1 yard, yd. 5 quarters Also, = 1 ell. The draper's rod, one yard long, is divided according to this measure ; though in practice, fractions (six- teenths) of a yard are more commonly used. Vn. Land Measure is used for measuring land. Sur- veyors use a chain for this purpose, called Gunter's chain, 22 yards (or 4 poles) long, and divided into 100 parts or links. = 1 chain of 100 Iks. = 1 furlong. Note.— The link = 7|f inches, 22 yards 10 chains VI TABLES OF MONEY, WEIGHT, AND MEASURES. SURFACE. VIII. Square Measure, sometimes called superficial measure, is used for measuring surface or area. 144 sq. in. = 1 sq. ft. 9 sq. ft. = 1 sq. yd. 30J sq. yd. = 1 sq. po. (or perch, per.) 40 sq. po. = 1 rood, ro. 4 roods = 1 acre, ac. 640 acres = 1 sq. ml. Still used for special purposes are the following measures : — 100 sq. feet ■= 1 square of flooring. ^'llq%o: ""'} = ^ rod of brickwork. 36 sq. yd- =.1 rood of building. Land-surveyors, as stated above, use the chain of 100 links, though they express the result of their measure- ments in this table :— 10,000 square links = 1 square chain. 10 square chains = 1 acre. SOLIDITY. IX. Cubic Measure is used for measuring the contents of solid hotlics, e.g., masses of stones or earth (hence often called solid mea- sure), or of bodies which have the shape of solids, e.g., rooms, cis- tenis, etc. 1728 cubic in. = 1 cubic ft. 27 cubic ft. = 1 cubic yd. Shipping is measured by tonnage, timber by loads, and general goods sometimes by barrel-bulk, thus : — 42 cub. ft. = 1 ton shipping, T. sh. 40 cub. ft. rough ) timber V = 1 load, lo. 60 do. hewn j 5 cub. ft =1 barrel-bulk, B.B. CAPACITY. X. Measure of Capacity- is used for the measurement of liquids, and also of dry goods, like grain, etc. 4 gills, gi. = 1 pint, pt. 2 pints = 1 quart, qt. 4 quarts = 1 gallon, gal. 2 gallons = 1 peck, pk. 4 pecks = 1 bushel, bu. 8 bushels = 1 quarter, qr. The peck, bushel, and quarter are used for dry goods only. For wine and beer, casks of various sizes are used, of which the most common are — FOR WINE. The puncheon = 84 gal. The pipe = 126 gal. The tun = 252 gal. FOR BEER. The kilderkin = 18 gal. The barrel = 36 gal. The hogshead, lihd. = 54 gal. But these casks are not standard measures, and vary in their capacity. The imperial gallon contains 277*274 cubic inches. TIME. XI. Measure of Time. 60 seconds, sec. = 1 minute, min. 60 minutes = 1 hour, ho. 24 hours = 1 day, da. 7 days = 1 week, wk. 52 wks. 1 day, or ) _ 365 days j- - >ear, yr. 366 days = 1 leap year. 100 years = 1 century. The year is divided into 12 calendar months : — January 31 days July 31 days February 28 August 31 March 31 September 30 April 80 October 31 May 31 November 30 June 30 December 31 Every year (with very rare excep- tions) whose number is divisible by 4, is a leap year ; in which February has 29 days. Thirty days have September, April, June, and November : All the rest have thirty-one. Excepting Febraary alone, Which has but twenty-eight days clear. And twenty-nine in each leap year. The lunar month = 29 da. 12 ho. 44 min. The solar year = 365 da. 5 ho. 48 min. 48 sec, i.e., nearly 3C5 days 6 hours (the Julian year). Quarterly Terms. In England. Lady-Day, . March 25. Midsummer, . June 24. Michaelmas, . Sept. 29. Christmas, . Dec. 25. In Scotland. Candleinas, . Feb. 2. Whitsunday, . May 15. . Lammas, . Aug. 1. Martinmas, . Nov. 11. TABLES OF MONEY, WEIGHT, AND MEASURES. VU The centuries are reckoned, among Christian nations, in numerical order from the birth of our Lord (called the Christian era) : thus the years 1 to 99 are the first century, 100 to 199 the second, and so on. This is the nine- teenth century. Any particular year, e.g., 1864, is denoted 1864 a.d., i.e.. Anno Domini, in the year of our Lord. The years before the birth of our Lord are reckoned back in order from that event: thus 1460 a.c, means Ante Christum, or before Christ. INCLINATION. XXL Angular Measure is used for measuring the angle or inclination of one line to another. 60 seconds, " =1 minute, ' 60' = 1 degree, " 90° = 1 right angle, L 860° = 1 circle, The following Tables are subjoined for reference : — Paper Measure. 24 sheets = 1 quire, qu. 20 quires = 1 ream, re. 21^ quires = 1 perfect ream. Cloth Measure. 5 quarters = 1 English ell. 3 quarters = 1 Flemish ell, Fl. E. 6 quarters = 1 French ell, Fr. E. 37 inches = 1 Scotch ell, S. E. Apothecaries' Weight. OLD MEASURE. 20 grains, gr. = 1 scruple, ^ 3 scruples = 1 drachm, 3 8 drachms = 1 ounce Troy, ? 12 ounces = 1 ft Troy. NEW MEASURE (1862). 437^ grains = 1 ounce Avoir. Apothecaries' Fluid Measure. 60 minims, TT^ = 1 fluid drachm, /. 3 3 fl. drachms = 1 fluid ounce, /. § 16 ounces = 1 ft 20 ounces = 1 pint, O 8 pints = 1 gallon, C FOEEIGN MONEY. United States. 10 cents = 1 dime. 10 dimes = 1 dollar, $ 1 dollar = 4s. 2d. France. 100 centimes = 1 franc. 1 franc = 9^d. nearly. Canada. Accounts are kept in £ s. d. currency, of which £1 = 16s. 8d. sterling. East Indies. 16 annas = 1 rupee. 1 rupee = Is. lOJd. OLD SCOTCH MONEY AND MEASURES still recognised in Scotland for certain purposes. Money. 1 shilling Scots = Id. sterling. £1 Scots = Is. 8d. do. being one-twelfth of the same names sterling. 1 merk = Is. l^d. Lineal Measure. 37 inches = 1 ell. 6 ells = 1 fall. 4 falls = 1 chain. 1 chain = 1| Imp. chain nearly. Square Measure. 36 sq. ells = 1 square falL 40 sq. falls = 1 rood. 4 roods = 1 acre. 1 acre = IJ Imp. acre nearly. LicLuid Measure. 4 gills = 1 mutchkin. 2 mutchkins = 1 chopin. 2 chopins = 1 pint. 8 pints = 1 gallon. 1 gallon = 3 Imp. gallons nearly. Dry Measure. 4 pecks = 1 firlot. 4 firlots = 1 boll. 10 bolls = 1 chalder. The Wheat Firlot was nearly equal to au Imp. bushel (= "998 bush. ) ; the Barley Firlot nearly equal to 1^ bush. (= 1-456 bush.) The Boll weighs 140 lb Avoir. PROPOSED DECIMAL COINAGE. 1 mil = one thousandth part of £1, or = id. less ^^d. 10 mils = 1 cent, one-hundredth of £1. 10 cents = 1 florin, one-tenth of £1. 10 florins =£1. NUMERATION AND NOTATION. Numbers of One Place. ■ One finger and one finger make two fingers. Two fingers and one finger make three fingers. Three fingers and one finger make four fingers. Four fingers and one finger make five fingers. Five fingers and one finger make six fingers. Six fingers and one finger make seven fingers. S&ven fingers and one finger make eight fingers. Eight fingers and one finger make nine fingers. Bf,^ One, two, three, four, five, six, seven, eight, nine, are the names of numbers used in counting. The naming of numbers is called Numeration. One, three, five, seven, nine, are called odd numbers. Two, four, six, eight, are called ei;e7t numbers. These nine numbers mean so many ones, or ^mits as they are called ; thus two means two ones or two units, three means three ones or three units, and so on. EXERCISE I. JBf. 1. Repeat the table of units, as given above. 2. Repeat it, using balls, marbles, boys, etc., instead of fingers. 3. Repeat it with the numbers alone, thus, '' one and one are two." 4. Count from one up to nine, and from nine back to one. 5. Count the odd numbers from one to nine ; from nine to one. 6. Count the even numbers from two to eight ; from eight to two. 7. Name the two numbers next above five, eight, three, etc.^ 8. Name the two numbers next below six, nine, four, etc. 9. Hold up three fingers, five, seven, etc. 10. How many wheels has a cart ? How many halfpence in a penny ? How many pence in a threepenny-piece ? How many letters in the word '^ dog" ? How many legs has a cow? etc. 11. If I have four pence and give one away, how many do I keep ? If I have six marbles, and get one from James, how many have I ? etc. The nine numbers are denoted by signs or figures, thus : — one, two, three, four, five, six, seven, eight, nine, 1 23 4 5 6 78 9 The figuring of numbers is called their Notation. 1 Bf. means that the ball-frame may be used for illustration. 2 Etc. means that various other c^uestions of the same kind may be given. 10 NUMERATION AND NOTATION. EXERCISE II. 1. Write down the figures— (1.) even along ; (2.) up and down. 2. Name the numbers in Ex. iv. sect. 16. 3. Write down the figures for the same numbers, i 3. Numbers of Two Places. If I count nine on my fingers, I find one finger over. Niiie fingers and one finger make ten fingers ; which is the whole number of them. If I wish to count beyond ten, I must begin again and go round a second time ; that will give me two-times ten or two tens. Three times round will give three-times ten or three te7is ; and so on, up to nine-times round, which will give nine-times ten or nuie tens. One ten is called Ten, denoted by 10. Two tens are „ Twenty, „ , 20. Three tens „ Thirtijj ,, 30. Four tens „ Fort^JJ „ 40. Five tens „ Fifty, „ 50. Six tens „ Sixty, „ 60. Seven tens „ Seventy, „ 70. Eight tens „ Eighty, „ 80. Nine tens „ Ninety, „ 90. The tens are numbers of two "places. They are denoted by the figures for the units with a cipher on the right. The value of a figure is increased ten times by its being written in the second place from the right : thus 3 denotes three units, but 30 denotes three tens. Hence the notation we use is called the decimal ^ notation. The cipher is used to fill up the first or right-hand place, when that place contains no units or nothing ; hence it is commonly called nought or nothing. It is never used alone. EXERCISE III. 1. Repeat the table of tens ; backwards ; by odds ; by evens. 2. Count the tens. 3. Name the tens next above forty, sixty, etc.; next below thirty, eighty, etc. 4. How many fingers have six boys ? eight boys ? etc. Bf. 5. How many boys together have thirty lingers ? seventy? etc. Bf. 6. How many units in eight tens ? six tens '{ etc. 7. How many tens in thirty units ? in seventy imits ? etc. 1 Either from the copy or to dictation. The teacher may vary the exercise by having the figures pointed out on the board from columns written by him- self. 2 From the Latiji word decern, ten. eleven, der loted by 11 twelve, jj 12 thirteen, jj 13 fourteen, » 14 fifteen, j> 15 sixteen. J? • 16 seventeen, jj 17 eighteen. » 18 nineteen, J) 19 NUMERATION AND NOTATION. 11 8. If I have ninety marbles and give away ten, how many do I keep ? If I have seventy, and get ten more, and other ten, how many have I ? etc. 9. Write down the figures for the tens below each other. 10. Name the numbers, Ex. vi. sect. 17, Nos. 1, 2. 11. Write down the figui'es for these numbers. One ten and two units , One ten and three units , One ten and four units , One ten and five units , One ten and six units , One ten and seven units ,, One ten and eight units ,. One ten and nine units ,. The tens-units are also numbers of two places ; the first being the units' place, the second the tens' place. The names of the numbers from 13 to 19 are formed by put- ting the number of the units before that of the tens ; thus thirteen is three and ten, fourteen is four and ten, etc. The names of all the other numbers of two places are formed by putting the number of the tens before that of the units ; thus Two tens and one are called twenty -one, denoted by 21 Two tens and two „ twenty-two, „ . 22 Etc. etc. etc. Three tens and one „ thirty-one, „ . 31 Three tens and two „ thirty-tivo, „ , 32 Etc. etc. etc. When numbers of two places are written below each other, units are written below units, and tens below tens. EXERCISE IV. 1. Repeat the table of tens-units fron ten to twenty, from twenty to thirty, etc. 2. Count the tens-units from ten to twenty, from twenty to thirty, etc. 3. If one boy holds up the fingers of his right hand, and other three boys all their fingers, how many fingers are up ? how many if another boy holds up his? if another? if one boy removes his ? etc., Bf. 4. If I hold up seven fingers, how many girls must hold up all their fingers to make twenty-seven ? to make thirty-seven ? etc., Bf. 5. Count by tens from thirty-one, from forty-two, etc. Count by tens hack from ninety-eight, eighty-seven, etc. 6. How many are 1 ten and 4 ? 