. ARITHMETIC M AN ^ LIBRARY OF THE UNIVERSITY OF CALIFORNIA. GIFT OF Class Pacific Theological Seminary, PRACTICAL BUSINESS ARITHMETIC, COMMON SCHOOLS AND ACADEMIES. INCLUDING A GBEAT VARIETY OF PKOMISCUOUS EXAMPLES. BY WHITMAN PECK, A.M., ItFTHOR OF THE PROMISCUOUS EXERCISES IK ANDREW'S LATIN LESSONS (REVISED EDITION.) NEW YOEK: J. W. SCHEEMEEHOEN & CO., PUBLISHEES, 14 BOND STEEET. 1868. V? Entered, according to Act of Congress, in the year 1868, by W. PECK, In the Clerk's Office of the District Court of the United States, for the Southern District of New York. H. C. STOOTHOFF, STEAM BOOK AND JOB PBINTER, 39 and 41 Centre St., N. T. PREFACE. THE distinguishing feature of this Arithmetic, that which has chiefly led to its publication, is its containing, in addition to the ex- amples under each rule, a large number of " Promiscuous Examples " under several different rules. No two of these together being alike, pupils need to think how each one is to be done independently of another, instead of only doing all like one already done in the book, or by their teacher. They can often do page after page of examples as commonly arranged under their respective rules, though they eould not do much simpler examples as they are apt to occur in prac- tical business. Hence men often say, that their knowledge of Arith- metic, when they commenced business, consisted in little more than knowing how to add, subtract, multiply and divide, when directed to do so in an arithmetic, or by their teacher. This defect, it is be- lieved, will be remedied by the repeated use of the promiscuous ex- amples in this book. They are so classified and arranged that each "Exercise" requires the application of what has been previously studied in some portion of the book. The author having found such exercises almost indispensable in teaching arithmetic, has thought it would be a great convenience to teachers, to have an arithmetic containing a large number of promiscuous examples. Many experienced teachers, also, all with whom he has consulted, have confirmed him in this opinion. Hence, though there is per- haps too great a variety of arithmetics already in use, this is offered to the public. It is believed, too, that this arithmetic contains in one book, all the most important matter usually found in an arithmetical series, in which much the same matter is repeated in different books, thus greatly increasing the expense, without any real advantage. The first part, including the Fundamental Rules, is adapted to children beginning to study arithmetic after having received a little oral in- struction; and they are advanced so gradually, that they will be apt to learn this part thoroughly before they reach Compound Numbers and Fractious 111900 PREFACE. Most of the examples in this arithmetic are designedly short, that less time may be consumed in the operations, and more be devoted to learning the principles and their applications. They are, also, so simple that most pupils may be expected to do them, witla a little as- sistance in some cases, without requiring too much of the teacher's time in explaining what they seldom understand or remember. Some more difficult examples, designed for advanced pupils, will be found at the end of the book, and it is designed to publish in another book many more such examples, and some principles of Higher Arith- metic omitted in this, which, however, is sufficient to fit persons for the practical business of life. The author, also, thinks that he has greatly simplified the study of arithmetic by reducing the number of its rules. He applies the Bules for Reduction of Compound Numbers to Reduction of Frac- tional Compound Numbers (common and decimal,) and the rules of Percentage to all its various applications, such as Commission, Brokerage, Stocks, Profit and Loss, etc., etc. Suggestions to Teachers. Pupils should be required to explain fully the examples in arith- metic, at least enough of them to show that they thoroughly under- stand them. At first, they will need to use the blackboard or then- slates, but they should also learn to give the explanations mentally, omitting the numbers if too large to be thus calculated, but naming them at each step as they proceed. If they can do this beforehand, they need not be required to perform operations with which they are already perfectly familiar. In this way they will study mental as well as written arithmetic. Though the Promiscuous Examples are numerous, some pupils may need to do them repeatedly, in order to become as familiar as they ought to be with the practical application of what they have previously studied. Others may not need to do them all. One or two exercises at a time may be sufficient. After a few days, give them one or two more similar exercises, and continue to do this from time to time till the principles and rules are permanently fixed in their minds. The rules are designed to aid pupils in making their own rules, rather than to be verbally committed to memory. They should learn to perform all arithmetical operations, and explain them, inde- pendently of the rules in books. CONTENTS. NUMBER NOTATION (Roman) . . Arabic NUMERATION FUNDAMENTAL RULES PAGE , 7 , 7 8 10 14 ADDITION 15 SUBTRACTION 22 MULTIPLICATION 27 By Composite Numbers 35 DIVISION 38 Short 39 Long 45 By Composite Numbers 47 General Principles 49 PROMISCUOUS EXAMPLES in Ad- dition, Subtraction, Multi- plication and Division 50 UNITED STATES MONET 56 Table Aliquot Parts 57 Promiscuous Examples. . 68 Bills ' 74 COMPOUND NUMBERS 77 MONEY English or Sterling 77 WEIGHTS Troy, Table 77 Avoirdupois, Table 78 Apothecaries, Table 78 Miscellane's, Table 78 MEASURES Cloth, Table 78 Long, Table 79 Surveyor's, Table 79 PAGE MEASURES Square, Table ... 79 Cubic, Table ... 80 Wine, Table.... 81 Beer, Table .... 82 Dry, Table 82 Time, Table ... 82 Circular, Table . 84 Miscellaneous Table of Units, &c., Paper, Books 84 REDUCTION of Compound Numbers 85 Examples 8895 Promiscuous Examples 95105 Addition of Compound Numbers 105 Subtraction of Compound Numbers 106 Multiplication of Com- pound Numbers 108 Division of Compound Numbers 109 Longitude and Time 110 Promiscuous Examples. . . 112 Cancellation 115 Prime and Composite Num- bers 116 Greatest Common Divisor 118 Least Common Multiple . . 119 FRACTIONS 121 Common 122 Reduction of . . . 125 CONTENTS. Addition of 131 Subtraction of 132 Multiplication of 134 Division of 136 Pronrscuous Examples 139146 DECIMAL FEACTIONS 146 Addition of 148 Subtraction of 150 Multiplication of 151 Division of 153 Promiscuous Examples . . . 156 Beduction of Common Fractions to Decimals . . 159 Beduction of Decimal Frac- tions to Common 160 Fractional Compound Num- bers 161 Promiscuous Examples . . . 164 Promiscuous Examples in Common and Decimal Fractions 166 DUODECIMALS 177 ANALYSIS 180 PEECENTAGE 184 Commission 190 Brokerage 191 Stocks 191 Gold 192 Insurance 192 Profit and Loss 1 Interest 198 Partial Payments 203 PAGE Compound Interest 209 Discount 211 Bank 213 Taxes 215 Duties 217 Exchange 218 Partnership 223 Promiscuous Examples in the various applications of Percentage 226236 EQUATION OF PAYMENTS 236 KEDUCTION OF CUBBENCIES . . 241 EATIO 244 PROPOETION Compound 2 Conjoined 250 ALLIGATION 251 INVOLUTION 2 EVOLUTION Square Koot 255 Cube Boot PEOGBESSION Arithmetical.. 264 Geometrical . . 256 MENSUEATION 267 PEOMISCUOUS EXAMPLES U. S. Money and Com- pound Numbers 271 Fractions, Common and Decimal 274 Percentage and its applica- tions 279 Miscellaneous Eules 284 ARITHMETIC Article 1. Arithmetic is the science of numbers. It teaches their nature and use. Number is one or more things, or Units ; as one, two, three ; the number of pupils in a class is four, five, &c. Abstract numbers are numbers not applied to any par- ticular thing ; as one, two, five, &c. Concrete numbers are numbers applied to particular things ; as five men, ten cents. NOTATION. Art 2. Notation is the method of writing numbers. There are two methods, the Roman, introduced by the ancient Romans, and the Arabic, introduced by the Arabians, which is chiefly used in Arithmetic. Art. 3* The Roman method uses letters for numbers ; as, I, one; V, five; X, ten; L, fifty; C, one hundred; D, five hundred; M, one thousand. These seven letters repeated or united express all other numbers. If a letter is repeated, its value is multiplied as many times ; as, II, (two times one,) two; XX, twenty; XXX, thirty. 8 NOTATION. If a letter is written before another of greater value, its value is subtracted from that of the greater ; but if written after another of greater value it is added; as, IV, four; VI, six; IX, nine; XI, eleven. A small line ( ) over a letter multiplies its value a thousand times ; as, V, five thousand. TABLE OF ROMAN LETTERS USED FOB NUMBERS. I. One. IX. Nine. LXXX. Eighty. IE. Two. X. Ten. XG. Ninety. m. Three. XX. Twenty. C. One hundred. IV. Four. XXX. Thirty. CO. Two hundred. V. Five. XL. Forty. D. Five hundred. VI. Six. L. Fifty. M. One thousand VII. Seven. LX. Sixty. V. Five thousand. VIH. Eight. LXX. Seventy. Art. 4. The Arabic Notation uses the following ten figures for numbers : (Written] O. /. 2. 3. 6. 5. 6. /. (9. #. Naught or o;Qe ^ Q three, four. five. six. seven, eight, nine. Cipher. (Printed) 0. 1. 2. 3. 4 5. 6. 7. 8. 9. These figures, except the cipher, are called Digits* A figure written alone, or on the right hand of a number, has only its simple value; as, 1, one; 2, two; 5, five, &c. A figure written before another has ten times its simple value; also when prefixed to two others it has one hun- dred times its simple value. Hence figures increase in value ten fold from right to left. NOTATION. 9 10 (ten and naught) ten. 20 (two tens) twenty. 11 (ten and one) eleven. 21 (2 tens and 1) twenty-one. 12 (ten and two) twelve. 30 (three tens) thirty. &c. 13 (ten and three) thirteen. 40 (four tens) forty, &c. 14 (ten and four) fourteen. 50 (five tens) fifty, &c. 15 (ten and five) fifteen. 60 (six tens) sixty, &c. 16 (ten and six) sixteen. 70 (seven tens) seventy, &c. 17 (ten and seven) seventeen. 80 (eight tens) eighty, &c. 18 (ten and eight) eighteen. 90 (nine tens) ninety, &c. 19 (ten and nine) nineteen. 100 ( ten tens) one hundred, &c. In all the numbers from 10 19 the figure 1 is used for ten. In the number 11 the figure 1 is used for ten and one ; and in 111 it is used for one hundred, ten and one. Next to hundreds are thousands, tens of thousands, hundreds of thousands, millions, &c., as in the following French method, which is chiefly used. FRENCH NOTATION AND NUMERATION TABLE. Next to trillions are quadrillions, quintillions, sextillions, sept.il- lions, octillions, nonillions, decillions, &c. In this table numbers are divided into periods of three figures each, beginning at the right hand, the 1st units, the 2d thousands, the 3d millions, &c. 10 NUMEBATION. ENGLISH NOTATION AND NUMERATION TABLE. 11 CM O O rj QQ 5 H H fi H 20987 65432 Periods. RULE FOR NOTATION. Leaving space enough on the right for as many periods, of three figures each, as the number will contain, begin at the left hand, and write the number belonging to' each period, filling the vacant places with ciphers. EXAMPLE. Write two millions, seventy-five thousand, three hundred and five. There will be two periods on the right of millions. Write 2 in the millions' period, 075 in the thousands' period, and 305 in the last or units' period ; thus, 2,075,305. NUMERATION. Art, 5. Numeration is reading numbers. Small numbers are easily read by repeating the name of each figure as it is written. In reading a large num- ber observe the following RULE. Consider the number as divided into periods of three figures each, beginning at the right hand ; then, begin- NUMERATION. 11 ning at the left hand, read each period as if it stood alone, adding its name, except that of the last ; thus, The number 1,230,987,654,321, is read one trillion, two hundred and thirty billions, nine hundred and eighty-seven millions, six hun- dred and fifty-four thousand, three hundred and twenty-one. EXEKCISES IN NUMERATION. Bead the following numbers down and across the page. It will be best for pupils to write them first, if they have not learned to do so readily and plainly. 10 28 30 48 13 25 33 45 16 22 36 42 19 27 39 47 11 24 31 44 14 21 34 41 17 29 37 49 12 26 32 46 15 23 35 43 50 61 72 80 91 56 68 76 88 98 59 65 73 85 95 51 62 79 82 92 54 67 70 87 97 57 64 74 84 94 52 60 77 81 99 55 69 75 89 96 58 66 78 86 93 18 20 38 40 53 63 71 83 90 210 228 234 389 465 328 333 456 598 55C 430 550 - 672 761 891 445 678 789 890 901 543 785 876 983 779 671 872 963 753 861 655 741 833 922 766 990 985 742 888 1000 1234 2345 3456 4567 5678 10000 23456 34567 45678 56789 67890 100000 345678 456789 567890 678901 789012 1000000 4567890 5678901 6789012 7890123 8901234 10000000 123456789 2345678901 34567890123 1000000000 12345678901 345678901234 4567890123456 12,345,678,908,765,432,102,468. 12 EXERCISES IN NOTATION. IN NOTATION. Write all the numbers from Ten to twenty-five. Fifty to seventy-five. Twenty-five to fifty. Seventy-five to one hundred. Write One hundred and ten. Five hundred and sixty-seven. Two hundred and eleven. Six hundred and seventy-eight. Three hundred and one. Seven hundred and eighty-nine. Four hundred and twenty. Eight hundred and eight. Five hundred and sixty-seven. Nine hundred and ninety. Six hundred and seventy-nine. Ten hundred and twenty. Eight hundred and ninety. Twelve hundred and eleven. Nine hundred and thirty-four. Sixteen hundred and seventeen. Ten hundred and eleven. Eighteen hundred and ninety. Eleven hundred and twenty. Nine hundred and seventy-five. Twelve hundred and fifty-five. Eight hundred and sixty-four. Fifteen hundred and sixty-two. Seven hundred and fifty-three. Nine hundred and eighty -six. Six hundred and forty-two. Six hundred and fifty-four. Five hundred and thirty one. Three hundred and twenty-one. Four hundred and twenty. One hundred and twenty-three. Three hundred and one. Four hundred and fifty-six. Two hundred and three. Seven hundred and eight. Three hundred and fourteen. Nine hundred and ten. Four hundred and twenty-four. Two hundred and eleven. Five hundred and fifteen. Three hundred and forty-five. Six hundred and ten. Four hundred and fifty-six. Seven hundred and twelve. One hundred. Two thousand. Thirty thousand. Four hundred thousand. Five millions. Six hundred and six. Seven thousand eight hundred and nine. NOTATION. 18 Eighty thousand and ninety. Nine hundred thousand and one hundred. Ten million, eleven thousand and twelve. Thirteen hundred and fourteen. Fifteen thousand, one hundred and two. Three hundred thousand and four. Sixty million, seventy thousand and eight hundred. One hundred and ten millions, two hundred and thirty-four thousand, four hundred and five. Two hundred and thirty-four. Five thousand, six hundred and seventy-eight. Ninety thousand and seventeen. Three hundred thousand, five hundred and seven. Eleven millions, one hundred and five thousand. Five hundred millions, seven thousand and eighty-one. Seventy-five thousand, three hundred and forty. Eight hundred thousand, two hundred and five. Nine thousand, seven hundred and fifty-three. Three millions, four hundred and thirty-two. Twelve millions, eleven thousand and nine hundred. One hundred and twenty millions, seventeen thousand, six hundred and seven. Six thousand, seven hundred and thirty-one. Seven hundred and forty-eight. Sixty-eight thousand, four hundred and fifty-one. Thirty-nine millions, nine hundred and twelve thousand, three hundred and ninety-six. Seven hundred and fifty thousand, five hundred and sixty- three. Forty-six thousand, five hundred and four. Twelve hundred and ninety-seven. Two thousand, five hundred and sixty-six. Four millions, five hundred and four thousand, three hun- dred and twenty-two. Twenty-five thousand, seven hundred and thirty-eight. One thousand, four hundred and thirty-three. FUNDAMENTAL RULES. Five millions, three hundred and one thousand, seven hun- dred and ninety-five. The following are not designed for very young pupils. Write One billion, two hundred and thirty-four millions, five thou- sand and seven hundred. Three trillions, twenty-five billions, three hundred and four millions, forty-five thousand, six hundred and seventy-four. Twenty billions, four hundred and twelve millions, sixty- five thousand and thirty-two. Four hundred trillions, seventy- seven billions, seven hun- dred and seven millions, nine thousand, five hundred and sixty-three. Four quadrillions and five hundred trillions. Five quintillions and sixty-eight trillions. Six sextillions and five hundred quintillions. Seventy billions. Eighty trillions. Ninety quadrillions. One hundred quintillions, two hundred and ten quadrillions, thirty-five trillions, seven hundred billions and sixty-four millions. Fifteen sextillions, five hundred and sixty quintillions, four hundred and twenty-five trillions. FUNDAMENTAL RULES. Art. 6. Arithmetic teaches the use of numbers in four principal ways, viz : Addition, Subtraction, Multiplica- tion, and Division, called the Fundamental Rules of Arithmetic. ADDITION. 15 ADDITION. j^rt. 7. Addition is uniting two or more numbers in one. The number thus found, or the answer, is called the Sum or Amount. Simple Addition is uniting like numbers, or numbers of the same name, in one ; as, 3 apples added to 4 apples are 7 apples. Unlike numbers cannot be added ; as 3 apples and 4 pears are neither 7 apples nor 7 pears. Addition is often expressed by the sign (-J-) Plus, placed between numbers to be added. [The sign (=) of equality placed between numbers, shows that they are equal. ] ILLUSTRATION. The sum or amount of 2 added to 3 is equal to 5 ; 2+3=5. ADDITION TABLE. [This table is promiscuously arranged. The answers are not given, because it is better that pupils should learn them by thinking for themselves, and not have them for reference. They should be able to recite them perfectly and promptly.] H! i i.S ! 2120323 2203130 Sums 1 2 133042414344 323414340424 052545153550 515350525456 626461636566 163656267606 173757727476 727476773577 16 ADDITION OF UNITS. Add J 2 8 4 8 6 8 8 8 8 5 8 7 1 8 3 8 5 8 7 8 8 4 8 6 8 3 9 5 9 7 9 9 4 9 6 9 8 9 4 9 6 9 8 9 9 5 9 7 9 ADDITION OF UNITS. MENTAL EXERCISES. How many boys are 2 boys and 1 more ? 1+2 ? 2+2 ?, 3+1 ? 2+3 ? 3+4 ? 4+2 ? 3+3 ? 4+3 ? 4+4 ? How many girls are 5 girls and 2 more ? 2+5 ? 3+5 ? 5+4? 5+3? 5+5? 6+3? 4+6? 6+5? 6+6? How many men are 7 men and 3 more ? 3+7 ? 7+4 ? 5+7 ? 7+6 ? 7+7 ? 8+3 ? 4+9 ? 5+8 ? 9+5 ? 6+8 ? 8+8? 7+9? 9+9? How many women are 8 women and 2 more ? 3+8 ? 9+4 ? 8+5? 6+9? 8+7? 9+6? EXAMPLES FOE THE SLATE OE BLACKBOARD. 4 EXAMPLE 1. Add 4, 3, 2, 3, 5, and 2. 1 Process. 2 and 5 are 7, and 3 are 10, and 2 are 12, and 3 are 15, and 4 are 19 . Name only the result K of each addition ; as 7, 10, 12, 15, 19. ^ Ans. 19 EULE. Write the numbers under one another; draw a line underneath^ and under it write the sum or amount. EXAMPLES. (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) 2 3 4 5 4 3 2 4 5 4 2 3 3 4 5 1 3 3 2 4 5 5 3 4 4 3 3 5 1 3 2 4 5 3 4 3 3 4 4 2 2 3 2 4 5 4 4 2 4 2 2 5 3 3 2 4 5 5 3 3 2 1 4 3 4 3 2 4 5 3 2 4 3 1 1 5 4 2 3 5 5 4 3 5 ADDITION. 17 <14) (15) (16) (17) (18) (19) (20) (21) (22) (23) (24) (25) 4 5 1 2 3 1 2 3 4 5 6 7 3 1 5 4 4 2 3 2 1 1 1 2 4 5 1 2 5 3 4 3 4 5 6 7 2 2 4 3 2 4 5 4 2 2 2 1 4 5 1 2 2 5 6 3 4 5 6 7 1 3 3 4 1 6 7 5 3 3 3 3 4 5 5 4 5 7 8 3 4 5 6 7 6 7 7 4 4 4 5 (26) (27) (28) (29) (30) (31) (32) (33) (34) (35) (36) (37) 1 2 3 4 5 6 7 1 2 3 7 1 7 6 4 4 6 6 6 2 3 7 7 2 2 3 5 5 7 6 7 3 4 5 7 3 7 6 6 5 5 6 6 4 4 3 7 4 3 4 7 6 6 6 7 5 6 7 7 5 7 6 6 6 7 6 6 6 7 3 7 6 4 5 4 5 4 6 7 7 6 7 7 7 7 2 5 3 6 6 4 6 7 7 8 (38) (39) (40) (41) (42) (- 43) (44 ) (4 >) (46; 1 (47) (48) (49) (50) 2 3 4 5 6 7 8 9 7 8 9 6 5 3 4 5 4 7 8 9 8 6 4 3 8 4 5 6 6 8 9 8 9 4 8 9 3 5 6 7 8 9 8 1 7 7 3 6 7 6 7 8 9 7 9 9 3 5 3 9 7 8 9 2 8 1 8 6 2 7 9 8 8 8 3 9 8 2 4 6 8 6 9 9 8 7 4 7 9 8 9 8 7 3 8 ADDITION OF UNITS, TENS, HUNDREDS, &o. MENTAJJ EXEECISES. How many lambs are 10 lambs and 1 more ? 10+3 ? 10+5 ? 10+7 ? 10+9 ? 10+2 ? 10+4 ? 10+6 ? 10+8 ? 11+1? 11+2? 11+4? 11+6? 11+8? 11+3? 11+5? 11+7 ? 11+9 ? How many sheep are 12 sheep and 1 more ? 12+3 ? 12+5 ? 12+7? 12+9? 12+2? 12+4? 12+6? 12+8-? 13+3? 13+4? 13+6? 13+8? 13+5? 13+7? 13+9? 18 KULE OF ADDITION. How many horses are 14 horses and 2 more ? 14+4 ? 14+6? 14+8? 14+3? 14+5? 14+7? 14+9? 15+2? 15+4? 15+6? 15+8? 15+3? 15+5? 15+7? 15+9? 16+2 ? 16+5 ? 16+8 ? How many cows are 17 cows and 3 more ? 17+5 ? 17+7 ? 17+9? 17+4? 17+6? 17+8? 18+4? 18+6? 18+8? 18+9? 18+7? 13+5? 19+3? 19+5? 19+7? 19+9? 19+6 ? 19+8 ? EXAMPLES FOR THE SLATE. EXAMPLE 51. Add 123, 234, 345, and 456. Process. Write the numbers thus, Ans. 1158 Add the right hand column 6+5+4+3=18 units, or 1 ten and 8 units. Write 8 under units and add 1 to the tens. Add the second column 1+5+4+3+2=15 tens, 1 hundred and 5 tens. Write 5 under tens and add 1 to hundreds. Thus proceed. RULE. Write the numbers under one another, so that all the right-hand figures shall be in the same column, and the others in proper order, tens next to units, &e. Beginning at the right hand, add each column separately. If the sum consists of only one figure, write it under the column ; but if it consists of two or more, write only the right hand figure and carry or add the others to the next column if there is any; otherwise write both figures. PROOF. Add the same columns downward. Figures of different local value cannot be added ; 2 tens and 3 units are neither 5 tens (50) r*or 5, but 23. EXAMPLES. (52) (53) (54) (55) (56) (57) (58) (59) (60) (61) (62) (63) (64) 23 34 45 50 01 12 23 32 45 33 44 55 34 34 51 01 12 23 45 44 34 54 44 44 55 53 45 02 23 34 34 30 54 22 45 55 44 55 45 23 34 45 51 50 44 32 10 54 22 44 55 34 34 51 04 23 12 32 45 54 45 11 44 55 53 ADDITION. 19 (65) (66) (67) (68) (69) (70) (71) (72) (73) (74) (75) (76) .(77) (78) 12 23 34 45 56 67 78 89 90 66 77 88 99 89 34 45 56 67 78 89 90 01 12 66 77 88 99 98 56 67 78 89 90 01 12 23 34 66 77 88 99 79 78 89 90 01 12 23 34 45 56 66 77 88 99 96 90091823 34^556677866 77889945 (79) (80) (81) (82) (83) (84) (85) (86) (87) (88) 123 423 345 456 567 678 789 890 901 910 456 456 678 789 890 901 012 123 234 315 789 789 901 '012 123 234 345 456 567 678 012 045 234 345 456 567 678 789 890 910 345 678 567 678 789 890 901 012 123 234 678 997 891 901 012 123 234 345 456 567 (89) (90) (91) (92) (93) (94) (95) (96) (97) 123 234 345 431 543 654 764 876 987 456 567 678 098 210 321 325 543 654 789 890 901 765 987 098 109 210 321 123 123 234 321 654 765 876 987 098 456 564 567 098 321 432 543 654 765 789 897 890 765 098 109 210 321 431 123 135 123 432 765 876 987 098 098 456 678 456 109 431 543 632 765 726 (98) (99) (100) (101) (102) (103) (104) (105) (106) 789 890 901 678 567 456 345 231 123 012 123 234 901 890 789 678 456 345 345 456 567 234 123 012 901 678 567 678 789 90 567 456 334 223 890 789 901 012 123 890 789 455 344 012 901 234 345 456 123 Oil 667 556 234 123 567 678 789 456 223 788 677 456 345 890 901 012 789 334 990 889 678 507 123 234 345 012 455 Oil 901 890 789 456 567 678 345 667 223 334 123 901 EXAMPLES FOE THE SLATE. MENTAL EXERCISES. How many chickens are 20 chickens and 2 more ? 20 -{- 4 ? 20+7 ? 20+9 ? 21+3 ? 21+5 ? 21+7 ? 21+9 ? 22+4 ? 22+6 ? 22+8 ? 23+3 ? 23+4 ? 23+7 ? 23+9 ? 23+6 ? How many ducks are 24 ducks and 3 more ? 24+6 ? 24+9 ? 25+4 ? 25+7 ? 25+9 ? 26+8 ? 27+4 ? 28+7 ? 29+5 ? 30+7 ? 31+8 ? 32+6 ? 33+5 ? 36+9 ? 39+8 ? 41+5 ? 43+7 ? 46+8 ? 47+7? 49+8? 51+7? 55+5? 57+6? 58+7? How many pigeons are 63 pigeons and 9 more? 63+7 ? 64+5 ? 65+7 ? 66+6 ? 67+7 ? 68+8 ? 69+9 ? 70+7 ? 75+6 ? 76+8 ? 77+8? 79+4? 79+6? 79+7? 81+9? 83+ ? 84+9? 86+7? 88+8 ? 90+9 ? 91+7 ? 93+8 ? 97+5 ? 98+7 ? 99+9 ? How many quails are 10 quails and 10 more ? 10 + 12 ? 10+17? 10+19? 11+13? 11+15? 12+11? 13+12? 14+13? 15+12? 16+11? 17+13? 18+15? 19+11? 19+14? 19+16? 19+19 ? EXAMPLES FOB THE SLATE. (107) (108) (109) (110) (111) (112) 12345 23456 34567 45678 56789 67899 67890 78901 89012 90987 98765 12344 11223 23456 34567 65432 56789 45677 34455 78901 89012 10123 91023 78900 6778 23456 34567 45678 34567 01234 89900 78901 89012 90876 78901 56789 11223 23456 34567 54321 12345 98765 34455 78901 89012 09876 67890 54321 (113) (114) (115) (116) (117) (118) 34682 46820 65431 76432 61234 5064 57931 06842 76543 67675 45678 785 24680 3697 87654 56567 54321 9543 13564 568 98764 78789 76428 6748 2805 7634 09876 97973 85947 97054 3579 86420 1234 46467 64758 7865 46820 13579 765 33590 81927 753 97531 24680 12076 45678 79635 39 ADDITION. 21 119. 65340 + 6731 + 748 + 68451+396+7503+46075 + 1290+ 25738+46803. 120. 54268+405+1708 + 43671+72049 + 492+1760+25357+ 1434+84162. 121. 246768 + 21380 + 4075 + 126849 + 257 + 1305 + 24350+ 439871+40306+601734. 122. 3947+73845+300901+499091+45131+564429+484292+ 178737+58072+65344+194532+758+14. Find the sum of the following numbers : EXAMPLE (123.) One hundred and eighteen thousand, nine hundred and forty-eight. One thousand, one hundred and ninety-two. Two millions, eight hundred and sixteen thousand, seven hundred and sixty. Ninety thousand, four hundred and forty -five. One hundred and twenty-eight thousand. One million, one hundred and forty-three thousand, eight hundred and twelve. Twenty-four thousand, six hundred and sixty-four. Three millions, two hundred and forty-three thousand. EXAMPLE (124.) Twenty-seven millions, six hundred and nineteen thousand^ eight hundred and sixty-six. Three hundred and fifty-four thousand, seven hundred and ninety-seven. Two minions, two hundred and ninety thousand, three hun- dred and sixty-three. Nine hundred and thirty thousand, four hundred and thirty. Four hundred thousand. One hundred and twenty-seven millions, seven hundred and seventy-eight thousand, nine hundred and eighty-one. One million, four hundred and twenty-one thousand, six hundred and sixty-one. Eight hundred and sixty-nine thousand. ^ 22 EXAMPLES. EXAMPLE (125.) Sixty millions, seven hundred and eight thousand, five hun- dred and two. Two millions, nine hundred and thirty-seven thousand and sixty-six. Sixty-one thousand. One million, six hundred and twenty-five thousand. EXAMPLE (126.) Two millions, one hundred and twenty thousand, three hun- dred and ninety-seven. Eighty-six thousand. Two millions and six hundred thousand. Four hundred and fifty thousand. PBACTICAIj EXAMPLES. 127. A farmer has sold at different times 30, 45, 48, 50, 56, and 63 bushels of oats ; how many bushels altogether ? 128. A family consumed in a year 6 loads of coal, weighing as foUows : 1250, 1168, 987, 1076, 879, and 1275 pounds; how many pounds in all ? 129. A merchant has bought 5 cases of muslin, containing respectively the following number of yards : 963, 897, 985, 1005, and 889 ; how many yards in all ? 130. The population of the New England States in 1850 was respectively 583,000, 318,000, 314,000, 995,000, 148,000, and 371,000, what was the whole population of New England ? SUBTRACTION. Art, 8. Subtraction is taking a less number from a greater. The number thus found is called the Differ- ence or Remainder. The number to be subtracted is called the Subtra- SUBTRACTION. 23 hend, and the number from whicH it is to be taken the Minuend. Simple Subtraction is taking one number from another of the same name. Subtraction is sometimes expressed by the sign ( ) minus* The number written after the sign is to be sub- tracted from the one before it ; as 5 3=2 ; three sub- tracted from 5 leaves 2. SUBTRACTION TABLE. [It is promiscuously arranged and without answers.] From 123412342342 Take 1 2 1 1 2 1 1 2 1 Remainder 1 1 &c. 454564565676 122313411234 787878787878 132537425466 9 10 9 10 9 10 9 10 9 10 9 10 234567823456 11 12 11 12 11 12 11 12 11 12 11 12 694725839876 13 14 13 14 13 14 13 14 13 14 13 14 987654328769 15 16 15 16 15 16 15 16 15 16 15 16 896798754346 24 MENTAL EXERCISES. Prom 17 18 17* 18 17 18 17 18 17 18 17 17 Take 9 6 3 8 5 7 8 9 6 5 4 3 19 18 17 16 19 18 17 16 15 14 18 19 978986989897 MENTAL EXEKCISES. In a class of 2 girls how many will be left if 1 girl leave it ? 2 girls ? How many are 21 ? 22 ? In a class of 3 boys how many will be left if 2 boys leave it? 1 boy ? 3 boys ? How many are 32 ? 31 ? 33 ? Edward has 5 apples and his sister 3, how many more has he than his sister ? How many are 5 4 ? 5 2 ? 5 5 ? 5 3 ? 51 ? Mary had 6 cakes and gave 3 to her brother, how many had she left ? How many are 65 ? 