2 tens and 6 ? 4 tens and 7 ? etc, 7. What tens and units make up 18, 27, 33, 47 ? etc. 8. Figure from ten to twenty, twenty to thirty, etc. 5. 6. 12 NUMERATION AND NOTATION. 9. Figiire 2 tens below 2 units, 3 tens below 3 units, etc. ; 9 nnits below 9 tens, 8 units below 8 tens, etc, 10. Name tlie numbers in Ex. vi. sect. 17, No. 3-25. 11. Write down, or tell in order, the figures for these numbers. Numbers of Three Places. Nine tens and one ten make ten tens. As we put ten units together, and call them one-ten, so we put the ten-tens together and call them one hundred. Bf. One hundred is denoted by . 100 Two hundreds „ . . 200 Three hundreds „ . . 300, and so on. The hundreds are mxmhQx^ of three places. They are denoted by the figures for the units with two ciphers on the right. The value of a figure is increased a hundred times by its being written in the third place; thus 3 denotes three units, but 300 denotes three hundreds. The two ciphers are used to fill up the first and second places, when these places contain no units and no tens. EXERCISE V. 1. Count the hundreds, backwards, by odds, by evens. 2. Name the numbers in Ex. ix. sect. 19, Nos. 1, 2. 3. Tell in order the figures in these numbers. 4. How many tens in 100, 500, 800 ? etc. 5. How many hundreds in 10 tens, 70 tens ? etc. 6. Figure the hundreds in an up-and-down line. 7. Figure 1 hund. below 1 ten, 2 hund. below 2 tens, etc. 9 tens below 9 hund., 8 tens below 8 hund., etc. 8. Figure 1 h. below 1 1. below 1 u. — 2h. below 2 1. below 2 u. etc. 9 u. below 9 1. below 9 h.— 8 u. below 8 1. below 8 h. etc. 9. Write down the figures for the numbers Quest. 2. Numbers consisting of hundreds, tens, and units are also numbers of three 2)lciccs ; the first being the units' place, the second the tens' place, and the third the hundreds' place. Their names are formed by combining, in their order, the number of the hundreds, the number of the tens, and the number of the units. Thus — 146 denotes 1 h. 4 t. 6 u., and is called one hundred and forty-six. 270 „ 2 h. 7 t. u., „ two hundred and seventy. 804 „ 8 h. 1. 4 u., „ eight hundred and four. Where there are no units, or no tens, these are omitted in the names, as in the last two numbers. When numbers are written in column, the same places must be kept below each other. NUMERATION AND NOTATION. 13 EXERCISE VI. 1. Count from one hundred to nine hundred and ninety by tens, and from nine hundred and ninety to one hundred by tens. 2. Count from two hundred and forty to two hundred and fifty. „ five hundred and sixty to five hundred and seventy, etc. 3. Name the numbers in Ex. ix. sect. 19, No. 3-25. 4. Tell in their order the figures in these numbers. 5. Figure below each other two hundred and twenty-two, two hundred and two, two hundred and twentj', two hundred, twenty, two :— etc. Repeat the same, beginning with the units. 6. Figure the numbers in Quest. 3. 7, Numbers of One Period. All numbers of one, two, or three places — that is, all num- bers from 1 to 999 — are numbers of one period. Numbers of one place may be written with their period completed by putting two ciphers to the left hand. Thus, since 6 units is the same as hundreds tens 6 units, the number 6 may be written 006, and read no hundred and six. Numbers of two places may be written with their period completed by putting one cipher to the left hand. Thus, since 6 tens 5 units is the same as hundreds 6 tens 5 units, the number 65 may be written 065, and read no hundred and sixty-five. A cipher placed to the left hand of any figure does not alter its 'place, nor, consequently, its value. EXERCISE VII. 1. What are the numbers whose figures in order are three, two, one ; four, nothing, six ; six, four ; seven, two, nothing, ? etc. 2. What figures in order denote two hundred, two hundred and six, five hundred and thirty-two ? etc. 3. What are these numbers made np of ?— Ex. ix. sect. 19. 4. Figure their several parts in order below each other ? . 5. Point out the tens' place in them ? units' place ? hundreds' ? 6. What numbers are made up of these parts, 3 h. 2 1. 6 u. ? 4 h. Ot. 7u.? 7h. 4t. Ou. ? 8h. 4u. ? etc. 7. Read these numbers, 7, 17, 20, 34, etc. (1.) as they stand ; (2.) with their periods filled up ? 8. Read these numbers, 008—080—800—088—880—80, etc. 9. Take any number, as 5. What does it denote Avith one nought before it ? with two ? with one after it ? with two ? with one before and one after it ? Which nought increases its value ten times ? which leaves it unaltered ? What two noughts increase its value one hundred times ? what two leave it unaltered ? What two increase its value ten times ? etc. 14 NUMEKATION AND NOTATION. 10. Write the numbers, eight, ten, twenty-five, etc. (1.) as incoltl* plete periods ; (2. ) as completed periods. ^ 11. Write in figures : fifty-three, thirty-seven, ninety-four, one hundred and seventy, four hundred and sixty-nine, eight hundred and eight, seven hundred and fourteen, seventy-eight, two hundred and eighteen, five hundred and five, six hundred and sixty, three hundred and thirty-three, nine hundred and forty one, five hundred and sixteen, etc. %* When tlie pi'jiil has obtained 'perfect facility in reading and writing Tiwm- tcrs of one period, he may proceed with their addition, subtraction, and multi- plication, returning afterwards to tJie notation of larger numbers. 8. Numbers of Two Periods. Nitie hundreds and one hundred make ten hundreds. As we put ten tens together and call them one hundred, so we put the ten hundreds together and call them one thousand. One thousand is denoted by . . . 1,000 Two thousands, .... 2,000 Ten thousands, .... 10,000 Eleven thousands, .... 11,000 One hundred thousand, . . . 100,000 Three hundred and forty-seven thousand, . 247,000 Any number of thousands is written as if it were units, with three ciphers on the right. If the number contain also hundreds-tens-units, these are written in place of the cyphers. Thus — One thousand five hundred is denoted by . 1,500 Two thousand six hundred and thirty, . 2,630 Ten thousand four hundred and twenty-five, 10,425 Eleven thousand seven hundred and eight, . 11,708 One hundred thousand one hundred and thirty, 100,130 Three and forty-seven thousand three hundred and forty seven, .... 347,347 Every number of thousands has from four to six places, forming two periods. The first period containing the hundreds -tens- units, if there are any ; the second the thousands. %* The two periods are often separated by a comma, as above, to prevent mistakes in reading numbers ; but by practice the pupil vill soon be able to do without it. EXERCISE VIII. 1. Read the numbers, Ex. x. sect. 20. 2. Write to dictation the numbers in same Exercise. 3. In 501274 (or any of the numbers in same Exercise), how many thousands ? hundreds ? tens of thousands ? units ? hundred thousands ? tens? 1 Counters may be used to aid the pupil in writing numbers of one period ; see Note, section 9. NUMERATION AND NOTATION. 15 4. In 347029 (or any of the numbers in same Exercise), what does the 3 denote ? the 9 ? ? 7 ? 4 ? 2 ? 5. What figures in order denote six thousand three hundred ? or any of the numbers in the same Exercise ? 6. What numbers are denoted by the following sets of figures in order, 4, 2, 4, 8 ? 8, 0, 7, 9, 2 ? 3, 6, 5, 2, 0, 1 ? etc. Numbers of Three Periods. Nine hundred thousands and one hundred thousands make a thousand thousands, which we call one Million, One million is denoted by . . . 1,000,000 Two millions are 2,000,000 Ten millions, 10,000,000 Eleven millions, 11,000)000 One hundred millions, .... 100,000,000 Three hundred and forty-seven millions, . 347,000,000 Any number of millions is written as if it w^ere units, with six ciphers to the right. If the number contain also thousands, hundreds, tens, and units, these are written in place of the ciphers, thus :— - One million five hundred thousand is denoted by 1,500,000 Two millions six hundred and thirty thousand, 2,630,000 Ten millions four hundred and twenty-five thousand, 10,000,000 Eleven millions seven hundred and eight thousand five hundred and ten, .... 11,708,510 One hundred millions one hundred thousand and one hundred, 100,100,100 Three hund. and forty-seven mills, three hund. and forty-seven thousand, three hun. and forty-seven, 347,347,347 Every number of millions has from seven to nine places, forming three periods; the first called the units' period, the second the thousands', and the third the millions'. EXERCISE IX. 1. Read the numbers, Ex. xii. sect. 21. 2. In 243,076,549 (or any of the above numbers), how many hun- dreds ? tens of thousands ? tens of millions ? units ? hundreds of thou- sands ? etc. 3. In 804395276 (or any of the above numbers), what does the 5 de- note? 4? 8? 0? 6? 7? etc. 4. Wliat figures in order denote seven millions and thirty thousand, or any of the above numbers ? 5. What is denoted by the 1st place, 2d period ? 2d place, 1st period ? 1st place, 3d period ? 2d period ? 1st period ? 3d place, 1st period ? etc.i 6. Write to dictation the numbers, Ex. xii. sect. 21. 1 This questioning nicay be continued with the help of three periods of counters; thus ••• ••• ••• These may be also advantageously used in the following exercises in dicta- 1 6 NUMERATION AND NOTATION. 10. Numbers of more than Three Periods. Nine hundred millions and one hundred millions make a thousand millions. One thousand millions are denoted by 1,000,000,000 Ten thousand millions, . . . 10,000,000,000 A hundred thousand millions, . . 100,000,000,000 Thousands of millions are written as if they were thousands, and six ciphers are added. If there are also millions, thousands, and units, these are written in place of the ciphers, thus : — One thousand two hundred and thirty millions is 1,230,000,000 Ten thousand five hundred and sixteen millions, five hundred and sixteen thousand, . 10,516,516,000 One hund. and thirty-seven thous., one hund. and thirty-seven mills., one hund. and thirty-seven thousand one hundred and thirty-seven, 137,137,137,137 Every number of thousands of millions contains from ten to twelve places, forming four periods ; which may be separated by commas, as above. StiU larger numbers may be expressed by a fifth period, com- mencing at a million of millions, or, as it is called, a Billion ; or even a sixth period for thousands of billions, thus : — B. M. U. 137,137,137,137,137,137 But numbers of more than three periods rarely occur. U^ Appendix on the Roman Notation. Numbers are sometimes denoted by another set of characters, called HoTuanA These are seven in number, thus : — 1 is denoted by the letter I, 5 by V, 10 by X, 50 by L, 100 by C, 500 by D, and 1000 by M. EXERCISE X. 1. Name the letters, with the numbers they denote. 2. Write down the letters, with the numbers they denote. tion. Thus the pupil may be asked to read 28 14 7, or to write numbers in that way in the first instance, and then to supply the necessary ciphers. 1 So called from having been used in the ancient Roman notation. The ordinary characters are often spoken of as the Arabic, from having come to us through the Arabs. ADDITION. 17 jL^m To denote otlier numbers, these seven characters are combined in two ways — First, a character /oZZov;m<7 another of greater or equal value adds thereto its own value ; thus VI denotes 5 + 1, or 6. Second, a character preceding another of greater value subtracts therefrom its own value ; thus IV denotes 5 — 1, or 4. The only numbers which are denoted by subtraction are the units next imder V and X, and the tens next under L and C ; thus 4 is de- noted by IV, 9 by IX, 40 by XL, and 90 by XC. All the rest are denoted by addition. I 1 X 10 XI 11 C 100 CX 110 M 1000 II 2 XX 20 XII 12 CC 200 CXX 120 MC 1100 III 3 XXX 30 XIII 13 CCC 300 CXXIV 124 MCC 1200 IV 4 XL 40 XIV 14 CCCC 400 CXLIX 149 MD 1500 V 5 L 50 XV 15 D 500 CCXXX 230 MDLXIV 1564 VI 6 LX 60 XLI 41 DC 600 CCCLXI 361 MDCX 1610 VII 7JLXX 70 XLII 42 DCC 700 DXC 590 MDCXCII1692 VIII s'lXXX 80 XLIII 43 DCCC 800 DCCIII 703 MDCCC 1800 IX 9 XC 90 etc. DCCCC 900 etc. MM 2000 The Roman characters are now used only to denote numbers, e.g., the chapters of a book, the hours on the clock, the houses in a street, and the years ; never to calculate with. EXERCISE XL 1. What numbers are denoted by V, X, IV, XX, XXII, XL, etc. ? 2. Name, or write down, letters for the numbers, Ex. iv. sect. 16. 3. Name, or write down, letters for the numbers, Ex. vi. sect. 17. 4. Name, or write down, letters for the numbers, Ex. ix. sect. 19. 6. Do. do. 1250, 1365, 1473, 1582, 1624, 1738, 1806, 1835, 1864. 13. ADDITION. Ex. — Of four flocks of sheep, one contained 35, the second 29, the third 50, and the fourth 47. They were put into one field ; how many sheep were there in all ? Here we have to find one number as large as four given numbers together. The number to be found is called the sum. The sum is got by adding the four given numbers together. The process of adding is called addition; and — when the things to be added are of one kind, as here — simple addition^ The sign of addition is + (plus) : thus 1 + 1 are 2. "We cannot find the sum of the above four numbers at once jr_ -^^ they are too large. We must therefore add them in parts ; ^ for which purpose we must learn the addition of the first nine ~ ' ' numbers. .^ ^^ r^ o3r 18 ADDltlOK. 14. Addition Table. %* This Table should be learnt first in lines even along ; thus, 1 and 1 aro 2 ; 2 and 1 are 3, etc. ; afterwards in lines up and down. Bf. 1 1 1 1 and i2 and '3 and l4 and 5 and 6 and |7and Sand 9 and 1 are 2 1 are 3 1 are 41 are 5 1 are 6 lare 7,1 are 81 are 9 lare 10 2... 32... 42... 52... 62... 7 2... 8 2... 9,2... 10 2 ... 11 3... 43... 53... 63... 73 ... 8 3... 93... 103 ... 11 3 ... 12 4... 5:4... 6 4... 7 4... 84... 9 4 ... 10 4 ... ll|4 ... 12 4 ... 13 5... 6'5... 7|5... 85... 95 ... 10 5 ... 115 ... 125 ... 13 5 ... 14 6 ... 7,6 ... 86 ... 96 ...10,6 ...11 6 ... 126 ... 136 ... 14l6 ... 15| 7... 8 7... 97... 107... 117... 12 7 ... 137 ... 14 7 ... 15 7 ... 16 8... 9 8... 10;8 ... 118 ...128 ...13 8 ... 148 ... 15 8 ... 16 8 ... 17 9... 109... 11,9 ... 123 ... 139 ... 14 1 1 i 9... 159 ... 16 9 ... 17 9 ... 18 EXERCISE T. Bf. 1. Repeat tlie several lines of the table even along ; backwards ; by Ddds and evens. 2. Repeat the several lines up and down in the same orders. 3. 5 and 6 are — ? 8 and 3 are — ? 4 and 9 are — ? etc. 4. 2 + 3 + 5 are — ? 6 + 3 + 8 are — ? etc.^ 5. 2 + 4 + 3 + 7 are — ? 5 + 2 + 2 + 6 are — ? etc. i 6. 2 books and 3 books are — ? I have 5d. and John 4d., how much have we both? John had 3 marbles; if he bought 6 and gained 7, how many has he now ? etc. 7. Write down the columns of the table in order. lO, If one of the numbers to be added contains tens and units, add the units as if they were alone, and prefix the number of tens. Thus — 11 and 1 are 12 ; 12 and 1 are 13 ; 13 and 1 are 14. 11 and 2 are 13 ; 12 and 2 are 14 ; 13 and 2 are 15. Etc. etc. etc. EXERCISE II. 1. Repeat the several lines of this table from 11 to 19, (1.) even along, (2.) up and down. 2. Repeat a similar table for 21-29, 31-39, etc. 3. 11 and 4 are — ? 17 and 8 are — ? etc. 4. 5 -f 19 + 4 are — ? 17 + 6 + 5 are — ? etc. 5. 16 -I- 7 -f 2 -f 4 are — ? 13 + 4 + 9 + 6 are — ? etc. 6. Write down any line of this Table in order. EXERCISE III. Count forward from 1, 2, 3, 4, 5, 6, 7, 8, 9 by twos, then t3y threes, fours, etc., up to nines. 1 In Ques. 4, the sum of the first two numbers, and in Ques. 6, the sum of the first three, should not exceed nine. ADDITION. 19 16. Addition of Numbers of One Place. Ex. — John had 8 marbles, James had 4, William had 7, and Henry 5 ; how many had they amongst them ? 8 We can find the sum of these small numbers 4 without writing ; but if we wish to write down 7 the process, we set the numbers below each other, 5 and add step by step, thus — • — (5 and 7 are) 12 ; (and 4 are) 16 ; (and 8 are) 24— 24 which is the sum required. \* The words within parentheses may be used for some time by the pupil, but should be omitted at the earliest moment he can do without them. The addition may be proved to be correct by adding the column downwards from the top. The sum of any series of numbers is the same in whatever order they are added. EXERCISE IV (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) (14) (15) (16) (17) (IS) (1.) 8 9 2 6 8 3 5 6 7 2 1 4 5 6 7 8 9 2 (2.) 7 4 5 1 6 2 9 7 2 1 3 6 5 9 4 (3.) 5 7 7 5 4 1 8 7 5 6 9 2 5 4 1 3 2 (4.) 6 6 8 4 8 6 4 9 4 3 2 1 8 6 9 3 1 (5.) 4 9 3 5 3 8 7 2 1 5 6 8 5 6 9 8 (6.) 3 8 1 2 4 8 3 6 9 4 5 9 1 7 3 4 (7.) 2 5 7 2 9 5 8 6 5 3 1 2 3 4 1 (8. 9 4 6 3 7 6 2 4 9 3 4 1 5 7 3 (9. 1 5 9 5 4 3 2 1 1 2 3 4 5 6 7 8 (10. 5 1 4 4 4 5 6 7 8 9 8 7 6 5 4 3 2 1 (11. 4 5 3 5 1 2 3 4 5 G 7 8 9 4 3 8 (12. 8 9 2 2 6 1 5 3 4 7 4 3 8 5 2 2 9 (13. 7 6 5 1 7 5 6 4 3 9 2 1 8 4 6 3 8 (14. > 3 2 7 2 6 4 5 4 9 1 2 3 5 8 2 9 (15. 5 6 4 1 7 3 6 5 4 2 9 2 1 1 7 (16. ) 8 7 7 9 1 4 7 8 2 1 5 3 9 4 3 6 6 4 (17. ) 2 6 8 3 6 7 5 4 8 6 2 1 4 7 2 5 (18. ) 9 5 3 4 8 2 4 3 2 6 9 5 4 3 2 1 (19. ) 5 1 5 6 1 2 4 6 8 2 4 6 8 2 4 6 (20. ) 7 8 4 7 1 3 5 7 9 1 3 5 7 9 1 3 5 7 (21. ) 4 2 2 9 4 2 3 6 2 4 7 3 5 8 4 6 9 5 (22. ) 4 3 5 4 3 4 7 8 9 5 6 8 2 1 (23. ) 6 6 1 1 3 4 9 6 8 2 1 4 3 7 7 6 4 (24. ) 8 9 7 2 5 6 2 1 4 9 3 2 8 6 4 3 2 (25. ) 1 8 8 3 4 5 8 9 1 4 7 6 8 5 1 2 3 %* These numbers may be added in parts of columns, or in whole columns, up— down— from left to right— from right to left. And the pupil should work at them a little every day till he attains expertness in adding. 20 ADDITION. J- • • Addition of Numbers of Two Places. The Table given, sect. 14, serves also for the addition of tens, thus : — If 1 and 1 are 2, 1 ten and 1 ten are 2 tens, or 10 and 10 are 20. 2 and 1 are 3, 2 tens and 1 ten are 3 tens, or 20 and 10 are 30. Etc. etc. etc. EXEKCISE V. Perform Ex. i. Quests. 1-5, with tens. Ex. — Of four flocks of sheep one contained 35, the second 20, the third 50, and the fourth 47. They were put into one field : how many sheep were there in all 1 Set the numbers below each other in their places. Then in the units* column : (7 and 9 are) 16, (and 5 are) 21 (units ; set down) 1 (in the units' place), and carry 2 (tens to the tens column). Next, in the tens column : (2 and 4 are) 6, (and 5 are) 11, (and 2 are) 13, rand 13 are) 16 (tens. Set down the) 6 (in the) tens (column), and (the ten tens as) 1 hundred (in the hundreds' column). 35 29 60 47 161 EXERCISE VI. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 20 70 46 23 14 22 34 54 72 29 13 27 41 64 39 30 40 50 13 43 19 96 34 49 64 70 91 27 36 44 40 7 64 14 50 47 94 18 81 49 17 9 4 51 63 50 20 36 29 69 90 25 60 70 80 90 40 56 4 20 60 60 45 56 24 47 18 26 43 31 83 7 91 54 7 40 9 69 73 38 58 37 48 62 15 24 19 9 48 17 70 80 92 20 41 64 74 51 64 82 39 24 47 64 8 80 10 87 34 76 92 82 27 39 51 63 75 87 99 9 90 5 71 47 92 10 45 14 17 20 23 6 9 2 49 30 50 25 56 85 86 37 35 38 41 44 47 50 63 80 40 30 34 81 24 48 29 94 91 87 84 62 59 72 27 50 40 28 73 37 35 15 62 59 18 60 53 27 9 93 16. 76+18+37+9 + 11+24+32+47+3+16+28+76 + 49+60. 17. 22+80+6 + 12+15+93+27+36 + 48+51+70+10+29+8. 18. 37 + 45+15 + 7+1+27+39 + 82+99+4+54+37+10+29. 19. 28 + 57 + 3 + 30+17+37+90+25 + 41 + 8+59+32+87+40. 20. 29+5 + 16+34+64+72+19+7+38+64+28 + 11 + 58+38. 21. 18+90+21+7+9+8+15+27+47+50+62+71 + 89+69. 22. 30+54+4+23+93 + 47 + 50+41 + 39+8+17+28 + 60. 23. 16 + 84+17 + 30 + 85 + 74+32+91 + 11 + 22+50+5 + 15+66. 24. 93+9+8 + 17 + 29 + 40+57+85+36 + 21+73+17+76+82. 25. 87+63+20+6 + 9 + 14+65+89+53+28 + 70+38+67+2. ADDITION. 21 EXERCISE VII. 1. lO + llare — ? 104-12are — ? 10+13are — ? 10 + 21 are — ? etc. 2. 20+11 are — ? 20+12 are — ? 20+13 are — ? 20+21 are — ? etc. 3. 30 + llare— ? 30 + 12are — ? 30 + 13are— ? 30 + 21 are — ? etc. 4. 40 + llare — ? 40+12are — ? 40 + 13are — ? 40+21 are — ? etc. 5. Add the remaining tens in a similar way. 6.50 + 25are — ? 20 + 18are — ? 40 + 29 are — ? etc. 7. 22+15 are — ? 34 + 18 are — ? 75+24 are — ? etc. %* In this last question, it is easier to add the tens first; thus : 34+18 are 4 tens and 12, that is 52. Addition of Numbers of One or more Periods. The table given, section 14, serves also for the addition of hundreds, thousands, etc. ; thus, If 1 and 1 are 2, 1 h. and 1 h. are 2 hs., or 100 and 100 are 200. 2 and 1 are 3, 2 h. and 1 h. are 3 hs., or 200 and 100 are 300. Etc. etc. etc. EXERCISE VIII. Perform Ex. i. Questions 1-5, with hundreds. JEx. — Four heaps of bricks were lying in a field. The first con- tained 208 bricks, the second 349, the third 160, and the fourth 87 ; how many bricks were there in all ? Set the numbers below each other in their places. In the units' column — (7 and 9 are) 16, (and 8 are) 23 (units ; set down) 3 (in the units' 208 place), and carry 2 (tens). 349 In the tens' column— (2 and 8 are) 10, (and 6 160 are) 16, (and 4 are) 20 ; (set down) (in the 87 tens' place) and carry 2 (hundreds). In the hundreds' column — (2 and 1 are) 3, (and 804 3 are) 6, (and 2 are) 8, (set down 8 in the hundreds' place). Sum, 804. %* After some practice in adding, the words within parentheses sliould be omitted. Rule. — Set the numbers below each other in their places ; and add the columns in their order from the units, carrying the tens. 22 19. 2 ADDITION. EXERCISE IX. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 100 200 418 524 638 793 814 701 649 547 890 736 300 500 296 615 800 215 427 693 524 64 47 624 500 900 306 500 524 300 324 414 700 147 562 93 700 100 851 924 357 618 650 710 810 291 50 14 900 300 628 705 184 509 379 327 81 17 900 257 400 800 435 396 225 493 800 967 47 364 73 39 600 600 200 527 604 215 930 413 913 84 654 572 800 700 753 713 593 336 247 258 27 913 209 809 20. 13. 365 + 210+93 + 27+110 + 345+563+207 + 824+85 + 127. 14. 241+56 + 37+256+357 + 842+506+37+81 + 190+429. 15. 306 + 194 + 516 + 70 + 7 + 829 + 593 + 601 + 72 + 720 +18. 16. 501 + 600 + 60 + 372+144+11 + 111 + 29 + 360+306+71. 17. 76 + 706 + 760 + 370 + 307 + 37 + 377 + 84+804+840 + 9. 18. 275 + 360 + 910 + 989 + 724+57+507+37 + 7+190+273. 19. 188 + 560+108+506+56 + 15 + 7 + 180+18+56 + 566. 20. 673 + 840+737 + 928 + 517 + 349 + 210+500 + 618 + 819. 21. 307 + 509 + 910 + 117 + 250+638 + 356+951 + 117 + 89 22. 15 + 27+119+94+101 + 709 + 364 + 87 + 2 + 370+241. 23. 293+18 + 573+194+346 + 504 + 673 + 936 + 19+207. 24. 64 + 604 + 406 + 600 + 640 + 460 + 46 + 83 + 803 + 830. 25. 199 + 96 + 737 + 307 + 516 + 93 + 7 + 16+738 + 259 + 59. EXERCISE X. 1.1 2. 3. 4. 5. 6. 7. 8. 9. 1,000 1000 5000 7000 1896 4567 8456 2408 9406 2,000 1100 500 700 1304 8432 7349 5493 1250 4,000 1200 4000 70 1940 9064 9118 9621 6430 6,000 1300 40 7 1284 2345 2565 8504 8094 8,000 1400 800 600 1700 7298 3894 7632 5432 9,000 1500 9000 4000 1676 5934 5248 4562 8006 7,000 1600 5000 900 1864 6309 7348 3901 9210 5,000 1700 600 6000 1547 7124 9176 2008 5090 10. 11. 12. 13. 14. : 15. 16. 17. 3476 2930 8046 10,000 30,000 70000 80000 27,300 593 456 810 30,000 40,000 30000 500 34,000 24 3948 9 50,000 70,000 6000 60 26,900 896 27 9421 90,000 80,000 200 50000 84,200 7208 639 39 80,000 10,000 8000 9000 53,700 5009 7204 840 40,000 30,000 90000 40 85,600 648 408 7240 20,000 60,000 600 30000 28,400 8 3072 384 50,000 50,000 50000 700 61,060 1 See Note, Ex. viii. p. 7. ADDITION. 23 18. 19. 20. 21. 22. 23. 24. 43,214 73059 83426 29070 45623 82472 19465 28,970 84320 34924 50846 72020 846 3947 36,429 92000 85241 63147 93647 9701 64 82,456 84372 12345 94621 804 35624 94702 93,484 50028 66666 80403 9562 256 876 21,086 90200 93002 70002 93 7 5724 73,481 89301 47020 70020 84756 9470 12730 18,498 56238 26. 13076 70200 7250 85064 9400 25. 27. 28. 29. 30. 