63? 61? 66? 64? 62? Anna has 7 good marks at school and 4 bad ones, how many more good ones has she than bad ones ? How many are 73 ? 75 ? 71 ? 77 ? 76 ? 74 ? John had 8 oranges and has given away all except 5, how many has he given away ? How many are 85 ? 87? 83 ? 86 ? 82 ? 8-4 ? Thomas has 9 cents and his brother 7, how many more has he than his brother ? How many are 96 ? 93 ? 97 ? 94 ? 95 ? EXAMPLES FOB THE STATE OB BLACKBOABD. EXAMPLE 1. From 5698 subtract 3245. Process. 5 units from 8 units leave 3 units ; 4 tens from 9 tens leave 5 tens, &c. Ans. 2453 Ex. 2 From 7653 subtract 4865. Process. Since 5 cannot be taken from 3, add one of the 5 tens, or 10 to the 3 ; then 5 from 13 leave 8. For the same reason add one of the 6 hundreds, or 10 to the Ans. 2988 4 tens left, and take 6 from 14, &c. SUBTRACTION. ZO Ex. 3. From 6004 subtract 3125. Process. Though there are no tens in the upper num- ber, it contains at least 10 units ; therefore add ten to the 4 and take 5 from 14, and consider that only 9 of the Ans. 2579 10 tens which the number, contains remain in the tens place ; or the result is the same if 1 is added or earned to the next lower figure, the two numbers being equally increased. RULE. Write the less number under the greater, so that the right hand figure of each shall be under each other, and draw a line beneath them. Begin at the right hand and subtract each figure of the lower number from the one directly above it, and write the difference beneath it. If the upper figure is less than the lower, add ten to it, then subtract and carry one to the next lower, or subtract one from the next upper figure. PROOF. Add the remainder to the less number, and if the sum is equal to the greater, the subtraction is correct. EXAMPLES. W (5) (6) (7) (8) 654321 765432 876543 987654 987653 331201 234201 145312 431212 534443 (9) (10) (11) (12) (13) 767676 878787 989898 888888 999999 423122 534231 675643 544332 497531 (14) 7532175321 4826084356 (15) 8642066420 3736393035 (16) 9753195713 2846098648 (17) 1023678415 897534706 (18) 2125374859 958763465 (19) 7685946354 5869370819 (20) 8796059384 3948126893 (21) 9281736471 2736453673 2 26 EXAMPLES. (22) (23) (24) (25) 3435796354 4356534798 5768743276 65439S7672 1527697616 2785374689 2378679863 5734689778 (26) (27) (28) (29) 7569354978 8954978654 9872579846 7963864201 3678567889 5379889910 6743287938 3576383712 (30) (31) (32) (33) 6743876549 9576854378 4759813548 8743645211 4371968732 2768907569 1964679473 1170927802 (34) (35) (36) (37) 5478698472 3789589756 6802468024 7791357915 3579137951 4749454347 8642042864 7894013498 (38) (39) (40) (41) 6547328964 2785679546 5473143256 8321731214 1758916407 5732157321 9543682410 5437691607 (42) (43) (44) (45) 4002007100 870345076 5002070022 6070005000 3754327819 3700072001 8573000012 4781973607 46. 75694 3590 = what number ? 47. 87532 7615 = 48. 987563 43215= " 49. 107923 36001= " 50. 100000 999 = " 51. 845067 71389= " 52. 1789543 76508= " 53. 200100 54321= " 54. 500000 99887= " 55. 6000000 887766 = " 56. From ten thousand and five, subtract seven thousand five hundred and seven. 57. From twenty-five thousand, one hundred and six, sub- tract ten thousand, six hundred and seventy-nine. MULTIPLICATION. 27 58. From one hundred thousand and fifty-eight, subtract fifteen thousand, three hundred and ninety. 59. From one million, subtract seven thousand and five. 60. From ten millions, ten thousand and ten, subtract five hundred thousand, five hundred and five. PBACTICAL EXAMPLES. 61. A man purchased a farm for 6750 dollars, and sold it for 9000 dollars, how much did he gain ? 62. Mt. Washington, N. H., is 6285 feet high ; Mt. Mans- field, Vt., 4,279 feet ; how much higher is Mt. Washington than Mt. Mansfield ? 63. The Missouri River is 3100 miles long, and the Missis- sippi 2500 ; how much longer is the former than the latter ? 64. The population of London in 1850 was two millions^ three hundred and sixty-three thousand ; that of Paris one million, fifty-three thousand and two hundred ; what was the difference ? 65. A man was born in the year 1780 and died in 1859, how old was he when he died ? 66. North America contains eight millions of square miles ; Europe, three millions five hundred thousand; how many more square miles are there in North America than in Europe ? MULTIPLICATION. Art. 9. Multiplication is finding a number, equal to a given number repeated by addition as many times as there are units in another given number ; as, 4 times 6 are 24, or 6+6+6+6=24. The number to be multiplied is called the Multiplicand* The number by which the other is multiplied is called the Multiplier* The answer, or number found by multi- plication, is called the Product* 28 MULTIPLICATION. Both the multiplicand and multiplier are called Factors* because they make the product. Simple multiplication is that in which the multiplicand is of only one name. The sign of multiplication is (X) an oblique cross. ILLUSTRATION. 5 times 6 are 30 : 6X5=30. The multiplicand is 6 ; the multiplier 5 ; and the product 30. 5 and 6 are also factors. The multiplicand must properly be of the same name as the pro- duct or answer required, though it is often more convenient to make the larger number the multiplicand, while the result is the same. The number of marks below is the same, whether they are arranged in five groups of six each, or six groups of five each, or all in one group ; thus Both 6X5 and 5X6=30. But if 6 men can reap a field in 5 days, and it be required to find how many men could reap it in one day, the multiplicand will be 6 men, and the product 30 men ; while if it be required to find in how many days one man could reap it, the multiplicand will be 5 days, and the product 30 days. The multiplier, though often used as a concrete number applied to some particular things, is properly only an abstract number. In the illustration of the preceding remark, 6 men multiplied by 5 days, or 6 days multiplied by 6 men, would be absurd ; but 6 men multiplied by 5, which is the same as the number of days, are 30 men ; and 5 days multiplied by 6, which is the same as the number of men, are 30 days. MUI/TIPUCATION TABLE. 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 36 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 30 10 40 10 50 10 60 10 70 11 " 22 11 33 11 44 11 " 55 11 66 11 " 77 12 " 24 12 36 12 " 48 12 " 60 12 " 72 12 " 84 MULTIPLICATION. 29 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 10 4 44 4 48 5 40 5 45 5 50 5 5 60 6 48 6 54 6 GO 6 66 6 72 7 56 7 63 7 70 7 77 7 8 64 8 72 8 80 8 88 8 96 9 72 9 81 9 90 9 99 9 10S 10 80 10 90 10 100 10 110 10 1-20 11 88 11 99 11 110 11 121 11 132 12 96 12 108 12 120 12 132 12 144 The multiplication table promiscuously arranged without the products : Multiplicands 12 9 11 8 10 7 4 6 3 5 2 Multipliers !?1110_9_8_7_6_5_4_3J2 ANOTHER FORM. Multiplies 47 10 258 11 0369 12 Multipliers J* _j?_?_2_2_?__?_?J?_J5_? 147 10 258 11 0369 12 147 10 258 11 0369 12 147 10 258 11 0369 12 147 10 258 11 0369 12 147 10 258 11 0369 12 147 10 258 11 0369 12 8989898989898 30 MULTIPLICATION. Multpl'dl 47 10 258 11 0369 12 Mulp'rs_9 _8_9_89_89898989 147 10 258 11 0369 12 147 10 258 11 0369 12 H ^ v j^ 11 12 10 jl JiJ 10 11 12 10 11 147 10 258 11 0369 12 1? 19 1^ 1? i? Ji 1? 1? 11 J^ J!^ \H J3 [Let pupils write the above table and the respective products on their slates ; also recite the products without being written till they can do so promptly without mistakes. Art. 10. Multiplication by one figure. MENTAL EXERCISES. At 3 cents each, how much will 2 oranges cost ? 4? 6? 3? 5? Answer. If 1 orange cost 3 cents, 2 oranges will cost 2 times or twice 3 cents, which are 6 cents. At 4 cents each, how much will 3 lemons cost ? 5 ? 2 ? 4 ? 6 ? 3 ? 5 ? At 5 cents each, how much will 4 melons cost ? 3 ? 5 ? 2 ? 4? 6? At 6 cents each, how much will 5 pine apples cost ? 3 ? 2 ? 4? 6? 5? At 7 cents a pint, how much will 2 pints of cherries cost ? 4? 6? 3? 5? At 8 cents a pint, how much will 3 pints of strawberries cost ? 5? 2? 4? 6? At 9 cents a quart, how much will 4 quarts of chestnuts cost ? 2? 6? 3? 5? At 10 cents a pound, how much will 5 pounds of sugar cost ? 3? 2? 4? 6? At 11 cents a pound, how much will 6 pounds of cheese cost? 4? 3? 5? 2? At 12 cents a pound, how much will 3 pounds of ginger cost ? 5? 2? 4? 6? MULTIPLICATION. 31 EXAMPLES FOE THE SLATE. EXAMPLE 1. Multiply 4326 by 5. Process. 5 times 6 are 30 ; write units and ) ( 4326 carry the three tens as in addition ; 5 times 2 > written -| ^ tens are 10 tens, and three carried are 13 tens ; ) ( Ans. 21630 write the 3 tens and carry the 1. Thus proceed. Proof. 4 times-}-l time the number are 5 times the number. Therefore multiply by 4, and ta the product add the multiplicand ; the result will be same as before if correct. Or repeat the process, inverting the order of the figures, as 6 times 4, 2 times 4, &c. PBOOF. 4326 4 17304 21630 RULE. Write the multiplier under the right-hand figure of the multiplicand, and draw a line under it. Begin at the right hand and multiply each figure of the multiplicand by the multiplier, carrying as in addition. PROOF. Multiply by a number 1 less than before, and to the product add the multiplicand. If the result is the same t it is correct. (20 "Write (3.) (4.) (5.) (6.) (7.) (8.) (9.) (10.) 135024 2 multiply (11.) (12.) (13.) (14.) (15.) (16.) (17.) (18.) 135024 3 by 2, (19.) (20.) (21.) (22.) (23.) (24.) (25.) (26.) 135024 4 thus and 246135 350246 461350 502461 613502 153042 264153 305264 each number 416305 520416 631502 310542 421653 532064 643105 520641 3, and 4 165320 510342 621435 205614 136205 103245 124356 235460 Multiply by 5 and 6 : (27.) (28.) (29.) (30.) (31.) (32.) 679867 579689 497867 938796 479685 596879 (33.) 689769 (34.) 786975 (35.) 879789 (36.) 787980 (37.) 979899 (38.) 676869 (39.) 486979 (40.) 579687 397869 278697 697987 (41.) (42.; (43.; (44.) 708090 32 MULTIPLICATION. MENTAL EXEECISES. At 2 shillings a yard, how mucli will 7 yards of calico cost ? 8? 9? 5? At 3 shillings a yard, how much will 8 yards of ribbon cost ? 7? 9? 6? At 4 shillings a yard, how much will 9 yards of silk cost ? 7? 4? 8? At 5 dollars a yard, how much will 7 yards of cloth cost ? 9? 6? 8? At 6 cents a skein, how much will 8 skeins of sewing silk cost? 6? 9? 7? 5? At 7 shillings a pound, how much will 9 pounds of wool cost? 7? 8? 4? At 8 dollars each, how much will 7 coats cost ? 9 ? 6 ? 8? 5? At 9 cents a yard, how much will 8 yards of muslin cost ? 8? 7? 9? 4V At 10 cents a spool, how much will 9 spools of cotton cost ? 7? 5? 8? 6? At 11 dollars each, how much will 7 shawls cost ? 5 ? 3 ? 9? 8? At 12 shillings a pair, how much will 8 pairs of gloves cost ? 6? 9? 7? 4? EXAMPLES FOB THE SLATE. (45.) 14725803 14725803 14725803 7 8 9 Write thus and multiply each number by 7, 8, and 9 : (46.) 69142580 (50.) 17452083 (54.) 12740853 (47.) 36914758 (51.) 61794250 (55.) 20859631 (48.) 69147280 (52.) 39641278 (56.) 31962745 (49.) 47258069 (53.) 50863194 (57.) 45270836 Art, 11. Multiplication by two or more figures. MENTAL EXEECISES. At 2 dollars each, how much will 10 caps cost ? 12 ? 11 ? 9? At 3 dollars each, how much will 11 hats cost ? 10 ? 12 ? 8 ? MULTIPLICATION. 33 At 4 dollars each, how much will 12 bonnets cost ? 10 ? 7? 11? At 5 shillings a pair, how much will 10 pairs of socks cost ? 12 ? 9 ? 11 ? At 6 shillings a yard, how much will 11 yards of linen cost ? 9? 12? 10? At 7 cents each, how much will 12 papers of pins cost ? 10 ? 8? 11? At 8 cents each, how much will 10 papers of needles cost ? 12? 8? 11? At 9 cents a dozen, how much will 11 dozen of buttons cost ? 9? 12? 10? At 10 cents a yard, how much will 12 yards of cambric cost ? 10? 7? 11? At 11 dollars each, how much will 11 shawls cost ? 8 ? 12 ? 10? At 12 dollars each, how much will 12 pieces of oilcloth cost ? 10? 6? 11? 9? 8? 7? EXAMPLES FOB THE SLATE. EXAMPLE 58. Multiply 172829 by 12. 17289 Process. --The same u:; multiplying by units or one TO figure; 12 times 9, *o. Write thus and multiply each number by 10, 11, and 12 : (59.) 96303527 (62.) 30S52741 (65.) 30691472 (60.) 85274196 (63.) 27419630 (66.) 25803691 (61.) 74196308 (64.) 19630852 (67.) 50642839 Art. 12, Multiplication by numbers greater than 12. EXAMPLE 68. Multiply 28357 by 234. Process. Multiplying by 234 is the same as multi- 28357 plying by 2 hundreds, 3 tens and 4. 4 times 28357 234 =113428 by the same process as before. 3 times 113428 2835785071, and since the multiplier is tens the pro- 85071 duct will be tens ; therefore write the right hand figure 56714 under tens. For a similar reason write the right hand figure of the product of 28357X2 under hundreds. Ans - Then the products of the number multiplied by 2 hundreds, 3 tens and 4 added together will be its product multiplied by 234. 2* MULTIPLICATION. BULK Write the multiplier under the multiplicand. Begin at the right hand, and multiply by each figure of the multiplier separately, writing under each the right-hand figure of its product : with the other figures in their proper order. Add the products of each figure together, and the product of the two numbers will be found. PROOF. Eepeat the multiplication, only inverting the order of the figures, as in the above example ; instead of 4 times 1 say 1 times 4. When the multiplier contains a cipher it may be passed over, but the right hand figure of the next product must be written, not un- der the cipher, but its own multiplier. (69.) (70.) (71.) (72.) (73.) (74.) (75.) (76.) (77.) (78.) (79.) 144X13 245X18 356X24 587X^5 789X56 890X67 901X78 234X89 987X99 876X98 765X87 (80.; (81.; (82.; (83; EXAMPLES. 1426X31 2536X42 3675X54 4879X65 (84.) 5098X76 (85.) 6109X87 (86.) 7432X98 (87.) 8987X89 (88.) 9786X78 (89.) 1067X99 (90.) 15789X125 (91.) 21478X234 (92.) 34890X345 (93.) 41256X456 (94.) 54675X567 (95.) 67812X678 (96.) 78569X789 (97.) 86453X897 (98.) 95387X901 (99.) 92896X802 Special Rules. Art. 13. Multiplication of numbers having ciphers on the right hand. Ex. 100. Multiply 245 by 100. Process. Write 245 with two ciphers (24500) on the right hand. This changes the local value of the figures the same as multiplying by 100. 245 100 000 000 245 Ans. 24500 MULTIPLICATION. 35 Ex. 101. Multiply 256 by 300. Process. 3 times 256 are 768, and two ciphers are to "be written after this product for the same reason as before. Ex. 102. Multiply 4600 by 32000. Process. 46X32=1472. Write five ciphers at the right hand, for the product of hundreds by thousands is hundreds of thousands by the general rule, 256 300 Ans. 76800 4600 32000 9200000 138 Ans. 147200000 RULE. When the multiplier is 10, 100, 1000, &c., write as many ciphers as it contains on the right hand of the mul- tiplicand. In other cases, write and multiply the other figures as if they had no ciphers on the right hand, and annex to the product as many ciphers as were not used in both numbers. EXAMPLES. (103.) 2250X 10 (109.) 1020000X 500 (104.) 3500X1200 (110.) 276000X 3 (105.) 4732X1000 (111.) 375X30100 (106.) 130X 51 (112.) 17020X 1000 (107.) 356X 100 (113.) 1000000x10000 (108.) 7000X 700 (114.) 1004000X10500 MULTIPLICATION BY COMPOSITE NUMBEKS. Art. 14. A Composite number is one which is the pro- duct of two other numbers ; as, 30 composed of 6 X 5. Ex. 115. Multiply 29 by 24. Process. Since 4 times 6=24, 4 tunes 6 times 29=24 times 29. 29 6 174 4 RULE. Find two or more numbers which being multiplied together will produce the given multiplier. Multiply the 36 MULTIPLICATION. multiplicand by one of them, and its product by another, till all the factors are used. (116.) 115X15 (117.) 123X16 (118.) 39X36 (119.) 162X18 EXAMPIiES. (120.) 126X27 (121.) 99X32 (122.) 265X35 (123.) 324X48 (124,) 87X63 (125.) 405X108 (126.) 3456789X 1019 (135.) 871496X2468 (127.) 9830291X 7305 (136.) 397684X6005 (128.) 5006284X 6635 (137.) 469537X3708 (129.) 4000059X 7239 (138.) 873576X8764 (130.) 873000X 1000 (139.) 468937X7056 (131.) 257000X 4000 (140.) 798600X8750 (132.) 5749362X 3827 (141.) 750000X9000 (133.) 4327000X 3500 (142.) 596875X9678 (134.) 13786926X85043 (143.) 658907X7869 (144.) 963852 7 1 4 Multiplicand. 475180 6 3 9 Multiplier. 145. Multiply seventy-six thousand by sixty-eight hundred and four. 146. Multiply nine million and eight thousand by five hun- dred thousand and sixty. 147. Multiply eighty-seven thousand, six hundred and three, by nine thousand, eight hundred and sixty-five. 148. Multiply eighty-three thousand, four hundred and fifty seven, by six thousand, eight hundred and thirty-five. 149. Multiply nine hundred and four thousand, by ten thou- sand and two hundred. 150. Multiply eighty thousand and six hundred, by seven thousand and two. 151. Multiply three million, two hundred and forty thou- sand, by three hundred and twenty-four thousand. 152. Multiply three hundred and four thousand and seven hundred, by ninety-seven thousand, six hundred and three. MULTIPLICATION. 37 153. Multiply eight million, six hundred and forty-three thousand, by nine thousand, two hundred and thirty. PBACTICAIi EXAMPLES. 154. At 4 dollars a yard, how much will 25 yards of cloth cost ? 155. How many yards in seven pieces of muslin if each piece contain 28 yards ? 156. At 9 dollars a barrel, how much will 124 barrels of flour cost ? 157. At 112 dollars an acre, how much will a farm contain- ing 270 acres cost ? 158. If a man travel 32 miles a day, how far will he travel in 24 days. 159. At 50 cents a bushel, how much will 136 bushels of apples cost ? 160. At 100 cents a day, how much will a laborer earn in 110 days ? 161. At 12 cents a pound, how much will 120 pounds of beef cost? 162. At $97 each, how much will 15 horses cost ? 163. At $10 a barrel, how much will 225 barrels of flour cost? 164. At $56 a head, how much will 25 cows cost ? 165. At $23 an acre, how much will 99 acres of land cost ? 166. In an orchard there are 84 rows of trees and 63 trees in each row ; how many trees are there ? 167. In a bale of sheeting there are 22 pieces, and in each piece 27 yards ; how many yards in all ? 168. In a hogshead there are 63 gallons ; how many gallons in 23 hhds. 169. In a box of calico there are 94 pieces, and each piece contains 35 yards ; how many yards in all ? 170. At $17 a barrel, how much will 211 barrels of molasses cost. Many more such examples will be found among the Promiscuous Examples, after Division, and U. S. Money. ? UNIVERSITY I 38 DIVISION. DIVISION. Art. 15. Division is finding either how often one number is contained in another ; or one of as many equal parts of a number as are expressed by another number ; thus, We find either that 5 is contained in 30 6 times ; or that 6 is one of 5 equal parts of 30. The number to be divided is called the Dividend; the dividing number, the Divisor ; the number found or the answer, the Quotient J and that part of the dividend less than the divisor, which is sometimes left after division, is called the Remainder. The sign of division is (--) a horizontal line between two dots, written after the dividend and before the divisor. Division may also be expressed by writing the divisor under the dividend. In this way the remainder becomes a part of the quotient at the right hand. Such an ex- pression is called a fraction; as | (read) one-half; , one-third ; f , two-thirds ; J, three fourths or quarters ; f, four-fifths, &c., &c. ILLUSTRATION. Divide 27 by 4. 27-H= or *- =6 and 3 left, or 6|. 27 is the dividend, 4 the divisor, 6| the quotient, or 6 the quotient and 3 the remainder. KEMABK 1. Dividing a number by 2, divides it into 2 equal parts each of which is called one-half (written 5;) dividing it by 3 divides it into 3 equal parts, each called one-third (-. ) So dividing by 4 gives one-fourth (,) by 5 one-fifth (,) by 6, 7, 10, 15, &c., gives' one-sixth ( , ) one-seventh ( |, ) one-tenth (-fa, ) one fifteenth (iV) &c. 2. The divisor and quotient correspond with the factors in multi- plication, and the dividend with the product. Hence 3. When one factor and the product is given, the other factor may be found by division. 4. One of the factors must be of the same name as the dividend, and one of as many equal parts of it as are expressed by the other. If the divisor is one of the equal parts, the quotient is the number of DIVISION. 39 parts ; but if the divisor is the number of parts, the quotient is one of the parts ; thus, 20 cts. -r- 5 cts. = 4 (equal parts); but 20 cts. ~ 4 (equal parts) = 5 cts. (one of the equal parts of 20). Art. 16. There are two methods of division, Short and Long. In short division the process is mental, and only the result written. This is used when the divisor is less than 12. In long division the process and result are both written. In dividing the divisor is usually \mtten before the dm- 3)15 dend, and the quotient in short division under it, thus, 5 In long division after it, thus : 13) || ( 2 Short Division. DIVISION TABLE WITHOUT THE QUOTIENTS. 3) 3 69121518212427 30^33^ 36 4)481216^20^242832364044 48^ 5)510^52025^30^35^40^45^5055 60 6 3 ^0^7548^ 108 10) ^0 50 _80 100 JA) 40 70 WO _30 60 90 120 11) ^2 55.88121114477110336699 132 12) _24 60 ^ 12 12_ 8 8 120 36 72108 144 MENTAL EXERCISES. At 2 cents each, how many peaches can be bought for 8 cents? 4? 12? 16? 20? 24? 28? 6? 10? 14? 18? 24? Process. If 2 cents will buy 1 peach, 8 cents will buy as many peaches as there are times 2 cts. in 8 cents, which are 4. Or if 2 cents will buy 1 peach, half as many as the cents, 8 cents will buy half of 8 peaches, which are 4. At 3 cents each, how many oranges can be bought for 6 cts ? 12? 18? 24? 30? 36? 9? 15? At 4 cents each, how many melons can be bought for 12 cents? 8? 16? 24? 32? 40? 48? 20? 28? 36? 44? At 5 shillings a pound, how many pounds of tea can be bought for 15 shillings ? 25? 35? 20? 30? 45? 55? 65? 40? 50? 60? EXAMPLES. 41 At 6 shillings a yard, how many yards of linen can be bought for 12 shillings ? 24? 36? 48? 60? 18? 30? 42? 66? 72? Divide the following numbers, and write the remain- ders over the divisors at the right hand of the quo- tient ; as, 6)38 ~6f Let the pupils also recite them without seeing the answers written : 2)511 1723 391521 7131925 3) 8_ 17 2735 514233211202837 4)11^21^3645717314113263749 5) 13_ 27_ 44 J38 8 22 j*7 3 H 23^ 48 63 6)1431 50 70 92744 62 20 25 _56 J4 7) 17_ 37 60j8irL29537423^ i(37 _89 8)1945 70 93133559 852735 75 101 9)2150 80 103 15 40 J55 j)7 JJ1 J59 84 112 10) 23 55 89 113 17^ 43 73 104 34 43 93 JL25 11)_25 _61 _90 _127 JL9 _47 _78 113 _36_48_jLOO ^31 12) 27 JO 100 13(5.?!_5^j3? 122 38 53 112 150 * MENTAL EXEBCISES. At 7 cents a quart, how many quarts of milk can be bought for 14 cents? 28? 35? 42? 56? 49? 63? 21? 70? At 8 shillings a bushel, how many bushels of corn can be bought for 24 shillings ? 40? 16? 32? 48? 64? 80? 56? 42 DIVISION. At 9 dollars a barrel, how many barrels of flour can be bought for 18 dollars ? 36? 27? 45? 63? 54? 72? 99? 81? At 10 shillings a bushel, how many bushels of wheat can be bought for 30 shillings ? 20? 50? 40? 70? 60? 90? 80? 100? At 11 cents a pound, how many pounds of rice can be bought for 33 cents ? 22 ? 44 ? 66 ? 55 ? 77 ? 99 ? 88 ? 110 ? At 12 cents a pound, how many pounds of coffee can be bought for 24 cents ? 48? 60? 36? 72? 96? 84? 120? If three apples cost 6 cents, how much will one apple cost ? Process. If three apples cost 6 cents 1 apple will cost as many cents as 3 is contained times in 6 ; or i of 6 cents, which is 2 cents. If 5 peaches cost 15 cts., how much will 1 peach cost ? If 7 melons cost 42 cts., how much will 1 melon cost ? If 9 bushels of chestnuts cost 36 dollars, how much will 1 bushel cost ? If 8 barrels of flour cost 72 dollars, how much will one bbl. cost? If 12 yards of cloth cost 84 dollars, how much will 1 yard oost? EXAMPLES FOB THE SLATE. EXAMPLE 1. Divide 4325 by 5. Process. Since it is not easy to divide so large a number at once, it is considered as separated into parts ; as 4000+300+20+5. If 5 were contained in 4, the first figure at the left of the quotient would be thousands ; but 5 is not contained in 4, therefore add 4000 to 300 and - divide the hundreds. 5 is contained in 43 hun- dred (4300) 8 hundred (800) times, and 3 hun- dred (300) remainder. Add the remainder to the 2 tens (20) and divide the tens. 5 is contained in 32 tens (320) 6 tens times and 2 tens remain- 300 20 5 Parts of the divi- dend. 800 ) Parts of 60 V the quo- 5 ) tient. Comm'nly written - 5)4325 Ans. "865 der, which add to 5 and divide. The last part of the quotient will be 5. The sum of all the parts is the quotient or answer required. The same result is obtained by the following process, commonly used : 5 into 438 and 3 remaining. Write the 8 under the last figure EXAMPLES. 43 divided, and prefixing the 3 remaining to the 2, divide 5 into 326 and 2 remaining. Write the 6 under the last figure divided, and thus proceed. RULE FOR SHORT DIVISION. Write the divivor at the left of the dividend, and draw a line under the latter. .Begin at the left, and divide each figure separately. Write each figure of the quotient under the last figure divided, and prefix the remainder, if there is any, to the next figure of the dividend and divide again. If there is a remainder of ter dividing all the figures of the dividend, write it, with the divisor under it, at the right hand of the quotient. If there is no other figure, a cipher must be written in any place except at the left hand. PROOF. Multiply the divisor by the quotient and add the remainder. The product should be like the dividend. (2.) 2)135024 3)135024 4)135024 Write thus, and divide each number by 2, 3, and 4. (3.) 246135 (4.) 350246 (5.) 461350 (6.) 502461 (7.) 613502 (8.) 153042 (9.) 264153 305264 (10. (11.) (12.) (13.) (14.) (15.) (16.) (17.) (18.) 416305 520416 631502 310542 421653 532064 643105 520641 (19.) 165320 (20.) 510342 (21.) 621435 (22.) 205614 (23.) 136205 (24.) 103245 (25.) 124356 (26.) 235460 Divide each number by 5 and 6. (27.) (28.) (29.) (30.) (31.) (32.) 679867 579689 497867 938796 479685 596879 (33.) 689769 (34.) 786975 (35.) 879789 (36.) 787980 (37.) 979899 (38.) 676869 (39.) (40.) (41.) (42.) (43.) (44.) 486979 579687 397869 278697 697987 708090 DIVISION. MENTAL EXERCISES. If 6 pounds of sugar cost 75 cents, how much will 1 pound cost? At 7 cents a pound how many pounds of fish can be bought for 25 cts. ? 31 cts. ? 37 cts. ? 45 cts. ? 65 cts. ? 72 cts. ? 90 cts. ? At 8 dollars a barrel, how many barrels of flour can be bought for $28 ? $18 ? $35 ? $50 ? $42 ? $60 ? $75 ? $83 ? $90 ? $100 ? If 12 pounds of rice cost 100 cts., how much will 1 Ib. cost ? At 12 cents a pound, how many pounds of starch can be bought for 30 cts. ? 40 cts. ? 50 cts. ? 64 cts. ? 70 cts. ? 80 cts. ? 90 cts.? 100 cts.? 110 cts.? 125 cts.? 150 cts.? At 9 dollars a ton, how many tons of coal can be bought for $100 ? $75 ? $67 ? $55 ? $48 ? $42 ? $37 ? $35 ? $94 ? If 11 tons of hay cost 125 dollars, what will 1 ton cost. At 10 dollars a yard, how many yards of cloth can be bought for $34 ? $45 ? $56 ? $64 ? $75 ? $84 ? $96 ? $125 ? EXAMPLES FOE THE SLATE. (45.) 7)14725803 8)14725803 Write thus, and divide by 7, 8, and 9. 69142580 36914758 69147280 47258069 58036914 72580369 80369147 9)14725803 (46.) (47.) (48.) (49.) (50.) (51.) (52.) (53.) (54.) (55.) (56.) (57.) (58.) 91472580 (61.; 17452083 61794250 39641278 50863194 75208639 42785301 83096417 94127528 (62.) (63.) (64.) (65.) (66.) (67.) (68,) (69.) 12740853 20859631 31962745 45270836 53081964 64195278 78526309 86304190 Divide the following numbers by 10, 11, and 12. (70.) (71.) (72.) (73.) (74.) (75.) 96308527 85274196 74196308 63085274 52741963 41963085 (76.) (77.) (78.) (79.) (80.) (81.) 30852741 27419630 19630852 93824605 82493057 71938246 (82.) (83.) (84.) (85.) (86.) (87.) 30691472 25803691 50642839 75039428 64283917 14725803 DIVISION. 