100,000 300,000 400000 648,724 910,317 542300 300,000 200,000 8000 720,720 843,256 272484 700,000 700,000 90 843,843 123,000 364862 800,000 60,000 900 920,000 456,700 127859 400,000 50,000 9000 647,000 506,840 780640 900,000 500 80000 564,300 920,100 827938 500,000 800,000 800000 734,310 800,701 910400 600,000 500,000 60000 173,094 308,452 478915 81. 843 + 2465 + 724 + 17 + 10934 + 59470 + 107 + 20094 + 800. 32. 927+250+3070+601 + 38 + 731 + 1456 + 1001 + 27 + 374. 33. 493 + 913 + 67 + 500 + 610 + 1100 + 1420 + 3706 + 3076 + 3760. 34. 39 + 280 + 563 + 730 + 525 + 3482+79 + 2496 + 7314+326 + 89. 35. 470 + 1493 + 293 + 674 + 825 + 300 + 93 + 1910 + 2564 + 836 + 932. 36. 9246 + 29805 + 367934 + 39 + 493 + 9 + 90 + 49321 + 7007. 37. 8439 + 7246 + 297 + 800 + 2094 + 73825 + 493 + 12345 + 936. 38. 4731 + 8472 + 938 + 76 + 3938 + 425 + 18 + 967 + 2005 + 6790. 39. 4901+829 + 736 + 90 + 894+3247 + 9694+8482+386. 40.7000+770+9382+54+504+5004+5040 + 5400+7054. 41.348 + 7+77+777 + 7777 + 77777+9+49 + 17248 + 34. 42. 2693 + 301 + 4 + 404+ 39456 + 327 + 999 + 45602 + 18. 43. 24962 + 37642 + 4936 + 2754 + 930 + 18500 + 2590 + 196. 44. 93642 + 80010 + 930 + 18275 + 60600 + 66000 + 60060. 45. 7285 + 93271 + 893 + 7249 + 90000 + 18506 + 375 + 9640. 46. 8546 + 2764 + 94681 + 27600 + 9300 + 71486 + 8206 + 9. 47. 45894 + 318 + 7462 + 80001 + 90309 + 7402 + 70906. 48. 437 + 938 + 94 + 7300 + 1805 + 72468 + 79005 + 9406 + 50. 49. 6293 + 946 + 8001 + 92465 + 716 + 24070 + 807 + 5005 + 397. 50. 5484 + 29367 + 937056 + 720000 + 804906 + 100000 + 9040. 51 . 249356 + 730854 + 272494 + 800800 + 549304 + 20400 + 701 . 62. 42836 + 90045 + 89362 + 5279 + 7264 + 7649 + 1200 + 937. 53 . 5000 + 50000 + 50 + 505 + 5050 + 5 + 555 + 55555 + 550. EXERCISE XI. Below the sum of the following numbers, write the uppermost, and add again ; below that sum write the second from the top, and add again ; continue the addition in this way till all the numbers are taken in, and find the sum. 24 ADDITION. 21. 1. 235 + 196 + 450+600 + 801. 2. 342+94 + 502 + 86 + 300. .3.279+50 + 116 + 270+207. 4. 100+50+322 + 901+626. 5. 736+941 + 257 + 509 + 316. 6. 241 + 80 + 173 + 428+299. 7.864 + 731 + 279+338+67. 8. 420+204+176 + 815+700. 9. 304+430 + 82+73+371. 10. 536 + 801 + 78 + 306 + 420. 11. 216 + 39 + 500 + 493 + 811. 12. 340+610 + 93 + 217 + 536. 13. 117 + 711 + 270 + 207 + 453. 14. 820 + 304+916 + 732+564. 15. 936 + 576 + 429 + 827 + 517. 16. 320+600 + 66 + 308 + 201. 17. 524+47 + 39+809 + 468. 18. 279+320 + 809+543+397. 1. 238946 72400 930 645046 8434 67 93248 100484 6. 8456729 8040506 3004005 3000400 2790364 8710800 6623938 7708804 2. 900500 2736 93 84293 701856 73900 2784 932048 7. 9203564 964383 728 92100 8056720 5296 931724 8403208 EXERCISE XII. 3. 1,000,000 3,000,000 8,000,000 4,000,000 6,000,000 7,000,000 9,000,000 2,000,000 8. 37,240,000 93,280,000 87,200,400 93,400,860 85,085,023 62,473,908 24,084,573 16,946,004 4. 8000000 800000 80000 8000 800 80 90000 7000000 9. 72,483,624 8,734,724 9,328 904,374 87,208,936 97,318 9,438,729 47,082,970 5. 3,564,236 2,564,304 2,197,629 8,469,038 7,382,01^3 2,946,904 3,842,460 8,080,808 10. 193,700,070 270,937,000 384,256,070 930,184,293 127,249,130 147,234,876 310,249,364 172,849,564 11. 1234567 + 7238049 + 3947246 + 8420800 + 9220000. 12. 8004930 + 12340 + 7248436 + 9436 + 87 + 72456 + 9384567. 13. 72483624 + 8734724 + 9328 + 904374 + 87208936. 14. 27007070 + 2700707 + 94302 + 734 + 85693 + 9438729. 16. 37248734 + 946432 + 87324 + 9256491 + 80724300. 16. 125000890 + 700700700 + 193299870 + 240019000. 17. 738456938+248724807 + 301234563 + 384965724. 18. 2000000 + 7304524 + 5428946 + 7289476 + 1 80050 + 72004. 19. 47849562 + 93859627 + 2507923 + 804974 + 2904 + 93006. 20. 192496924 + 534920815 + 8256293 + 79000600 + 180000018. EXERCISE XIII. 1. John has 38 marbles ; he buys 20 more, wins 17, and gets 11 from a friend. How many has he now ? 2. In a school, the first class has 15 scholars, the second 24, the third 27, the fourth 30, and the fifth 31. How many scholars are in the school ? 3. If I pay 8 shillings for bread, 14 shillings for tea, 7 shillings for sugar, and 11 shillings for butter and cheese ; how many shillings do I pay? SUBTRACTION. 25 4. In a wood there are 41 oak-trees, 18 firs, 63 beeclies, and 9 elms. How many trees in all ? 5. A traveller went 110 miles by train, 62 miles by steamer, 17 miles by coach, and then he had to walk 2 miles. What was the length of his journey? 6. England has 52 counties, Scotland 33, and Ireland 32. How many counties in the whole ? 7. A class of 26 i^upils receives 14 new ones. How many pupils has it now ? 8. Three apple-trees in a garden were shaken for fruit : if one gave 516 apples, and the other two 620 each, how many apples did they give in all ? 9. Three omnibuses started on a pleasure-trip : one carried 23 per- sons, the second 32, and the third 26. If 4 were taken up by the way, how many persons were there in the party ? 10. A grocer pays £140 for shop rent, £37 for taxes, £11 for rent of cellars, and he spends £75 on repairs. What is the whole expense ? 11. In a railway train there were 79 first-class passengers, 101 second-class, and 249 third-class. How many passengers in all ? 12. When will a boy born in 1855 be 69 years old ? 13. From Glasgow to Stirling is 30 miles, from Stirling to Perth 31, from Perth to Aberdeen 90. How far from Glasgow to Aberdeen ? 14. A merchant owes to one creditor £4275, to a second £531, to a third £300, and to a fourth £3005. How much does he owe ? 15. A basket of eggs contains 232, another contains 35 more than the first, and a third 101 more than the second. How many eggs in all? %* Only a few problems of the very simplest kind are presented at this stage : the pupil will be able to continue them to more advantage when he lias learnt the four elementary rules. See Ex. § 55. 23. SUBTRACTIOK Ex. — Of 689 trees in a park, 327 were cut down. How many remained standing ? Here we have to find the difference between two given num- bers, or what remains when the less is taken from the greater. The greater of the two numbers is called the Minuend, which means the number to be diminished ; the less is called the Suh- trahendy which means the number to be taken away. The number which remains is called the Difference or Ee- mainder. The process of finding it is Subtraction ; called, when the things are of one kind, as here, Simple Subtraction. The sig7i of Subtraction is — (minus) ; thus 2 — 1 is 1. We cannot find the difference between 689 and 327 at once ; the numbers are too large. We must, therefore, subtract them in parts ; for which purpose we must learn the subtraction of the first nine numbers. 2G SUBTRACTION. Subtraction Table. 1 from 2 from 3from| 4 from 5 from 6 from 7 from 8 from 9 from 2 is 1 3 is 1 4 is 1 5 is 1 6 is 1 7 is 1 8 is 1 9 is 1 10 is 1 3... 2 4... 2 5 ... 2 6 ... 2 7... 2 8... 2 9... 2 10... 2 11 ... 2 4... 3 5 ... 3 6 ... 3 7 ... 3 8... 3 9... 3 10 ... 3 11 ... 3 12... 3 5... 4 6... 4 7... 4 8... 4 9... 4 10 ... 4 11 ... 4 12 .. 4 13 ... 4 6 ... 5 7 ... 5 8... 5 9... 5 10... 5 11 ... 512 ... 5 13 ... 5 14... 5 7... G 8... 6 9 ... 610... 6 11 ... 6 12 ... 6 13 ... 6 14... 6 15 ... 6 8... 7| 9... 7 10 ... 7ill ... 7 12 ... 7 13 ... 7jl4 ... 7 15 ... 7 16 ... 7 9... 810... 8|11 ... 8112... 8 13 ... 8 14 ... 815 ... 8 16 ... 8 17 ... 8 10... 911 ... 912... 913... 914... 9 15 ... 91Q ... 9 17 ... 9 18 ... 9| w> EXERCISE I. 1. Repeat tlie several columns— backwards — by odds— by evens. 2. Subtract tbe units in each column from its highest number. 3. 3 from 8 leaves — ? 4 from 13 leaves — ? etc. 4. 9 less 2 less 3 is— ? 17-8-4 is —? etc. 5. To 7 add 3 and take away 4? 9+8-2-2 is — ? 6. From 5 books take 2, and how many remain? John had 6 marbles ; if he lost 3 and then 1, how many had he ? Jane has 7 pence ; if she gets 6 pence more and gives away fourpence, what has she now ? etc. 7. Write down the columns of the Table in order. 24-. Subtraction of Numbers of Two Places. The Table given above serves also for tbe subtraction of tens ; thus : — If 1 from 2 is 1, 1 ten from 2 tens is 1 ten, or 10 from 20 is 10. If 1 from 3 is 2, 1 ten from 3 tens, is 2 tens, or 10 from 30 is 20. Etc. etc. etc. If 2 from 3 is 1, 2 tens from 3 tens is 1 ten, or 20 from 30 is 10. If 2 from 4 is 2, 2 tens from 4 tens is 2 tens, or 20 from 40 is 20. Etc. etc. etc. EXERCISE II. Perform Ex. i. with tens instead of units. Ex.— A woman had 76 eggs in a basket ; if she sold 34, how many had she remaining ? Set down the subtrahend below the minuend in 76 its place ; then, subtract the places in their order. 34 4 from 6 is 2 units ; set down the 2 in its place. — 3 from 7 is 4 tens ; set down the 4 in its place. 42 Total difference, 42. SUBTKACTION. 27 To prove the result, add together the subtrahend and the -1^"^ difference ; the sum should be the minuend, since what is taken away from a nuuiber and what is left of it make up between them the whole number. EXERCISE III. (1.) 84 32 (2.) (3.) 56 76 24 36 (4.) 48 25 (5.) (6.) 59 37 32 21 (7.) (8.) 29 70 19 30 (9.) (10.) (11.) (12.) 86 91 64 73 20 31 20 52 13. 14. 15. 47-24 78-51 63-30 16. 17. 18. 39-19 40-20 93-63 19. 81- 20. 56- 21. 78- ■ 41 -36 ■47 22. 85-42 23. 71-31 24. 99-57 25. Though the minuend must always be greater than the sub- ' trahend, any place of the minuend except the highest may be less than the place below it of the subtrahend. Ex. — A teacher has 45 steel pens ; if he distributes 29 to his class, how many are over ? 9 from 5 cannot be taken ; change one of the tens into units, making 15 units in all ; 9 from 15 is 6 45 units, set down the 6 in its place. 29 2 from 3 (the 3 tens remaining) is 1 ten ; set down — the 1 in its place. 16 Total difference, 16. Rule.— Write the less number under the greater in its place ; subtract the columns in their order beginning with the units' ; change one of the next highest name w^hen necessary. Or thus,i 9 from 5 cannot be taken ; add 10 units to the 5, 45 ' making 15 in all ; 9 from 15 is 6 units. 29 Add 1 ten to the 2 tens ; 3 from 4 is 1 te7i. — Total difference, as before, 16. 16 In adding 10 units to the minuend and 1 ten to the subtra- hend, we have added the same number to both. This does not alter their difference ; but makes it easier to find, by keeping each place of the minuend greater than the place below it of the subtrahend. Rule.— Write the less number under the greater in its place ; subtract the columns in their order beginning with the units' ; add te7i to any place of the minuend which is less than the place below it of the subtrahend, and one to the next place of the subtrahend. 1 Both methods of subtraction are given ; the teacher may choose either. 28 SUBTRACTION. EXERCISE IV. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. a. 35 47 53 64 71 60 82 91 47 24 63 30 44 28 34 41 17 39 27 35 49 29 35 53 19 17 45 21 27 9 16 27 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 81. 32. h. 44 21 43 94 42 76 48 32 51 36 22 74 52 81 34 45 18 12 24 47 25 39 29 17 37 17 13 49 26 39 27 19 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44. 45. 40. 47. 84. c. 53 70 42 80 48 30 52 63 74 85 96 97 50 82 43 77 18 43 19 37 19 12 24 34 45 67 29 38 26 27 17 58 49. 66-17 57. 21-13 65. 74-16 73. 44-27 50. 47-28 58. 38-19 m. 81-25 74. 58-39 51. 23-14 59. 72-43 67. 62-37 75. 86-48 52. 55-27 60. 83-54 68. 50-23 76. 90-54 53. 70-34 61. 51-26 69. 27-9 77. 93-65 54. 84-27 62. 66-37 70. 34-15 78. 45-29 55. 95-46 63. 80-43 71. 53-27 79. 74-36 56. 60-24 64. 91-54 72. 67-39 80. 82-43 EXERCISE V. Perform the above exercise mentally. %♦ In doing .so, it is more convenient to subtract the tens first, and then tlie units ; thus in 35-17, 10 from 35 leaves 25, and 7 from 25 leaves 18. 26. Subtraction of Numbers of One or more Periods. The Table given, sect. 23, serves also for the subtraction of hundreds, thousands, &c. ; thus : If 1 from 2 is 1, 1 hund. from 2 hund. is 1 huiid., or 100 from 200 is 100. If 1 from 3 is 2, 1 hund. from 3 hund. is 2 hund., or 100 from 300 is 200. Etc. etc. etc. If 2 from 3 is 1, 2 hund. from 3 hund. is 1 hund., or 200 from 300 is 100. If 2 from 4 is 2, 2 hund. from 4 hund. is 2 hund., or 200 from 400 is 200. Etc. etc. etc. EXERCISE VI. Perform Ex. i. with hundreds instead of units. Ex. 1. Of 689 trees in a park, 327 were cut down : how many remained standing ? 