45 Art, 17. Long Division, EXAMPLE 1. Divide 97836 by 18. Process. The same as in short division, ex- Divsr.Divid'd.Quoti'nt cept the quotient is placed at the right hand, and 18)97386(5435^ the products, with the remainders, are written 90 under the parts of the dividend used. ~78 Quotient figures must often be found by trial, 72 and if the product of any trial figure and the di- -go visor, is greater than the part of the dividend used, that figure is too great ; on the other hand, if the remainder is greater than the divisor, the quotient figure is too small. Very often a quotient figure may be found, or 6 nearly so, by dividing the first figure or two in the dividend, by the first figure in the divisor, allowing more or less for carrying. RULE. Write the divisor at the left of the dividend, and leave a place for the quotient on the right. Find how many times the divisor is contained in the few- est left-hand figures of the dividend that contain it, and write the number in the quotient. Multiply the divisor by this quotient figure and write the product under that part of the dividend which was found to contain the divisor. Subtract the product from the figures above it ; to the remainder bring down the next figure in the dividend, and divide again. Continue dividing till all the figures of the dividend are used, and if, after bringing down a figure, the number thus formed is less than the divisor, write a cipher (0) in the quo- tient and bring down another figure. PBOOF. The same as in short division. EXAMPLES. (2.) (3.) (4.) 13)H7256( U)152768( 15)175265( DIVISION. Divide (5.) 3540 by 12 (17.) 5258 by 22 (29.) 158500 by 50 (6.) 8764 " 13 (18.) 10713 " 14 (30.) 140998 " 26 (7.) 5200 " 15 (19.) 11829 " 18 (31.) 116873 " 28 (8.) 5499 " 13 (20.) 11347 " 19 (32.) 127364 " 68 (9.) 7556 " 16 (21.) 39825 " 27 (33.) 295482 " 74 (10.) 4935 " 15 (22.) 48440 " 28 (34.) 316704 " 96 (11.) 9454 " 18 (23.) 37430 " 29 (35.) 190850 " 25 (12.) 6567 " 17 (24.) 13125 " 35 (86.) 151445 " 35 (13.) 5472 "16 (25.) 30825 " 45 (37.) 135050 " 37 (14.) 6614 " 19 (26.) 32806 " 47 (38.) 55168 " 64 (15.) 7348 " 17 (27.) 19608 " 43 (39.) 5748435 " 63 (16.) 9182 " 21 (28.) 41088 " 52 (40.) 6480752 " 96 (41.) 215045924 by 86 (44.) 562752060 by 64 (42.) 405147456 " 56 (45.) 7254231 " 82 (43.) 459932616 " 66 (46.) 1053990 " 63 Art, 18, Special Rules. Division by numbers with ciphers on the right hand. Ex. 47. Divide 4720 by 100. Process. Cut off two figures from the right 1,00)47,20=47^ hand of the dividend ; since this changes the value of the rest, the same as dividing by 100, 20 is the remainder unless regarded as a decimal fraction. Ex. 48. Divide 76900 by 300. Process. The same as before, but divide also by 3,00)769,00 3, the other factor in the divisor. 055111(1 3 % KULE. Cut off the ciphers on the right of the divisor, and and as many figures on the right of the dividend. Divide the rest of the dividend by the rest of the divisor, if greater than one, and if there is a remainder, annex to it'the figures cut off from the dividend for the true remainder. DIVISION BY COMPOSITE NUMBERS. EXAMPLES. Divide (49.) 14100 by 600 (53.) 364000 by 6400 (50.) 365400 5000 (54.) 48000 1600 (51.) 138000 1000 (55.) 170000 it 400 (52.) 36009 " 1200 (56.) 1000009 " 300 Art. 19. Division by Composite Numbers. Ex. 57. Divide 1732 by 24. Process. Dividing by 3 divides the number into one of three equal parts and 1 remainder ; and divid- ing this by 8 divides the number itself into one of 3 times 8 equal parts, and one remaining to each of the three parts, which added to 1, the first remainder, makes 4 remainder. 3)1732 8)577+1 72+4 EULE. Divide by one of the factors, and the quotient thus found by the other factor. If there are remainders, multiply the second one by the first factor, and add the first remainder. Proceed in the same way if there are three or more factors. EXAMPLES. Divide (58.) 2976 by 24 (70.) 186021 1 >y 148 (59.) 134120 " 56 (71.) 119753 " 156 (60.) 155145 " 54 (72.) 246813 ' 169 (61.) 105409 " 63 (73.) 2500000 " 200 (62.) 12148 " 35 (74.) 1435792 " 218 (63.) 1728 " 144 (75.) 1579248 ' 227 (64.) 476345 " 100 (76.) 1681357 ' 239 (65.) 567324 " 111 (77.) 1792460 4 244 (66.) 643142 " 115 (78.) 1864219 ' 256 (67.) 792468 " 125 (79.) 2004312 ' 260 (68.) 864219 " 131 (80.) 3600000 300 (69.) 975312 " 142 (81.) 1080000 " 1200 DIVISION. (82.) 700239 by 123 (90. ) 470205 by 215 (83.) 1883187 " 249 (91. ) 962984 " 276 (84.) 593583 " 241 (92. ) 197776 " 376 (85.) 1886826 " 314 (93. ) 255136 " 476 (86.) 478224 " 324 (94. ) 124488 " 342 (87.) 854661 " 347 (95. 5698546 " 829 (88.) 2539615 " 439 (96.) 5354320 " 635 (89.) 1200000 " 600 (97.) 2400000 " 800 (98.) 4874583 by 643 (109.) 7046606 by 898 (99.) 6079864 " 719 (110.) 7977489 " 923 (100.) 8264574 " 846 (111.) 3769248 " 948 (101.) 8095230 " 935 (112.) 3779008 " 548 (102.) 4674784 " 694 (113.) 4991875 " 625 (103.) 4663778 " 1246 (114.) 9691836 " 1234 (104.) 5332114 " 1234 (115.) 5237479 " 1823 (105.) 61142488 " 4136 (116.) 18219071 " 3001 (106.) 452491424 " 3143 (117.) 70287492 " 7117 (107.) 297396341 " 3047 (118.) 16736642 " 3497 (108.) 960000000 " 8000 (119.) 72000000 " 9000 (120.) 3013974002 by 3074 (121.) 25174363929 " 30243 (122.) 881137279449 " 90807 (123.) 153288487686 " 407091 ^ (124.) 49062139937803 " 7001009 (125.) 156000000000 " 520000 126. Divide one hundred and twenty-seven thousand, by three thousand, seven hundred and forty-six. 127. Divide four million, six hundred and sixty-three thous- and, seven hundred and seventy-eight, by three thousand, seven hundred and forty-three. 128. Divide ten million, two hundred and five thousand, seven hundred and twenty-one, by three thousand two hun- dred and forty-three. GENERAL PP.INCIPLES IN DIVISION. 49 129. Divide one hundred and forty-one thousand, by two thousand, three hundred and fifty. 130. Divide eight million, eight hundred and sixty thou- sand, and sixty, by one thousand and thirty. 131. Divide ninety-two million and eighty thousand, by one hundred and two. 132. Divide twenty-three million and forty thousand, by ninety-six hundred. 133. Divide two million, seven hundred and thirty-six thousand, three hundred and seventy, by three thousand and seven. PRACTICAL EXAMPLES. 134. If it would take 1 man 3540 days to build a house, how long will it take 12 men to build it ? 135. At $65 each, how many cattle can be bought for $38740. 136. If $38805 will buy 597 cattle, what is the price per head? 137. If 137 acres of land cost $17125, what is the price per acre? 138. At $125 an acre, how many acres of land can be bought for $17250. 139. At $37 each, how many cows can be bought for $14689. 140. If 396 cows cost $14652, what is the price per head. 141. At $23 each, how many coats can be bought for $5451. 142. If 235 coats cost $5405, what is the cost of each ? Many more such examples will be found among the Promiscuous Examples. Art, 20, General Principles in Division. 1. Multiplying the dividend, or dividing the divisor by any number, multiplies the quotient by that number ; thus 24^-4=6. (24X2)-r4=12, or(6X2). 24 =12. 50 PROMISCUOUS EXAMPLES. 2. Dividing the dividend, or multiplying the divisor, by any number divides the quotient by that number ; thus, 24-r4=6 ; ( 24-^2 )-r4=3, or (6-i-2); 24-^(4X2)= 3, or (6-f-2.) 3. Multiplying or dividing both the dividend and di- visor by the same number, does not alter the quotient ; thus, 24-f-4=6; (24X2)-KX2)=6 ; =6. Art, 21, Promiscuous Examples in Addition. Sub- traction, Multiplication and Division. [Pupils are now supposed to know how to add, subtract, multiply and divide. The following examples are designed to teach them when to apply the different rules. They should be fully explained by the pupils. ] MENTAIj EXEECISES. EXAMPUG 1. A boy had 25 cents, and his father gave him 25 more ; how many did he then have ? Answer. He had as many as the sum of 25 cts. added to 25 cts., which is 50 cts. 25+25=50. Ex. 2. A girl had 50 cents, and paid 25 of them for ribbon ; how many had she left ? Ans. She had as many as the difference between 25 and 50 cents ; or as the remainder after subtracting 25 from 50 cts., which is 25 cts. 5025=25. Ex. 3. John has 25 cents, and his brother has 3 times as many ; how many has his brother ? Ans. He has as many us 3 times 25 cts., which are 75 cents. 25X3=75. Ex. 4. Three boys have 75 cents to be divided equally be- tween them ; how many will each boy have ? Ans. Each one will have as many cents as there are times 3 in 75, or 4 of 75 cts., which is 25 cts. 75+3=25. MENTAL EXERCISES. 51 Ex. 5. If 75 cents be divided equally among some boys, and each one receive 25 cents, how many boys will there be ? Ans. There will be as many boys as there are times 25 cts. in 75 cts., which are 3 times. Therefore there will be 3 boys. 75-^-25=3. 6. At 4 cts. each, how many oranges can be bought for 32 cents. 7. At 4 cents each, how much will 9 oranges cost ? 8. If 12 oranges cost 48 cents, what is the price of each ? 9. A boy picked 20 quarts of chestnuts one day, and 14 quarts the next ; how many did he pick in the two days ? 10. A boy had 20 quarts of chestnuts, and sold 14 quarts of them ; how many had lie left ? 11. At 9 cents a quart, how many quarts of plums can be bought for 72 cents ? 12. At 9 cents a quart, how much will 12 quarts of plums cost? 13. If 10 quarts of plums cost 100 cents, what is the price per quart ? 14. James has agreed to pick 100 quarts of strawberries in 5 days ; how many must he pick each day ? 15. James has agreed to pick 144 quarts of strawberries ; how long will it take him if he pick 12 quarts each day ? 16. James picked 12 quarts of strawberries a day ; how many quarts did he pick in 8 days ? EXERCISE I. FOR THE SLATE OR BLACKBOARD. 1. A farmer has 327 sheep in one flock, and 258 in another ; how many has he in both ? 2. A farmer ha I 640 lambs, and has sold 325 ; how many has he left ? 3. A farmer has 1950 bushels of oats, and can carry to market 75 bushels at a load ; how many loads will there be ? 4. A farmer carried to market 25 loads of oats, and 75 bush- els at a load ; how many did he carry ? 5. A farmer has 1800 bushels of oats, and wishes to carry 52 PEOMISCUOUS EXAMPLES. them all to market in 24 loads ; how many bushels must he carry each time ? 6. If a horse eat 12 quarts of oats a day, how long will it take him to eat 1728 quarts ? 7. If a horse eat 12 quarts of oats a day, how many will he eat in 132 days ? 8. If a horse eat 1740 quarts of oats in 145 days, how many will he average each day ? 9. If each horse eat 12 quarts of oats a day, how many horses will eat 1800 quarts in the same time ? 10. If 100 horses eat 1000 quarts of oats' in a day, how many quarts on an average will each horse eat ? MENTAL EXERCISES. 1. Henry picked 36 quarts of cherries and his brother 27 quarts ; how many more did Henry pick than his brother ? 2. At 8 cents a quart, how many quarts of cherries can be bought for 80 cents ? 3. At 8 cents a quart, how much will 12 quarts of cherries cost? 4. If 11 quarts of cherries cost 99 cents, what is the price per quart ? 5. Henry picked 29 quarts of cherries and his brother 37, how many did they both pick ? 6. 'At 18 cents each, how many knives can be bought for 72 cents? * 7. At 18 cents each, how much will 5 knives cost ? 8. If 6 knives cost 90 cents, what is the price of each ? EXERCISE H. FOB THE SLATE OK BLACKBOARD. 11. A barrel of flour contains 196 pounds, how many pounds are there in 679 barrels ? 12. How many barrels of the same will contain 97412 pounds ? 13. If there are 97216 pounds of flour in 496 barrels, how many pounds are there in each ? 14. A flour dealer bought 5624 barrels of flour, and has since sold 3768 of them, how many has he left ? MENTAL EXERCISES. 53 15. At $14 a barrel, how many barrels of flour can be bought for $11578 ? 16. At $15 a barrel, how much will 3246 barrels of flour cost ? 17. If 45 barrels of flour cost $540, what is the price per barrel ? 18. If a ship sail 96 miles each day, how long will it take her to sail 2688 miles ? 19. If a ship sail 96 miles each day, how far will she sail in 27 days ? 20. If a ship sail 2784 miles in 29 days, how far will she sail on an average each day ? MENTAL EXERCISES. 1. If 56 yards of calico will make 7 dresses, how many yards will make 1 dress ? 2. If 9 yards of calico will make a dress, how many yards will make 8 dresses ? 3. If 9 yards of calico will make a dress, how many dresses will 45 yards make ? 4. Jane has 26 yards of ribbon for trimming her dress, and Anna 17, how many more has Jane than Anna ? 5. How many yards have both together ? 6. At $4 a week, how many weeks can a person board for $416? 7. At $6 a week, how much will 26 weeks' board cost ? 8. If 23 weeks' board cost $115, what is the price per week? EXEECISE m. FOR THE SLATE OR BLACKBOARD. 21. In a large hotel, 857 pounds of beef are consumed daily ; how many pounds will be consumed in 365 days. 22. If 1000 men consume 137970 pounds of beef in 365 days, how much will they consume in a day ? 23. If 378 pounds of beef be consumed daily, how long will it take to consume 137592 pounds ? 24. At $18 a barrel, how many barrels of sugar can be bought for $32166 ? 54 PKOMISCUOUS EXAMPLES. 25. At $18 a barrel, how much will 1700 barrels of sugar cost? 26. If 1700 barrels of sugar cost $32300, what is the price per barrel ? 27. A merchant bought 320 barrels of molasses for $4800 ; 2000 barrels for $28000 ; 1900 barrels for $29730 ; how much did they all cost ? 28. At $15 a barrel, how many barrels of molasses can be bought for $29730 ? 29. At $14 a barrel, how much will 2000 barrels of molasses cost? 30. If 320 barrels of molasses cost $4800, what is the price per barrel ? EXERCISE rv. 31. At $67 an acre, how many acres of land can be bought for $122878 ? 32. At $53 an acre, how much will 234 acres of land cost ? 33. If 872 acres of land cost $47088, what is the price per acre ? * 34. If a person travel 26 miles a day, how far will he travel in 14 days ? 35. If a person travel 52 miles a day, how long will it take him to travel 728 miles ? 36. If a person travel 390 miles in 15 days, at what rate per day does he travel ? 37. If a man travel 1020 miles the first week, and 965 the next, how far will he travel in the two weeks ? 38. If a man travel away from home 3400 miles, and 765 miles on his return, how far from home will he be ? 39. A manufacturer paid 19 journeymen $57 apiece, what was the amount paid ? 40. A manufacturer paid his journeymen $1140, and each one received $57, how many were there ? EXERCISE v. 41. If 235 barrels of mackerel cost $3055, what is the price per barrel ? MENTAL EXERCISES. 55 42. At $14 a ban-el, how much will 235 ban-els of mackerel cost? 43. At $13 a ban-el how many barrels of mackerel can be bought for $3042 ? 44. A speculator having $15000 lost $7000, and afterwards gained $9653, how much did he then have ? 45. At $19 each, how much will 346 overcoats cost ? 46. If 345 overcoats cost $6555, what is the price of each ? 47. At $18 each, how many overcoats can be bought for $6228. 48. At 13 cents a pound, how many pounds of cheese can be bought for 8775 cents. 49. At 24 cents a bushel, how much will 496 bushels of ap- ples cost ? 50. If a man travel 28 miles a day, how long will it take him to travel 4256 miles ? EXERCISE VI. 51. A huckster carried 6867 melons to market in 27 loads ; how many in each load ? 52. If a huckster carries 325 melons at a load, how many will he carry in 21 loads ? 53. A huckster carries 337 melons at a load ; in how many loads will he carry 8175 melons ? 54. A huckster took to market at one time 179 cabbages ; at another 268 ; at another 947 ; and at another 144. He finally sold 1000 and brought back the rest for his cattle ; how many did he bring back ? 55. At 63 cents a basket, how much will 325 baskets of peaches cost ? 56. At 65 cents a basket, how many baskets of peaches can be bought for $208 ? 57. If 325 baskets of peaches cost $195, what is the price per basket ? 58. What cost 45 cows, at $40 each ? 59. A carriage-maker sold 77 carriages, for $212 each ; how much did he receive for all of them ? 56 UNITED STATES MONEY. 60. A speculator bought 1400 acres of land, at $56 an acre, and selling it he gained $6600 ; for what did he sell it per acre ? EXERCISE VII. 61. John's arithmetic contains 296 pages, and he wishes to review it for examination in 18 days ; how many pages must he review each day ? 62. After reviewing it 14 days, how many pages would be left? 63. James reviewed 18 pages a day in the same arithmetic ; in how many days could he finish it ? 64. A country merchant went to New York to buy goods, and paid for them in cash 1215 dollars ; in notes 1238 dollars, in barter 2512 dollars ; all his expenses were 65 dollars ; he sold them for 6000 dollars ; what did he gain ? 65. A horse dealer having 2549 dollars, bought 21 horses, and after paying for them had 113 dollars left ; what was the average price of the horses ? 66. At 200 dollars each, how many horses can be bought for 3000 dollars ? 67. At 150 each, how much will 16 horses cost ? 68. If a family's expenses are 18312 dollars in 24 years ; how much do they average a year ? V UNITED STATES MONEY. Art. 22. United States or Federal Money is the legal money of the United States. It consists of Eagles, Dollars, Dimes, Cents, and Mills. Its Coins are in Cold. Double Eagle, Eagle, Half Eagle, Quarter Eagle, Three Dol- lars, and Dollar. Silver. Dollar, Half Dollar, Quarter Dollar, Dime, (Ten Cents,) Half Dime, Three Cents. United States Money is a species of compound numbers ; but may also be treated much like simple numbers, since it increases in the same ratio. UNITED STATES MONEY. 57 TABLE. 10 mills (m.) make 1 cent, (ct.) 10 cents make 1 dime. 10 dimes, or 100 cents make 1 dollar. ($.) 10 dollars make 1 eagle, Art, 23, Aliquot Parts of a number are such as will divide it without a remainder. ALIQUOT PAETS OF U. S. MONEY. 5 mills = % cent. 10 cents = fL dollar. cents = % dollar. 50 75 20 " = i 25 " = I Art, 24, NOTATION OF U. S. MONEY. EULE. Write the dollars as in simple numbers, with a point (.) on the right ; next to this, if there are cents and mills, write two figures or ciphers for cents, and then one figure or cipher for mills. EXAMPLES TO BE WRITTEN. 1. Ten dollars fifteen cents and seven mills. $10,157 2. Seven dollars seven cents and seven mills. 3. Sixty dollars and six mills. 4. Fifty dollars fifty cents and five mills. 5. Nine dollars six cents and eight mills. 6. Sixty-three dollars four cents and two mills. 7. One hundred dollars and twenty-five cents. 8. Two hundred and ten dollars and five mills. 9. Seventy-five dollars two cents and one mill. 10. Five hundred dollars and fifty cents. 11. Twelve and a quarter dollars. 12. Sixty-one dollars thirty-seven and a half cents. 13. Twenty and a half dollars. 14. Seventy -five and three quarter dollars. 15. One thousand dollars twelve and a half cents. 68 UNITED STATES MONEY. Art. 25* NUMERATION OF U. S. MONEY. RULE. Read the figures before the separating point as dol- lars ; the next two (if there are any] as cents, and the third as mills. EXAMPLES TO BE BEAD. 1. $12. 375. Twelve dollars, thirty-seven cents and five mills or half a cent. 2. $10.25 7. $250.043 12. $ 21.25 3. $ 9.375 8. $125.000 13. $202.458 4. $12. 9. $87.00 14. $405.50 5. $12.00 10. $121. 15. $700. 6. $12.000 11. $63.405 16. $700.00 Art* 26* REDUCTION OF U. S. MONEY. Reduction of U. S. Money is changing dollars to cents, and cents to mills, or mills to cents, and cents to dollars, &c. Since there are 100 cents in 1 dollar, and 10 mills in 1 cent, any number of dollars is equal to as many hundred cents or thousand mills ; thus, $1.=100 cts. =1000 m. $12. =1200 cts. =12000 m. $3.87=387 cts. =3870 m. $4.375=437 cts. 5 m.=4375 m. ; hence, RULES. To reduce dollars to cents, multiply by 100 or annex two ciphers. To reduce cents to mills, multiply by 10 or annex one cipher. To reduce dollars to mills, annex three ciphers. To reduce dollars and cents to cents, or dollars, cents, and mills to mills, remove the separating point. This is the same as reducing the dollars and adding the cents or mills. Again, since 10 mills make 1 cent, and 100 cents make 1 dol- lar, every 10 mills in any number make 1 cent, and every thousand mills or hundred cents make 1 dollar ; thus, 1000 m.=100 cts.=$l. 15000 m.=1500 cts. =$15. 250 cts.= $2.50. 6375 m.=637 cts. 5 m.=$6.375 ; hence, UNITED STATES MONEY. 59 BULES. To reduce mills to cents, divide by 10, or point off the right hand figure. To reduce cents to dollars, divide by 100, or point off two figures. To reduce mills to dollars, point off three figures. MENTAL EXERCISES. How many mills in 2 cents ? 3? 5? 8? 9? 10? 13? 16 ? 20 ? 23 ? 28 ? 31 ? 40 ? 56 ? 75 ? How many cents in $2? $4? $5? $7? $10? $15? $20? $24? $36? $42? $50? $75? $87? $90? $100? How many cents in 10 mills ? 20 ? 40 ? 50 ? 65 ? 24 ? 30? 36? 45? 50? 100? 210? 750? How many dollars in 200 cents ? 400? 500? 800? 1000? How many cents in 20 dimes ? 30 ? 50 ? 80 ? How many dimes in 20 cents ? 30 ? 50 ? 80 ? How many cents in $3 ? 20 dimes ? 30 mills ? $5 ? 50 mills ? 50 dimes ? $60 ? 60 dimes ? 60 mills ? How many dollars, &c., in 125 cents? 125 dimes? 2000 mills? 3250 mills? 375 cents ? How many mills in 25 cents and 3 mills ? 20 cts. 5 mills ? 7 cts. 5 mills ? 80 cts. 9 mills ? How many cents in $2.37 ? $6.25 ? 75 dimes ? 75 mills ? $75? EXAMPLES FOE THE SLATE. Reduce or change 1. $25 to cents. 2. 10250 cents to dollars, &c. 3. 250 mills to cents. 4. 1100 cents to mills. 5. 1100 cents to dollars. 6. $3.75 to mills. 7. $10.25 to cents. 8. 170 cents to miUs. 9. 170 mills to cents. 10. 170 cents to dollars. 11. $70 to mills. 12. $60 to cents. 13. $24. 25 to mills. 14. $30.375 to mills. ' 15. $35.50 to cents. 16. 75375 nulls to dollars. 17. 75375 cents to dollars. 18. 34000 mills to cents. 19. $45 to mills. 20. $250 to mills. 60 UNITED STATES MONEY. 30. 675 mills to cents. 31. $37 to cents. 32. $16. 37 to cents. 33. $21. 04 to mills. 34. 13405 cents to dollars. 35. 759 cents to mills. 36. 287 cents to doUars. 37. $300 to mills. 38. 1200 mills to dollars. 21. 1275 mills to dollars, c, 22. 1000 mills to cents. 23. 1000 cents to dollars. 24. 1375 mills to dollars, &c. 25. 1000 dimes to dollars. 26. $927.25 to cents. 27. 3760 cents to dollars. 28. 1275 cents to mills. 29. 1325 mills to dollars. Art, 27. Application of the Fundamental Rules to U, S, Money, Since numbers in U. S. money increase from right to left in a ten-fold ratio, the same as simple numbers, they may be added, subtracted, multiplied, and divided by nearly the same rules. Art, 28, ADDITION OF U. S. MONEY. EULE. Write the numbers so that the separating points will be under one another, and proceed as in simple Addition. EXAMPLE 1. Add $5.125, $17.062, $10.43, Process. $8.055, $15.706. \^ 2. Add $18.15, $24.45, $7.21, $9.38, $11.33. 17.06 2 3. Add $19.041, $17.315, $112.18, $75.873, 10.43 $60.50. ^l I 4. Add $44.76, $28.19, $18.657, $270.508, An8m ^^37~8 $87.60; $67.005. 5. Add $10.625, $112.35, $1.75, $11.875, $100, $17.37. 6. What is the sum of $21 lOcts., $17 4cts. 6m., $23 17cts. 3m., $19 18cts. 6m., $25, $16 8 cts., $15 5m.? 7. Add $200, $43.875, $56.937, $18.50. $12.315. 8. What is the amount of $304 50cts., $304 4m., $820 35cts. ADDITION OF U. S. MONEY. 61 9. What is the sum of $25 Sets., $40 21ots. 3m., $108 5m., $63 4cts., $312 let. 7m., $1000 15cts., $50 8m.? 10. Add $6 Gets. 3m. t $14 17cts., $21 Sets. 6m., $25 50cts., $17 8m., $100 lOcts. 3m., $1 let. 1m., $10 lOcts. 7m. 11. Add $5 4cts. 3m., $1 14cts., $98, $2 2m., Sots. 3m., $15 16cts. 4m. 12. What is the amount of $300, $4 4cts, $50 5m., $70 Tots., $45 5m. ? 13. What is the sum of 35 dollars 6 cents 7 mills, $11 4cts. 6m., $17 18cts. 9m., $400 83cts., $12 20cts. 2m.? 14. Add 3 dollars 12 cents 5 mills, $50 50cts., $300 6m., 75cts., $75 Tots. 5m., $201 3cts. 15. What is the sum of $18%, $12}, $6^, $%, $5^? (See Table of Aliquot Parts.) 16. What is the amount of 9 dollars 62} cents, 87)cts, $15^, $108 62)^ cts., $1 Sets, $27%, $63 12>^cts.? 17. Add 10} dollars, 87> cents, $105 62^cts., $16^ 37) cents, $21%. 18. Add 9 dollars 12} cents, $6 3m., $28 87>cts, $56 5cts. 5m. 19. What is the sum of $39, $109 12}cts., 5m, $5 5m., $1 20. What is the amount of $67 12) cts., $60%, $62^, $ Add the following numbers in U. S. money 21. Three hundred dollars and three cents, Three dollars and three mills, Five hundred dollars, Five hundred cents, Five hundred mills. 22. Eighty-five doUars, Sixty dollars sixty-two and a half cents, Thirty-seven and a half cents, Forty dollars four cents and five mills, Forty cents and four mills, Forty-four mills. 62 UNITED STATES MONEY. 23. Seventy dollars, Five dollars, eighty-seven and a half cents, Fifty dollars fifty cents and five mills, Six and three-quarter dollars, Five and a half dollars and a half cent. Ten and one quarter dollars. 24. Two dollars and two mills, Seven dollars eighty-seven and a half cents, Nine dollars thirteen cents and three mills, Sixty-seven dollars and eight mills, Four dollars and seventy-five cents, One and three-quarter dollars. 25. Seven dollars and eighty cents, Twelve dollars and twenty-five cents, Ten dollars and two mills, Sixty-five dollars, Six cents and five mills, One dollar one cent and one mill. 26. Ten eagles ten dollars ten dimes ten cents and ten 27. One half eagle one half dollar and one half cent. 28. Thirty-seven and a half dollars, Thirty-seven and a half cents, Twenty-four and three-quarter dollars, Six and a quarter dollars and a half cent, Twelve and a half cents. 29. A family has paid for beef $19.15, flour $17.375, butter $10.125, and sugar, $4.65, what is the amount ? 30. A farmer bought a horse for $135, a pair of oxen for $97.375, a cow for $35, and 20 sheep for $50, how much did he pay for them all ? 31. A young man bought a suit of clothes for $56, a watch for $87>, a watch chain for $12%, and a pair of gloves for cents, what did they all cost ? 32. A young lady bought a silk dress for $26, a shawl for $18%, a bonnet for $7^, and a pair of gaiters for $3.37>o> what lid they all cost ? SUBTRACTION OF U. S. MONEY. 63 Art. 29. SUBTEACTION OF U. S. MONEY. EULE. Write the numbers so thai the separating points mill be under each other, and proceed as in simple Subtraction. EXAMPLE 1. From $125 take $37.053. Proem. 2. From $39.25 take $16.246. 125. wTo 3. From $127.384 take $15.60. 37J05 3 4. From $95.28 take $45.183. Ans. $87.94 7 5. From $118.05 take $67.45. 6. From $95 take $33. 60. 7. From $25 take 25 cents. 8. From $100 take 100 cents. 9. From $10 take 10 cents. 10. From $1 take 1 cent. 11. From 1 cent take 1 mill. 12. From $5 take 5 mills. 13. A man paid $175 for a carriage, and $162}^ for a horse, how much more did he pay for the carriage than the horse ? 14. A merchant bought a hogshead of molasses for $26% and sold it for $35, how much did he gain ? 15. A young man sold his watch for $37^, and it cost $45. how much did he lose ? 16. A lady having $50, spent $27.62} in shopping, how much was left ? 17. A laborer has earned $100, and been paid $53.87^, how much is still due to him ? 18. A man owed $500, and has paid $263.62)^, how much does he still owe ? 19. A person having bought a bill of goods amounting to $7.12^, gave in payment a ten dollar bill, how much change did he receive ? 20. One man earns $1.62)^, and another $1% a day, how much more does one earn than the other ? 64 UNITED STATES MONEY. Art, 30, Multiplication of U, S, Money, EULE. Proceed as in simple Multiplication, and place the separating point as far from the right as it is in the multipli- cand, sometimes used as the multiplier. Process. $8.625 15 EXAMPLE 1. Multiply $8.625 by 15. 13125 8625 Ans. $129.375 Multiply (2.) $112.08 by 7 (4.) $94.375 " 9 (6.) $65. " 12 (8.) $100. " 100 (10.) 75cts. " 14 (12.) 12>cts" 10 (14.) $8^ " 5 (3.) $1.25 by 10 (5.) $12.50 " 100 (7.) $48.375 " 35 (9.) $10. " 100 (11.) 5 mis" 25 (13.) $12^ " 20 (15.) $18% " 1000 In multiplying U. S. money by 10, 100, &c. , it is sufficient to re- move the separating point as many places to the right as there are ciphers in the multiplier ; as $4.50><100= $450. 16. At $1.25 a bushel, how much will 20 bushels of corn cost ? 17. At 62)^ cents a bushel, what will 15 bushels of apples cost? 18. If an acre of land cost $87}^, how much will 100 acres cost? 19. If a cord of wood cost $6%, how much will 12 cords cost ? 20. At $5.62)^ a yard, how much will 3 yards of cloth cost ? 21. At 12> cents a quart, how much will 7 qts. of cherries cost? 22. If one doz. eggs are worth $^, how much are 100 doz. worth ? 23. If a pound of butter is worth 22> cts., what are 14 pounds worth ? 24. At $10% a ton, what are 10 tons of hay worth ? DIVISION OF U. S. MONEY. 65 Art, 31, Division of U, S. Money. RULE. Proceed as in simple Division, observing that either the divisor or quotient must be of the sam,e name as the divi- dend, reduced if necessary to cents or mills, and have the sep- arating point in the same place ; while the other has no sepa- rating point except in decimal fractions. The dividend is the price of the whole quantity. The price of the whole divided by the quantity gives the price of each part ; or The price of the whole divided by the price of each part gives the quantity. $5.00-^100 Ibs., etc.=5 cts. $5.00-f-.05 cts.=100 Ibs., etc. EXAMPLE 1. Divide $316.753 by 5. 5)$316.753 Process. Short Division Ans. $63.350+3 $ c. $ c.m. Ex. 2. Divide $225.50 by 18. 18)225.50(12.527+ Process. Long Division 4:0 If there is a remainder after dividing any given 36 number of dollars or cents, reduce the dollars to "95" cents and the cents to mills. 50~~ 36 140 mills. 126 U c.m. $ c.'mAns. E*.3.-Dmde too by 62^ cents = 625 625 H mills. 2250 Process. 1875 ~3750 3750 66 UNITED STATES MONEY. Divide Ex.4. $124.64 by $3.28, or into 38 equal parts. 5. $62.64 " $7.83, " 8 " 6. $108.837 " $12.093, " 9 7. $1862.42 " $35.14, " 53 8. $368.288 " $23.018, " 16 9. $2.125 " $0.125, " 17 10. $2.50 < $0.25, " 10 " To divide U. S. Money by 10, 100, Ac., it is sufficient to remove the separating point as many places to the left as there are ciphers in the divisor ; thus, $45-^-10G=$0.45. Ex. 11. At 12^ cents a pound, how many pounds of sugar can be bought for $2.00. 12. If 24 pounds of sugar cost $3.00, what is the price per pound ? 13. At 28 cents a pound, how many pounds of butter can be bought for $2.24? 14. If 9 pounds of butter cost $2.52, what is the price per pound? 15. At 65 cents a bushel, how many bushels of corn can be bought for $130'? 16. If 200 bushels of corn cost $120, what is the price per bushel ? 17. At 37 cents a bushel, how many bushels of oats can be bought for $21 ? 18. If 60 bushels of oats cost $24, what is the price per bushel ? 19. If a bushel of wheat cost $1.37^, how many bushels can be bought for $49.50 ? 20. If 40 bushels of wheat cost $55, what is the price of one bushel ? 21. At $li^ a day, in how many days can a man earn $25? 22. If a man earn $20 in 16 days, how much IB it a day ? ALIQUOT PARTS OF A DOLLAR 67 Art. 32. Aliquot Parts of a Dollar. (See Table of U. 8. Money.} When the price of anything is an aliquot part of $1, it shortens the operation, To divide by the number of parts, instead of multiplying by the number of cents, and to multiply instead of dividing. EXAMPLE 1. What will 49 yards of calico cost, at 25 cts. a yard ? Process. Since 1 yard costs 25 cts. ($,) 1 quar- 4)49 ter of a dollar, 49 yards will cost 1 quarter of 49 dollars. $12i=12.25 Ex. 2. How many yards of calico can be bought for $10, at 25 cts. a yard ? Process. Since 25 cents, or ($) will buy one 10 yard, $1 will buy 4 yards, and $10 will bay (4 times 4 10) 40 yards. 4$ EXAMPLES. 3. At 20 cts. each, -what will 400 writing books cost ? 4. At 25 cents each, how many writing books can be bought for $5 ? 5. At 33^ cts. a gallon, what cost 30 gallons of vinegar ? 6. At 33^ cts. a gallon, how many gallons of vinegar can be bought for $12 ? 7. At 50 cents a bushel, how many bushels of apples can be bought for $25 ? 8. At 50 cts. a bushel, how much will 100 bushels of apples cost? 9. At 12^ cts. a pound, how many pounds of rice can be bought for $6 ? 10. At 12-| cents a pound, how much will 16 pounds of rice cost? 11. If a boy earn 33}^ cts. a day, how much will he earn in 6 days ? 12. If a boy earn 33^ cts. a day, in how many days will he earn $1 ? 68 UNITED STATES MONEY. Art, 33, Price per Hundred or Thousand, When the given price is per hundred, call the dollars cents ; when per thousand, call them mills, which will be the price of one of the things specified. EXAMPLE 1. What will a bale of hay weighing 156 pounds cost at $1 per hundred ? Process. $1=100 cts., and 100 cts. per hundred pounds is 1 cent per pound, and 156X-01 $1.56. Ans. $1.56 Ex. 2. What cost 12500 shingles at $18 per thousand ? 3. What cost 7500 bricks at $9 per thousand ? 4. At $9 per thousand, how many bricks can be bought for $63? 5. What cost 56 pounds of flour at $4 per hundred ? 6. What cost 615 feet of pine boards, at $21 per thousand ? Examples of this kind, in which the price is cents, in- volve decimal fractions, but the process is the same. Art, 34, Promiscuous Examples in the Fundamental Rules, including U. S, Money. EXERCISE I. 1. A young man bought a horse for $150 ; a watch for $53.875 ; a suit of clothes for $46.937 ; a hat for $4.50 ; a pair of boots for $4.00, and some other things for $2.313 ; what was the amount ? 2. From $100 subtract $1, 1 cent and 1 mill. 3. At 25 cents a yard, how many yards of ribbon can be bought for $4. 4. At 25 cts. a yard, how much will 12 yards of ribbon cost ? 5. If 20 yards of ribbon cost $5, what is the price per yard? 6. At $8.05 a ton, how much will 20 tons of hay cost ? 7. At 34 cents a yard, how many yards of muslin can be bought for $30.26 ? UNITED STATES MONEY. 6 8. In a case of broadcloth there are 19 pieces, containing in all 437 yards ; how many yards in a piece on an average ? 9. A farmer carried to market 20 loads of oats, and each, load contained 75 bushels ; how many bushels in all ? 10. A farmer had 1200 bushels of wheat, and could carry 50 bushels at a load ; how many loads were there ? 11. A lady went shopping with $5 in her purse ; she paid 75 cents for a collar ; $1,50 for kid gloves; 50 cts. for ribbon and 25 cts. for needles and pins ; how much had she left ? 12. At Si. 60 a day, how much will a man earn in 40 days ? 13. At $1. 121 a bushel, how many bushels of wheat can be bought for $208. 14. At 431 cts. a bushel, what will 750 bushels of buckwheat cost? 15. A farmer owed a merchant $500, and paid him 435 bush- els of oats at 45 cts. a bushel ; how much does he still owe ? EXERCISE n. 16. At 7^ cts. a quart, how many quarts of cherries can be bought for $1.35 ? 17. If a clerk's salary is $800 a year, how much is it for each day he is employed in business (313 in the year) ? 18. At $2.25 a day, how much will a laborer earn in 313 days working time in a year ? 19. At $17.565 an acre, how many acres of land can b bought for $2722. 575? 20. If 3 men gain $1000, what is each one's equal share ? 21. A lady having $26, bought a silk dress for $13.10 ; a shawl for $6, and gloves for 75 cts ; how much had she left ? 22. What cost 8 pieces of calico, each containing 19 yards, at 23 cts. a yard ? 23. At $5.67 a yard, how many yards of cloth can be bought for $136.08 ? 24. If 168 lambs cost $451.92, what is the price of each ? 25. What cost 17 firkins of butter, each containing 51 pounds, at 14 cents, 7 mills per pound ? 70 FUNDAMENTAL RULES. EXERCISE TTT. 26. At $12 a barrel, how many barrels of flour can be bought for $1512 ? 27. If 670 pounds of cheese cost $87.10, what is the price per pound ? 28. What number multiplied by 9 will produce 315 ? 29. At $8 a ton, how many tons of coal can be bought for $1728 ? 30. A farmer sold his pork for $21.75, and received sugar, $3.75; molasses, $2.50; tea, $1.35; cheese, $1; pepper, 25 cts. ; ginger, 18 cts. ; the rest in cash ; how much cash did he receive ? 31. How much coffee at 13 cents a pound can be bought for $18.59 ? 32. At $1.43 a day, how much will a man earn in 312 days ? 33. If 54 bushels of wheat cost $67.50, what is the price per bushel ? 34. At $10 and 5 mills an acre, what will 150 acres of land cost? 35. If 17 bags of coffee, each weighing 51 pounds, cost $127.449, what is the price per pound ? EXERCISE IV. 36. If 137 shares of bank stock are worth $17125, what is a share worth ? 37. If 1000 men consume 856 pounds of beef in a day, how many pounds will last them 365 days ? 38. At $34 a barrel, how many barrels of sugar can be bought for $128.690 ? 39. A lady bought a cloak for $25.125. a muff for $12.375, a bonnet for $9.15, and gave the merchant a $50 bill ; how much change was due her ? 40. What cost 24 arithmetics, at $0.37 each ? 41. At $0.37^ each, how many arithmetics can be bought for $13.50 ? 42. If 18 arithmetics cost $6.75, what is the price of each ? 43. If a horse travel 34 miles a day, how far will he travel in 75 days ? UNITED STATES MONEY. 71 44. If a horse travel 34 miles a day, how long will it take Trim to travel 1700 miles ? 45. What cost 8 barrels of sugar containing 225 Ibs. each, at 6| cents a pound ? EXEECISE V. 46. A drover bought 397 cattle for $14689, what is the average price of each ? 47. In a pile there are 237 boards, each containing 23 square feet, how many square feet in all ? 48. A manufacturer has a contract for 273249 yards of calico to be made in 313 days ; how many yards must he average daily ? 49. A man bought a lot for $375, paid for building a house on it $750, and for improvements $160.87^ ; he then sold the place for $1500 ; what did he gain ? 50. A laborer was paid $23. 75 for 19 days' work ; how much did he have a day ? 51. At $1.12^ a day, how much will a man earn in 16 days ? 52. If 25 men earn $35.50 in a day, how much will they earn in 50 days ? 53. If 25 men earn $1775 in 50 days, how much will one man earn in the same time ? 54. At 37^ cents a bushel, how much corn can be bought for $58.50 ? " 55. Bought 20 pieces of muslin, each measuring 19 yards, for $87.40 ; what was the price per yard ? 56. A farmer paid his hired man $165 for 12 months, how much was it a month ? 57. A laborer dug 29 bushels of potatoes a day, how many did he dig in 42 days ? 58. A laborer agreed to dig 1212 bushels of potatoes in 42 days, how many must he dig a day on an average ? 59. Eeduce 37500 mills to dollars. 60. A merchant bought some cloth for 26 dollars and 3 cents, a bale of sheeting for 50 dollars and 90 cts., 20 pieces of calico for 49 dollars, 1 cent, 12 pieces of merino for $108.14, 10 72 FUNDAMENTAL KULES. pieces of silk for $77.25, 15 pieces of linen for $83.68, and 3 dozen kid gloves for $40 and 8. cents ; what was the amount of the bill ? 61. A young man having $40, paid $20.20 for a coat ; how much had he left ? 62. At $8.245 a yard, how much will 10 yards of cloth cost ? 63. At $6.25 a barrel, how many barrels of eggs can be bought for $256.25 ? 64. If 40 barrels of eggs cost $250, what is the price per barrel ? 65. Paid 16 men $516 for work at 75 cts. a day ; how many days did they work ? EXERCISE vn. 66. A steamship, after consuming 17,500 pounds of coal, has 30,000 pounds left ; how many had she at first ? 67. A steamship had 50000 pounds of coal, and has con- sumed 27035 pounds ; how "much remains ? 68. If a merchant gain $65 a day, how much will he gain in (313 business days) a year ? 69. If a merchant gain 18780 dollars a year, how much will be the average gain a day for 313 business days ? 70. If a merchant gain $50 a day, how long will it take him to gain $10000 ? 71. If a ship sail 75 miles a day, how far will she sail in 60 days? 72. If a ship sail 4500 miles in 50 days, how many miles will she average in a day ? 73. If a ship sail 60 miles a day, how long will it take her to sail 4500 miles ? 74. If a 150 quarts of cherries cost $9375, what is the price per quart ? 75. At $3.95 a bushel, how many bushels of timothy seed can be bought for $146.15 ? EXERCISE VUL 76. In a case of broadcloths there are 12 pieces, and in each piece 49 yards ; how many yards in all ? 77. In another case there are 12 pieces containing 564 yards, how many yards on an average in each piece ? UNITED STATES MONEY. 73 78. In another case each piece contains 45 yards, and there are in all 540 yards, how many pieces ? 79. A merchant has in cash $576.32, notes $135.375, flour $97.10, butter $57.19, for which he wishes to purchase goods amounting to $1000, the balance to remain on credit, how much will the balance be ? 80. If 360 laborers receive $405 a day, what will each one of them receive ? 81. At $7.625 a bushel, how many bushels of flaxseed can be bought for $1807.125 ? 82. At $6.625 a bushel, how much will 237 bushels of flax- seed cost ? 83. If 250 bushels of flaxseed cost $1312.50, what is the price per bushel ? 84. At $5. 72 a cord, what will 23 cords of wood cost ? 85. At $5.75 a cord, how many cords of wood can be bought for $138. EXERCISE IX. 86. A farmer sold 56 loads of hay, each weighing 1400 pounds ; how many pounds did they all weigh ? 87. A farmer received $700 for 56 loads of hay; what was the price of a load ? 88. Bought 18 barrels of sugar, each containing 235 pounds ; how many pounds in all ? 89. In 20 barrels of sugar there are 4115 pounds ; how many pounds will they average ? 90. If it would take 1 man 567 days to build a house, in how many days could 28 men build it ? 91. A merchant bought dry goods amounting to $5862.97, and groceries amounting to $1279.50; he paid in cash $4000 and gave notes for the balance ; what was the amount of the notes ? 92. Bought 75 books at $1.25 ; how much did they all cost ? 93. Bought 25 pairs of shoes for $23.75 ; what was the price of a pair ? 94. Paid $51.75 for 9 hats; what was the average price of each ? 95. What cost 23 cases of boots, at $37.52 a case ? 4 74 UNITED STATES MONEY. 96. At $0.375 each, how many books can be bought for $4.50 ? 97. What cost 128 barrels of sugar, at $18.96 a barrel ? 98. If 254 barrels of sugar cost $2407.92, what is the price of a barrel ? 99. At $63.75 an acre, how much will 200 acres of land cost ? 100. If 100 acres of land cost $6375, what is the price per acre ? Art, 35, Bills in U, S. Money, A bill is a written account of what is to be paid for, as goods, labor, &c. EXAMPLE I. NEW YOKE, June 1st. J. KING, Dr. To W. BKOWN. 51bs. of Tea, at 62) cts $3.125 8 " Coffee, at 15 ots 1.20 3 " Starch, at 12} cts 375 14 " Sugar. at 11 cts 1.54 6 gals. Molasses, 37^ cts 2.25 What is the amount ? Ans. $8.49 EXAMPLE n. 6 yds. of Cloth, at $4.37> $ 18 " Calico, at .21 10 . " Muslin, at .19 3 spools Cotton, at .09 5 sheets Wadding, at .12> What is the amount ? EXAMPLE m. 10 Ibs. of Sugar, at $0.16 5 " Tea, at 1.12) 17 " Butter, at .22^ 9 " Coffee, at .14 2bbls. Flour, at 9.50 What is the amount ? BILLS IN UNITED STATES MONET. 75 EXAMPLE IV. 9yds. Silk, at Si. 25 15 " Calico, at 20 " Muslin, at .21 7 " Gingham, at 6 skeins Silk, at .05 What is the amount ? EXAMPLE v. 175 bushels of Wheat, at 300 " Corn, at .81 625 " Oats, at 92 Buckwheat, at .56 , 112 Eye, at .75 . What is the amount ? EXAMPLE VI. 1250 bushels of Potatoes, at 625 " Turnips, at 172 " Carrots, at .35 85 " Beets, at .68 126 barrels of Apples, at What is the amount ? EXAMPLE vn. 8 yds. Merino, at $1.37> . . . 13 " Mns. de Laine, at .4A ... 11 " Alpaca, at .75 . ... 1 " Figured Satin, at 3.00 ... 9 " Col'd Cambric, at .12^... 14 " Drab Fringe, " .62)^..., What is the amount ? 76 COMPOUND NUMBERS. EXAMPLE VUL / at tea, at at 5 n&n naeice, at at .38 6 fiaiA What is the amount ? In like manner write out and find the amounts of the following bills: 9. T. White bought of L. Camp, 3 yards of cloth at $6.50 ; 2 yds. of cassimere, at $2.75 ; 5 yds. cambric, at 37)cts. ; 2 doz. buttons, at 12> cts. ; 3 skeins of silk, at 6^ cts. 10. W. Savage bought of L. Stearns, 2 bbls. of flour, at $9.50 ; 25 Ibs. of sugar, at 18% cts.; 10 Ibs. coffee, at 37} cts: ; 3 Ibs. tea, at $1. 12)^ ; 1 bar of soap, at 15 cts. 11. Mrs. Nelson bought of A. Halsted, 12 yards of silk, at $2^' ; 3 yds. satin, at $12) ; 6 yds. of cambric, at 16 cts. ; 3 pairs of hose, at 56)^ cts. ; 8 skeins of silk, at 4 cts. ; 1 doz. spools of cotton, at 62)^ cts. 12. S. Hoes bought of I. Ball & Co., 4 yards of cloth, at $6 ; 12 yds. of satinet, at 87) cts. ; 3 woolen shirts, a 6 collars, at 20 cts.; 1 pair gloves, at $1.50. COMPOUND NUMBERS. 77 COMPOUND NUMBERS. Art. 36. Compound Numbers are numbers having dif- ferent denominations, or names, under a general name, to express one quantity ; as, pounds, shillings, pence and farthings under English or Sterling Money, and pounds, ounces, etc., under Weights. The general names include Money "Weights and Meas- ures of different kinds ; the different denominations of which are exhibited in the following tables. Money. United States Money is a species of compound num- bers, but, being much like simple numbers, has been al- ready introduced. Art. 37. English or Sterling Money is the money used in England. It was formerly used in this country, and is still used to some extent, though its value has been changed, and varies in different States. TABLE. 4 farthings (far.) make 1 penny. (d.) 12 pence 1 shilling. (s.) 20 shillings ..1 ( . ( sovereign. v 21 shillings 1 guinea. 1 is valued at Weights. Art. 38, Troy Weight is used in weighing gold, silver, and gems. TABLE. 24 grains (gr. ) make 1 pennyweight, (pwt. ) 20 pennyweights 1 ounce. (oz.) 12 ounces 1 pound. (Ib.) 78 COMPOUND NUMBEBS. Art, 39. Avoirdupois Weight is used in weighing heavy and common articles. TABLE. 16 drams (df.) make 1 ounce. (oz.) 16 ounces 1 pound. (Ib.) 25 pounds 1 quarter. (qr. ) 4 quarters, or 100 Ibs 1 hundred-weight, (cwt.) 20 hundred-weight 1 ton. (T.) Formerly 28 Ibs. made 1 qr., and 112 Ibs. made 1 cwt. 1 Ib. (av.)=7000 grains (troy.) Art, 40. Apothecaries' Weight is used in weighing medicines at retail. The pound and ounce are the same as in Troy Weight. TABLE. 20 grains (gr.) make 1 scruple, (sc. or B) 3 scruples 1 dram. (dr. or 5) 8 drams 1 ounce. (oz. or ) 12 ounces 1 pound. (ft>.) Art. 41. MISCELLANEOUS TABLE OF WEIGHTS. 196 Ibs. make 1 barrel of flour. 200 1 barrel of beef, pork or fish. 56 1 firkin of butter. 60 1 bushel of wheat. 56 '. .1 bushel of rye or corn. 48 1 bushel of barley. 32 1 bushel of oats. Measures. Art, 42, Cloth Measure is used in measuring cloths, and other goods sold by the yard. TABLE. 234 inches (in.) make 1 nail. (na.) 4 nails 1 quarter, (qr.) 4 quarters 1 yard. (yd.) 1 qr.=> yd, 2 qr.=> yd, 3 qr.=% yd. TABLES OF MEASURE. 79 Also, 5 quarters make ____ 1 Ell English. (E. E.) 3 quarters ---- 1 Ell Flemish (E. Fl.) 6 quarters ---- 1 Ell French. (E. F. not used.) Art. 43. Long Measure is used in measuring distances, or lines extended in length, breadth, heighth and depth. TABLE. 12 inches (in.) make .......... 1 foot. (ft.) 3 feet .......... 1 yard. (yd.) 5 yds., or 16^ ft ........... 1 rod or pole, (rd.) 40 rods ........ . . 1 furlong. (fur.) 8 furlongs .......... 1 mile. (m.) 3 miles .......... 1 league. (lea.) ^degrees . ....'. ..... Also 6 feet make 1 fathom, (used in measuring deep water.) 160 rods make mile. 80 rods ^ mile. Art. 44. Surveyors' Measure is used in measuring roads and boundaries of land, &c., with chains. TABLE. 7^ inches make ............. 1 link. (li.) 25 links ............. 1 rod or pole. (P.) 4 poles ............. 1 chain. (ch.) 10 chains ............. 1 furlong. (fur.) 8 furlongs, or 80 chains ............. 1 mile. (m.) Art, 45. Square Measure is used in measuring sur- faces or areas, as land, floors, &c., in which both length and breadth are considered. TABLE. 144 square inches (sq. in.) make ---- 1 sq. foot. (sq. ft.) 9 square feet ---- 1 sq. yard. (sq. yd. ) 301 S q. yards, or 272^ sq. ft ..... 1 sq. rod or pole. (sq. rd.) 40 sq. rods ---- 1 rood. (B.) 4 roods, or 160 rods ---- 1 acre. (A.) 640 acres ---- 1 sq. mile. (sq. m.) 80 COMPOUND NUMBERS. This measure is directly applicable only to surfaces whose contents are known, as any number of acres, square rods, &c. Such surfaces, depending on the length of certain lines, are found by first using long measure. Squares and rectangles are included among them. A Square is a surface having four equal sides, and four equal angles, which, also, are right angles. If the sides of a square are each one inch in length (long measure) it is a square inch ; if the sides are each one foot, yard, rod or mile, it is a square foot, yard, rod or mile. If any square is divided into square feet, as in the i i i ' i annexed diagram, it contains as many rows of square \ feet as there are linear feet in one side, and the same 1 number of square feet in each row. Therefore if the I number of feet in each row be multiplied by the number of rows, or, (which is the same,) if the number of feet in one side be multiplied by itself, the product will be the number of square feet in the whole square. The same is true of yards, rods, &c. Hence the RULE. To find the contents or area of a square, multiply the length of any one of its sides by itself. There is no difference between a square foot and a foot square, but there is a difference between such expressions as 3 square feet, and 3 feet square. A square yard is 3 feet square, and contains 9 square feet, (see last diagram,) hence the difference between 3 feet square and 3 square feet is 6 square feet. The same is true of rods square and square rods, &c. A Rectangle is like a square, only it is longer than it is wide, and its contents are found by multiplying its length by its breadth, both being reduced if necessary to the same name. This diagram represents a rectangle 6 ft. long and 3 ft. wide contents 6X3=18, the number of square feet. Since, also, the product of two numbers di- vided by one of them gives the other, either side of a rectangle may be found by dividing the contents by the other side ; as, 18 sq. ft. -H* ft. (wide)=6 ft. long. Art, 46, Cubic Measure is used in measuring solids, in which length, breadth and thickness are considered. It is sometimes called Solid Measure. COMPOUND NUMBERS. 81 TABLE. 1728 cubic inches (CM. in.) make 1 cubic foot (cu, ft.) 27 cubic feet .......... 1 cubic yard (cu. yd.) 40 feet of round or ) 50 feet of hewn [ tim t>er 1 ton (T.) 16 cubic feet .......... 1 cord foot (0. ft.) 8 cord feet or ) 128 cubic feet } .......... l cord of wood ' A Cube is a solid having six equal and square sides. If the sides are each a square inch, the solid is a cubic or solid inch ; if the sides are each a square foot or yard, the solid is a cubic or solid foot or yard, &c. If the base of a cube be 4 feet square, it will con- tain 16 square feet, and a solid on it, 1 foot high, would contain 16 solid feet, 2 feet high 16X2= 32 solid feet; four feet high 16X4=64 solid feet ; hence RULE. The contents of a cube, or any regular solid, may be found by multiplying together the length, breadth and thickness. The contents and any two of the dimensions being given, the other dimension may be found by dividing the contents by the product of the given dimensions. Art. 47. Wine Measure is used in measuring liquids, except beer and ale. TABLE. 2 pints 1 Quart (ot ) 4 quarts 1 erallon (e*al } 314r gallons . I barrel (bbl ) 63 gallons (or 2 barrels) . . . . . .1 hogshead (hhd ) 2 hogsheads . . . . . 1 pipe (D } 2 pipes (or 4 hhds.) . .1 tun (T.) Also 42 fallons . . .1 tierce (tr ) 48 eallons . .1 puncheon ( pun.) A gallon contains 231 cubic inches 4* 82 COMPOUND NUMBERS. Art* 48. Beer Measure is used for measuring beer, and formerly milk. TABLE. 2 pints make 1 quart. 4 quarts 1 gallon. 36 gallons 1 barrel. 5i gallons 1 hogshead. A beer gallon contains 282 cubic inches. Art. 49. Dry Measure is used for measuring grain, fruit, vegetables, &c. TABLE. 2 pints (pt.) make 1 quart (qt.) 8 quarts 1 peck (pk.) 4 pecks 1 bushel (bu.) Also, 8 bushels make 1 quarter (q.) 36 bushels 1 chaldron. 32 bushels 1 chaldron (in U. S.) A bushel in form of a cylinder is 18 1 in. in diameter and 8 inches deep. It contains 2150|^ cubic inches. Art. 50. Time Measure is used in making divisions of time. TABLE. 60 seconds (sec.) make 1 minute (in.) 60 minutes 1 hour (h.) 24 hours 1 day (and night) (d.) 7 days 1 week (w.) 4 weeks or ) 1 month (mo.) 30 days i 12 months (calendar) or i 13 " (lunar) or (365d.nearly) f lvear - GrO The Solar year consists of 365 days, 6 hours (very nearly) in which the earth makes one revolution around the sun. The Civil year is 365 days. This makes a difference of one day in four years, which is added to the month of February, making it con- COMPOUND NUMBERS. 83 sist of 29 instead of 28 days. This occurs whenever the number of years is divisible by 4, which is Leap Year The addition of one day every Leap Year, makes about one day too much in 100 yeJars, hence it is not added during the year which completes a century, as A.D. 1800, A.D. 1900. The names of the months and the number of days in each, are as follows : January (2d Winter month,) 31 days. February 3d " 28 or 29 days. March 1st Spring 31 days. April 2d 30 May 3d " 31 June 1st Summer 30 July 2d < k 31 August 3d " 31 September 1st Fall 30 October 2d " 31 November 3d " 30 December 1st Winter 31 To assist in remembering the number of days in each month the following lines have been much used : Thirty days hath September, April, June, and November, February twenty-eight alone All the rest have thirty-one ; Except in Leap Year, then is the time When February has twenty-nine. TABLE, THE NUMBER OF DAYS FKOM ANY DAY OF ONE MONTH TO THE SAME DAY OF ANY OTHER MONTH IN THE SAME YEAR. FROM ANY DAY OF TO THE SAME DAY OF JAN. FEB. MAE. APL. MAY. J'NE. J'LY. AUG. SEP. OCT. NOV. DEC. JANUA'Y 367 31 59 90 120 151 181 212 243 273 304 334 FEBR'Y . 334 365 28 59 89 120 150 181 212 242 273 303 MARCH. 306 337 365 31 61 92 122 153 184 214 245 275 APBZL.. 275 306 334 365 30 61 91 122 153 183 214 244 MAT... 245 276 304 335 365 31 61 92 123 153 184 214 JUNE . . 214 245 273 304 344 365 30 61 92 122 153 183 JULY . . 184 215 243 274 304 335 365 31 62 92 123 153 AUGUST 153 184 212 243 273 304 334 365 31 62 92 122 SEPTM'K 122 153 181 212 242 273 303 334 365 30 61 91 OCTOB'R 92 123 151 182 212 243 273 304 335 365 31 61 NOVEM. 61 92 120 151 181 212 242 273 304 334 365 30 DECEM. 31 62 90 121 151 182 212 243 274 304 335 365 The number of days is opposite the first given month and under the other. 84 COMPOUND NUMBIiKS. Art. 51. Circular Measure is used in measuring cir- cles, as in finding the latitude and longitude of places or of heavenly bodies. A Circle is a figure bounded by a curved line, in which every point is equally dis- tant from the centre. The Circumference is the curved line. The Diameter is the straight line passing through the centre to opposite points in the circumfer- ence. A Radius is any straight line from the centre to the circumference. Circles are divided into 360 parts called degrees, vary- ing according to the size of the circles. TABLE. 60 seconds (") make 1 minute (') 60 minutes 1 degree () 30 degrees 1 sign (S.) 12 signs or 360 1 circle (C.) Art. 52. MISCELLANEOUS TABLE. 12 units or things make 1 dozen. 12 dozen 1 gross. 12 gross 1 great gross. 20 units or things 1 score. PAPEB. 24 sheets make 1 quire. 20 quires 1 ream. BOOKS. A sheet folded in two leaves makes a folio. " " four " " a quarto or 4to. " " eight " " an octavo or 8vo. " " twelve " " a duodecimo or 12mo. " " eighteen " " an ISrno. " " twenty-four" " a 24mo. COMPOUND NUMBERS. 85 Reduction, Art. 53. Reduction of Compound Numbers is chang- ing one denomination or name to another, without alter- ing its value, and is either descending or ascending. Art. 54, Beduction descending is changing a greater denomination to a less, as pounds to shillings, &c. Reduction ascending is changing a less denomination to a greater, as farthings to pence, &c. Examples illustrating reduction descending : EXAMPLE 1. Reduce 24 15s. 10)d. to farthings. Process. Reduction descending because it is s. d. f. changing greater denominations (pounds, &c.) to 24 15 10 2 a less (farthings.) _20 1. Change the pounds to shillings. Since there 95 ghJU'gg are 20s. in 1, there are in 24, 20 times as many j2 shillings, which are 480s., to which add the 15s. and the sum will be 495s. 595 J pence. 2. Change the shillings to pence. Since there are 12d. in Is., there are in 495s. 12 times as many Ans. 23802 fart'gs. pence, which are 5940d., to which add the 10 jd. and the sum will be 5950 or 5950d. 2far. 3. Change the pence to farthings. Since there are 4 far. in Id., there are in 5950d. 4 times as many farthings, which are 23800far., to which add the 2far. and the sum will be 23802far., the answer. Ex. 2. In 15 rods, 4 yds., 2 ft., 9 in., how many inches ? Process. Eeduction descending, as in the rds. yds. ft. in. preceding example. 15 4 2 9 1. Change the rods to yards. Since there 5k are 5 5 yards in one rod, in 15 rods there are ~~y 5s* times as many yards, which are 82 \ yds., 75 to which add the 4 yds., and the sum will be 86| yds. 8 ^ 7**> 2. Change the yards thus found into feet. rf Since there are 3 feet in 1 yard, in 865 yds. 261 i ft. there are 3 times as many feet, which are 12 259 2 ft, to which add the 2 ft., and the sum ^ m ~3i47~in will be 261i ft. 3. Change the feet to inches. Since there are 12 inches in 1 foot, in 261 i ft., there are 12 times as many inches, which are 3138 in., to which add the 9 in. , and the sum will be 3147 inches, the answer. * Since this is by a rule in Fractions, the teacher should explain the process in such cases, as it is explained in Fractions. 