7 from 9 is 2 units ; set down the 2 in its place. 689 2 from 8 is 6 tens ; set down the 6 in its place. 327 3 from 6 is 3 hund. ; set down the 3 in its place. Total difference, 362. 362 SUBTRACTION. 29 Ex. 2. How much greater is 6073 than 484 ? In this example, there is a cipher in the minuend, and the highest place of the minuend has no place below it in the subtrahend. 4 from 13 is 9 for the units' place. 6073 8 from 16 (changing one of the next highest name, 484 which is thousands) is 8 for the tens' place. 4 from 9 (the 9 hundreds remaining when the one 5589 thousand was changed) is 5 for the hundreds' place. from 5 is 5 for the thousands' place. Or thus : 4 from 13 is 9 for the units' place. 9 from 1 7 is 8 for the tens' place. 5 from 10 is 5 for the hundreds' place. 1 from 6 is 5 for the thousands' place. EXERCISE VII. 1. 2. 3. 4. 5. 6. 7. 8. 9. 796 454 805 403 909 100 483 857 564 150 724 203 769 960 637 456 500 415 10. 11. 12. 758-342 975-600 856-326 13. 576-420 14. 874-574 15. 716-516 EXERCISE VIII. 16. 7345-5135 17. 8500-7000 18. 2021-1020 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 547 635 248 429 511 924 700 801 540 707 800 600 219 427 154 274 364 519 451 605 229 593 209 405 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 713 256 391 420 706 300 98 301 279 ]07 401 208 535 328 297 316 802 732 198 49 541 342 194 94 25. 26. 27. 28. 29. 30. 31. 32. S3. 34. 3429 2763 5642 7396 4524 3804 5409 2790 8527 6050 5418 2980 4000 5040 6080 2534 3956 2500 9004 5084 35. 36. 37. 38. 39. 40. 41. 42. 43. 44. 7320 2496 8074 2094 7000 1943 859 3456 5484 2390 9302 1903 7549 1368 2008 5840 509 1123 7309 978 30 45. 407 --298 46. 630-450 47. 275-87 48. 116-58 49. 730-563 50. 805-496 SUBTRACTION^; 51. 357-192 52. 207-84 53. 476-189 54. 520-218 55. 600-315 56. 2809-939 57. 7340-2093 58. 9008-572 59. 1009-450 60. 7084-3921 61. 8000-1090 C2. 5009-3094 63. 9101-9011 64. 7308-5904 65. 8234-4731 66. 2890-1936 EXERCISE IX. \* In the following, find the first remainder less than the subtrahend. 28. 1. 402-86 2. 530-105 3. 736-209 4. 900-121 5. 437-99 6. 7. 215-67 600-143 8. 816-197 9. 701-156 10. 2760-672 11. 8207- 12. 6094- 13. 9400- 14. 8405- 15. 3091- 1938 856 2763 1504 750 16. 7463-1976 17. 5000-987 18. 5185-1978 19. 7320-2094 20. 9017-1853 1. 45060 29360 378923 194033 14. 2567283 730946 EXERCISE X. 2. 3. 4. 5. 38905 27936 84571 73021 19450 10007 25038 49950 45239 S 84901 29308 56402 934856 256094 10. 734085 508506 11. 400000 40401 13. 276408 120394 15. 45070134 29098040 16. 23900140 4015002 17. 50000014 6010305 18. 100200300 100199025 19. 25678-19341 20. 38056-9456 21. 45804-993 22. 50600-5600 23. 89476-4890 24.793246-45600 25.840800-524080 26. 608409- 27. 900000- 28. 2.57931- 29. 456890- 30. 8409302- 31.10000000- 32.57340506- 33. 34. 35. 93560 90909 80002 193456 36. 908567 37. 1001001 38. 8530205 73894219-25934764 170170170-7107100 59340947 -205G0724 123456789-98764532 10000000-100000 500500500-650650 \* In the following, find the first remainder less than the subtrahend. 39. 56030-9807 43. 60930-9493 47. 730294-165085 40. 10101-3427 41. 27092-5083 42. 47138-7509 29. 44. 127936-29647 45. 982401-109472 46. 273408-84279 EXERCISE XI. 48. 100901-10192 , 49. 605090-92071 50. 400000-101010 1. Count back by twos from 100, from 101. 2. Count back by threes from 102, from 101, from 100. 3. Count back by fours from 100, from 101, from 102, from 103. 4. Count back by fives from 100, 101, 102, 103, 104. 5. Count back by sixes from 102, 101, 100, 103, 104, 105. 6. Count back by sevens from 105, 104, 103, 102, 101, 100, 106. 7. Count back by eights from 100, 101, 102, 103, 104, 105, 106, 107. 8. Count back by nines from 108, 107, 106, 105, 104, 103, 102, 101. %* This and the following Ex. should be practised along with the foregoing. SUBTRACTION. 31 EXERCISE XII. 1. 3 + 2+9-5-4 + 1-3 + 8 + 2-7-1 + 3 + 6-4-5 + 5 + 6-8. 2. 7+4-3+5 + 7-5 + 9 + 6-7-3 + 9 + 4-9-1 + 8 + 5-7-3. 3. 15-8 + 9-4 + 9 + 5-3-7 + 4-5 + 10 + 20-11-7 + 4 + 8-5-10. 4. 22 + 8-11-4+8 + 4-7-1 + 9 + 4-3 + 2-7 + 9-4-3 + 6 + 7-6. 5. 40 + 3-7-4 + 20-7-9-4 + 8-7 + 6-8 + 10-5-7-2 + 8 + 10. 6. 14+3-9 + 10-6-2 + 20-5 + 9-7-10 + 11 + 11-8 + 13-4-12. 7. 36 + 9-4-8 + 2 + 5 + 9-12-10+7 + 9-4-8 + 5 + 4-6-8 + 20. 8. 19 + 9-5-8 + 7 + 10-5-11 + 20-11 + 9 + 4-5 + 5-4 + 9-7-4. 9. 28 + 10 + 7-12-10 + 6-7-4+3 + 1-9-8 + 11 + 5-7-9 + 8 + 4. I 10. 50 + 10-20 + 30 + 16-10-10 + 20 + 7-10 + 20 + 50-30-7-10. 11. 49-5 + 12-9 + 16-10 + 8 + 13-5-8-7 + 11 + 7 + 15-30-10 + 9. 12. 53 + 8-11 + 5 + 9-14-15 + 9 + 30-4-9 + 12+20-5 + 16-13-9. Etc. etc. etc. EXERCISE XIII. How many are 37-29 + 48-33 + 79-15 ? Here, instead of subtracting 29 from 37, tlien adding 48, and so on, it is shorter to add together the numbers which are +, then add to- gether the numbers which are — , and find the difference of the two sums, thus : — 37 -29 For it is the same thing whether, in find- + 48 - 33 ing 9 - 2 - 2, we say 2 from 9 is 7, 2 from + 79 - 15 7 is 5 ; or 2 and 2 are 4, 4 from 9 is 5. 164 -77 is 87 X. 125 + 37-84-10+76 + 53-101+56 + 279-184-45 + 293. 2. 74-40+51-9 + 29 + 16-19-5 + 36-27 + 40 + 11. 3. 18 + 15-10 + 40 + 36-19-14+23-39 + 20 + 16-19. 4. 56 + 20-43-27 + 39+24-31 + 64-45 + 21 + 10-34. 5. 90+45 + 16-49-51+6 -15 + 39-60 + 49 + 53--19. 6. 36-19+53-29 + 36-24-11-9 + 64 + 17-24-9 + 14. 7. 49 + 36-29-14+20 + 36-18-9 + 25+84-59-27+40. 8. 74+52-63-10 + 29 + 37-45-37 + 22-51 + 69-19 + 26. 9. 192-56-14+58 + 213-191 + 64-49 + 346-154-48 + 90. 10. 724 - 593 + 824 - 48 + 93 + 702 - 500 + 293 - 59 - 73 + 256 - 100. 11. 50004 - 8456 - 401 + 4592 + 9400 - 10100 + 734 - 809. 12. 29340 - 4560 - 9390 + 7248 - 15600 + 93402 - 56840. 30. EXERCISE XIV. 1. A woman went to market with a basket of eggs containing 342 : if she sold 192, how many did she bring back ? 2. John has 95 nuts, but gives 37 to William. How many does he keep ? 3. A teacher gives out pens to a class of 60 scholars, but the box nas only 37. How many does he want ? 4. A cheese weighs 78 pounds. How much heavier is it than an- other which weighs only 47 pounds ? 5. A tradesman owes £260, but he has only £137. How much does he require to pay his debts ? 32 SUBTRACTION. 6. A cask of siisjar contains 539 pounds' weight. How much must be sold to leave 257 pounds ? 7. J ames has 24 marbles, and his brother gives him 37. How many- must he buy to make up 100 ? 8. If a school has 374 scholars, of whom 27 are in the first class, and 32 in the second ; how many are in the other classes together ? 9. A green-grocer received a basket of apples and pears, 264 in all : 157 were apples ; how many were pears ? 10. A house is worth £520, biit it will cost £84 to repair it. How much should it be sold for ? 11. Edinburgh to Dunbar is 29 miles, and Edinburgh to Berwick is 57 miles. How far from Dunbar to Berwick ? 12. A tradesman earns 16s. a week, and spends 13s. How much does he save in four weeks ? 13. A farmer had in his yard 31 fowls, 17 geese, 24 turkeys, and his ducks made up the entire number of his poultry to 87. How many ducks had he ? 14. How much of 385 yards remains if 93 yards be cut away from the piece? How often may 93 yards be cut away, and what will remain ? 15. A train started with 374 passengers. At the first station 16 went out and 9 came in ; at the second, 11 went out and 25 came in ; at the third, 3 went out. How many passengers left the train at the terminus ?— See Ex. § 55. 31. MULTIPLICATION. Ex. — ^Five boxes of oranges contained 120 each, how many oranges were there in all ? Here we have to find a number equal to 120 repeated 5 times. We could find that by adding 125 to itself 5 times ; but a shorter way is to multijjly 125 by 5. The number to be repeated is called the multiplicand. The number of times it is to be repeated, multiplier. Both are sometimes called the . . factors. The result is called the . . . jjroduct The process is called multiplication ; and, when the multipli- cand is of one kind as here, simple multiplication. The sign of multiplication is X {multiplied by) ; thus 2X2 are 4. We cannot find how much 5 times 125 is by one step ; the multiplicand is too large. We must therefore do it in parts ; for which purpose we must learn the multiplication of the first nine numbers. 32. MULTIPLICATION. Multiplication Table. 33 2 times 3 times 4 times 5 times 6 times 7 times 1 are 2 1 are 3 1 are 4 1 are 5 1 are 6 1 are 7 2 ... 4 2 . .. 6 2 ... 8 2 . .. 10 2 . .. 12 2 ... 14 3 ... 6 3 . .. 9 3 ... 12 3 . .. 15 3 . .. 18 3 ... 21 4 ... 8 4 . .. 12 4 ... 16 4 . .. 20 4 . .. 24 4 ... 28 5 ... 10 5 . .. 15 5 ... 20 5 . .. 25 5 . .. 30 5 ... 35 6 ... 12 6 . .. 18 6 ... 24 6 . .. 30 6 . .. 36 6 ... 42 7 ... 14 7 . .. 21 7 ... 28 7 . .. 35 7 . .. 42 7 ... 49 8 ... 16 8 . .. 24 8 ... 32 8 . .. 40 8 . .. 48 8 ... 56 9 ... 18 9 , .. 27 9 ... 36 9 . .. 45 9 . .. 54 9 ... 63 10 ... 20|10 . .. 3010 ... 40 10 . .. 5010 . .. 60 10 ... 70 11 ... 22ill . .. 3311 ... 44 11 . .. 5511 . .. 66 11 ... 77 12 ... 2412 . .. 3612 1 ... 48 12 . .. 60|r2 . .. 72 12 ... 84 8 times 9 times 10 times 11 times 12 times 1 are 8 1 are 9 1 are 10 1 are 11 1 are 12 2 ... 16 2 ... 18 2 ... 20 2 .. 22 2 ... 24 3 ... 24 3 .. 27 3 . .. 30 3 .. 33 3 . .. 36 4 ... 32 4 .. 36 4 . .. 40 4 .. 44 4 . .. 48 5 ... 40 5 .. 45 5 . .. 50 5 .. 55 5 . .. 60 6 ... 48 6 .. 54 6 . .. 60 6 .. 66 6 . .. 72 7 ... 66 7 .. 63 7 . .. 70 7 .. 77 7 . .. 84 8 ... 64 8 .. 72 8 . .. 80 8 .. 88 8 . .. 96 9 ... 72 9 .. 81 9 . .. 90 9 .. 99 9 . .. 108 10 ... 80 10 .. 90 10 . .. 100 10 .. 110 10 . .. 120 11 ... 88 11 .. 99 11 . .. 110 11 .. 121 11 . .. 132 12 ... 96 12 .. 108 12 . .. 120 12 .. 132 12 . .. 144 ♦»* This Table should be learnt first in lines even along, then in lines up and down. The pupil should practise it daily till he has it thoroughly at command. EXERCISE. I. Bf. 1. Repeat the several lines even along ; backwards ; by odds ; by evens. 2. Repeat the lines up and down ; backwards ; by odds ; by evens. 3. 4 times 5 are — ? 6 times 9 are — ? 8 times 7 are — ? etc. 5 times 4 are — ? 9 times 6 are — ? 7 times 8 are — ? etc. 4. How many fingers have 8 boys ? How many wheels have 9 carts ? How many days have seven weeks ? How many farthings have four pence ? How many units in 5 tens ? How many marbles have 9 boys with 11 each? What cost 6 oranges at 2 pence each? 7 fowls at 3 shillings each ? etc. 5. Name two factors of 18, 24, 96, etc. 6. How many times 7 is 63 ? 21 ? 70 ? etc. 7. 36 is 9 times — ? 72 is 6 times — ? etc. 8. 3 times 6 + 2 are — ? 5 times 8 with 9 added are — ? etc. 4 times 12 less 9 are — ? 7 times 5 - 6 are — ? etc. 9. 2 times 4 and 3 times that are — ? etc. 6 multiplied twice by 2 are — ? etc. 10. Write down the several columns of the Table, c 34 34 MULTIPLICATION. ^Om The Table given above serves also for the multiplication of tens, hundreds, etc. Thus — If 2 times 1 are 2, 2 times 1 ten are 2 tens, or 2 times 10 are 20. If 2 times 2 are 4, 2 times 2 tens are 4 tens, or 2 times 20 are 40. Etc. etc. etc. If 3 times 3 are 9, 3 times 3 tens are 9 tens, or 3 times 30 are 90. Etc. etc. etc. EXERCISE II. Perform Ex. i. with tens in the multiplicand. If 2 times 1 are 2, 2 times 1 h. are 2 h., or 2 times 100 are 200. If 2 times 2 are 4, 2 times 2 h. are 4 h., or 2 times 200 are 400. Etc. etc. etc. EXERCISE III. Perform Ex. i. with hundreds in the multiplicand. Multiplication by Units. Ex. — Five boxes of oranges contained 125 each, how many oranges were there in all ? Set the multiplier below the multiplicand in its place ; then, multiplying each place in its order, 6 times 5 are 25 units ; set down 5 units and carry 2 tens. 125 5 times 2 are 10, and 2 are 12 tens ; set down 5 2 tens and carry 1 hundred. 5 times 1 are 5, and 1 are 6 hundreds. 