86 COMPOUND NUMBERS. RULE. In reduction descending, multiply the greatest de- nomination by that number which it takes of the next less to make one of this greater, and add to the product any given number of the same name. Proceed thus till the required denomination is found. EXAMPLES ILLUSTRATING REDUCTION ASCENDING. Ex. 3. Reduce 23802 farthings to pounds, &c. Process. Keduction ascending, changing 4)23802 far. a less denomination (farthings) to greater 12)5950 d -4-2far (pounds, &c.) 1. Change the farthings to pence. Since 20)49.5s.+10. 4 farthings make 1 penny, there are as 24-f-15s.-{-10d. many pence in 23802 far. as there are times 4 farthings, which are 5950 times, and 2 farthings remaining. There- fore 23802far. are 5950d. 2far. 2. Change the pence thus found to shillings. Since 12 pence make 1 shilling, there are as many shillings in 5950d. as there are times 12d. which are 495 times, and lOd. rem. Therefore 5950d. are 495s. lOd . 3. Change the shillings to pounds. Since there are 20 shillings in 1 pound, there are as many pounds in 495s. as there are times 20s. which are 24 times, and 15s. rem. Therefore 495s. are 24 15s. Also 23802far. are 24 15s. 10d. Ex. 4. In 3147 inches, how many rods, &c. Process. Keduction ascending as in the 12)3147 in. preceding example. 3)262 ft.+3 in. 1. Change the inches to feet. Since 12 in. make 1 ft., there are as many feet in 5 *) 8 J yds.+ Lft. 3147 in. as there are times 12 in., which are 262 times, and 3 in. rem. Therefore 11)174 3147 in. are 262 ft. 3 in. T5 rds.+9 half yds. 2. Change the feet thus found to yards. /^ y( j g ^ f t g j n \ Since 3 ft- make 1 yd., there are as many Ans , 15 ^8,4 yds. 2 ft. 9in. yds. in 262 ft. as there are times 3 ft., which are 87 times. Therefore 262 ft. are 87 yds. 1 ft. 3. Change the yards to rods. Since 5 yds. make 1 rod, there are as many rods in 87 yds. as there are times 5 yds. which are 15 times, * and 4| yds. (=4 yds. 1 ft. 6 in. ) rem. Therefore 87 yards are 15 rods, 4 yds. 1 ft. 6 in. And 3147 in. are 15 rods, 4 yds. 2 ft. 9 in. * By a rule in Fractions. ENGLISH MONEY. 87 KULE. In reduction ascending, divide the given denomi- nation by that number which it takes of this denomination to make one of the next less. Proceed thus until the greatest denomination required is found, which, with the remainders, in regular order, will be the answer. Art, 55. English or Sterling Money, MENTAL EXERCISES. How many farthings in 3 pence ? Process. Since there are 4far. in 1 penny, in 3d. there are 4 times as many farthings as pence, which are 12 farthings ; or there are 4 times 3 far. How many farthings in 5d ? 7 ? 9 ? 11 ? 4 ? 6 ? 8? 10 ? 12 ? in 1 shilling V 1 pound ? In 50 farthings how many pence ? Process. Since 4 farthings make 1 penny, in 50 farthings there are as many pence as there are times 4 farthings, which are 12 times, and 2 farthings remaining. Therefore 50 far. are 12d. 2far. How many pence in 8 farthings ? 16 ? 24 ? 32 ? 40 ? 48? 10? 20? 25? 30? 42? 50? How many pence in 2 shillings ? 4? 6? 8? 10? 5? 7? 9? 12? in 1 pound? How many shillings in 24 pence? 48? 72? 36? 60? 96? 30? 40? 56? 63? 81? 100? 144? How many shillings in 2 pounds ? 4? 6? 9? 3? 5? 7? 10? How many pounds in 40 shillings ? 60 ? 80 ? 50 ? 65 ? 72? 84? 100? 120? How many shillings in 2 guineas ? 4? 6? 3? 5? 7? 10? How many guineas in 42 shillings ? 63 ? 84 ? 50 ? 65 ? 76? How many farthings in 1 shilling ? 2 ? 3 ? How many pence in 1 pound ? 2 ? 3 ? How many shillings in 48 far. ? 96 ? 88 COMPOUND NUMBEES. EXAMPLES FOR THE SLATE, ETC. Reduce (5.) 17 2s. 6d. to pence. (6.) 11 15s. 4d. to farthings. (7.) 15 18s. lOd. to pence. (8.) 24 12s. 8d. to far. (9.) 45 8s. 7>d. to far. (10.) 8 4s. to pence. (11.) 10s. 9d. Sfartofar. (12.) 1813s. 8d. to far. (18.) 12 14s. to shillings. (14.) lid. 2far. to farthings. (15.) 4 Gd. to farthings. (16.) 5 15s. to pence. (17.) 10s. 3far. to far. (18.) 10 0d. to far. Reduce (33.) 1510s. 9d. to pence. (34.) 23470far. to pence. (35.) 18s. 7d. 3far. to far. (36.) 1340d. to pounds. (19.) 23475far. to pounds, &c. (20.) 3756d. to pounds, &c. (21.) 7620s. to pounds, &c. (22.) 4325far. to shillings, &c. (23.) 2754d. toshiUings, &c. (24.) 3240far. to pence, &c. (25.) 4360s. to pounds, &c. (26.) 1345d to pounds, &c. (27.) 7563far. toshiUings, &c. (28.) 2560id. to pounds. (29.) 1781s. to pounds, &c. (30.) 156M. to shillings, &c. (31.) 1682far. to pence. &c. (32.) 5146|d. to pounds, &c. (37.) 12560far. to pounds, &c. (38. ) 16 4s. to pence. (39.) 1260s. to pounds. (40.) 17s. 6^d. to far. &c. Art. 56. Troy Weight. MENTAL EXERCISES. How many grains in 2 pennyweights ? 4 ? 6? 3? 5? 7? 9? 10? How many pennyweights in 48 grains? 72? 96? 30? 56? 81? How many pennyweights in 2 ounces ? 4? 6? 3? 5? 8? 10? 7? 12? How many ounces in 40 pennyweights ? 60 ? 100 ? 50 ? 75 ? 110 ? COMPOUND NUMBERS. 89 How many ounces in 2 pounds ? 4? 6? 8? 3? 7? 10? 5? 9? 12? How many pounds in 24 ounces ? 36 ? 48 ? 72 ? 96 ? 144? 30? 56? 68? 112? EXAMPLES FOB THE SLATE. Reduce (41.) 59125 gr. to oz. (42.) lOlbs. 10 oz. lOpwts. to pwts. (43.) 6743 pwts. to Ibs. (44.) 1000 Ibs. to pwts. (45.) 1000 pwts. tooz. (46.) 62 Ibs. 5oz. to pwts. (47.) 2840 grs. to oz. (48.) 2840 Ibs. tooz. (49.) 12 Ibs. 15 pwts. to grains. (50.) 2895 pwts. to Ibs. Art. 57. Avoirdupois Weight. MENTAL EXERCISES. How many drams in 2 ounces ?4?6?8?3?5?7?9?10? How many ounces in 32 drams ? 48 ? 64 ? 36 ? 50 ? 75 ? How many ounces in 2 pounds ? 4?6?8?3? 5?7? 10? 9? 11? How many pounds in 32 ounces ? 48 ? 64 ? 30 ? 54 ? 75 ? How many pounds in 2 quarters ?4?3?8?5?10? How many quarters in 50 pounds V 75 ? 100 ? 40 ? 80 ? 108 ? How many quarters in 2 hundred weight ?3?5?7?4?6? 9? 12? How many hundred weight in 8 quarters ? 16 ? 24 ? 12 ? 20? 10? 15? 25? How many hundred weight in 2 tons? 4? 6? 3? 5? 7? 10? How many tons in 40 hundred weight ? 60 ? 80 ? 100 ? 50? 75? EXAMPLES FOB THE SLATE. Reduce (51. ) 3 cwt. to pounds. (52.) 4815 Ibs. to cwts. (53.) 2748000 dr. to tons. (54.) 4 T. 15 Ibs. to pounds. (55.) 67200 oz. to cwt. (56.) 15 cwfc. 1 qr. 12 oz. to oz. (57.) 22500 Ibs. to tons. (58.) 3 T. 15 cwt. 16 Ibs. to Ibs. 90 COMPOUND NUMBERS. (59.) 4 cwt. 3 qrs. to pounds. (60.) 1120 oz. to pounds. {61.) 1225 Ibs. to drams. (62.) 1750 Ibs. tocwts. (63.) 2000 Ibs. to ounces. (64.) 4000 Ibs. tocwts. (65.) 28000 Ibs. to tons. Art. 58, Apothecaries' Weight. MENTAL EXERCISES. How many grains in 2 scruples ? 4 ? 6 ? 3 ? 5?8?7? 9? 10? How many scruples in 40 grains ? 60? 100? 50? 65? 87? 112? How many scruples in 3 drams ? 5? 7? 2 ? 4 ? 6 ? 9 ? 11 ? How many drams in 2 ounces ?4? 6? 3? 7? 9? 8? 10? 12? How many ounces in 3 pounds ? 2? 4? 6?5?7?10? 9? 12? EXAMPLES FOB THE SLATE. Reduce (66.) 47230 gr. to ounces. (67.) 5375 Ibs. to ounces. (68) 74376 dr. to pounds. (69.) 10752 gr. to ounces. (70.) 1728 Ibs. to ounces. (71.) 1730 dr. to ounces. (72.) 6 Ib. 4 oz. to drams. (73.) 5000 gr. to ounces, &c. Art, 59, Cloth Measure. MENTAL EXERCISES. How many inches in 2 nails ? 4? 6? 3? 5? 7? 10? 12? How many nails in 5 inches ? 10 ? 15 ? 20 ? 6 ? 12 ? How many nails in 2 quarters ? 4? 3? 5? 7? 6? 8? 11? How many quarters in 3 yards ? 5? 8? 10? 12? How many yards in 8 quarters ? 16 ? 20 ? 12 ? 9 ? 15 ? 18 ? 21? 24? EXAMPLES FOR THE SLATE. Reduce (74.) 128yds. to E. E. (75.) 75yds. to E. Fl. (76.) 764 qrs. to yds. (77.) 10 yds. 3 qrs. to nails. (78.) 11^ yds. to E. E. (79.) 16> yds. to quarters. (80.) 20 yds. 3 qrs. to inched. (81.) 6 E. E. to E. F. COMPOUND NUMBERS. 91 Art. 60. Long Measure, MENTAL EXERCISES. How many inches in 3 ft.? 5? 8? 7? 9? 11? 12? How many feet in 24 inches ? 36? 48? 60? 30? 56? 63? 72? 108? 144? How many feet in 3 yards ?2?4?6?5?7?9?10?12? How many yards in 6 ft.? 9? 15? 18? 24? How many yards in 3 rods ? 6? 5? 7? 10? How many feet in 2 rods ? 4 ? 3 ? 5 ? How many rods in 3 furlongs ? 4 ? 6 ? 5 ? How many miles in 16 furlongs ? 24 ? 40 ? 32 ? 56 ? 72 ? 64 ? 10? 20? 25? EXAMPLES FOR THE SLATE. Beduce (82.) 103 in. to yards. (83.) 7 yds. 10 in. to inches. (84.) 3840 rods to miles. (85. ) 7 M. 6 fur. 30 rods to rods. (86.) 2910 rds. to miles. (87.) 5 yds. 1 in. to inches. Art. 61. Square Measure. MENTAL EXERCISES. How many square feet in 2 square yards ? 4? 6? 8? 3? 5 ? 7 ? 10 ? 12 ? How many square yards in 18 square feet ? 27 ? 54 ? 36 ? 63? 99? 20? 44? 56? How many square feet on a board 1 ft. wide and 12 ft. long (12 ft. by 1 ft. ?) in a door 6 ft. high and 3 ft. wide (6 ft. by 3 ft. ?) in a room 12 ft. by 11 ft. ? What is the length of a board 2 ft. wide and containing 24 square feet ? a pane of glass 9 inches wide and containing 108 square inches ? of a room 12 feet wide and containing 180 square feet ? of a field 10 rods wide and containing 1 acre ? What is the difference between 4 square feet and 4 feet square ? 5 square rods and 5 rods square ? 92 COMPOUND NUMBERS. EXAMPLES EOR THE STjATE. Reduce (88.) 243 sq. rods, to acres, &c. (89.) 8 sq. yds to sq. in. (90.) 4176 sq. in. to sq. yds. (91.) 24000 sq. rds. to acres. (92.) 1 sq. M. to sq. rods. (93.) 71680 sq. rds. to acres. (94. ) 16 A. 18 sq. rds. to sq. rds. (95.) 78436 ft. to acres. 96. How many square inches in a window pane 15 in. by 12 in.? 97. How many sq. ft. in a looking glass 36 in. long and 24 in. wide (36 in. by 24 in. ?) 98. How many sq. ft. in a floor 12 ft. by 10 ft. ? 99. How many sq. yds. in a floor 18 ft. by 15 ft. ? 100. How many sq. rods in a field 40 rds. by 20 rds. ? 101. How many acres in a field 45 rds. by 30 rds. ? 102. How many acres in a farm ^ M. by 60 rds. ? 103. What is the length of a window pane 15 in. wide and containing 300 square inches ? 104. What is the width of a looking glass 60 in. long and containing 15 square feet ? 105. What is the length of a room 12 ft. wide and contain- ing 192 square feet ? 106. What is the width of a room which requires 30 sq. yds. of carpeting, and is 18 feet long ? 107. How many sq. yds. in 6 M. 7 sq. rods ? 108. How many square miles in 6400 acres ? 109. How many sq. yds. of paper will cover the walls of a room 24 feet by 20, and 12 feet high ; and how many square yards of ceiling are there ? 110. How many square miles in 92160 acres ? 111. How many yards of muslin 3 qrs. wide, will line a quilt 8 feet square ? Art. 62, Cubic Measure. Reduce 112. 4 cord, 16 cub. feet to cub. feet. 113. 1 cub. yd., 10 cub. feet to cub. inches. 114. 34 cords, 64 cub. feet to feet. COMPOUND NUMB 115. 31104 cub. inches to cub. feet. 116. 3584 cub. feet to cords. 117. 442368 cub. inches to cords. 118. How many cubic feet of wood in a load of wood 8 ft. long, 4 ft. wide, and 5 ft. high ? 119. How many cubic yards in a cellar 24 ft. long, 15 ft. wide, and 6 ft. deep ? 120. How many cords of wood in a pile 40 ft. long, 8 ft. wide, and 6 ft. high ? 121. How many bricks will it take to lay the foundation of a house 32 ft. by 30, the height of the foundation being 8 ft. and the thickness 1 ft. ; the bricks being 8 in. long, 4 in. wide, and 2 in. thick ? Art. 63, Wine Measure. MENTAL EXERCISES. How many gills in 3 pints ? 5 ? 7 ? 10 ? How many pints in 8 gills ? 16 ? 20 ? 32 ? How many pints in 2 quarts ? 4? 6? 8? 12? How many quarts in 3 gallons ? 5? 7? 9? 12? How many quarts in 4 pints ? 8 ? 10 ? 16 ? 22 ? How many gallons in 8 quarts ? 12 ? 20 ? 25 ? How many gallons in 2 barrels ? 2 hogsheads ? EXAMPLES FOB THE SLATE. Reduce (122.) 2 hhd. 2 qt. 1 pt. to gills. (123.) 17 gaL 2 qts. to pints. (124.) 140 pts. to gallons, &c. (125.) 40736 gills to tuns, &c. Reduce by Beer Measure (126.) 3 bbls. 16 gals, to pts. | (127.) 1730 pts. to barrels, &c. 128. How many gills in 1 hogshead of wine ? 129. How many pints in 1 barrel of beer ? 130. How many quarts in 5 hhds., 31 } gals, of vinegar ? 94 COMPOUND NUMBEKS. 131. How many quarts in 3 hhds. of ale ? 132. How many hhds., &c., in 1200 gals. ? 133. How many barrels in 850 gals, of beer ? Art. 64. Dry Measure. MENTAL EXEKCISES. How many pints in 4 quarts ? 6? 9? 12? 15? How many quarts in 4 pints ? 6? 12? 9? 15? How many pecks in 16 quarts ? 24 ? 40 ? 56 ? How many pecks in 2 bushels ? 5 ? 3 ? 7 ? How many bushels in 16 pecks ? 32 ? 48 ? 72 ? EXAMPLES FOB THE SLATE. Reduce (134.) 12 bus. 3pks. to qts. (135.) 3 pks. 1 pt. to pints. (136.) 15 bu. 6 qts. to qts. (137.) 4^ pks. to pints. (138.) 1000 quarts to bushels. (139,) 560 pecks to bushels. (140.) 32140 pints to pecks. (141.) 2764 qts. to bushels. 142. In 100 bushels how many quarts ? 143. In 1200 pints how many pecks ? Art. 65. Time. MENTAL EXEKCISES. How many seconds in 2 minutes ? 4 ? 6 ? How many minutes in 120 seconds ? 180 ? How many minutes in 3 hours ? 5 ? 7 ? 10 ? How many days in 2 weeks ? 4 ? 7 ? 12 ? How many weeks in 3 months ? 5 ? 7 ? 10 ? How many years in 24 months ? 36 ? 48 ? 65 ? EXAMPLES FOB THE SLATE, ETC. Reduce 144. 2 hours, 3 minutes, 50 seconds, to seconds. 145. 3 days, 30 minutes, to minutes. 146. 1 year, 6 months, 27 days to hours. 147. 3 years, 2 weeks, 10 hours to seconds. EEDUCTION. 95 148. 9 months, 15 days, 10 minutes to minutes. 149. 1 solar year to seconds. 150. 12000 minutes to days, &c. 151. 99840 seconds to days, &c. 152. 7200 seconds to hours. 153. 56000 minutes to months. 154. 450000 seconds to days. 155. 12500 days to years. 156. How many hours in 5 years, 6 months, 3 weeks, 4 days ? 157. How many days in 365000 seconds ? 158. How many seconds in 1 day ? 66. Circular Measure. Reduce 159. 10 S. 15, 45', 30" to seconds. 160. 3000000 seconds to signs. 161. 18, 36', 12" to seconds. 162. 25000" to degrees. Art, 67, Promiscuous Examples in Reduction of Compound Numbers. EXERCISE I. 1. How many farthings in 5 10s. 6d. ? 2. How many pounds in 3540 pwt. ? 3. How many pounds in 3 T. 7 cwt. 3 qr. ? 4. How many pounds in 7580 scruples. ? 5. How many nails in 25 yd. 3 qr. ? 6. How many miles in 500000 ft. ? 7. How many sq. yards in 5 A. 1 R. 12 rd. ? 8. How many sq. yards in 350 000 sq. in. ? 9. How many sq. inches in a looking glass 30 in. long and 1 8 in. wide ? 96 REDUCTION. 10. How many cubic feet in 6 C. 72 ft. ? 11. How many cubic feet in a load of wood 8 feet long, 4 feet wide, 5 feet high ? 12. How many gallons in 12548 gills ? 13. How many gallons in 3 T. 2 hhd. 21 gal. ? 14. How many quarts in 9 hhd. 15 gal. 3 qt. of beer ? 15. How many pints in 1 bu. 4 qts. ? 16. How many days in 325760 seconds ? 17. How many seconds in 14 16' 15" ? EXERCISE n. 18. How many pounds in 2540 farthings. 19. How many pennyweights in 12 pounds, 9 ounces ? 20. How many hundredweight in 36456 drams ? 21. How many grains in 7 pounds, 8 ounces, 1 scruple ? 22. How many yards in 1650 nails ? 23. How many yards in 5 miles, 3 furlongs, 4 yards ? 24. How many acres in a field 69 rods long, 45 rods wide ? 25. How many cords of wood in a pile 5 rods long, 8 feet wide and 6 feet high ? 26. How many pints in 5 hhd. 36 gallons, 1 quart, 1 pint ? 27. How many barrels in 5000 quarts of beer ? 28. How many bushels in 3540 quarts ? 29. How many minutes in 3 years, 4 months, 1 week, 2 days ? 30. How many degrees in 9863 seconds of a circle ? EXERCISE m. 31. How many pence in 6 12s. 4d. 32. How many pounds in 75603 grains of silver ? 33. How many pounds in 5 T. 10 cwt. 20 pounds ? 34. How many pounds in 7856 grains of calomel ? 35. How many yards in 6 ells English ? 36. How many yards in 432 inches ? 37. How many yards in 7 miles 12 rods ? 38. How many acres in 5 square miles ? 39. How many cubic feet in the walls of a brick house 42 feet long, 30 feet wide, 25 feet high, the walls being 1 foot thick ? 40. How many barrels in 6481 pints ? REDUCTION. 97 41. How many pints in 25 barrels of beer ? 42. How many quarts in 15 bushels, 3 pecks ? 43. How many days in 72000 minutes ? 44. How many minutes in 27 12' ? EXERCISE rv. 45. How many shillings in 260 farthings ? 46. How many shillings in 12 10s. ? 47. How many ounces in 12 Ibs. 10 oz. of gold ? 48. How many ounces in 12 Ibs. 10 oz. of tea ? 49. How many ounces in 856 scruples ? 50. How many ells English in 5 yards ? 51. How many inches in 1 M. 32 rods ? 52. How many square inches in 5 rods square ? in 5 square rods ? 53. How many cubic feet in the walls of a stone house, 30 feet long, 26 feet wide, and 24 feet high, the walls being 1 foot thick ? 54. How many gallons in 1 pipe, 1 hhd. 1 barrel ? 55. How many barrels in 648 gallons of beer ? 56. How many bushels in 3840 quarts ? 57. How many days in 5 years ? 58. How many signs in 1800' ? EXEECISE v. 59. How many pence in 10 10s. ? 60. How many pence in 500 farthings ? 61. How many pounds in 5672 pwt. ? 62. How many pounds in 4 T. 16 cwt. 21 Ibs, ? 63. How many pounds in 6742 gr. of camphor ? 64. How many yards in 672 nails ? 65. How many rods in 3>^ miles ? 66. How many rods in 10,672 inch ? 67. How many acres in a field containing 320 sq. rods ? 68. How many acres in a field 160 rods sq. ? 69. How many cubic feet in a ditch around a garden 6 rods long and 4 rods wide, the ditch being 2 feet wide and 3 feet deep? 5 98 REDUCTION. 70. How many gallons in 4728 gills ? 71. How many gallons in 2 hhd. 1 barrel of beer ? 72. How many pints in 15 bushels 6 quarts ? 73. How many months in 273456 minutes ? 74. How many seconds in 27 30' ? EXEECISE VI. 75. How many farthings in 18 7s. 10}d. ? 76. How many grains in 11 ounces of gold ? 77. How many hundredweight in 12760 ounces of flour ? 78. How many grains in 56 ounces of morphine ? 79. How many ells English in 10 ells Flemish ? 80. How many feet in 312 inches ? 81. How many feet in 275 rods ? 82. How many square inches of gilding will cover the frame of a looking glass 36 inches long, 20 in. wide, the width of the frame being 4 inches ? 83. How many cords of wood in pile 64 feet long, 4 feet wide and 5 feet high ? 84. How many quarts in 9 hhds. 3 qts. ? 85. How many barrels in 3472 qts. of beer ? 86. How many pecks 712 pints ? 87. How many seconds in 1 month ? 88. How many degrees in 56312 seconds of a circle ? EXEECISE vn. 89. How many shillings in 187^ pence ? 90. How many ounces in 6512 grains of silver ? 91. How many ounces in 11 cwt. 12 Ibs. ? 92. How many ounces in 678 grains of laudanum ? 93. How many nails in 1% yards ? 94. How many rods in 12^ miles ? 95. How many rods in 1428 feet ? 96. How many acres in a town 5 miles square ? 97. How many acres in 6 square miles ? 98. How many solid feet in 72 tons of hewn timber ? 99. How many bogheads in 6854.gallons ? 100. How many half -pints in 2 barrels of beer ? 101. How many pints in 36 bushels ? PROMISCUOUS EXAMPLES. 99 102. How many hours in 36000 seconds ? 103. How many minutes in *> a circle ? Promiscuous Examples in Reduction of Compound Numbers occurring in business, &c, EXERCISE Vm. 104. A silversmith made a gold cup weighing 8 oz, 10 pwt. ; what did it cost at 5 cts. a grain ? 105. A manufacturer bought 7 cwt. 16 Ibs. of wool ; what did it cost at 37 1 .< cts. a pound ? 106. A druggist sold 5 dr. 2 sc. morphine ; what was the amount at 8 mills a grain ? 107. If a tailor uses 4 yds. 2 qrs. of cloth in making an over- coat, how many can he make from 29 yards ? 108. A contractor agreed to make a road for $325.50 a mile ; what was the price per rod ? 109. A farmer sold 3 acres 1 K. of land for 45 cts. a square rod ; what did he receive for it ? 110. At $5 a cord, what costs 1 cord foot of wood ? 111. A merchant bought 15 gallons 2 quarts of oil at 1 shil- ling a pint ; what did it all cost ? 112. If 3 bushels 4 quarts of salt cost $4, what is the price per pint ? 113. If a man's income is $1200 a year, how much is it per day? 114. If a star move from W. to E. at the rate of 10' 30" daily, how long will it be in completing a circle ? 115. A stationer sells paper for 20 cents a quire ; what will 3 reams cost ? EXEECISE IX. 116. A silversmith paid $300 for 1 Ib. of gold ; what was the price per grain ? 117. A blacksmith used 8 ounces of iron in making a horse shoe ; how many did he make from 60 pounds ? 100 REDUCTION. 118. Paid a druggist $1.25 for 4 ounces of jalap ; what was the price per grain ? 119. A merchant tailor made 12 vests, each containing 3 qrs. of silk velvet ; how many yards did he use ? 120. A telegraphic company paid $1250 for wire at 3 cents a yard; how many miles did it extend ? 121. A man bought a field 150 rods long and 75 rods wide, at $64 an acre ; what did it cost ? 122. A ditcher agreed to dig a ditch 60 rods long, 2 ft. wide, and 3 ft. deep, for $44 ; what did he receive for each solid yard ? 123. A brewer sold 5 hogsheads, 15 gallons of ale at 4 pence a quart ; what did he receive for it ? 124. A woman sold a quantity of blackberries for $6.25, at 6 cents a quart ; how many bushels did she sell ? 125. A laborer earned $56 in chopping wood at 75 cents a day; how many months did he work ? 126. The apparent motion of the sun being one circle in the heavens a year, how many seconds does it move daily ? 127. What will 6 gross of buttons cost at 8 pence a dozen ? EXEECISE x. 128. If a silversmith uses 8 ounces of silver in making a cup, how many cups can he make from 30 pounds ? 129. If a family consumes 2 pounds 12 ounces of flour daily, how long will 1 barrel Kst them ? 130. How many powders, each composed of 4 grains, can be made from a mixture of medicine weighing 1 Ib. 6 oz. 2 dr. 1 sc. ? 131. How many vests, each containing 3 quarters, can be made from a piece of valentia measuring 16> yards ? 132. At 12)^ cents a foot, how much will a lead pipe cost extending from a house to a spring ^ mile distant ? 133. How many panes of glass, 8 inches by 10 inches, are in a box containing 100 square feet ? 134. What is a pile of wood worth which is 112 feet long, 4 feet wide, and 6 feet high, at $4^ a cord ? 135. How many bottles, containing 3 pints, each can be filled, from a hogshead of wine ? PROMISCUOUS EXAMPLES. 101 136. At 6 cts. a quart, how much will 5 1 ^ bushels of salt cost ? 137. If a carriage wheel turn round once in passing over 8 feet 9 inches, how many times will it turn in going 3% miles ? 138. If a clock tick 4 times in a second, how many times does it tick in a day ? 139. A stationer sells paper for $4 a ream ; how much is it a quire ? EXEECISE XI. 140. At 4d. a grain, how many pounds sterling will a gold cup cost weighing 7 oz. 10 pwt. ? 141. If a pound of wool cost 4s. 6d., how many pounds sterling will 8 cwt. 3 qrs. cost ? 142. At >d. a grain, how much will 2 oz. 5 dr. 2 sc. of qui- nine cost '? 143. At lOd. a foot, how much will a gas pipe cost measur- ing 3 rds. 4yds.? 144. At 10s. 6>d. a square yard of oil cloth, how much will it cost to cover a hall floor 30 feet by 6 feet ? 145. At 8d. a solid foot, how much will a ton of hewn tim- ber cost ? 146. If a pint of Port wine cost 7s. 6d., how much will 10) gallons cost ? 147. If a laborer earn 6d. an hour, how much will he earn in a month of working time (6 days a week and 10 hours a day ?) 148. If 10 gross of buttons cost $6, what is the price per doz. ? EXERCISE XII. 149. If a silversmith buy 1 pound of silver for $40, and sell it for 1 cent a grain, how much will he gain ? 150. If a merchant buy 10 cwt. 2 qrs. of sugar for $115.50, for what must he sell it to gain 1 cent, a pound. 151. If a druggist pay 8 mills a grain for calomel and sell it for 9 mills, how much will he gain on a pound ? 152. A contractor agreed to make a road 7^ miles long for $2500 and paid his laborers $1 a rod ; how much did he gain ? 153. A speculator bought city lots containing lj^ acres for 102 BEDUCTION. $50,000, and sold them for $1 a square foot ; how much did he gain? 154. A job mason agreed to build the foundation of a house 32 feet long, 24 feet wide, the foundation to be 9 feet high, 18 inches thick, for $12, but was obliged to pay his journeymen 1 cent a solid foot ; how much did he lose ? 155. A merchant bought 3 hhds. of molasses for 28 7s. and sold it for 1 shilling a quart, how much did he gain ? 156. A merchant in selling peas at 20 cents a quart gains 5 cents; what did they cost a bushel ? 157. If a watch lose 2 seconds an hour, how much will it lose in a week ? EXEKCISE XTTT. 158. A silversmith paid $175.68 for gold ore at three cents a grain ; how much did he buy ? 159. A merchant bought 1 T. 1 cwt. of rice at 9d. a pound ; what did it all cost ? 160. A druggist sells opium for 4d. a scruple ; how many ounces can be bought for 2 10s. ? 161. How many suits of clothes, each containing 5 yards 3 quarters ; can be cut from 69 yards ? 162. How much will it cost to make a wire fence 4 wires high, around a field 40 rods long and 32 rods wide at 3 cents a yard ? 163. What will an acre of ground cost at 75 cents a square yard ? 164. How many cubic inches in a block of marble 1 yard long, 2 feet wide, 18 inches thick ? 165. What will a hogshead of vinegar cost at 5 mills a gill ? 166. What will 2 bushels 3 pecks of plums cost at 4 cent* a pint ? 167. If a ship sail 12 miles an hour, how far will she sail in 2 months, 1 week, 3 days. 168. If a planet move 1 30' a day ; how long will it be in moving through each sign of the Zodiac ? 169. If one gross of steel pens cost 72d. , what does 1 pen cost ? PROMISCUOUS EXAMPLES. 103 EXERCISE XIV. 170. If 24 English Bibles cost 22 guineas, 6s., what is the price of each ? 171. A silver dollar contains 412^ grains ; how many dollars can be made from 25 Ibs. 15 pwt. 15 gr. of silver ? 172. In 6 tons 14 cwt. 3 qrs. 9 Ibs. of butter, how many firkins ? 173 . How many acres in a field 430 rods long and 132 feet wide ? 174. How many yards of muslin 3 quarters wide will line 10 yards of merino 1| yards wide ? 175. How many sheets of tin 15 inches by 12 inches, will cover the roof of a house 40 feet long and 25 feet wide, the rafters on each side being 16 feet long ? 176. How many blocks of granite 3 feet long, 21 inches wide, and 18 inches thick, will it take to build the walls of a church 80 feet long, 62 feet wide, 36 feet high, the walls being 18 inches thick ? 177. If a barrel of ale cost $7.25, what will be gained by re- tailing it a 4 cents a pint ? 178. How long would it take a locomotive to travel 3000 miles, at the rate of 3 rods a second ? 179. If a ream of paper cost $4 and be sold for 25 cents a quire, how much would be gained ? 180. At 5 cents a dozen, how many gross of buttons can be bought for $3 ? EXEKOISE XV. 181. How many yards of carpeting 2 feet wide, will cover a room 24 feet by 18 feet ? 182 . How many pieces of paper 9 yards long, 15 inches wide, will cover the sides of a room 18 feet long 15 feet wide, and 11 feet high, there being one door 6 feet 3 inches by 2 feet 9 inches, and two windows 5 feet by 4 feet ? 183. How many square yards of plastering in the, same room ? 184. How much will it cost to pave a sidewalk 10 rods long, 10 feet wide, at $10 a square rod ? 185. If a piece of land containing 6 acres be divided into 104 EEDUCTION. building lots 8 rods long and 2 rods wide, and each be sold for $300, for how much will it all be sold ? 186. How many shingles will cover the roof of a house 30 feet long, the rafters on each side being 15 feet long, allowing one shingle for every 20 square inches ? 187. How many bricks will it require to build a chimney, averaging 4 feet by 3 feet and 30 feet high, the walls being 8 inches thick, and the bricks 8 inches long, 4 inches wide and 2 inches thick ? EXERCISE XVI. 188. How many spoons, each weighing 2 ounce 10 pwt., can be made from 5 Ibs. of silver ? 189. What will 6 hogsheads of beer cost at 3 cents a quart ? 190. What will 6 hogsheads of wine cost at 1 shilling a pint ? 191. How many bushels of nuts can be bought for $20 at 4 cents a quart ? 192. What will it cost to plaster a room 30 feet long, 20 feet wide and 8 feet high, at 25 cents a square yard, there being two doors 7 feet by 4 feet, and two windows 5 feet by 3 feet, and a mop board 6 inches wide ? 193. What is the value of a pile of wood 108 feet long, 6 feet high, and 4 feet wide, at $7 a cord ? 194. How many rods of fence will inclose a tract of land 1 mile long and 180 rods wide ? 195. How many acres in a town 5 miles square ? 196. How many acres in a town 6 miles long and 3 miles wide ? 197. How much will it cost to carpet a room 24 feet long, 18 feet wide, the carpet being 2 feet wide and the price $1 a yard ? 198. What will it cost to dig a cellar 40 feet long 32 feet wide, and 8 feet deep at 25 cents a cubic yard ? 199. What will it cost to dig a ditch around a garden (out- side) which is 6 rods long and 4 rods wide, the ditch to be 3 feet deep and 2 feet wide, at -| cent a cubic foot ? EXERCISE XVH. 200. How many yards of muslin 3 quarters wide, will it take to line 21 yards of silk 2 quarters 2 nails wide ? COMPOUND NUMBERS. 105 201. How much will it cost to paper a room 27 feet long, 18 feet wide and 12 feet high, each piece of paper containing 9 yards, 1 foot 6 inches wide and costing 30 cents ? 202. How many bricks will pave a side-walk 120 rods long, 12 feet wide, each brick being 8 inches long and 4 inches wide ? 203. How high must a load of wood be to contain a cord, if it is 9 feet long and 4 feet wide ? 204. How wide must a field be to contain six acres, if it is 80 rods long ? 205. How many bricks in a pile 20 feet long, 12 feet wide, and 10 feet high, each brick being 8 inches long 4 inches wide and 2 inches thick ? 206. How many bricks of the same size in a furnace chimney 60 feet high, having four equal sides, averaging 5 feet, and the walls being 8 inches thick ? The Fundamental Rules Applied to Compound Numbers. Art. 68. ADDITION OF COMPOUND NUMBEKS. EXAMPLE 1. What is the sum of 9 10s. 7d. ; 7 11s. 4d. Ifar. ; 2 18s. 7d. ; 4s. 9d. 3 far. s. d. far. Process. First, add the farthings, the sum of 9 10 7 which is 6 farthings (by reduction ascending) Id. 7 11 4 1 2 farthings. Write the 2 farthings under the 2 18 7 2 column of farthings, and add or carry the Id. to 14 9 3 the pence in the next column. Add and reduce (if possible) the other denominations in the same Ans. 20 15 4 2 manner. RULE. Write the numbers of the same denomination, or name under one another in order. Add each denomination, beginning with the least, as in simple numbers, and reduce the sum to the next greater denomination. Write the re- mainder under the column added, and carry the quotient to the next column. 106 REDUCTION. PROOF As in simple Addition. In writing compound numbers, if any denomination is wanting, supply its place with a cipher. EXAMPLES. 2. Add 5 Ibs. 10 oz. 12 pwt. 9 gr ; 1 Ib. 11 oz. 13 pwt. 7 gr ; 6 Ibs. 9 oz. 7 pwt. 14 gr. ; 3 Ibs. 6 oz. 12 pwt. 12 gr. 3. Add 9 Ibs. 8 oz. 17 gr. ; 8 Ibs. 10 pwt. 12 gr. ; 7 oz. 12 pwt. 15 gr. ; 4 Ibs. 5 oz. 21 pwt. 4. What is the sum of 7 T. 19 cwt. 1 qr. 15 Ibs. ; 10 T. 8 cwt. 2 qrs. 18 Ib. ; 12 T. 10 cwt. 3 qrs. 14 Ibs. ; 5 T. 11 cwt. 1 qr. 9 Ibs. ? 5. Add 5 cwt. 3 qrs. 16 Ibs. 12 oz. 10 dr. ; 7 cwt. 1 qr. 18 Ibs. 14 oz. ; 9 cwt. 8 Ibs. 10 oz. 10 d. ; 3 qrs. 18 Ibs. 9 oz. 8 dr. 6. Add 3 Ibs. 2 oz. 4 dr. 2 sc. 10 gr. ; 4 Ibs. 8 oz. 5 dr. 15 gr. ; 6 Ibs. 11 oz. 7 dr. 2 sc. ; 2 oz. 4 dr. 1 sc. 15 gr. 7. What is the sum of 12 yd. 2 qr. 2 na. ; 8 yd. 3 qr. 1 na. 4 yd. 1 qr. 2 na. ; 6 yd. 2 qr. 2 na. ? 8. Add 12 m. 6 fur. 18 rod. 2 ft. ; 10 m. 4 fur. 16 rod. 10 ft, ; 9 m. 7 fur. 15 rod. 9 ft. ; 7 m. 5 fur. 10 rod. 12 ft. 9. How many acres in four fi Ms, the first containing 6 A. 3 E. 25 rod. ; the second 5 A. 1 E. 30 rds. ; the third 4 A. 2 E. 35 rds. ; the fourth 7 A. 3 E. 28 rd. ? 10. How many cords in three piles of wood measuring as follows. 8 C. 24 ft. ; 7 C. 64 ft. ; 9 C. 84 ft. ? 11. What is the sum of 10 hhd. 42 gal. 1 qt. 1 pt. ; 9 hhd. 30 gal. 1 pt. 2 giUs ; 54 gal. 3 qt. 1 pt. 2 gills ; 6 hhd. 25 gal. 2 qt. 1 pt. ? 12. Add 5 y. 4 mo. 12 d. 16 m. 20 sec. ; 4 y . 3 mo. 15 d. 45 m. ; 6 y. 6 mo. 20 d. 14 m. ; 6 m. 6d. 6 h. 40 sec. ; 7 y. 28 d. 20 m. 10 sec. 13. What is the difference between 45 16' 28" north latitude and 30 43' 32" south latitude ? Art. 69, SUBTRACTION OF COMPOUND NUMBERS. EXAMPLE 1. From 3 hhds. 18 gals. 1 qt. 1 pt. 1 gi., take 1 hhd. 31 gals. 2 qts. 1 pt. 3 gi. SUBTRACTION OF COMPOUND NUMBERS. 107 Process. Since 3 gills cannot be taken from hhd.gal.qt.pt.gi. 1 gill, add 1 pint reduced to 4 gills, to the 1 3 18 1 1 1 gill, then subtract the 3 gills, and write the re- 1 31 2 1 3 mainder, 2 gills, underneath, also carry the 1 m 1 49 2 1 2 pint to the next denomination. Thus proceed. RULE. Write the less compound number under the greater, so that numbers of the same name will be under one another. Subtract each denomination as in simple numbers. If any number is greater than the one above it, add to the upper number 1 of the next denomination reduced to the same name, then subtract and carry 1 to the next denomination. PKOOF .4s in simple Subtraction. EXAMPLES. 2. From 11 12s. 8d. 2 far., subtract 4 15s. 6d. 3far. 3. From 161bs. 10 oz. 14pwts. 20grs., subtract 81bs. 8 oz. 20 pwts. 10 grs. 4. What is the difference between 17 T. 11 cwt. 1 qr. 10 Iba. and 10 T. 15 cwt. 3 qr. 6 Ibs. ? 5. What is the difference between 16 Ibs. 8 oz. 6 ^dr. 2 sc. and 8 Ibs. 4 oz. 6 dr. 12 grs. ? 6. From 14 yds. 1 qr. 2 ua. subtract 8 yds. 2 qrs. 3 na. 7. From 127 A. 1 B. 30 rds. subtract 56 A. 2 E. 24 rds. 8. What is the difference between 4 y. 5 mo. 10 d. 6 h. and 1 y. 9 mo. 21 d. 45 m. ? 9. What is the difference between 20 3 12' 24" and 5 15' 45'? Art 70. To find the time between two dates. Ex. 10. How old was a person, born May 15, 1820, when he died, Oct. 21, 1856 ? Process. (October is the 10th and May Died 1856 10 21 the 5th month. 1 Born 1820 5 15 Age ~~36 5 6 Ans. EULE. Subtract the first date from the last, numbering the months in their order. 30 days is considered a month, and 12 months a year. 108 REDUCTION. EXAMPLES. 11. A note dated Jan. 1, 1860, was due July 15, 1861 ; for what time was it given ? 12. A father was born May 16, 1815, and his son June 23, 1855 ; how much older is the father than his son ? 13. A note dated Aug. 15, 1865 was due May 1, 1866 ; for what time was it given ? 14. A man was born April 12, 1828, and his wife Sept. 21, 1833 ; how much older is the man than his wife ? 15. On the 19th of June, 1860, a man gave his note to be paid July 1, 1801 ; how long was the time ? Art. 71. MULTIPLICATION or COMPOUND NUMBEBS. EXAMPLE 1. Bought 5 barrels of molasses, each containing 34 gal. 3 qt. 1 pt. 2 gills ; how much did they all contain ? Process. The same as in addition, except hhd.gal.qt.pt.gi. we multiply instead of adding each denomi- 34 3 1 2 nation. Ans. 2 46 2 1~2 RULE. Multiply each denomination separately, as in sim- ple multiplication, beginning with the least, and reduce the pro- duct to the next greater denomination. Write the remainder under the number multiplied, and carry the quotient to the next denomination. PROOF. The same as in simple Multiplication. The multiplier is properly an abstract number, though it may be of a particular denomination or name. EXAMPLES. 2. Multiply 3 oz. 12 pwts. 18 grs. by 5. 3. Multiply 14 cwt. 1 qr. 15 Ibs. by 6. 4. Multiply 4oz. 3 drs. 2 sc. 10 grs. by 7. 5. Multiply 3 yds. 1 qr. 2 na. by 8. 6. Multiply 2 m. 3 fur. 9 rods, 8 ft. by 9. 7. Multiply 26 A. 2 B. 20 rods by 1*0. 8. Multiply 5 0. 112 ft. by 11. 9. Multiply 7 hhds. 45 gals. 2 qts. by 12. 10. 8 y, 6 mo. 7 d. 30 min. by 15. 11. 12 12' 12" by 8. DIVISION OF COMPOUND NUMBERS. 109 Art. 72. DIVISION OF COMPOUND NUMBERS. CASE 1. To divide a compound number into any number of equal parts. EXAMPLE 1. Divide 12 cwt. 2 qr. 18 Ibs. equally among 5 persons. Process. 12 cwt. -^-5=2 cwt. and 2 cwt. 5)12 cwt. 2 qrs. 18 Ibs. remainder; write the quotient under cwt. 2 " 2 " 3f " and reduce the remainder to quarters ; 2 cwt ; X4=8 qrs., to which add the 2 qrs. and the sum will be 10 qrs., which divide and thus proceed. RULE. Divide each denomination separately, beginning with the greatest, and write the quotient under it. Reduce the remainder to the next less denomination, to which add it and divide again. PROOF. The same as in Simple Division. Ex. Divide 48 T. 10 cwt. by 25. Process. The same except by long division. T. cwts. 25)48 10(1 T. 25 23 20 25)470(18 cwt. 25 Ans. 1 T. 18 cwt. 3 qr. 5 Ib. EXAMPLES. 3. Divide 100 15s. lOd. by 4. 4. Divide 10 Ibs. 8 oz. 10 pwt. by 6. 5. Divide 12 cwt. 1 qr. 14 Ibs. by 9. 200 ~W 4 25)80(3 qrs. '75 5 25 25)125(5 Ibs. 125 110 REDUCTION. 6. Divide 4 Ibs. 6 oz. 4 dr. 2 scr. by 11. 7. Divide 15 yds. 2 qrs. 3 na. by 15, 8. Divide 16 m. 3 fur. 20 rd. 12 ft. by 21. 9. Divide 124 A. 3 B. 150 rd. by 100. 10. Divide 50 C. 72 ft. by 16. 11. Divide 1200 bu. 3 pk. by 175. 12. Divide 9 hhd. by 11. CASE 2. To divide one compound number by another. Ex. 13. Divide 7 10s. among as many persons as possible, giving to each 12s. 6d. Process. Reduce both the dividend and divisor to pence, and then divide. Since this is performed by reduction and simple division it is not usually included in Conpound Division. Art, 73, Special Application of Compound Numbers to Longitude and Time. Since the apparent motion of the sun around the earth is in a circle of 360 in 24 hours 360-f 24 or 15=1 hour of time. 1=JL hour or 4 minutes of time. 15_:_60 or 15' =1 minute of time. 1' _L minute or 4 seconds of time. 15' -H>0 or 15 "I second of time. Art, 74, To FIND THE DIFFERENCE OF TIME WHEN THE DIFFERENCE OF LONGITUDE is KNOWN. EXAMPLE 1. The difference of longitude between two places is 18 35' ; what is the difference of time ? Process.- 1st, since 15 C =1 hour, 18=1 hour 15)18 35' 0" and 3 remainder; 3=180, to which add the 35' ib. 14 m> 20 sec. and the sum will be 215' ; and since 15'=1 min. 215'=14 niin. and 5' remainder, which reduced to (") and divided, is equal to 20 sec. Process. 2d, since l'=4 sec. 35'=140 sec. or 2 18 35' min. 20 sec. And since 1:=4 min. 18=72 min. to _ which add the 2 min. and the sum will be 74 min. = j jj 14 m 20 sec 1 h. 14 min. Therefore 18 35'=1 h. 14 min. 20 sec. SPECIAL APPLICATION OF COMPOUND NUMBERS. Ill RULE. Divide the longitude by 15 as in Compound Num- bers and the quotient of degrees will be hours, of minutes (' ) minutes of time, and of seconds (" ) seconds of time. Or Multiply the longitude by 4, and the product of min- utes (' ) will be seconds of time, and of degrees minutes of time, which may be reduced to hours. EXAMPLES. 2. The difference of longitude between Boston and London is 71 4'; what is the difference of time ? 3. The difference of longitude between Washington and London is 76 53 ; what is the difference of time ? 4. The difference of lou ntude between New York and San Francisco is 51 26'; what is difference of time ? When the longitudes of two places are given, their difference may be found by subtraction when they are both East or West ; by addition when one is East and the other West. Art, 75, To FIND THE DIFFERENCE OF LONGITUDE WHEN THE DIFFERENCE OF TIME is GIVEN. EXAMPLE 5. The difference of time between Philadelphia and Cincinnati is 37 min. 20" ; what is the difference of longi- tude? m. sec. Process. 1st, since 1 sec. =15" of long. 20 sec.= 37 20 300" =5' and since 1 min. =15' of long. 37 min. =555 15 min., to which add the 5' and the sum will be 560'= ns 9 2(F~"~ 9 20' m. sec. Process. 2d, since 4 min.=l of long. 37 min. 4)37 20 =9 and 1 min. rem., 1 min. =60 sec. and 20 sec. A^ 9~2(F~ more are 80 sec., and since 4 sec.=l' of long. 80 sec. =20' of long. Therefore 37 min. 20 sec. =9 20' of long. RULE. Multiply the time by 15, as in Compound Numbers, and the product of seconds of time will be ( " ) seconds of longi- tude, of minutes (') minutes of longitude, and of hours, of degrees of longitude. Or Multiply the hours by 15 to find degrees, and divide the minutes by 4 to find more degrees, and the seconds by 4 to find minutes of longitude. 112 PROMISCUOUS EXAMPLES. EXAMPLES. 6. The difference of time between Washington and Cincin- nati is 29 minutes 36 seconds ; what is the difference of long- tude ? 7. The difference of time between Washington and London is 5 hours, 8 minutes, 4 seconds; what is the difference of longitude ? 8. The difference of time between New York and San Francisco is 3 hours, 25 minutes, 44 seconds ; what is the difference of longitude ? Art, 76, Promiscuous Examples in Addition, Sub- traction, Multiplication, and Division of Compound Numbers. EXEBCISE I. 1. A man when married was 20 years, 9 months, 12 days old ; he and his wife lived together 21 years, 4 months, 15 days, and he lived after the death of his wife 12 years, 8 months, 3 days ; what was his age when he died ? 2. A railroad when completed is to be 224 miles long. The part already finished is 109 miles, 3 furlongs 18 rods; how much remains to be made ? 3. A druggist sold 9 boxes of medicine, each weighing 3 ounces, 4 drams, 2 scruples ; what was the whole weight ? 4. A wholesale grocer shipped 1 ton of sugar in 9 barrels of equal size ; how much was there in each barrel ? 5. A note dated July 10, 1860, was due January 1, 1862 ; how long was the time ? 6. The longitude of Jerusalem is 35 32 East, and that of Baltimore 76 37' West, what is the difference of time ? 7. The difference of time between Boston and London is 4 hours 44 minutes 32 seconds, what is the difference of longi- tude ? EXEECISE n. 8. Bought 4 pieces of cloth measuring as follows 23 yards, COMPOUND NUMBERS. 113 3 quarters, 2 nails ; 27 yards, 2 quarters, 1 nail ; 25 yards, 20 quarters, 3 nails; how many yards in all of them ? 9. A party started on a journey of 315 miles. After travelling 156 miles, 4 furlongs, 20 rods, how much farther had they to travel ? 10. A farm is divided into 16 fields, and each field contains 9 acres, 130 square rods ; how large is the farm ? 11. A man wishes to draw 12 cords of wood in 14 loads; how much must he draw in each load ? 12. A note was dated June 15, 1861, and was paid June 1st, 1862 ; how long was the time ? 13. The difference of longitude between Washington and London is 77, what is the difference of time ? 14. The difference of time between New York and San Fran- cisco is 3 hours, 25 minutes, 44 seconds ; what is the difference of longitude ? EXERCISE HE. 15. A farmer carried to market four loads of corn; in the first 36 bushels, 2 pecks, 4 quarts ; in the second 42 bushels, 3 pecks, 1 quart ; in the third 38 bushels, 1 peck, 3 quarts ; in the fourth 40 bushels, 6 quarts ; how many bushels in all ? 16. A merchant has sold from a hogshead of molasses 38 gallons, 1 quart, 1 pint ; how much is unsold ? 17. What is the weight of a dozen silver spoons, each weigh- ing 6 ounces, 8 pennyweights, 10 grains ? 18. If a pound of rhubarb be divided into 100 doses ; how much will a dose contain ? 19. John Q. Adams, was born July 11, 1767 ; what was his age when he died, February 23, 1848 ? 20. What is the difference in the time of two places, one in longitude 56 30', and the other in longitude 125 20', both East? 21. When it is noon at Greenwich England, a captain of a ship finds that is 4 o'clock P.M. where he is ; in what longitude is he ? EXERCISE TV. 22. Two men building a stone fence, built the first day 5 114 PROMISCUOUS EXAMPLES. rods, 8 feet, 9 inches ; the second 6 rods, 4 feet, 6 inches ; the third 6 rods ; and the fourth they finished it, building 4 rods 8 inches ; how long was the fence ? 23. A merchant tailor bought a piece of cloth supposed to contain 27 yards, 2 quarters, but on measuring it he found that it contained only 25 yards, 3 quarters ; how much did it fall short ? 24. A lumber merchant sold 10 loads of hewn timber, each containing 1 ton, 32 feet, 1600 inches ; how much did he sell ? 25. A farmer having 256 acres of land, divided it equally among his seven children ; how much did he give each ? 26. A note was dated May 16, 1858, and was due October 1st, 1860 ; how long was the time ? 27. What is the difference of time in two places, one 36 30 East longitude, and the other 40 15' West longitude ? 28. A sea captain found that it was 20 minutes past 9 o'clock where he was, when it was noon at Greenwich, England. In what direction from Greenwich was he, and in what longitude ? 29. Bought four loads of coal, weighing as follows : 18 cwt. 3 qrs. 20 Ibs. ; 16 cwt. 1 qr. 12 Ibs. ; 19 cwt. 21 Ibs. ; 17 cwt. 2 qrs. 9 Ibs. ; 18 cwt. 3 qrs. ; how much in all ? 30. A merchant bought 12 hogsheads, 36 gallons of oil, and sold 7 hogsheads, 42 gallons, 3 quarts, how many had he left ? 31. At 1 12s. 8d. a yard, how much will 20 yards of cloth cost ? 32. If 18 yards of cloth cost 28, what is the price per yard ? 33. Washington was born February 22, 1732 ; what was his age when he died, December 14, 1799 ? 34. The longitude of Boston is 71 4', and that of St. Louis is 90 15, what is the difference of time ? 35. A sea captain having sailed from Philadelphia, longitude 75 10 , finds his watch 1 hour 20 minutes slower than the time where he is; supposing his watch has kept good time, in what longitude is he ? CANCELLATION. 115 CANCELLATION, Art. 77. Cancellation is the omission of equal factors in the dividend and divisor, or any corresponding terms, for the purpose of shortening the operation. It is on the principle that dividing both the dividend and divisor by the same number, does not alter the quotient. (Art. 20, 3. ) EXAMPLE 1. Multiply 12 by 4, and divide the product by 4. Process. 12X4=48 and 48-^4=12. We therefore omit both the multiplication and division, or cancel the =12 4, by drawing a line across it, in both the dividend and ** divisor. Ex. 2. Bought 12 oranges at 4 cents each, and gave in ex- change for them 8 quarts of nuts ; what was the price of the nuts per quart ? Process. 12X4=48, the price of all the oranges, and 6 48-f-8=6 the price of the nuts per quart, but we cancel X-ttyJL the factor 4 by drawing a line across it in the dividend -: =6 and dividing the divisor 8 by it, which gives the same result ; we also cancel the 2. % KULE. Write the factors of the divisor under those of the dividend, and omit equal factors in both the dividend and di- visor, or corresponding terms, drawing a line across them, then proceed as in other similar operations. Since cancelling is the same as dividing, the place of a cancelled factor must be supplied by 1 if there is no other factor left. EXAMPLES. Cancel equal factors iu (3.) 2X3X4X6X7 (5.) 4X9XHX16 8X4X5 X 2 3X 22 X 5 (4) 4X6X8X10 (6.) 2X13X28 2X3X4X 2 26X14 7. Multiply together the factors 8, 12, 15 and 18, and divide the product by- the factors 4, 5 and 9. 116 CANCELLATION. 8. Multiply the factors 6, 9, 11 and 12, and divide the pro- duct by the factors 3, 4, 3, 4. 9. Multiply the factors 9, 14, 15, 16, and divide the product by the factors 2, 3, 7, 5, 8, 4. 10. Multiply the factors 11, 24, 22, 30, and divide the pro- duct by the factors 4, 5, 44, 33. 11. Bought 25 firkins of butter, each weighing 54 Ibs. , at 30 cts. a pound, and paid for them in exchange, potatoes in barrels, 3 bushels in each, at 75 cts. a bushel ; how many bar- rels ? 12. How many boxes of starch, each containing 20 Ibs., at 12 cts. a pound, will pay for 10 sacks of corn, each containing 3 bushels, at 72 cts. a bushel ? 13. How many bags of coffee, each weighing 60 Ibs. , at 30 cts. a pound, can be bought for 20 firkins of butter, each contain- ing 90 Ibs., at 25 cts. a pound ? 14. How many barrels of sugar, each weighing 225 pounds, at 16 cts. a pound, can be bought for 25 barrels of flour at 8 dollars a barrel ? Art, 78. Properties of Numbers. A number is either odd or even ; odd when it cannot be exactly divided by 2, and even when it can be , as 1, 3, 5, 13, &c., are odd numbers ; 2, 4, 8, 12, &c., are even numbers. Numbers are, also, either prime or composite. A. prime number is one which is not the product of any numbers greater than 1, or of any number into itself ; as 3, 13, 127. A composite number is the product of other numbers greater than 1 ; as 4=2X2 ; 12-4X3 ; 150-10X5X3. PKIME NUMBERS. 117 Art. 79. To resolve Composite Numbers into Prime Numbers or Factors. Small composite numbers may be thus resolved by in- spection ; thus, the prime factors of 6 are 3 and 2 ; of 15 5 and 3. The prime factors of larger numbers are found by trial. EXAMPLE 1. What are the prime factors in 42 ? Process. Beginning with the least prime number, 2, we find that it is a factor in 42, and 21 another factor ; and by further trial we find that 3 is a factor in 21, and 7 another factor. Hence 2, 3, and 7, being all prime numbers, are the prime factors in 42. BULE. Divide the given number by any prime number, and the quotient in the same manner, till it is found to be also a prime number. The several divisors, and the last quotient will be the prime factors of the given number. [Any one familiar with the multiplication or division table can easily distinguish prime numbers less than 144 from composite num- bers. To find other prime numbers less than 1000, reference may be made to the following] TABLE OF PRIME NUMBERS. 2)42 3)21 7 149 193 241 293 353 409 461 509 571 617 661 727 773 829 ^3 947 151 197 251 307 359 419 463 521 577 619 673 733 787 839 887 953 157 199 257 311 367 421 467 523 587 631 677 739 797 853 907 967 163 211 263 313 373 431 479 541 593 641 683 743 809 857 911 971 167 223 269 3171379 433 487 547 599 643 691 751 811 859 919 977 173 227 271 331 383 439 491 557 601 647 701 757 821 863 929 983 179 229 277 337 389 443 499 563 607 653 709 761 823 877 937 991 181 233 281 347 397 449 503 569 613 659 719 769 827 881 941 997 191 239 283 349 401 457 EXAMPLES. What are the prime factors in (2.) 8 (3.) 10 (4) U (5.) 21 (6.) 115 (7.) 35 (8.) 32 (9.) 230 (10.) 256 (11.) 288 (12.) 720 (13.) 1728 118 THE GKEATEST COMMON DIVISOR. The Greatest Common Divisor. Art, 80, The Greatest Common Divisor (g. c. a.) of two or more numbers is the greatest number that will divide each of them without a remainder ; as 3 is the g. c. d. of 6 and 9. Art, 81, To FIND THE GREATEST COMMON DIVISOR OF ANY NUMBERS. This may be often done by inspection ; thus the g. c. d. of 18 and 27 is evidently 9. Or the greatest common divisor may be found by trial. , Ex. 1. What is the g. c. d. of 96, 144, 1728. Process. We find by trial that 12 is a common 12)96 144 1728 divisor of the given numbers ; also that 4 is a 4^3 J2 144 common divisor of the quotients, and that there is no other common divisor. Since dividing by 12 and 4 is the same as dividing by 48, the greatest common divisor is 48- RULE. Divide the given numbers by any number that will divide each of them without a remainder, and the quo- tients in the same manner, till the last common divisor is found ; the product of the several common divisors will be the greatest common divisor. The products of the prime factors common to all the given numbers will also be the g. c. d. The common method of finding the g. c. d., has been to divide the greater of two numbers by the less, and the last divisor by the last remainder, till nothing remains. The last divisor is the g. c. d. Pro- ceed in the same manner with the divisor thus found and another number. Ex. 2. What is the greatest common divisor of 48, 114,. 132? Process. Any divisor of 48 is a divisor of 48 ^Q^ 2 48X2=96. Therefore any divisor of 48 and 114 is a divisor of 114 and 96, also of their 18)48(2 difference, 18. For the same reason it is a 36 divisor of 12, the next remainder, and 6, the ~~12)18(1 divisor of 12, is the g. c. d. Proceed in the 12 same way with 6 and 132. "6)12(2 6)132 12 22 LEAST COMMON MULTIPLE. 119 This method depends on the principle, tii.it the divisor of any number is the divisor of any product of it, and (he divisor of any two numbers is also the divisor of their sum or difference. EXAMPLES. What is the greatest common divisor of (3.) 85 105 (7.) 72 84 (11.) 12 18 24 (40 90 152 (8.) 84 96 (12.) 18 32 40 (5.) 100 150 (9.) 98 112 (13.) 44 66 88 (6.) 108 114 (10.) 100 120 (14.) 81 90 117 The Least Common Multiple. Art* 82. The multiple of any number is the product of that number multiplied by another number ; as 15 i& the multiple of 5 by 3 or 3 by 5. The Common Multiple of two or more numbers is the same or common product of each multiplied by other numbers ; as, 36 is the common multiple of 12 by 3 ; 9 by 4 ; 6 by 6 ; 4 by 9 ; 3 by 12. The Least Common Multiple (1. c. in. ) of two or more numbers, is the least common product of each by other numbers ; as, 24 is the 1. c. m. of 12 (by 2) ; 8 (by 3; ; 6 (by 4). Art. 83. To FIND THE LEAST COMMON MULTIPLE OF Two OR MORE NUMBERS. Ex. 1. What is the 1. c. m. of 4, 6, 12. Process. By inspection we find that 12 is a common multiple of 4 (by 3 , and 6 (by 2), and there can be no less multiple of itself ' r therefore it is the least common multiple of the given numbers. RULE I. When the largest given number is a common multiple of the others, it is the least common multiple. Ex. 2. What is the least common multiple of 6, 8, 12 ? Process.- -By inspection we find that 12X2 will be a common multiple of 6 (X4) and 8 (X3). Therefore it is the 1. c. m. RULE H. When the largest number is not already a com- mon multiple of the othei*s, multiply it by the least. number 120 LEAST COMMON MULTIPLE. thai ivill evidently make it such, and the product will be the least common multiple. If it is not known by what number the largest given number must be multiplied to make it a common multiple of the others, the 1. c. m. may be found by the following method. The common multiple of 6, 8, 12 (Ex. 2) is 6X8X12= ) 8X6X12= f 576. But 6, a common factor in two of the numbers 12X6X M (6 and 12) , and 4, a common factor in 8 and 12, are common factors in the multipliers of the given numbers, and therefore may be can- 6X2X2- ) celled, leaving 8X1X3= Y 24 the 1. c. m. 12X1X2= ) Since each product is the same, it is not necessary to repeat the process, and it makes no difference whether the common factors be cancelled in the multipliers of the given numbers, or in the numbers themselves ; hence 2o? Process. Cancel the 6 because it is a factor ( in 12, and 4 the greatest factor in 8, because it is J $ 12 also a factor in 12. Then multiply the given num- ( 2X12=24 Ans. ber, 12, by the factor 2, and the product, 24, is the least common multiple. RULE IH. Cancel any number or greatest factor that is also a factor in another number, and the product of the re- maining numbers and factors will be the least common multiple. Sometimes two factors in the same number may be omitted when they are also factors in different numbers. If there is no common factor, the product of the given numbers will be the least common multiple. The common method has been the following : 2)6 8 12 2)3 4 6 3)3 2 3 TT1 Ans. RULE IV. Divide the numbers by any prime number that will divide two or more of them without a remainder ; place the quotients and undivided numbers in another line ; divide these also in tfie same manner, and continue the process till no two THE GREATEST COMMON DIVISOR. 121 numbers can be thus divided. Then multiply together all the divisors and undivided numbers. EXAMPLES. What is the 1. c. m. of (3.) 4, 8, 9? (4.) 6, 9, 12 ? (5.) 9, 15, 18 ? (6.) 10, 20, 30? (7.) 5, 10, 12, 24? (8.) 6, 12, 14, 28? (9.) 8, 14, 21, 30? (10.) 9, 11, 22, 27? Art. 84, Promiscuous Examples in Properties of Numbers. 1. What are the prime factors in 24 ? 35 ? 196 ? 2. What is the g. c. d. of 9, 18, 24, 30 ? 3. What is the 1. c. m. of 8, 12, 20, 32 ? 4. What are the prime factors in 14 ? 26 ? 34 ? 5. What is the g. c. d. of 21, 28, 35, 70 ? 6. What is the 1. c. m. of 9, 11, 18, 22 ? 7. There are three rooms respectively, 12, 18 and 24 feet wide ; how wide may be the widest oil-cloth that will exactly fit them all without being cut ? 8. There are three horses running a circuit ; the first can complete it in 10 minutes, the second in 12, and the third in 15 ; if they start together and keep running, how long will it be before they will be together again ? 9. There is a garden the sides of which are respectively, 112, 126, 140 and 154 feet ^what must be the length of the longest boards that will fence it without being cut ? 10. Three ferry boats, starting together, are making regular trips to different points and back ; the first in 20, the second in 25, and the third in 30 minutes ; how long -will it be before they will all return together ? FRACTIONS. Art. 85. A Fraction is one or more equal parts of a number or thing, and expresses division. 122 FRACTIONS If a number or thing is divided into two equal parts, one of them is called one-half ; if divided into three equal parts one of them is one-third, two of them two- thirds ; if divided into four equal parts, one of them is called one-fourth, or one-quarter. If the number thus divided is not expressed, it is un- derstood to be one, or a single thing ; thus in the line AE, A o is one-half, A B one-quarter ; A D three-quarters ; A F one-third ; A a two-thirds of the line AE. E Art. 86. Fractions are of two kinds, Common and Decimal. Common fractions are such as express any number of equal parts ; as, one-hall, two-thirds. Decimal fractions are such as express only one or more of ten equal parts, or ten times ten, &o.; as, one-tenth,, three-hundredths, five-thousandths, &c. Common Fractions. Art. 87. Common fractions are usually written with two numbers, one the above other, with a line between them ; as, J, one-half ; f , two-thirds. The number above the line is called the Numerator, and the number below the line the Denominator $ both together are called the Terms of a fraction. The Denominator shows the number of equal parts in- to which anything is divided, and corresponds with the divisor in division ; also gives name to the fraction. FRACTIONS. 