625 Product, 625. Rule. — To multiply by units, multiply each place of the multiplicand in order, carrying tens. The answer may be proved by adding the multiplicand to itself 5 times ; the sum should be the same as the product. Or we may multiply by 4, the number next below the multiplier, and add the multiplicand to the product. EXERCISE IV. 1. Multiply the following numbers by 2, 3, etc., to 12, in order :— 13 21 31 41 51 61 71 81 91 52 53 54 55 56 57 58 59 60 14 22 32 42 15 23 33 43 16 24 34 44 17 25 35 45 18 26 36 46 19 27 37 47 20 28 38 48 29 39 49 30 40 50 62 72 82 92 63 73 83 93 64 74 84 ' 94 65 75 85 95 66 76 86 96 67 77 87 97 68 78 88 98 69 79 89 99 70 80 90 100 MULTIPLICATION. 35 2. Multiply the several columns mentally. 3. 2 times 27 are — ? 3 times 32 are — ? 4 times 48 are — ? etc. 4. Multiply the following numbers by 2, 3, etc., to 12, in order : — 1. Ill 11. 893 21. 2461 31. 24682 41. 34194 2. 222 12. 248 22. 5382 32. 74394 42. 21384 3. 333 13. 604 23. 2081 33. 31208 43. 75689 4. 444 14. 573 24. 4095 34. 24295 44. 38472 6. 555 15. 421 25. 2496 35. 19064 45. 29319 6. 666 16. 298 26. 5162 36. 70538 46. 82964 7. 777 17. 157 27. 7349 37. 25819 47. 70109 8. 888 18. 820 28. 8210 38. 39147 48. 10840 9. 999 19. 659 29. 9347 39. 16731 49. 30028 10. 427 20. 416 30. 1924 40. 42858 50. 90084 \* This exercise is designed to be performed orally from the book as well as on slate. 35^ Multiplication by Factors. ^x.— Multiply 248 by 24. Since 24 is 6 times 4, we multiply by 24, if 248 * we multiply first by 6, and then that product by q 4 ; thus :— — — The result may be proved by multiplying by ^ 3 and 8, or by 2 and 12 ; which are also factors Z. of 24, and which should therefore give the same 5952 product. A number like 24 which is made up of factors (other than 1) is called a composite number. A number like 7, 11, or 23, which is not made up of factors, is called a prime number. Multiplication by two factors may be used in the case of all composite multipliers between 12 and 144. Practice in multiplying will show the pupil that three factors may often be used for a multiplier with advantage ; thus, 252=4X7X9. EXERCISE V. Multiply, using factors : — 1. 536x14, 15, 21, 22. 6. 4732x77, 81, 84. 2. 270 X 25, 27, 28, 32. 7. 2096 x 88, 96, 99. 3. 905x33, 42, 44, 45. 8. 8405x108, 121, 132. 4. 827 X 54, 55, 56, 63. 9. 7289 x 144, 160, 270. 5. 638x63, 66, 72. 10. 8175x420, 840. 11. 3497 X 16, 18, 48, 72, in two ways. 12. 7302 x24, 36, in three ways. 36 MULTIPLICATION. 36. Multiplication by more than One Place. A cipher annexed to the right of a figure increases its value 10 times, that is, multiplies it by 10. Therefore, to multiply by 2 tens or 20, multiply by 2, and annex the cipher ; to mul- tiply by 30, multiply by 3, and annex the cipher ; and so on. Similarly to multiply by 200, multiply by 2, and annex two ciphers ; to multiply by 300, multiply by 3, and annex two ciphers ; and so on. Hule. — To multiply by tens, hundreds, etc., multiply by the left-hand figure, and annex the ciphers. EXERCISE VI. 1. Multiply the columns in Ex. iv. by 20, 40, 50, 90. 2. Multiply the same columns by 300; 600, 700, 800. fj I , Ex. — A book contains 356 pages, and each page 237 words : how many words are in the book ? Set the multiplier below the multiplicand in its 356 place ; then multiplying by the 7 units, ^37 we have ..... 2492 Multiplying by the 3 tens, we have . . 10680 Multiplying by the 2 hundreds, we have . 71200 Product by whole mutiplier is . . 84372 The result may be proved by interchanging the multiplier and multiplicand, that is, multiplying 237 by 356 ; which will give the same product. Rule. — To multiply by a number of several places, multiply by each place in order from the units, and add the several products. **♦ The pupil may by and by omit the ciphers, denoting the tens and hundreds in the second and third lines of multiplication ; being careful to place the right-hand figure of each line exactly under that place of the multiplier which gives it. Should there be a cipher in the tens, or some higher place in the multi- plier, it is simply passed over in multiplying. EXERCISE VII. 1. 2364 X 29, 37, 43. 5. 8256 x 17, 93, 49. 9. 40001 x 81, 28, 34. 2. 4328 X 39, 61, 86. 6. 6439 x 38, 57, 61. 10. 73000 x 47, 59, 92. 3. 5936 X 28, 46, 59. 7. 20480 x 71, 43, 53. 11. 90000 x 27, 64, 79. 4. 9320 X 19, 73, 31 . 8. 30093 x 98, 83, 78. 12. 70091 x 75, 88, 99. MULTIPLICATION. 37 EXERCISE VIII. 1. 35627x183, 2. 47231x245, 3. 93086x240, 4. 23456x409, 5. 73610x930, 6. 85093x418, 7. 72170x936, 8. 37293x904, 9. 80050x629, 297, 403. 318, 721. 825, 649. 207, 308. 470, 290. 738, 562. 259, 816. 506, 801. 350, 680. 10. 90000x456, 11. 70700x843, 12. 90280 X 706, 13. 456789x297, 14. 724936x840, 15. 459630x364, 16. 536298x230, 17. 210830x821, 18. 914567x439, 789, 910. 529, 365. 604, 209. 399, 536. 908, 273. 814, 518. 563, 720. 913, 713. 546, 208. 1. 500606 X 2. 730000 X 3. 700000 X 4. 830830 X 5. 308070 X 6. 934764 X 7. 621930 X 8. 493628 X 9. 840300 X 10. 621934 X 11. 493002 X 5423, 6106. 2936, 8492. 4028, 5003. 6300, 7240. 8740, 5007. 23418, 93125. 19728, 73465. 27368, 93480. 19030, 80807. 70029, 54309. 56721, 12765. EXERCISE 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. IX. 2389745 6348576 2930840 7394900 8002006 7802058 4932096 7007007 3489493 9000000 4290000 X4567, 7394, 6270. X7321, 8492, 1029. X 6080, 5090, 7200. X 8936, 2009, 5900. X 7290, 5718, 3290. X 35467, 29631. X 84932, 94629. X 93021, 80709. X 29100, 2810L X 73500, 82090. X 80972, 50608. 1. 25473809 2. 73890496 3. 90900900 4. 25608709 5. 70409360 6. 49328914 7. 82483949 8. 72340090 9. 53042485 10. 73249000 X 258956, X 483921, X 259671, X 408506, X 273093, X 506090, X 210000, X 724801, X 493094, EXERCISE 817456. 293185. 798491. 930850. 129608. 709080. 930039. 520936. 891172. 249056. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. X. 490562001 293904510 710842930 256849361 209209209 600040068 394620100 824904561 296382173 493084095 X 362987, X 450813, X 293050, X 259928, X 123456, X 900405, X 736493, X 437285, X 555555, X 828561, 450893. 920854. 493096. 936190. 789012. 908550. 856190. 737292. 505050. 400800. 38. ScLuares and Cubes. A figure like this, which has 4 rows of counters, .... each containing 4, is called a square. The .... number of counters we see by counting to be 16 ; that is, the number even along (4) multi- plied by the number up and down (4). Bf. Similarly 7 rows of trees with 7 trees in each would be a square of 49 ; 10 lines of soldiers with 10 soldiers in each line would be a square of 100. When any number is multiplied by itself, the product is called the square or second power of that number. The square of 4 is denoted 4^. 39. 40. 38 MULTIPLICATION. EXERCISE XI. 1. Repeat the squares of 1, 2, 3, i, etc., up to 12. 2. Find the squares of 13, 14, 15, 16, 17, 18, 19, 20. 3. Find the squares of these numbers : — 1. 784 5. 3456 9. 23456 13. 75423 17. 50005 2. 937 6. 2930 10. 90307 14. 20056 18. 728946 3. 508 7. 4500 11. 58126 15. 90030 19. 809407 4. 610 8. 7000 12. 37000 16. 80705 20. 916738 When a number is multiplied twice by itself, the product is called the cube or third power of that number ; thus 4X4X4i=: 64. The cube of 4 is denoted 4^. *^^* This may be illustrated by a small cube of wood, or, better still, by a box of such cubes. EXERCISE XIL 1. What are the cubes of 1, 2, 3, etc., up to 10? 2. Find the cubes of these numbers : — 1. 789 4. 4506 7. 12000 10. 67809 2. 405 5. 5730 8. 37100 11. 40506 3. 628 6. 9825 9. 24089 12. 12345 EXERCISE XIII. 1. How many eggs in 16 boxes, each having 96 ? 2. How many pupils in a school which has 7 classes of 23 each ? 3. How many hours in 36 days ? 4. How many pence in 47 half-crowns ? 5. How many oranges, at 15 for a shilling, will 25s. buy ? 6. How long a journey shall I make in 27 days, at 18 miles a day ? 7. How many yards of linen in 387 pieces, each 35 yards ? 8. How many bottles in 45 dozen and 5 ? 9. How many pages in a yearly volume, of which a monthly part has 96 ? 10. What cost a railway 49 miles long, at £4500 a mile ? 11. A postman delivers 29 letters each morning and evening for a week ; how many did he deliver in all ? 12. A pipe pours into a cistern daily 13410 gallons water ; how many gallons will it pour in during November ? 13. A house of five storeys has seven windows in each, and twelve panes of glass in each window ; how many panes of glass are there in all? 14. Three men, in business together, receive £672 each of the profits at the end of the first year ; what were the whole profits ? 15. If a baker reckons 13 to a dozen, how many biscuits does he count to 136 dozen ? 16. A merchant's ofiice occupies 43 clerks at £2 a week each, and 24 at £3 ; what sum is required in a year for their wages ? 17. There are 129 trees in the side of a square plantation ; how many trees has the plantation ? 41. DIVISION. DIVISION. 39 Ex. — A box of eggs, containing 852, is to be divided amongst a number of families, each getting 6 ; how many f^imilies will be served ? Here we have to find how often 6 is contained in 852. We could find that by subtracting 6 from 852 successively till nothing remains, and then counting the number of 6's we have got, but a shorter way is to divide 852 by 6. The number to be divided is called the dividend. The dividing number is called the divisor. The number of times the divisor is contained in the dividend is called the quotient. The process of dividing is called division ; and, where the dividend is of one kind as here, simple division. The sign of division is -4- (divided by) ; thus, 4 -f- 2 is 2. We cannot find how often 9 is contained in 243 by one step ; the dividend is too large for that. We must therefore do it in parts, for which purpose we must learn the division of the first nine numbers. 42. Division Table. 2 in 2 is 4 ... 6 ... 8 ... 10 ... 12 ... 14 ... 16 ... 18 ... 20 ... 22 ... 24 ... 3 in 3 is 1 6 ... 2 6 in 6 is 12 18 24 80 36 42 48 54 60 66 72 16 24 32 40 48 56 64 72 80 ... 10 ... 11 ... 12 9 in 9 is 18 ... 27 ... 45 54 63 72 81 90 99 108 1 2 3 4 5 6 7 8 9 10 11 12 10 in 10 is 20 ... 30 ... 40 ... 50 ... 60 ... 70 ... 80 ... 90 ... 100 110 120 1 2 3 4 5 6 7 8 9 10 11 12 11 in 11 is 22 ... 33 ... 44 ... 55 ... 66 ... 77 ... 88 ... 99 ... 110 ... 121 ... 132 ... 12 in 1 12 is 2 24 ... 3 36 ... 4 48 ... 5 60 ... 6 72 ... / 84 ... 8 96 ... 9 108 ... 10 120 ... 11 132 ... 12 144 ... . 10 11 12 40 DIVISION. EXERCISE I. Bf. 1. Repeat tlie lines of this Table up-and-down ; backwards ; by odds ; by evens. 2. Repeat the lines even along in the same way. 3. 2 in 8 is — ? 5 in 35 is —? 9 in 72 is - ? etc. 4 in 8 is — ? 7 in 35 is — ? 8 in 72 is — ? etc. 4. How many pence in 8 farthings ? Divide 15 shillings among 5 persons. Divide 40 marbles among 8 boys. How many oranges at 2d. each can I buy with 16 pence ? etc. 5. Write down the several columns of the Table. 40a This Table serves also for the division of tens, hundreds, etc. Thus— If 2 in 2 is 1, 2 in 2 tens is 1 ten, or 2 in 20 is 10. If 2 in 4 is 2, 2 in 4 tens is 2 tens, or 2 in 40 is 20. Etc. etc. etc. If 3 in 3 is 1, 3 in 3 tens is 1 ten, or 3 in 30 is 10. Etc. etc. etc. EXERCISE II. Perform Ex. i., Nos. 1, 2, 3, with tens in the dividend. If 2 in 2 is 1, 2 in 2 hunds. is 1 hund., or 2 in 200 is 100. If 2 in 4 is 2, 2 in 4 hunds. is 2 hund., or 2 in 400 is 200. Etc. etc. etc. EXERCISE III. Perform Ex. i., Nos. 1, 2, 3, with hundreds in the dividend. ^^, Division by Numbers of One Place. Ex. — How often is 3 contained in 963 ? Place the divisor to the left of the dividend. 3 in 9 hundreds is 3 hundreds. 3 ) 963 3 in 6 tens is 2 tens. 321 3 in 3 units is 1 unit. Quotient, 321. EXERCISE IV. Divide — 1. By 2 : 86, 128, 420, 642, 864, 4806, 6428. 2. By 3 : 63, 96, 123, 249, 630, 963, 6093. 3. By 4 : 84, 168, 244, 488, 804, 884, 4084. 4. By 5 : 105, 155, 250, 355, 505, 4550, 5035. 5. By 6 : 126, 246, 306, 426, 5460, 6048, 12660. 6. By 7 : 147, 217, 357, 714, 6377, 7063. 7. By 8 : 168, 248, 320, 880, 1608, 5680. 8. By 9 : 189, 279, 540, 3609, 4599, 8190. ( DIVISION. 41 4 5- The places of the dividend do not often contain the divisor evenly ; there is generally a remainder. EXERCISE V. • 2 in 3 is 1 and 1 over ; in 5 is — ? in 7 is — ? etc. 3 in 4 IS 1 and 1 over ; in 5 is — ? in 7 is — ? etc. 4 in 5 is 1 and 1 over ; in 6 is — ? in 7 is — ? etc. 5 in 6 is 1 and 1 over ; in 7 is — ? in 8 is — ? etc. %* The exercise should be continued up to ] 2 as divisor. 46. ^X' 2. — A box of eggs, containing 852, is to be divided amongst a number of families, each getting 6 ; how many families will be served ? Set the divisor to the left of the dividend. Then 6)852 6 in 8 hundreds is 1 hundred and 2 hundreds over ; 142 set down the 1 in its place, and change the 2 hun- dreds into tens, making 25 in all. 6 in 25 tens is 4 tens and 1 ten over ; set down the 4 in its place, and change the 1 ten into units, making 12 in all. 6 in 12 units is 2 units. Quotient, 142. Rule.— To divide by a number of one place, divide the places of the dividend in order from the highest, carrying the tens. The result may be proved by multiplying the quotient by the divisor ; the product should be the dividend. EXERCISE VI. Divide 1. By 2 : 98, 258, 374, 454, 526, 598, 638, 694, 738, 876, 938, 972. 2. By 3 : 87, 378, 465, 471, 513, 582, 648, 657, 726, 735, 879, 978. 3. By 4 : 96, 492, 536, 548, 620, Q76, 768, 792, 860, 892, 948, 956. 4. By 5 : 565, 590, 675, 680, 745, 775, 865, 880, 930, 975, 7345. 5. By 6 : 150, 672, 726, 744, 804, 852, 918, 990, 6834, 8526, 8730. 6. By 7 : 161, 798, 805, 875, 910, 987, 7847, 7952, 8596, 8764, 9233. 7. By 8 : 256, 896, 960, 992, 8976, 9544, 1896, 1944, 2888, 3976. 8. By 9 : 144, 252, 423, 603, 828, 1026, 2160, 3267, 5040, 6543, 7038. 9. By 10 : 730, 840, 9320, 4500, 7310, 2030. 10. By 11 : 748, 396, 594, 286, 7942, 8503, 25894, 92477, 56089. ] 11. By 12 : 348, 564, 936, 3888, 5737, 20928, 3708, 94020, 67308. 47. Ex.— How often is 6 contained in 24295 ? Dividing as before, there is a remainder of one 6)24295 after dividing the units. This is annexed to the 40491 quotient with the divisor below in the form ^, which denotes one-sixth^ or the sixth part of one. 42 DIVISION. In multiplying the quotient in this case by the divisor to prove the result, the remainder must be added to the product : thus, 4049 X 6+ 1=24295. 48. EXERCISE VII. ivide— 1. 2. 3. 4. 5. 6. 1. By 2, 345 467 931 857 1129 2525 2. By 3, 472 305 721 922 2684 7055 3. By 4, 105 653 437 829 5634 8631 4. By 5, 732 482 911 573 8421 7018 5. By 6, 515 833 791 273 5927 6381 6. By 7, 452 635 134 608 3210 7962 7. By 8, 123 537 817 909 4561 8347 8. By 9, 258 316 501 823 7082 1293 9. By 10, 137 259 533 471 2563 9327 10. By 11, 564 800 601 942 3874 6088 11. By 12, 373 529 705 637 1949 2009 Division by Factors. In dividing by any composite number up to 144, we may get the quotient by dividing by its two factors successively. E.g., in dividing an apple into 4 parts, we first divide it into 2 parts, then each of these again into 2 parts. Ex. — Divide 3568 marbles into parcels of 24. The factors of 24 are 6 and 4. 6 Dividing first by 6, we have for quotient 4 3568 49. /iviuuig lust uy D, we iiave lor quotient 4 594 4 \ 594 (parcels of 6), and 4 (marbles) over. i^g o [ 1^ Dividing next by 4, we have for quotient ^ 148 (parcels of 4 sixes or 24's) and 2 (parcels of 6) over. Adding now the second remainder (2 parcels of 6, or 12 marbles) to the first (4 marbles), we have for total re- mainder 16 marbles : 6X2+4 = 16. Hence, to get the real remainder, multiply the first divisor by the second remainder, and add the first remainder to the product. If there be no second remainder, the first is the real one. EXERCISE VIII. 1. 23456-^14, 15, 21, 22 6. 905036-4-84, 88, 96 2. 37095-^25, 27, 28, 32 7, 249076-^99, 108 3. 90851-^33, 42, 44, 45 8. 593250-r-120, 132, 144 4. 84379-4-54, 55, 56, 63 9. 731105-^16, 18, 48, 72, in two ways. 5. 65927-^66, 77, 81 10. 847644-^24, 36, in three ways. Division by more than one Place. As a cipher annexed to the right of a figure multiplies it by 10, so a cipher removed from the right of a figure divides the number by 10 : thus, 20 -^ 10 = 2. DIVISION. 43 If the dividend do not end in a cipher, then the figure in the units' place is removed for a remainder : thus, 21-^-10= 2^^ If the divisor contain more tens than one, as 30, divide hrst by 10 as one factor, and then by the other factor, 3 ; that is, remove the units' place of the dividend for the remainder, and divide by the second factor, carrying what is over in this divi- sion to the remainder. Thus, 63 -4- 20 = 3^ ; 73 —• 20 = 3 Jf . To divide by a number of hundreds, remove the two last ciphers of the dividend, or the two last figures of it, for re- mainder, in a similar way. Thus, 200 -j- 100 = 2 ; 564 -- 200 = 21M. EXER CISE IX. 3 by 10, 30, 50, 70, 90— 1. 370 7. 1200 13. 2474 19. 32814 2. 290 8. 6600 14. 3935 20. 56732 3. 835 9. 8800 15. 5066 21. 83940 4. 672 10. 7000 16. 7317 22. 50761 5. 425 11. 4800 17. 8058 23. 69005 6. 901 12. 6300 18. 9720 24. 85436 EXERCISE X. Divide by 200, 400, 600, 800, examples 7-24 in last Exercise. ^a;.— How often is 234 contained in 849726 ? 234 in 8 or in 84 cannot be taken, but 234)849726(3631^ in 849 (thousands) is 3 (thousands), 702 and 147 (thousands) over. Set down the 3 in the thousands' place of the 1477 quotient, and carry the 147 to the 1404 hundreds' place, making the next part of the dividend 1477 (hun- 732 dreds) in all. 702 234 in 1477 (hunds.) is 6 (hundred), and 73 (hunds.) over. Set down 306 the 6 (hunds.) in its place in the 234 quotient, and carry the 73 (hunds.) ■ • to the tens' place, making the next 72 part of the dividend 732 (tens) in all. 234 in 732 (tens) is 3 (tens), and 30 (tens) over. Set down the 3 (tens) in its place in the quotient, and carry the 30 (tens) to the units' place, making the next part of the divi- dend 306 (units) in all. 234 in 306 (units) is 1 (unit), and 72 (units) over. Set the 1 (unit) in its place in the quotient. The 72 units are remainder. 44 DIVISION. This form of division, which is required when the divisor contains more than one place, is known as Long Division. EXERCISE XI. 1. 370374-25, 37, 43. 2. 29835-f-34, 49, 51. 3. 73632^47, 93, 39. 4. 80294-T-19, 26, 41. 5. 90000-^73, 61, 17. 6. 50032-^29, 53, 98. 7. 17918-^13, 34, 82. 8. 473204-38, 91, 47. 9. 209714-67, 82, 93. 10. 54280-^23, 46, 85. EXERCISE XII. 1. 45682^ 2. 40936H 3. 23843-^ 4. 89040-i 5. 90000H 6. 12384^ 7. 73027H 8. 29041H 9. 92881H 10. 79948H -251, 183, 342. -301, 457, 631. -113, 911, 564. -824, 159, 296. -457, 734, 825. -391, 516, 364. -801, 709, 208. -257, 314, 846. -934, 652, 293. -418, 506, 853. 11. 560802- 12. 293544- 13. 858841- 14. 485361- 15. 934110- 16. 500800- 17. 700000- 18. 205806- 19. 934165- 20. 714408- ^293, 791, 846. =-151, 258, 174. ^-325, 291, 397. ^851, 702, 813. f-561, 582, 738. ^921, 309, 257. ^416, 526, 736. f-901, 754, 815. -297, 358, 492. T-824, 964, 708. EXERCISE XIII. 1. 74893184 2. 29348214 3. 7348640-^ 4. 35848164 5. 60845164 6. 54031444 7. 72561544 8. 91446684 9. 82717594 10. 91939324 11. 57338064 12. 63092706 13. 72491840 14. 20018414- 15. 92100625 37, 74, 89. 41, 73, 97. 594, 416, 607. •208, 541, 732. -2342, 5684. -9348, 2571. -3040, 8009. -9401, 5008. -3075, 4908. •5671, 2943. 4-5437, 3024, 9902. ^2931, 4708, 5004. -^3040, 8009, 5231. ■f-7298, 6804, 7734. 4-5136, 1984, 2875. 16. 800000004-8345, 6205, 7095. 17. 538054484-4001, 8936, 9027. 18. 73006924-7506, 9324. 19. 90000004-8931, 7295. 20. 82035704-4583, 9308. 21. 2568903684-28, 79, 39. 22. 9314562044-17, 47, 82. 23. 2490860224-457, 329, 704. 24. 3036067964-293, 718, 274. 25. 7240880434-8561, 2793. 26. 3659057804-5006, 2918. 27. 8543724004-9300, 8540. 28. 2936001704-2005, 7009. 29. 8759127804-3054, 4090. 30. 2934000004-7200, 5090. 51. To find an Average. Ex. — A boy gets 23 marks on Monday, 17 on Tuesday, 28 on Wednesday, 31 on Thursday, 25 on Friday, and 14 on Saturday : what is his average number of marks daily for the week? Here the sum of his marks for the whole week is 138. There is a certain number of marks, which had he got every day of the week, the sum of his marks at the end of the week would DIVISION. 45 have been the same as it is now. That is the number we wish to find. The average of a series of numbers is that number which, if repeated as often as there are numbers, will amount to their sum. It is found by dividing the sum of the numbers by their number ; thus 138-r 6 = 23. EXERCISE XIV. Find the average of the following numbers : — 1. 27, 37, 42, 50, 22, 24. 6. 2738, 3624, 3001, 2. 13, 49, 35, 64, 53, 42. 7. 937, 1001, 1100, 1010, 1110. 3. 93, 87, 59, Q7, 73. 8. 856, 1533, 930, 1399. 4. 29, 30, 37, 32, 33. 9. 8973, 10704, 9320, 14976, 9999. 5. 125, 250, 315, 193. 10. 27345, 73421, 85648, 79286. 52. Fractional Multipliers and Divisors. H Ex. — A train runs 27 miles an hour for 14f hours ; what distance will it go in the time ? The distance is 27 miles repeated 14 times and f a time ; which is got by multiplying 27 by 14|. To multiply by }, multiply by 3 and divide the product by 4. Then in multiplying by 14, the right-hand figure of the first line, being units, is set in the units* place. The number J, which is less than 1 is called a fraction. If one is divided into 2 equal parts, each is called a half ; if into 3, each is called a third ; if into 4, a fourth ; and so on. A fraction is denoted by two numbers, the one written below the other ; thus one-half is written ^, one-third J, one-fourth | ; if more than one part be taken, the upper figure denotes how many, thus three-fourths is written j. The number 14 j, which consists of a whole number and a fraction, is called a mixed number, EXERCISE XV. 1. Find one-half of 38, 57, 108, 265, 798, 6357. 2. One-third of 51, 252, 254, 768, 784, 8472. 3. One-fourth of 56, 92, 94, 397, 3828, 8927. 4. Multiply by « : 85, 101, 357, 456, 2456, 7530. 5. Multiply by | : 84, 356, 537, 933, 1272, 7000. 6. 3456x4A, 6f, 15^, 275, 139*, 308|. 7. 98582 xlOJ, 200^, 750f, 409f, 30|, 5^. 46 DIVISION. Ex.— Row often is 29j contained in 9384 ? The numbers cannot conveniently be 29j 9384 used for divisor and dividend as they 4 4 stand. 117 )37536(320A't Multiply both by 4, the fraction in the 351 divisor being three-Zo-i^r^/^s. This will 040 give a new divisor and dividend four times 034 greater than those given ; but which will be free from fractions, and will give the same quotient. 96 53. EXERCISE XVI. 1. 3482V3i, 6§, 8^. 6. 900536H-12J, 74^ 256^. 2. 8506-^4i, 5i, 9 J. 7. 852079-^5.1, 30^, 3651. 3. 72584-^27i,"o4i 79|. 8. 205930-^15^ 85f , 365^. 4. 59321-^191, 68^, 128.i 9. 7305267-^29^ 217|, 8342^. 6. 80999-^15^, 265, 94g. 10. 45067824-i-14i 58^, 100^ Multiplication and Division Combined. £x. — What number results from multiplying 57 by 16, and dividing by 24 1 To nmltiply by 16 is the same as to multiply by 2 and then by 8 ; and to divide by 24 is the same as to divide by 3 and and then by 8. We may strike out the 8 from both terms ; since to multiply a number by 8 and then to divide it by 8 leaves it unaltered. So that — 2 57 X 16 __ 57j s. d. Take the next less which can, that 24 14 7jx4 is, 64. Find the price of 64 chests 8 (8X8), and add the price of 4 197 \Q 10 price of s chests, chests ; for 64 = 8 X 8+4. 8 The price of 64 chests is found 3582 14 8 » 64 chests, as above : the price of 4, by mul- gg jg 5 „ 4 chests tiplying the price of one (first line) ^^^^^^^^ by 4 ; the price of 68 by adding the price of 64 and the price of 4 together. Other factors which might be used are 9x7 + 5 and 10 X 6 + 8, either of which pairs may be taken to prove the result. Rule. — ^When the multiplier is not above 144, and cannot be resolved into two factors under 12, multiply by the factors of the next less number which has them, and add the product of the multiplicand by the difierence between that number and the multiplier. It is advisable to take factors for the number next above the multiplier, when that number exceeds it only by 1, and then subtract the excess; thus, 39 = 10X4 — 1. In the present case we might have taken 68 = 10x7 — 2. EXERCISE V. 1. £2, 14, 2f X 13, 17, 19, 24, 29, 31. 