123 The Numerator shows how many of the equal parts are taken, or it may be a number divided by the denomina- tor, and the same as the dividend in division. . In the fraction f, the denominator (3) shows that something is considered as divided into three equal parts, and the numerator (2) shows that there are two such parts, or two things thus divided. The traction expresses either two-thirds of one, or one-third of two ; as, one-third of two dollars, or two-thirds of one dollar. Artt 88, Common fractions are usually divided into Simple, (either proper or improper,) Compound, Complex, and Mixed Numbers. A Simple fraction has but one numerator and one de- nominator, both whole numbers. It is called a Proper fraction when its numerator is less than its denominator, and Improper when its numerator is equal to or greater than its denominator ; as, 1, (proper) f , or f- , (improper. ) A Compound fraction is a fraction of a fraction ; as fof f. A Complex fraction is one whose numerator or denomi- f 3 nator is a fraction or mixed number ; as, |y, ^i- A Mixed Number is a whole number and fraction written together ; as 3f . It will be found convenient, also, to divide common fractions into two other kinds, Like and Unlike. Like fractions are such as have the same name or com- mon denominator ; Unlike, such as have different denom- inators ; as, i, f , I are Like fractions ; f, , f- are Unlike. Art. 89. The Value of a fraction is the quotient of the numerator divided by the denominator ; as, the value of f is 2 ; of f is 1. The value of a proper fraction is less than 1, and therefore can only be expressed in the form of a fraction ; as f . 124 FRACTIONS. The value of the equal parts of a fraction depends on the denominator, or the number of parts. The less the denominator the greater the value of each part ; the greater the denominator the less the value of each part ; thus i is more than i. f than I. Art* 90 Since the numerator of a fraction corres- ponds with the dividend in division, the denominator with the divisor, and the value with the quotient, the follow- ing propositions are of frequent application in fractions. I. Multiplying the numerator or dividing the denominator by any number, multiplies the value of the fraction by that number ; thus ; 6X4 ==8 (2X4). == II. Dividing the numerator or multiplying the denominator by any number, divides the value of the fraction by that num- ber ; thus 2 == 2 HE. Multiplying or dividing both numerator and deno- minator by the same number does not alter the value of a fraction; thus EXEBCISES. Read, name the kind, and explain the terms, of the follow- ing fractions : K t, f , f , i- of f , 3i, f , f , |, I , JgS I, 6f , If, f of ?. 4 Of Also, mention some of the same that are like, and others that are unlike. Write and describe the following fractions : Three-fifths, seven-ninths, nine-fourths, eleven-quarters, two-thirds of four-ninths, nine and a half, five-sevenths, sev- FRACTIONS. 125 en-thirds, four-fifths of seven-eighths, six and three-quarters, 3-sevenths, 5-ninths, 8-etevenths, 10- twelfths. Bead the following in order according to the value of their equal parts, beginning with the greatest. Kepeat the same, beginning with the least : t, A, i, i A. i, -/o, I, H, I, A, f, i A, Write a fraction of each kind repeatedly till it can be done correctly. MENTAL EXERCISES. If an apple or anything is divided into two equal parts, what is each part called ? If divided into three equal parts ? 4? 5? 6? 7? 8? 9? 10? 11? 12? 13? 15? 18? 20? 25? 50? 65? 84? 100? If an orange or anything is divided into 12 equal parts, what is one of them called ? 3 of them ? 6? 9? 4? 7? 10? 12? Which is the greater, -^ or ? Why ? f or f ? | or or? or? or? or Art. 91. REDUCTION or FRACTIONS. Reduction of fractions is changing their form without altering their value ; as, |=|, t=|, f=2|. Fractions are reduced to their simplest form when they are reduced to simple and proper fractions in their lowest terms, whole or mixed numbers ; thus the simplest form of |f is i ; of f is 2 ; of | is 1| ; f of Hs &. CASE I. Artt 92. To reduce a fraction to its lowest terms or simplest form. A fraction is in its lowest terms, when its value is expressed by the least numbers possible ; thus -& reduced to its lowest terms is ." EXAMPLE 1. Reduce if to its lowest terms. The fraction |-| can be reduced to lower terms because both its numerator and denominator can be divided by 2 and 3, or 6, which (Prop. HI. Art. 90) does not alter its value. 126 FARCTIONS. Process. Dividing both terms by 3, the result is 3)18_2)6_3 f, which may also be divided by 2, the result being -54 07 I, the lowest terms. Or dividing f by 6 the result is the same. or 6)_18 = 3 24 4 RULE. Divide the numerator and denominator by any number that will divide them both without a remainder, and continue dividing till the lowest terms are found. Or divide both terms by their greatest common divisor. EXAMPLES. (2.) (3.) (5.) M if (6.) (7.) (8.) (90 M M if (ii.) (12.) Hi m (15.) (160 (17.) tff CASE H. Art. 93. To reduce an improper fraction to a whole or mixed number, or its simplest form. MENTAL EXEBCISES. In 2 half dollars or 2 halves ( ) of one dollar, how many dollars? Inf? |? f? f? f? tf? f? |? f? f? f? ? V? |? |? |? |? V? f? f? f? ^? EXAMPLES FOR THE SLATE. EXAMPLE 18. Reduce ^- to a whole number. Process. Since the denominator (3) shows the number of 27 equal parts in 1, and the numerator (27) how many such 3 parts there are, the fraction ^ is as many times 1 as 3 is contained times in 27, which are 9 times. Therefore a whole number. 2. Reduce Process. 18 to a mixed number. _ ~~3 a mixed number. RULE. Divide the numerator by the denominator, and the quotient will be the whole or mixed number. FRACTIONS. 127 (19.) Jj (20.) \ (21.) \ 34. In 35. In 36. In 37. In EXAMPLES. (22.) J> I (25.) ag. (28.) (23.) * (26.) -^ (29.) (24.) ^ I (27.) 3 (30.) of a mile, how many miles ? of a pound how many pounds ? of a yard how many yards ? of a bushel how many bushels ? CASE HI. (31.) (32.) (33.) Art* 94. To reduce whole or mixed numbers to improper fractions. MENTAL EXERCISES. In 1 apple or anything how many halves ? [Let the answer be first oral, then written.] How many thirds ? fourths ? quarters ? fifths ? &c. In 3 apples ? 5 ? 7 ? 8 ? &c. In \y z apples how many halves? 2^? 4^? 6^? 7}? In IK apples how many thirds ? 2%? 3^? 5%? 7^? UK? 12%? In 1% apples how many quarters ? 2^? 3%? 5^? 5 33 -=- EXAMPLES FOE THE SLATE. 38. Beduce 64 to an improper fraction. Process. -Since 1= 6=6 times f=AA to which add f and the sum will be ^ u4ns. Or, Since there are f in 1, in 6 there are 5 times as many fifths as there are times 1, and 5 times 6 are 30. Therefore 6=^- to which add f, and the sum will be a 5 a , Ans. EXAMPLE 39. Beduce 8 to an improper fraction. Process. Since 1=1, 8=8 times |=f. RULE. To reduce a mixed number to an improper fraction, multiply the whole number by the denominator of the frac- tion, add the numerator, and write the denominator under the sum. To reduce a whole number to an improper fraction, write 1 under it with a line between them. 128 FRACTIONS. EXAMPLES. (40.) 6J (41.) 7| (42.) 8} (43.) 9f (44.) 10 (45.) Ill (46.) 12f (47.) 15f (48.) 18| (49.) 25 (50.) 30^ (51.) 52. In 3 pounds, how many fifths of a pound ? 53. In 11| hours, how many thirds of an hour ? 54. In 12 rods, how many ninths of a rod ? CASE IV. Art. 95. To reduce unlike fractions (having different de- nominators,) to like fractions (having a common denom- inator. ) EXAMPLE 55. James has % of an orange, John %, Henry %; how can these unequal parts be divided into equal parts ? Since the denominator of a fraction shows the number of equal parts into which anything is divided, fractions having different de- nominators must be reduced to a common denominator, that all the parts may be equal. Process. To reduce f f to a common denominator, we multiply the numerator and denominator of each fraction by all the other de- nominators which (Prop. III. Art. 90) does not alter their value ; thus The fractions , f, f-, are thus reduced to equivalent fractions f , f , f , having a common denominator. Therefore if , f , f , of an orange be divided respectively into 12, 16, 18 equal parts all the parts will be equal, and each part will be -^ of an orange. RULE I. Multiply the numerator and denominator of each fraction by all the other denominators. Since the denominators thus found will be the same, after one is found it may be taken for the common denominator. EXAMPLES. (56.) iff (59.) fff (62.) J f f (57.) if? (60.) $f (63.) f J A (58.) Ill (61.) I | A (64.) | f f Art. 96. To reduce unlike fractions to equivalent like frac- tions having the least common denominator. FRACTIONS. 129 The fractions , f, (Ex. 55, ) reduced to a common denominator, are f , f , if , but these may be reduced to lower terms, so that the common denominator will be less, as &, f 2 , & . 12 is the least common denominator, because the fractions cannot be reduced to lower terms and still have a common denominator. To find the least common denominator of , f , f , we multiply the numerator and denominator of the fraction by 6, f by 4, by 3, because we observe that the fractions thus multiplied will have a common denominator, the least that can be found, and become , d 2 , A, A, Ans. RULE n. Multiply the numerator and denominator of each fraction by the least number that mil produce a common de- nominator. If such a number cannot be conveniently found by in- spection, find the least common multiple of the denomin- ators, and divide it by the denominator of each fraction. The least common multiple will be the least common de- nominator, therefore the denominators need not be multiplied. Before reducing fractions to a common denominator reduce them to their lowest terms. EXAMPLES. (65.) f | (68.) | i H (71.) f f i (66.) iff (69.) i I A (72.) f A (67.) iff (70.) f f | (73.) iff When there are no two denominators of the fractions to be reduced, which can be divided by the same number, the first rule for finding the common denominator must be used. EXAMPLES UNDER BOTH BULES. (74.) Vi * (77.) I | M (80.) | | (75.) f | f (78.) I | i| (81.) A & & (76.) 1 f | (79.) i ^ A (82.) | ^ | The object of reducing fractions to a common denominator, is to prepare them for addition and subtraction. Art, 97. Reduction of Compound Fractions is prop- erly included in Multiplication of Fractions, and Reduc- tion of Complex Fractions in Division of Fractions. 130 FRACTIONS. Art, 98. Promiscuous Examples in Reduction of Fractions. 1. Beduce to its lowest terms, or simplest form. 2. Reduce ^ to a mixed number, or its simplest form. 3. Eeduce 9 to an improper fraction. 4. Reduce ^, it, ^, to like fractions or a common denomin- ator. 5. Reduce ^^ to its simplest form. 6. Reduce 12| to an improper fraction. 7. Reduce ^ to its simplest form. 8. Reduce f , f , -|, 4 to like fractions. 9. Reduce \ 8 to a mixed number. 10. Reduce ff to its lowest terms. 11. Reduce 13f to an improper fraction. 12. Reduce f , -&, f , 3 to like fractions. 13. Reduce f f to its simplest form. 14. Reduce ^ to its simplest form. 15. Reduce 14^ to an improper fraction. 16. Reduce f , T %, f , 6 to like fractions. 17. Reduce ff to its lowest terms. 18. Reduce f f to a whole number. 19. Reduce 15^ T to an improper fraction. 20. Reduce f , f , f , 5 to like fractions. 21. Reduce f , $, ^, f to like fractions. 22. Reduce 15f , to an improper fraction. 23. Reduce ^ to its simplest form. 24. Reduce ^| to its simplest form. 25. Reduce f , f , f , 2 to like fractions. 26. Reduce f , f , f , 1 to like fractions. 27. Reduce 3 / to a mixed number. 28. Reduce 9f to an improper fraction. 29. Reduce**^ to a whole number. 30. Reduce , f , -$y, 7 to like fractions. 31. Reduce f to its lowest terms. 32. Reduce , ^, f , 3 to like fractions. ADDITION OF FRACTIONS. 131 33. Eeduce 16^ to an improper fraction. 34. Keduce ^fi to a mixed number. 35. Reduce f , f , f , -^ to like fractions. 36. Reduce f , f , ^ 4 to like fractions. 37. Reduce f|f| to its lowest terms. 38. Reduce *f to its simplest form. 39. Reduce f , f , fj, 3 to like fractions. 40. Reduce f, 1 1, -^ to like fractions. 41. Reduce |, &, - , to like fractions. Addition of Fractions. Art. 99. Addition of Fractions is finding their sum. EXAMPLE 1. Add , f, f , f . Since , f, H=IH. ^S=i ^ns. 24~~12 RULE. Reduce the fractions to like fractions or a common ^Denominator, add their numerators, and under the sum write the common denominator. 1. Whole and mixed numbers may be reduced to improper fractions and then added like other fractions : but it better to add the fractional parts separately, and the whole numbers to the result. 2. After adding, reduce the sum to its simplest form. 132 SUBTRACTION OF FRACTIONS. EXAMPLES. (3.) Add} J (10.) (5.; (9.) Add f (12.) 3^ f 2} (13.) " 2| 31 f 1 (14.) " & i A 3 (7.) " f 2| f (80 " f f J 15. How many dollars will pay for a coat worth $12}, pants $9f , vest $41, hat $5}, and a pair of boots $6 ? 16. How many yards in five pieces of cloth measuring 27^, 25f , 261, -241, 22$ ? 17. How many pounds of tea in six packages weighing 6, 54, 3, 4f , 21, i ? pound ? 18. How many pounds of butter in four tubs weighing 18|, 16}, 20ft, 19f ? 19. How many hundred weight of sugar in five barrels weighing 2}, l^fc, 2f, 1}, 2f cwt? 20. How many tons in four loads of coal weighing 1^, 1|, *M, l^W tons ? Subtraction of Fractions. Art. 100, Subtraction of Fractions, is finding the difference between two fractions or a fraction and a whole number. Ex. 1. From f take f . Since f and f are like fractions, they can be subtracted by taking the less numerator from the greater, the same as taking 3 cents from 4 cents. Process. f f=} Ans. Ex. 2. From f take }. Since f and $ are unlike fractions, they cannot be subtracted as they are, any more than 1 cent can be taken from 3 dollars and leave 2 dol- lars or 2 cents. Process. - Eeduce f, to like fractions j a 2 , ^, and substract ^ from' fa. The remainder, &, is the answer. SUBTRACTION OF FRACTIONS. 133 RULE. Reduce the fractions to like fractions or a common denominator, subtract the numerators, and under the remain- der write the common denominator. 1. After subtraction reduce the remainder to its simplest form. 2 . Whole or mixed numbers may be reduced to improper fractions and then subtracted, but it is better to subtract the fractional parts separately, thus Ex. 3. From 4 subtract f . Process. Since there is no fraction from which to take f, 4 take it from 1 (a part of the 4)=. f=f and since 1 has been used in the operation, carry as in subtraction of whole g? numbers. Ex. 4. From 8^ subtract 3f . Process. Reduce ^ and f to like fractions, -fa, , and since 8 cannot be subtracted from-/ 4 -, addl (a part of the 8)=fto 3 5 \, and from their sum | f take , and the remainder will be ^\, "TT carry 1, &c. EXAMPLES. (5.) From f subtract f (6.) " | " | (7.) " * " f (8.) M " f (9.) " M " A (10.) " fts " f (11.) From 4| subtract 2 (12.) " 5f " 31 (13.) " lOf " 3 T (14.) " 9^ " 4f (15.) "14 " Cf (16.) " 18f " 9^ 17. If a man have $6, and spend $3f , how many dollars will lie have left ? 18. If a piece of cloth contain 27 yds., and Of yds. be cut off, how many yards will be left ? 19. If I buy 25 Ibs. of sugar and use 12-j 6 ^ Ibs., how much will be left ? 20. If a man on a journey of 65^ miles, has traveled 48f miles, how many more miles has he to travel ? 21. If there are 37 gals, of wine in a cask, how many will be left after 15 gals, shall have been drawn from it ? 22. If a boy study 5| hours and play 2 hours, how many more will he study than play ? 134 MULTIPLICATION OF FRACTIONS. Multiplication of Fractions. CASE I. Art. 101 1 Multiplying a fraction by a fraction. Multiplying any number by is finding of it, which is the same as dividing it by 2 ; multiplying by % is finding ^ of it, the same as dividing it by 3, and of a number multiplied by 2, gives f of it, &c. EXAMPLE 1. Multiply f by f . Process. - 5 _X- =- Therefore *LX- = ^=- , 8 4 32 8 4 32 32 !LyJ? = 15 (Prop. I. II. Art 90.) 84 32 Ex. 2. Multiply f by ^. ^01 Process. Cancel 4 and 9, factors common to the nu- X = ~7 merators and denominators, before multiplying. 9 ^9 4 4 CASE n. Art. 102. Multiplying a fraction by a whole number, or a whole number by a fraction. Ex. 3. Multiply % by 3. Process. 3 times =V or f (P. I. Art. 90)=2. Ex. 4. Multiply 3 by . Process. 3X^=1 or | ; therefore 3X1= V or =2. The product of |X^ is the same as that of 3X$, and since 3=f this case may be included in Case I. CASE HI. Art. 103. Multiplying mixed numbers. Ex. 5. Multiply 4 by 3|. 1st Process. 4|X3* 2d Process. = 4X3= 12 The products added are 17|^. Ans. By the first process the mixed numbers are reduced REDUCTION OF COMPOUND FRACTIONS. 135 to improper fractions and then multiplied as in Case I. ; by the second process, the whole numbers and fractions are multiplied as in Case EL KULE. Reduce whole or mixed numbers to improper frac- tions, cancel all factors common to the numerators and the de- nominators, then multiply the remaining numerators together for a new numerator and the remaining denominators for a new denominator. A whole number may be multiplied as the numerator of an im- proper fraction without being reduced. Art, 104, Reduction of Compound Fractions is pro- perly included in multiplication of fractions : f of f=fxfHr ; f of f of 2jc=fxiX=H=i&- MENTAL EXERCISES. If an apple cost ^ cent, what will ^ of an apple cost ? ? ? Process. If an apple cost which is | cent. cent, i of an apple will i of ^ cent, At cent each, what will 2 apples cost ?4?5?8?11?12? If an orange cost 4 cents, what will ^ of an orange cost ? f ? f ? f? ? What will 3 plums cost at J cent each? 4? 5? 6? 8? 11? 12 ? If a melon cost 6 cents, what will ^ a melon cost ? f ? ? i? f ? A? If a peach cost f of a cent, what will of it cost ? ? ? f? f? I? EXAMPLES FOB THE SLATE, ETC. Multiply (13.) 6 by | (6.) 4 by f (7.) | by 3 (8.) fby | (9.) 31 by 4 (10.) 5 by2f (11.) 6| by 3 (12.)iof3by2J {14.) (15.) (16.) (17.) (18.) (19.) 4|by5 6 by 7 8^by5| 7byfof6 (20.) (21.) 8 by f i by 16 (22.) A b J A (23.) 6| by 3 (24.) (25.) 7 by4f 9f by 6f 136 DIVISION OF FRACTIONS. 27. At $| a bushel, what will 6 bushels of potatoes cost ? 13? 27? 28. At $5 a bushel, what will -| bushel of clover seed cost ? f? f? 29. At $f a bushel, what | of a bushel of corn cost ? f ? ? 30. At $4 a bushel, what will 2 bushels of timothy seed cost? 3? 5? 31. At $4 a bushel, what will 3f bushels of timothy seed cost? 5? 6f? 32. At $4 a bushel, what will 2| bushels of timothy seed cost? 4? 6f ? 33. At $6 a bushel, what will f of 3| bushels of clover seed cost? Division of Fractions, CASE I. Art. 105. Dividing a fraction by a whole number. Ex. 1. Divide f by 3. Process. H-3=* or ft (P. II. Art. 90.) RULE. Divide the numerator by the whole number when it can be done without a remainder, otherwise multiply the de- nominator by the whole number. A mixed number may be reduced to an improper fraction and then divided ; or the integral part may be divided first, and the remainder afterwards ; thus 9^-4-4=^=^fi or 2 and a remainder 1 which di- vided by 4 equals f-^-4=f . (2.) Divide by 2 (3.) Divide f by 3 (4) Divide ^ by 4 EXAMPLES. (5.) A by 5 (6.) if by 6 (7.) V by 100 (8.) if by 7 (9.) ^ by 8 (10.) f by 8 FRACTIONS. 137 CASE II. Art. 106. Dividing by a fraction. Ex. 11. Divide f by f . Process. -|-^1=, and is contained in 5 2 5 $ 5 any number 3 times more than 1 is, there- TT-T-Q -\/ -_ i~=\ \ fore (U-=3 times f -^ , but is con- tained in a number ^ as many times as ^, therefore, -^-f=-5-xt If the same as multiplying f by the divisor inverted. Ex. 12. Divide 6 by . or 6X1, the divisor Process. 6=$, therefore, 6-^f=f-^f= inverted. Mixed numbers may be reduced to improper fractious and then divided. KULE. Multiply the dividend by the divisor inverted. Another method is to reduce the fractions to a common denomina- tor and divide their numerators. EXAMPLES. Divide (13.) f by f (14.) f by f (15.) 6 by | (16.) 4fby | (17.) 7 by 21 (18.) }iby3f (19.) 5fby2f (20.) f by A (21.) 16 by f (22.) ! by (23.) 100 by (24) f| by (25.) by (26.) 12 by (27.) * by (28.) f by (29.) 7| by $ (30.) 16 by 51 (31.) f off by* (32.) 31 by | of |f (33.) | of 5J by If of 8 (34.) 6fby2f MENTAL EXERCISES. If f of a yard of silk cost &, what is the price per yard ? of S=$A and 1 yd. Process. If f yd. cost S will cost 4 times ,/ 4 -:=f f=$ yd. will cost Ans. If 3 yards of silk cost $2^, what is the price per yard ? If 4 yards of silk cost $3f what is the price per yard ? 138 DIVISION OF FRACTIONS. If f of a yard of silk cost $2, what is the price per yard ? At f of a cent each, how many pears can be bought for 12 cents ? At $| a bushel, how many bushels of oats can be bought for $| ? $14 ? #3} ? What part of $1 will 1 quart of nuts cost if 5 qts. cost $ ? What will a pound of candy cost, if f of a pound cost 8 cents ? What will a pound of candy cost if 4| Ibs. cost $ ? EXAMPLES FOE THE SLATE. 35. If 31 yds. of silk cost $6, what is the price per yard ? 36. If 3f Ibs. of tea cost $2^-, what is the price per pound ? 37. What will a yard of muslin cost if 26f yards cost |10 ? 38. What costs a yard of cloth if f of a yard cost $2f ? 39. If 3 yards of cloth cost $15f , what is it a yard ? 40. If 3 yards of cloth cost $15 : what is it a yard ? 41. If 9 quarts of cherries cost $J, what part of $1 will 1 quart cost ? 42. What costs 1 bushel of corn if ^ of a bushel costs $ ? 43. If 12 bushels of wheat cost $31, what is the price per bushel ? 44. At $lf a bushel, how many bushels of wheat can be bought for $26 ? 45. If 111 ibs. of rice cost $1|, what is the price per pound ? 46. At 10| cts. a pound how many pounds of rice can be bought for 87-| cents ? 47. At $f a- pound, how much butter will $6 purchase ? 48. If 22 Ibs. of butter cost $7^, what is the price per pound ? 49. At 16 cts. a pound, how many pounds of sugar can be bought for $2. 33^? Art, 107, Reduction of Complex Fractions to simple fractions is properly included in division of fractions, the numerators being the same as dividends and the denomi- nators the same as divisors, PROMISCUOUS EXAMPLES. 139 EXAMPLES. 3 50. Keduce JL to a simple fraction. i 3.5 3 V 2 -J-H -.Xj- 2 51 Eeduce 1 * 3- ? . I 7' li' . 8 2 53. From -f subtract f . From | subtract i 4 i 4 I H 54. Multiply I by}; | by || by |. 55. Divide | by 1; | by } ; | by i. Promiscuous Examples in Common Fractions. EXEECISE I. 1. Add f , ft, i f ; 2J, f , f of |, 6J ; i of 2fc f , 3, 4|. 2. From | subtract f ; 9g 1^ ; 60 45f ; 16| | of 7^. 3. Multiply ft by 4 ; 3|Xf 5 5XA ; I of 6f X5f ; 4. Divide 14 by 7 ; 8^-f ; 10 J-^5i J I of f-h| of 3^. 5. Add |, ^, f, | ; of 9f, f, 3, 4 ; 5^, ^ $ of |, 6. 6. From ^ subtract | ; 7^ ^ ; | of 16 2^4, ; 120 96J. 7. Mul. 11 by ^ ; 7><8f. ; f of ^Vf of 3. 8. Div. 12 by f ; ^^4 ; 16f -^ of f ; 22f -f-7. 9. Add f , A, 11, ^ ; 4i, 21, 3f , i of f ; | of 18^-, 7*, 8. 10. From T % subtract f ; 1261 ; f of f of f . 11. Multiply i^ by f ; 6f X4 ; 14Xf ; |- of 2Xf of 6f . 12. Divide by | ; 24^-3^ ; 9|^3 ; f of 140 FRACTIONS. 13. Add H, &,*,#; 6f , 4, 5f, f of 5i ; f of f , f of 12, J of 6J. 14. From ^ subtract f ; 14f 4 ; f of 4 of &. 15. Multiply ,& by H ; 8f X6 ; 12Xf ; f of HXf of T \ of 4|. 16. Divide f by f4 ; 38-^-9 ; 24f ^-6 ; 15 4i of lOf . 17. Add |, f, if, ; 8J, 16, 10, ^ ; f of ft, f of 6f ,10f. 18. From Jf subtract t \; 20 Sf ; 11-L 6^; }$ J of f . 19. Multiply f by ^ ; 15X/o 5 tf X? ; $ of |f X3J. 20. Divide A by f ; 16-fi ; f ^6 ; 17^- of 13J. EXEBCISE n. 21. How many yards , |, |, and | of a yard ? 22. From 8| subtract 2. 23. What will 11 apples cost, at ^ of a cent each ? 24. How many pounds of sugar at 7^ cents a pound, can be bought for 48| cents ? 25. At 5 cents a mile how much will it cost to travel 20 miles ? 26. At 7f cts. a qt., how many quarts of cherries can be bought for 76 cents ? 27. Five pieces of muslin contain the following numbers of yards, 18f , 15^, 14, 16|, and f of 16 ; how many yards in all the pieces ? 28. A piece of cloth measured 22 yds., of which f remain ; how many yards would be left if 5f yards more should be cut off? 29. At $1 a head, how many sheep can be bought for $56f ? 30. What will 12^ yds. of ribbon cost at 6 cts. a yard ? EXERCISE ni. 31. How many are 6, of f , 7, and 3 ? 32. From 7^ subtract f of }f 33. At $11 a bushel how much wheat can be bought for $50 ? 34. At $1 a bushel, how much cost 72 bushels of wheat ? 35. At ISf cts. a pound, how much cost 18} Ibs. of butter ? 36. At 22^ cts. a pound, how many pounds of butter can be bought for $5.37$ ? PROMISCUOUS EXAMPLES. 141 37. Four cheese weigh as follows : 25|, 26, 24f, and f of 254 pounds ; how much do all of them weigh ? 38. From a piece of cloth which contained 27f yards have been cut off at different times. 3, 4, 5f , and % yds. ; how many yards are left ? 39. At 621 c ts. a pound, how much will 7 Ibs. of tea cost ? 40. At G2-| cts. a pound, how much tea can be bought for $56^? 41. How many pounds are 3, 2^, and f pounds ? 42. From 7^ yards of ribbon 2| yards have been cut off, how many are left ? 43. How much will 9| Ibs. of sugar cost at 7^ cts. a pound ? 44. At 1^ cts. a pound how much meal can be bought for 13^ cts.? 45. At 15 cents a yard, how much cost 37f yards of muslin ? 46. At 62^ cts. a bushel, how many bushels of corn can be bought for $7.811 ? 47. Four pieces of calico measure as follows : 22|, 20, 21^, and f of 25 yds. , how many yards in all of them ? 48. A piece of cassimere contained 16f yards, of which ^ re- main ; if 3^ yards more should be cut off, how many would be left ? 49. At $6f a cord, how much will 8f cords of wood cost ? 50. At 74^ cts. a bushel, how many bushels of potatoes can be bought for $4.841 ? 51. How many are 6, of f , 1, f of f , 2| ? 52. From 3| yards of linen 2^ yards have been but off, how many remain ? 53. At $1^ a day what will a man receive for labor in 26 days? 54. At 9| cts. a pound, how many pounds of rice can be bought for 53f cts. ? 55. At $f a bushel, how much will 10 bushels of corn cost ? 56. At $36 an acre, how much will $ of an acre of land cost ? 142 FRACTIONS. 57. How many yards f of a yard wide will line 25 yards of cloth I yard wide ? 58. From a hogshead of molasses 7f gal. have deen drawn, how many are left ? 59. At 5 cents a pound, how much will 9$ pounds of fish cost? 60. At 9^ cts. a pound, how much cheese can be bought for $1.42$? EXEKCISE VI. 61. How many are 5, of -fa, of f , and 3? 62. From 4 take -j^. 63. At 3 cts. a yard, how much will 100 yds. of tape cost ? 64. At f of a dollar a yard, how many yards of flannel can be bought for |2ii? 65. 91 cts. a yard, what will 15 yds. of muslin cost ? 66. If 7| pounds of tea cost $4&, what is the price per pound ? 67. Six pieces of ribbon measure as follows : 9, 8^, lOf , 7, 5 yds. ; how many yards in all the pieces ? 68. From a piece of linen containing 11^ yds. have been cut off at different times, f, , 1 of \ yds., how many are left ? 69. At 121 c ts. a dozen, what cost 7| dozen eggs ? 70. If 13f bushels of apples cost $5.17|, what is the price of a bushel ? EXEECISE VH. 71. How many bushels are 5, 3J, 7 of -&, and f of f bush- els ? 72. From 7} take f of f . 73. At 16| cts. a yard, what cost lOf yards of calico ? 74. Paid $2.53 for 6f bushels of turnips, what is the price per bushel ? 75. At 7 cts. a pound, what cost 75^ Ibs. of sugar ? 76. At $12f a ton, how many tons of coal can be bought for $47ff? 77. Four pieces of tape measure as follows : 23 J, 22^, 24,. and f of 24 yds. ; how many yards in all of them ? PROMISCUOUS EXAMPLES. 143 78. From a cheese weighing 26 pounds have been sold to different persons 5^, 4, 3f , and 6 pounds ; how much is left ? 79. At 9| cts. a pound, what cost 14| Ibs. of codfish? 80. Paid $4.41it for 18 yds. of muslin ; what was the price i>c-r yard ? EXEBCISE VIII. 81. At $9i a barrel, how much cost 7 bbls. of flour ? 