2. £7, 10, 91 X 34, 38, 43, 51, 68, 61. 3. £13, 8, 5| X 62, 69, 74, 78, 82, 87. 4. £34, 3, 2i X 91, 94, 101, 106, 117, 122. 6. £60, 0, 9^ X 129, 135, 142, 145. 6. £79, 18, 6^ X 67, 79, 46, 39, 89, 105. 58 MONEY. 64. Multipliers of Three Places. -EJic.— Find the price of 457 chests at £24, 14s. 7Jd. per chest. 50 400 £24 14 7i X 7 = 10 £247 6 0^X5 = 10 £173 12 2| 1236 10 2J £2475 5X4 = 9900 1 8 Total product, £11310 4 2j „ 457 „ Kule.— Multiply by factors for 100 (10X10). Then multi- ply the multiplicand by the number of units in the multiplier, ten times the multiplicand by the number of tens in it, and a hundred times the multiplier by the number of hundreds in it : add these three products for the total product. EXERCISE VI. 1. £9, 13, 7h X 257, 381, 473. 7. £59, 7, H x 915, 638, 187. 2. £13, 10, §ix319, 459, 542. 8. £73, 8, 10^x562, 784, 268. 3. £19, 8, 5;^ X 417, 534, 629. 9. £83, 15, 7lx 400, 701, 511. 4. £23, 10, Of X 566, 671, 713. 10. £89, 0, 5h x208, 962, 609. 6. £31, 19, 4jx647, 738, 825. 11. £93, 14, 2f x354, 849, 276. 6. £43, 1, 111 X 724, 850, 993. 12. £109, 7, 9^ x 417, 651, 767. Multipliers of Four Places. The same method is used for multiplying by thousands. Rule.— Multiply by factors for 1000 (10 X lOx 10). Then multiply the multiplicand and the successive products by the several places of the multiplier in order, beginning with the units' place ; add these products for the total product. EXERCISE VII. 1. £13, 18, 51x1924, 2438. 4. £57, 10, 7| x6234, 7941. 2. £19, 5, 10} X 2741, 3925. 5. £69, 5, 8^ x 8301, 9042. 3. £27, 3, 4| X 4837, 5529. 6. £124, 15, 6|x 4520, 6009. * * These products are obtained more easily by practice. QQ COMPOUND DIVISION. Ex. 1. — Divide £93, 15s. 9|d. equally among 7 persons : what is the share of each ? Write the divisor and dividend as in simple division. COMPOUND DIVISION. 59 Then 7 in £93 is ^13 and £2 over; set down the £13, and carry 7)93 15 9| the £2 to the shillings, making 33 7~TTJf 55s. in all. 7 in 55 is 7s. and 6s. over ; set down the 7s. and carry the 6s. to the pence, making 81 d. in all. 7 in 81 is lid. and 4d. over ; set down the lid. and carry the 4d. to the farthings, making 19 farthings in all. 7 in 19 is 2 farth. and 5 farth. over ; set down the 2 farth. and, as the division is now finished, there is a remainder of 5 farthings, divided thus, f . Quotient, .£13, 7s. ll^f. The result may be proved by multiplying the quotient by the divisor, and adding the remainder, which wUl give the dividend. Ex. 2. — Divide the same sum kiqo ik qs equally among 28 persons. ' |^£_L5 !li. Resolve the divisor into its two 4 | 13 7 112 +5 / ^gf, factors (7x4), and divide by each 3 6 ll|+2 ) in succession. Quotient, £3, 6s. ll|Jf. The result may be proved by reversing the order of factors in dividing, or by multiplying the product by the divisor. Rule.— When the divisor is not above 12, divide each name by it in order, beginning at the highest, and carry the remainder to the next lower. When the divisor is not above 144, and has two factors neither above 12, divide in the same way by each factor in succession. The division of a quantity of several names, as here, is called compound division. 66. EXERCISE I. 1. 2d. 3d. 5d. 6d. 7d. etc. -^2, 4. 10. 1/3,1/6, 1/9, 2/, 2/3, etc. -^6, 12. 2. Ud. 3d. ^d. 6d. 7p. etc.-^3, 6. 11. 1/0|, 1/2, l/M, im, etc.-^7. 3. l|d. 2.id. 3fd. 5d. 6|d. etc.-^5. 12. 1/1^, 1/4^, 1/7^ 1/10, etc. -Ml. 4. ip. 3id. 5|d. 7d. etc.-^7. 13. £1, £1, 4, £1, 8, etc.-^2, 4, 8. 5. 2d. 4d. 6d. 8d. etc.-^8. 14. £l,2/6,£l,5/6,£l,8/6,etc.-T-3,9. 6. 2id. 4^d. 6|d. 9d. etc.-^9d. 15. £1, £1, 5, £1, 10, etc.-^5, 10. 7. l/,l/2, 1/4, 1/8,1/10,2/, etc.-h2,4. 16. £1,4, £1,10, £1,16, etc. ^6, 12. 8. llil/l.i,l/3|,l/6,l/8ietc.H-3,9. 17. £1, 1, £1, 4/6, £1, 8, etc.-^7. 9. 1/0^, 1/3, 1/5^, 1/8, 1/lOh, etc.H-5. 18. £1, 2, £1, 7/6, £1,13, etc. -Ml. 60 MONEY. EXERCISE II. 1. £8 19 73^2, 3, i, 5. 13. £89 14 10f4-14, 15, 21. 2. 7 5|^3, 4, 5, 6. 14. 91 2 8W24, 27, 22. 3. 19 10 3|-^4, 5, 6, 7. 15. 156 17 3|^25, 28, 100. 4. 27 15 61-^5, 6, 7, 8. 16. 193 5 -^30, 32, 108. 5. 79 1 ll|-^6, 7, 8, 9. 17. 279 6 10i-^84, 96, 99. 6. 54 Oi-^7, 8, 9, 10. 18. 309 1 4|-r-80, 81, 35. 7. 60 5 7i-f-8, 9, 10, 11. 19. 600 10 10|-^77, 72, 121. 8. 86 14 9|-f-9, 10, 11, 12. 20. 793 15 6^-^70, 64, 18. 9. 43 6 lll-f-10, 11, 12, 7. 21. 72 5 6|-^56, 63, 16. 10. 37 18 l|^ll, 12, 5, 9. 22. 68 7 3^-r-48, 50, 144. 11. 5 17 5 -^12, 6, 7, 10. 23. 81 19 0^-^42, 44, 132. 12. 3 12 9^4-7, 9, 4, 5. 24. 59 2 7|i^36, 40, 33. 67. Divisors of Two or more Places. ^a:.— Divide ^93, 15s. 9|d. 43)93 15 9|(2 3 7^8 among 43 persons. 86 ~y Rule. — Divide each name 20 in order by the divisor, be- ~)155 s ginninor at the highest ; and j29 carry each remainder to the — ^^ next lower name. , „ )321 d. 301 20 4 )83f. 43 40 V The 40 farthings over are written in the quotient with the di\isor "below them, as ^. EXERCISE III. 1. £567 10 3^-^29, 37, 53, 71, 83. 2. 734 18 5 -^19, 41, 67, 86, 91. 3. 392 15 4i-^52, 23, 47, 95, 13. 4. 78 2 llJ-r-124, 213, 352, 793, 61. 5. 27 18 0|^225, 538, 401, 191, 17. 6. 115 10|-^115, 116, 237, 73, 85. 7. 1897 14 31^372, 416, 509, 1000, 1937. 8. 2700 18 0|-^562, 57, 829, 900, 2340. 9. 8035 17 5l-f-1256, 4073, 236, 800, 158. 10. 5682 11 31-^721, 1356, 2943, 673, 78. 11. 73582 14 71^2905, 7238, 825, 34, 304. 12. 290732 9 l|-^59, 97, 652, 8905, 4005. 68. COMPOUND DIVISION. 61 Fractional Multipliers. Ex.—Wha^t cost 8j packages if 1 £5 17 9i package cost £5j 17s. O^d. ? 8} Multiply first by the fraction (|), then 4)17 13 3| by the whole number (8). Add the ~~^ o ^3 3 products. 47 g 2 £51 10 5|i EXERCISE IV. 1. £7, 10, 3^ X 71, 9h, llf. 5. £91, 15, 6i x 73^, 59^, 91f . 2. £14, 15, 7i X 4, 6|, 8^. 6. £256, 14, lOx 29|, 13i, 631 3. £24, 19, 3 X 15^, 27J, 36f . 7. £509, 8, 3f x 231^ 4501, 671^ 4. £71, 5, 11^ X 49J, 84i, 100^. 8. £891, 11, 1| x 307|, 593|, 713|. Fractional Divisors. Ex.—U 17} yards cost £9, 18s. lOjd., what is that per yard? We have to divide the 17} £9 18 10^ whole price by the num- 4 4 ber of yards to get the "yT )39 15 6 (lis. 21 f^ price of one yard. 20 Multiply both divisor 795 and dividend by 4 to re- 7I move the fraction from the divisor. 85 71 14 12 >174 142 32 4_ )128 71 57f. EXERCISE V. 69. 1. £7, 10, ll|-h5i, 6|, n. 5. £58, 16, 1^1301, 200^, 563^. 2. £11, 14, 5l^8i, 15i 491. 6. £251, 17, 4^^1171, 352|, 401*. 3. £29, 5, Oi^l8l, 291, 634. 7. £309, 19, 2h-308|, 510^, 713j. 4. £36, 7, 2|-T-2l|, 87|, 52^. 8. £643, 0, 5^831, 173^, 824^. Money Divisors. Ex.— How often is £6, 13s. Gjd. contained in £39, 14s. 7}d. ? 62 MONEY. Rule. — Reduce divisor and dividend to the same name, and proceed as in simple division — £39, 14, 7|^£5, 13, 6i=38143f.-f-5449=7. EXERCISE VI. %* To be performed after reduction has been learnt. 1. £27, 17, 34-£6, 3, 10. 2. £137, 8, 9-^£8, 19, 4^. 3. £361, 2, 9f -^£72, 4, 6^ 4. £2090, 0, 7J-^£81, 0, 9i 5. £459, 18, 2^£24, 17, S} 6. £63, 8, 0f-^£21, 2, 8^ 7. £671, 10, l-^£47, 19, 3^. 8. £268, 10, 3^£100, 9, 1C)|. 9. £675, 19, 3f-T-£75, 2, 11. 10. £870, 0, 5|^£39, 18, 5^. 70. EEDUOTIOK MONEY OF ACCOUNT TABLE I. From a Higher to a Lower Name. Ex. — In £7, 13s. 3|d., how many farthings ? We cannot change this sum to farthings by one step, as it is too large ; we must therefore do it in parts, changing first the pounds to shillings, then the shillings to pence, then the pence to farthings. Thus, to change the pounds to shillings, since there are 20/ for every pound, there will be 20 times as many shillings as pounds ; multiply 7 by 20, and add the 13/ already in the sum, making 153 sh. To change the shillings to pence, since there are 12d. in every shilling, there will be 12 times as many pence as shil- lings ; multiply 153 by 12, and add the 3d. already in the sum, making 1839d. To change the pence to farthings, since there are 4 farth. in every penny, there will be 4 times as many farthings as pence ; multiply 1839 by 4, and add the 3 farth. already in the sum, making 7359f. in all. Rule. — Multiply each name, in order from the highest, by the number of the next lower which it contains, adding to each product the number of the lower in the given sum. £ s. 7 13 20 153 12 1839 4_ 7359 d. 3| 3| sli. in the sum. ;| pence in the sum. farth. in the sum. » 71. REDUCTION. 63 The process of changing from one nanae to another is called Beduction. The result may be proved by changing back the farthings to pounds ; dividing by the same numbers by which we have multi- plied. If £7, 13s. 3|d., when changed to farthings gives 7359f., 7359 farthings, when changed to pounds, must give £7, 13s. 3|d. (See sect. 71.) EXERCISE I. 1. How many farthings in l|d., l.^d., Ifd., 2d., 2;^d., etc., to 12d. ? 2. How many pence in 1/1, 1/2, etc., 2/1, 2/2, etc., to 20 ? 3. How many sliillings in £1, Is. ; £1, 5s., etc. ; £2 ; £2, 7s ; £10? EXERCISE II. (1.) To pence-£75; £352; £1001 ; £2450 ; £23, 10s ; £179, 17s. ; £305, 19s. ; £5024, 15s. ; £734, 17s. 4d. ; £809, 10s. 8d. ; £2702, Os. lid.; £6304, Is. 7d. (2.) To lialfpence-5/, 7/, 13/, 8/2, 18/3, 14/7^, 53/8^, £15, £23, 17s., £27, 9s. lOd., £150, Os. 7id., £207, 19s. O^d. (3.) To farthings- 4/, 9/, 24/, 37/, 3/4^, 11/9^, 19/1^ 15/0|, 29/10-1, 72/8i, 13/9i, 194/Oi. (4.) To farthings — 1. £93. 5. £39, 17. 9. £4, 17, 10. 13. £922, 10, 0^. 2. £201. 6. £125, 8. 10. £172, 0, 0|. 14. £507, 19, 11^. 3. £485. 7. £709, 10. 11. £250, 0, 04. 15. £1854, 0, 3. 4. £7392. 8. £4890, 19. 12. £793, 15, 11^. 16. £3000, 10, 10|. From a Lower to a Higher Name. Ex. — To what sum of money are 37227 farthings equivalent ? Here we have to change the farthings to the highest name. We cannot do this at one step, as the number is too large ; we must therefore do it by several steps, first changing the farthings to pence, then the pence to shillings, thien the shil- lings to pounds, thus : — • To change for the far- 4 37227 things to pence, since it 12 takes 4 farthings to make 1 penny, there be only 2(0 9306 X = pence in the sum. 77(5 6f = shillings, etc. in sura. one-fourth as many pence ^^^ ^^ 6 j=pounds, etc. in sum. as farthings ; which is got by dividing the number of pence by 4, giving 9306|d. To change the pence to shillings, since it takes 12 pence to make 1 shilling, there will be only one-twelfth as many shil- lings as pence ; which is got by dividing by 12, giving 7753. ejd. 64 MONEY. To change the shillings to pounds, since it takes 20 shillings to make 1 pound, there will be only one-twentieth as many shillings as pounds ; which is got by dividing by 20, giving ^37, 15s. 6|d. Hule. — To change a sum of money from a lower to a higher name : — Divide by the number of the lower contained in the next higher, and so on till the required name be reached. The result may be proved by changing back the pounds, shillings, and pence to farthings. If 37227f., when changed, give £37, 15s. 6|d., so must £37, 15s. 6|d., when changed back again to farthings, give 37227f. EXERCISE III. 1. How many pence in 4 f. 5, 6, 7, etc., to 48 f. 2. How many shillings in 12d., 13d., etc., 24d., 25d., etc., to 240d. 3. How many £ in 20/, 40/, etc., 21/, 22/, etc., 30/, 31/, etc., to 200/. EXERCISE IV. 1. To shillings from farthings- 912, 1344, 1680, 2352, 737, 501, 1079, 1893, 600, 903, 1807, 2356. 2. To shillings from halfpence— 360, 432, 552, 768, 247, 301, 423, 593, 827, 1327, 1613, 2597. 3. To pounds from pence— 6480, 2376, 4800, 11040, 35721, 60089, 23459, 45930, 49087, 780923. 56421, 93000. 4. To pounds from farthings- 23496, 39408, 45082, 69857, 289508, 543306, 60085, 932092, 1000000, 2456793, 4560000, 5369480. $A102 UNIVERSITY OF CALIFORNIA LIBRARY ?IHi dO