82. At $9 a barrel, how many barrels of flour can be bought 83. If 11 barrels of flour cost $100|, what is the price per barrel ? 84. How many pounds of sugar are f , ^, 3, and 4 Ibs. ? 85. From 13 subtract 9f 86. At 5f cts. a pound, what cost 76f Ibs. of iron ? 87. At 5f cts. a pound, how many pounds of iron can be bought for $455f ? 88. If 72^ Ibs. of iron cost $4.62-|, what is the price per pound ? EXEBCISE IX. 89. At 12^ cts. a pound, how much lard can be bought for $5.311? 90. At 12 cts. a pound, how much will 16lbs. of lard cost? 91. If 18f Ibs. of lard cost $2.061 cts., what is the price per pound ? 92. A boy paid 58^ cts. for an arithmetic, 18f cts. for a slate, 9 cents for a sponge, 4 cts. for a lead pencil, and 12^ cts. for paper, what did he pay for all of them ? 93. A girl bought a handkerchief for 33^ cts. and gave the merchant 37^ cents ; how much change should she have re- ceived ? 94. If one person consume 8| Ibs. of beef in a week, how many persons would consume 5874 Ibs. in the same time ? 95. If one person consume 8 Ibs. of beef in a week, how much would a family of 9 consume in the same time ? 96. If a family of 9 persons consume 73f Ibs. of beef in a week, how many pounds will one person consume in the same time? 144 FRACTIONS. EXERCISE X. 97. If 7f Ibs. of tea cost $9 ;I 7 , what is the price per pound" 7 98. At $1.18 a pound, how much will 6| Ibs. of tea cost ? 99. At $lf a pound, how many pounds of tea can be bought for $8 ? 100. Bought of the butcher on Monday, 6-^ Ibs. of beef, on Tuesday 5^ Ibs, on Wednesday 4 Ibs. , on Thursday 7| Ibs., on Friday 3^ Ibs. , and on Saturday 12 Ibs. , how many pounds during the week ? 101. A piece of beef weighed 11| Ibs. ; after being divided the smaller piece weighed 4^ Ibs. ; how much did the larger weigh ? 102. If a tub of lard contains 48^- Ibs., how many tubs will 582 Ibs. fill ? 103. If ll^tubs of lard contain 618f Ibs. of lard, how many pounds must there be in a tiib ? 104. If 1 tub will contain 46^f Ibs. of lard, how many pounds will 12 tubs hold ? EXEECISE XI. 105. At 43f cts. a bushel, how many bushels of oats can be bought for for $25 ? 106. At 43f cts. a bushel, how much will 47f bushels of oats cost? 107. If 7f bushels of oats cost $3.22|, what is the price per bushel ? 108. A horse ate 4 bushels one week ; 5^ the second ; 4| the third ; 5 the fourth ; how many bushels did he eat in the four weeks ? 109. A man bought 27f bushels of oats for his horse, and fed him f of 9f bushels ; how many had he left ? 110. If a horse eat ^ of a bushel of oats in a day, in how many days will he eat 18f bushels ? 111. If a horse eat | of a bushel of oats in a day, how many will he eat in 18| days ? 112. If a horse eat 5f bushels of oats in 10-| days, how many does he eat in a day ? PltOMISCUOUS EXAMPLES. 145 EXEECISE XII. 113. If a meadow yield 2 tons of hay on an acre, and the whole of it yield 13^ tons, how many acres does it contain ? 114. If a meadow yields 0.2f tons of hay on an acre, and contains 6f acres, how much will the whole of it yield ? 115. If a meadow containing 7f acres yield 15^ tons of hay, how much does it yield on an acre ? 116. A farmer has a load of hay in bales weighing as fol- low : 14:7\, 156, 139, 144-$,, 153, 146 Ibs. ; what is the weight of the whole load ? 117. A farmer having a load of hay weighing 24$ cwt., and finding it too heavy for his team, took off 5f cwt. ; what was then the weight of his load ? 118. At $9f a ton, how many tons of hay can be bought for $9871 ? 119. At $9 a ton, how much will 50 tons of hay cost ? 120. If 86 tons of hay cost $S62, what is the price per ton? EXEKCISE XIH. 121. If | of 21 Ibs. of soap cost f of $2J, what is the price per pound ? 122. If a pound of soap cost 8f cts. , how much will 14 times 1 Ibs. cost ? 123. If f of 21| Ibs. of starch cost f of f of 3f dollars, what is the price per pound ? 124. A man having a farm of 675| acres, gave | of f of it to his son, and f of the remainder to his daughter ; how many acres did he give to both of them ? 125. If H of 45 gals, of vinegar cost $3^, what is the price per gallon ? 126. If 1 gallon of vinegar cost 31^ cts., how much cost 11 times it of a gallon ? 127. If a man's expenses are 30 times of a dollar a month, how much will they be for of 1 month ? 128. If a man's expenses are f of 30 times f of a dollar for | of a month, how much are they for a month ? 7 146 DECIMAL FRACTIONS. DECIMAL FRACTIONS. Art, 109, Decimal Fractions are such as express one or more of 10 equal parts of anything, or of some mul- tiple of 10, by itself ; as, 100, 1000, etc. The Denominator, not usually written, is always 10, 100, or 1000, etc.; when written the fraction is also common. Since the value of figures decreases in a tenfold ratio from left to right, if, beginning at the left hand, we di- vide the value of any figure by 10, we find the value of the same figure in the next place, as in 111 or 222, 1(00) -4-10=1(0), 2(00)-4-10=2(0) etc. For the same reason if we continue dividing by 10 we shall find the value of the same figure repeated any number of times after the whole number ; but it will be less than 1, and therefore a fraction. To distinguish the whole number from the fraction, a decimal point (.) is placed between them, thus 111.111, &c., 222.2222, &c. In these numbers the value of the first decimal figure next to the point is 1-^10=^ ; 2-^10=1%, and of the next 1 i r -^10= T ir , ^j-^10 j^y, etc., but they are usually read as one numerator having a common denominator the same as that of the last decimal figure ; thus .11 is read as rife, .222 as ^fifr, etc. Artt 110. NUMERATION TABLE. "Whole numbers. Decimals. 1234567. 6784. DECIMAL FEACTIONS. 147 Bead, 1 million, 234 thousand, 567, and (decimal) 235 thou- sand, 784 inillionths. .2 is read 2 tenths ^ .23 is read 23 hundredths ^ . 234 is read 234 thousandths = .2345 is read 2,345 ten thousandths = . 23456 is read... 23,456 hundred thous. = .234567 is read. .234,567 millionths = Art. 110. RULE FOR BEADING DECIMALS. Read as whole numbers and add the name or denominator of the last decimal figure. The denominator is always 1, with as many ciphers annexed as there are decimals ; .25=-^. 025= T f TJ . Prefixing a cipher to a decimal divides its value by 10, because it multiplies its denominator by 10 without changing the numerator ; as, .5=^, but .05=ythj 5 annexing a cipher does not. alter the value, because it is the same as multiplying both the numerator and de- nominator of a fraction by 10 ; as .5=,^. 5Q=-f'o i 5=~Q- To distinguish the whole number from the decimal in reading, use the word decimal before the decimal expression, or when the numbers are concrete, read them as such (dollars, yards; &c.) EXAMPLES TO BE BEAD. 25; .6; 2.5; .025; .3; .36; 3.6; .036; 12.5; 125.; 1.25; 125; .0125; 136.; 13.6; 1.36; .136; 00136; 147.; .147; 1.47; 14.7; .0147; .00147; 2356.; .2356; 23.56; 2.356; .235.6; .002356; 200. 02; 100.;. 001; 1728.; 1.728; 172.8; .1728; 17.28; .001728; 2500.25; .07; .067; 4.37; 21.21; 300.03; 40.4; 4.04; .404; .000404; .0005; 31.0031; .310031; 3.10031. Art. 111. RULE FOR WRITING DECIMALS. Write what would be the numerator of a common fraction as a whole number, and prefix, if necessary, ciphers till the right hand figure is in its proper place, and then the decimal point. EXAMPLES TO BE WRITTEN. Fifteen ; fifteen hundredths ; fifteen thousandths ; 15 mil- lionths ; (15.; .15; 015; .000015;) 5 tenths ; 5 thousandths ; 5 millionths ; 2 and 5 tenths ; 25 thousandths ; 25 hundred 148 DECIMAL FRACTIONS. thousandths ; 333 thousandths ; 33 thousandths ; 3 thousandths; 27 and 27 thousandths ; 3 and 45 hundred thousandths ; 36 hundredths ; 3 tenths and 6 hundredths ; 356 thousandths ; 3 tenths, 5 hundredths and 6 thousandths ; 28 hundredths ; 128 ten thousandths ; 8 millionths ; 7 hundred thousandths ; 4 and 4 tenths ; 200 and 2 hundredths ; 3000 and 3 thousandths ; 45 millionths ; 175 ten thousandths. (See examples in Addition, &c.) Art* 112* APPLICATION OF THE FUNDAMENTAL, RULES TO DECIMALS. Since decimals increase from right to left in a tenfold ratio, the same as whole numbers, to which they are com- monly annexed, they may be added, subtracted, multi- plied, and divided by the same rules except in a few par- ticulars. Art. 113, Addition of Decimals, EXAMPLE 1. Add 6 and 5 tenths ; 10 and 1 hundredth ; 250 and 25 thousands ; 144 and 265 thousandths ; 14 and 4 tenths ; 144 thousandths. Process. 6.5 10.01 250.025 144.265 14.4 .144 Ans. 425.344" RULE. Write the numbers so that all the decimal points will be under one another ; atid as in whole numbers and place the decimal point in the sum under the others. Pupils are supposed to be familiar with addition, and the follow- ing examples are chiefly designed to exercise them in writing deci- mals. If they find the correct answers they will probably have writ- ten the numbers correctly. ADDITION OF DECIMALS. 149 EXAMPLES. 2. Add 2 and 17 hundredths ; 13 and 6 thousandths ; 12 and 145 thousandths ; 10 and 93 thousandths ; 17 and 81 ten thousandths ; 75 and 708 hundred thousandths ; 16 and 456 thousandths. 3. Add 26 and 5001 ten thousandths; 37 and 604 thousandths; 8 and 77 hundredths ; 15 and 708 thousandths; 98 and 7 tenths ; 1.99 hundredths ; 18 and 45 thousandths. 4. Add 61 and 4 thousandths ; 4 and 7 hundredths ; 329 and 8 tenths ; 47 and 39 hundredths ; 731 and 96 thousandths ; 5 and 5 ten thousandths ; 6 and 8 tenths. 5. Add 42 and 8 hundredths; ; 521 and 28 thousandths ; 63 and 125 ten thousandths ; 108 and 215 thousandths ; 14 and 25 hundredths ; 9 and 5 tenths ; 18 and 23 hundredths ; 110 and 11 thousandths. 6. Add 17 and 55 thousandths ; 4 and 81 hundredths ; 90 and 1935 ten thousandths ; 77 and 85 hundred thousandths ; 24 and 106 ten thousandths ; 35 and 7 tenths. 7. Add 12 and 5 tenths ; 13 and 65 hundredths ; 114 and 25 thousandths ; 46 and 121 ten thousandths ; 64 and 7 tenths ; 127 and 18 thousandths ; 43 and 9 hundredths ; 102 and 6 tenths. 8. Add 6 and 157 thousandths ; 18 and 225 ten thousandths ; 172 and 16 hundredths; 27 and 81 thousandths; 9 and 23 hundred thousandths ; 13 and 27 thousandths ; 6 and 12 hun- dredths. 9. Add 12 and 9 thousandths ; 125 and 8 tenths ; 245 ten thousandths ; 249 ; 63 and 63 hundredths. 10. Add 17 ; 17 hundreds ; 17 thousandths ; 17 hundredths ; 1 and 7 tenths. 11. Add 16 thousand ; 16 hundreds ; 160; 16 ; 1 and 6 tenths; 16 hundredths ; 16 thousandths. 12. What is the sum of 8 and 75 hundredths; 60 and 7 tenths ; 12 and 5 thousandths ; 180 and 27 hundredths ; 29 and 21 thousandths ; 3 and 15 hundredths ? 13. What is the sum of 3 thousand and 6 hundredths ; 2 150 DECIMAL FRACTIONS. hundred and 45 thousandths ; 10 and 1 hundredth ; 4 thousand and 6 hundredths ; 4 hundred and 6 thousandths ? 14. What is the sum of 10 and 1 tenth ; 100 and 1 hun- dredth ; 1000 and 1 thousandth ; 200 and 2 thousandths ; 20 and 2 hundredths ? 15. What is the sum of 25 tenths ; 126 hundredths ; 1354 thousandths ; 13579 ten thousandths ; 2468 thousandths ; 357 hundredths ? Art. 114, Subtraction of Decimals. EXAMPLE 1. From 123 and 56 hundredths, subtract 12 and 156 ten thousandths. Process. 123.56 12.0156 Ans. 111.4444 RULE. Write the numbers so that the decimal points will be under each other, subtract as in whole numbers, and place the decimal point in the remainder under the others. If there is no figure directly above the one to be subtracted, con- sider the place as filled with a cipher. EXAMPLES. 2. From 1. subtract 1 tenth. 3. From 20. subtract 2 hundredths. 4. From 250. subtract 25 thousandths. 5. From 1356. subtract 356 ten thousandth. 6. From 23464. subtract 3464 hundred thousandths. 7. From 100,000. subtract 100 thousandths. 8. From 1,000,000. subtract 1 millionth. 9. From 8 tenths subtract 4 hundredths. 10. From 75 hundredths subtract 75 thousandths. 11. From 56 thousandths subtract 56 ten thousandths. 12. From 5 tenths subtract 5 thousandths. 13. From 5 hundredths subtract 5 hundred thousandths. 14. From 5 thousandths subtract 5 millionths. 15. From 5 subtract 3 tenths. MULTIPLICATION OF DECIMALS. 151 "What is the difference between 16. 10. and .01 ? 17. 5. and .5 ? 18. 600 and 6 hundredths ? 19. 7000 and 7 thousandths ? 20. 80000 and 8 ten thousandths ? 21. 3 and .3 ? 22. 3 tens and 3 tenths ? 23. 3 hundreds and 3 hundredths ? 24. 3 thousands and 3 thousandths ? 25. 3 millions and 3 millionths ? Art. 115. Multiplication of Decimals. General Principle. The denominator (understood) of any product of decimals is the product of their denom- inators ; thus the product of tenths and tenths is hun- dredths (iVXiV^Tihy); the product of tenths and hun- dredths is thousandths, (i 1 ijXTou=roW) &c. t &c. EXAMPLE 1. Multiply 1.25 by .125. Process The same as in whole numbers, except 1.25 five figures are pointed off in the product for deci- .125 mals, because the product of hundredths and thous- g25 andths is hundred thousandths. 1. 25=l-, s - ( f j=|^f ; OKA .125=-^% ; and I^XiV&^iWo^^. 15625, Ana. 125 Ans. .15625 BULE. Multiply as in whole numbers, and point off from the right of the product as many figures for decimals as there are decimals in both the multiplicand and multiplier ; or so that the denominator (understood) of the product of the de- cimals shall be the product of their denominators. If there be not figures enough in the product, prefix ciphers. Ex. 2. Multiply .256 by 100. Process. .256X100=25.600 or simply remove the decimal point two places to the right. 152 MULTIPLICATION OF DECIMALS. SPECIAL RULE. To multiply by 10, 100, &c. : remove the decimal point as many places to the right as there are ciphers in the multiplier, annexing ciphers if necessary. EXAMPLES. [Let pupils write the following examples with the respective pro- ducts on their slates ; also recite the answers without having them written, till they can do so readily and without mistakes.] Multiplies Multipliei Products mds . 'S. . . . 3 3 2 2 .3 .2 .06 .06 .05 .03 .2 .006 7 6 .03 .02 .3 .3 .03 .3 J009 .07 .06 6 76 .0006 7 .7 .6 .6 .09 .07 .6 6 6 5 .5 .6 .5 .06 .5 8 7 10 9 8 .7 10 .9 .7 1.0 .08 .10 .9 .08 .07 .10 11 11 11 .09 10 1.0 .10 .09 1.1 1.0 .08 .11 .10 12 11 12 1.1 12 .11 .12 1.1 .12 .11 12 12 12 12 1.2 .12 .12 1.2 .12 .12 3 2 tenths. 4 tenths. 5 hund'ths. 6 hund'ths. 7 hund'ths. 3 tenths. 4 tenths. 5 hund'ths. 6 thous'ths. 4 5 tenths. 6 hund'ths. 7 hund'ths. 8 hund'ths. 3 tenths. 4 tenths. 5 tenths. 6 hund'ths. 7 thous'ths. 5 4 tenths. 6 5 tenths. 6 tenths. 7 hund'ths. 8 hund'ths. 9 hund'ths. 5 tenths. 6 tenths. 7 hund'ths. 8 thous'ths. 7 tenths. 6 tenths. 8 hund'ths. 7 tenths. 9 hund'ths. 10 hund'ths. 8 hund'ths. 9 thous'ths. 12 thousandths. 11 tenths. 12 ten thousandths. 11 hundredths. 12 millionths. 11 thousandths. EXAMPLES FOE THE SLATE, ETC. 3. Multiply 21 and 6 tenths by 3 and 6 hundredths. 4. 156 and 25 thousandths by 2 and 75 hundredths. 6. 50 and 5 hundredths by 2 and 16 thousandths. DIVISION OF DECIMALS. 153 6. 175 thousandths by 100. 7. 22 ten thousandths by 11 hundredths. 8. 18 by 256 thousandths. 9. 6 and 5 tenths by 65 hundredths. 10. 325 thousandths by 50. 11. 672 ten thousandths by 25. 12. 1 millionth by 1000. 13. 100 by 1 thousandth. 14. 5 thousandths by 4 thousandths. 15. 125 milHonths by 1,000,000. 16. 275 and 275 thousandths by 25 and 25 hundredths. In U. S. Money cents and mills are decimal fractions of a dollar ; 1 cent=$.01 ; 25 cents=$.2o ; 50 cents=$.5 ; 1 mill=:$.001 ; 5 mills=$.005, &c. 17. At 12 cts. a yard, how much will 16. 5 yds of calico cost ? 18. At 75 cts. a bushel, how much will 18. 25 bushels of com cost? 19. At 62 cts. 5 m. a gallon, how much will 20.5 gals, of mo- lasses cost ? 20. At $8.625 a ton, what cost 6.5 tons of coal ? 21. At $100 an acre, what cost 63.75 acres of land ? 22. At $9.625 a barrel, what cost 20 barrels of flour ? 23. At $0.1875 a pound, what cost 37.5 Ibs. of lard ? 24. At $.025 a mile, how much will it cost to travel 100 miles? Art. 116. Division of Decimals, General Principle. The denominator (understood) of the dividend divided by that of the divisor is the de- nominator of the quotient ; thus thousandths divided by tenths are hundredths, and hundredths divided by tenths are tenths dolo '. IO=TOU > i&o ' i 1 o=Ar) &c. EXAMPLE 1. Divide 15.625 by .25. 7* 154 DIVISION OF DECIMALS. Process. The same as in whole numbers, ex- ,25)15.625(62.5 cept one figure is pointed off in the quotient be- 15 Ans. cause thousandths divided by hundredths, are ^3 tenths ; 50 15625 25 ,1 12^ and 625_. ?0 '- EULE. Divide as in whole numbers, and from the right of the quotient, point off as many figures for decimals as the de- cimals in the dividend exceed those in the divisor ; or, so that the denominator (understood) in the quotient, shall be the quo- tient of the denominator in the dividend divided by that of the divisor. If there are not figures enough in the quotient, prefix ciphers to the decimals, or annex them to whole numbers. If there are not as many decimals in the dividend as in the divisor, annex ciphers. If other ciphers are annexed to the remainder, they must be considered as filling decimal places in the dividend. Ex. 2. Divide 3.25 by 1-0. Process. Remove the point two places to the left, Ans. .0325 prefixing a cipher where a figure is wanting, which is the same as dividing by 100. SPECIAL EULE. To divide decimals by 10, 100, &c., re- move the decimal point as many places to the left as there are ciphers in the divisor, prefixing ciphers if necessary. EXAMPLES. [Let the pupils write the following examples with the respective quotients, on their slates ; also recite the answers without seeing them till they can do so readily and without mistakes.] 2)6 .2)6 .2).06 .2). 006 .02). 0006 .3). 09 .3). 009 .03). 0009 4)24 .4)2.4 .4).24 .4). 024. .04). 024. .004).Q24 5)30 .5)30 .05)30 6)42 .6)42 .6^4.2 7)5_6 .7)56 .7)5.6 .07). 56 .8)72 .8). 72 9)90 .9)90 .09)_.90_ 8).96 .8)9.6 .08).96 DIVISION OF DECIMALS. 155 11)132 .11)132 11)13.2 1.1)132 .11)1.32 12)144 .12)144 12)14.4 1.2)144 .12)1.44 Divide 6 tenths by 2 tenths. 6 hundredths by 2 tenths. 6 thousandths by 2 tenths. 9 (whole number) by 3 hun'hs 9 hundredths by 3 hund'ths. 9 thousandths by 3 tenths. 12 thousands by 4 hund'ths. 12 hundredths by 4 tenths. 16 tenths by 4 tenths. 16 thousandths by 4 hund'ths. 24 hundredths by 4 hund'ths. 32 tenths by 8 tenths. 36 thousandths by 9 hund'ths. 45 hundredths by 5 tenths. 56 millionths by 7 thous'ndths. 64 millionths by 8 hundredths. EXAMPLES FOE THE SLATE. 3. Divide 1728 by 12, .12, 1.2, .012 4. Divide .1728 by 12, .12, 1.2, .012 5. Divide 17.28 by 12, .12, 1.2, .012 6. Divide 1.728 by 12, .12, 1.2, .012 7. Divide 172. 8 by 12, .12, 1.2, .012 8. Divide 1728 by 144, .144, 1.44, 14.4 9. Divide 17.28 by 144, .144, 1.44, 14.4 10. Divide 1.728 by 144, .144, 1.44, 14.4 11. Divide 172.8 by 144, .144, 1.44, 14.4 Divide 12. 34 and 11 hundredths by 9 and 8 tenths. 13. 125 and 18 thousandths by 5 and 25 hundredths. 14. 40 and 215 ten thousandths by 8 and 5 tenths. 15. 25 ten thousandths by 25 hundredths. 16. 1 thousandth by 1000. 17. 63 and 63 hundredths by 7 tenths. 18. 125 millionths by 500. 19. At $5. 75 a cord, how many cords of wood can be bought for $50 ? 20. If 7.5 cords of wood cost $38.775, what is the price of a cord? 21. If 75.97 acres of land cost $2696.935, what is the price of an acre ? 156 DECIMAL FRACTIONS. 22. At $35.50 an acre, how many acres of land can be bought for $1348. 4625. 23. If 12.75 yards of cloth cost $97.5375, what is the price per yard ? 24. At 7. 65 a yard, how many yards of cloth can be bought for $91.80 ? 25. At $7.375 a ton, how many tons of coal can be bought for $176. 63125? 26. If 100 tons of coal cost $737.50, what is the price per ton? Art, 117, Promiscuous Examples in Decimal Fractions, including U, S, Money. EXEECISE I. 1. Bought tea for 1 dollar 25 cts. ; sugar, 3 dollars 37 cts. 5 m. ; starch, 50 cts. ; molasses, 2 dol. 9 cts. ; ginger, 18 cts. 5 m. What is the amount of the bill ? 2. A man having $10,000, paid $6,521.875 for a farm ; how much had he left ? 3. At $1.875 a bushel, how much will 12.5 bushels of wheat cost? 4. If 16.25 bushels of wheat cost $28.4375, what is the price per bushel ? 5. At $1.75 a bushel, how many bushels of wheat can be bought for $30 ? 6. How many hundred weight of sugar in five barrels weigh- ing 2 cwt. 75 hundredths ; 2 cwt. 875 thousandths ; 3 cwt. 4 hundredths ; 2 cwt. 5 tenths ; 3 cwt. 1275 ten thousandths ? 7. A merchant having a barrel of sugar weighing 2 cwt. has sold from it 1 cwt. 375 thousandths ; how much of it is left ? 8. If 2.5 yards of cloth will make a coat, how many yards will make 15 coats ? 9. If 2 375 yards of cloth will make a coat, how many coats will 28.5 yards make ? 10. If 37.5 yds. of cloth will make 15 coats, how many yards will make one coat ? PROMISCUOUS EXAMPLES. 157 EXERCISE H. 11. Bought a barrel of iiour for 10 dol. 50 cts. ; a bushel of timothy seed for 4 dol. 61; cts. 5 m. ; a cheese for 3 dol. 6 cts. 5 m. ; a box of soap for 8 dollars, and a broom for 38 cts. ; what is the amount of the bill ? 12. A lady having purchased a bill of goods amounting to 4 dollars 6 cts. , gave the clerk a 10 dollar bill ; how much change was due her ?. 13. At $1.37 a pound, how much will 3.25 pounds of tea cost ? 14. At $1.375 a pound, how many pounds of tea can be bought for $5 ? 15. If 4.75 pounds of tea cost $6.00, what is the price per pound ? 16. How many hundred weight of meal in 4 bags weighing 1 cwt. 1875 ten thousandths ; 1 cwt. 57 hundredths ; 1 cwt. ; .98 cwt. ? 17. A miller ground 6 cwt. of meal and sold 3 cwt. 125 thou- sandths ; how much of it was left ? 18. If a barrel contain 2.75 bushels of potatoes, how many bushels will there be in 12.5 barrels ? 19. If a barrel will contain 2.5 bushels of potatoes, how many barrels will contain 10.75 bushels ? 20. If 100 bushels of potatoes be put in 36.5 barrels, how many must each barrel contain ? EXEECISE HI. 21. Bought a load of potatoes for $15.75 ; of turnips for $7.375 ; of carrots for 9 dollars, 8 cts. 5m.; of beets for $10 ; to what did they all amount ? 22. Exchanged a horse worth $225, for a pair of oxen worth $180.625, and the rest in cash ; what was the amount of cash? 23. At $6.375 a yard, how much will 3.75 yds. of cloth cost? 24. At $6.625 a yard, how much cloth can be bought for $25? 25. If 3.8 yds. of cloth cost $20.5, what is the price per yard? 158 DECIMAL FKACTIONS. 26. How many acres of land in five fields measuring 4 A. 27 hundredths ; 6 A. 28 thousandths ; 5 A. ; 7 A. 7 hundredths ; 6 A. 5 tenths ? 27. If a field measures 10 A. 5 hundredths, and 5 A. 5 tenths be fenced off from it, how much of it will be left ? 28. If a field contains 6.54 acres, how many acres will there be in 10 fields of the same size ? 29. If 8 fields of equal size contain 45.75 acres, how many acres in each ? 30. If a field contain 6.45 acres, how many such fields will contain 58.05 acres ? EXEKCISE IV. 31. Bought a carpet for $12 ; matting, $6 ; carpet binding, 50 cts. ; tacks, 6 cts. ; what was the amount of the bill ? 32. A clerk's salary is $1,000 a year, and his expenses $656.- 625. how much can he lay up ? 33. If a clerk receives $1,000 a year (313 working days), how much is it a day ? 34. If a man earn $2.626 a day, how much will he earn in 300.5 days ? 35. If a man earn $572.5 in 312.75 days, how much does he earn in a day ? 36. How many pounds in 4 hams weighing 20 and 2 tenths pounds ; 21 and 28 hundredths ; 19 Ibs. ; 18 and 25 thousandths? 37. A ham weighed 22.5 Ibs. ; after being smoked it weighed 19 Ibs. 8125 ten thousandths ; how much weight had it lost ? 38. If each firkin of butter contains 54.75 Ibs., how many pounds will there be in 10. 5 firkins ? 39. If a firkin will hold 54.24 Ibs. of butter, how many fir- kins will hold 1000 Ibs ? 40. If 512. 25 pounds of butter be packed in 9 .75 firkins, how many pounds will there be^in each ? EXERCISE v. 41. If a man's income is $5,000 a year, and his expenses $10'a day, how much will he hav.e left ? 42. If a man's income is $5,000 a year (365 days), how much may he expend, and lay up $5.00 every day ? REDUCTION OF COMMON FK ACTIONS. 159 43. If a man lay up $2.000 a year (or 313 working days), how much will he lay up each day on an average ? 44. How much butter in four tubs, weighing 16.5 Ibs. ; 20 Ibs. ; 18.25 Ibs. ; 19 and 25 thousandths pounds ? 45. If each load of coal weigh 18. 75 cwt. , how much will 20 loads weigh ? 46. If a man draw with his team 20.5 cwt. of coal in 24 loads, how much must he draw at a load ? 47. If a man draw 16. 75 cwt. at a load, how many loads will 251.25 cwt. make ? 48. If a man chop 3.25 cords of wood in a day, how long will it take him to chop 30 cords ? Art* 118. REDUCTION OF COMMON FRACTIONS TO DECI- MALS. EXAMPLE 1. Eeduce f to a decimal. Process. Since ^=3-f-4 and 4 is not contained in 3 a whole number of times, we find the value of 3 in the first place of decimals to be 3.0 (30 tenths) the same as the 4)3.00 numerator, with a cipher annexed, which divided by 4, ^ ns ' "75 is .7-)-. 2 remainder, but .2=. 20 the same as annexing another cipher to the numerator, and .20-^-4=. 05, which added to . 7=. 75, Ans. RULE. Add ciphers to the numerator and divide by the denominator. Ex. 2. Reduce ^ to a decimal. Protess. 7)1.000000 or 7)1.000 "142857+ Tl43 In this Arithmetic, whenever a decimal, as in the above example, will extend to more than four places, three places will be considered sufficiently accurate if it is to be used afterwards, but if the figure in the fourth place would be more than five, 1 will be added to the figure in the third or thousandths place ; as .143 for .14283. Ex. 3. Reduce to a decimal. Process. 3)LOOOOO() or 3)1.0 .333333 3 2. A decimal, consisting of the same figure repeated (as in the above example), or several figures, is called a repetend or circulating decimal, and is distinguished as such by a (.) dot over the first and 160 DECIMAL FRACTIONS. last figures repeated. If only a part of the decimal is repeated it is called a mixed repetend. Art, 119, KEDUCTION OF DECIMAL TO COMMON FRAC- TIONS. EXAMPLE 1. Eeduce .375 to a common fraction. Process. .375=^, which reduced to its low- 25)375 515 3 ost terms is f. 1000 15 8 KULE. Erase the decimal point, and write, for the denom- inator of the common fraction, 1 with as many ciphers an- nexed as there are are decimal figures ; then reduce the frac- tion to its lowest terms. Ex. 2. Eeduce .3 or 333333, &o., to a common fraction Process. Since .3=.3^ ( reduced to a decimal) and 3 . 3 I or f , is & less than 3 (or *\ .3 is -fj less than .3 ; but if 3 ^9~3 the denominator of -,% be diminished in the same proportion the value of the fraction is not altered. Therefore, 3 or -$j=$, which equals i. If the repetend consist of more than one figure the denominator of the common fraction will be as many nines as there are figures repeated ; thus, .123=^=^. "When the decimal is a repetend write for the denominator of the common fraction as many 9's as there are figures in the repetend. Ex. 3. Eeduce .16 to a common fraction. When only a part of the decimal is repeated it is called a mixed decimal, and is the same as a mixed number in a complex fraction ; thus : .lfe=.lf==_li=$ft=k Ans. 10 EXAMPLES. Reduce the common fractions to decimals, and the decimals to common fractions. CM *; f; I; I (2.) .5; .25; .75; .125 (3.) f; 1; f; A (4.) .375; .625; .875; .025 (50 ; H; ; M (6.) .8; .16; .075; .225 (7.) ; &; i; (8.) .6; 027; .123; .16 FRACTIONAL COMPOUND NUMBERS. 161 Fractional Compound Numbers. Art. 120. Fractional compound numbers are com- pound numbers in the form of fractions ; as jB| ; j cwt. ; I gal. GENERAL RULE. Proceed, as in similar cases, in com- pound whole numbers, using the rules for fractions when necessary or convenient. Special rules will also be given in some cases, but it is better to understand and apply the general rule. CASE I. Art. 121. Compound numbers reduced to fractional com- pound numbers ; reduction ascending. EXAMPLE 1. Eeduce 10s. 6d. to the fraction of a pound. Process. By common fractions. 6d. 12)6 d. 12 = -A-s. =is., to which add or OOMO& - omiftl prefix the 10s. 2 the sum 10is.^20= 'tt ^^ $k- This common fraction reduced ^240 tft 40 BW to a decimal is .525. Process. By decimal fractions, 6d.-^-12= 12)6.0 6.0d.-^12=.5s., to which add or prefix the 20)10~5 10s. 'The sum 10.5s.-:-20 = .525 = V$fc ' ajL _ 2 "ftoo \ SPECIAL RULE. Beginning with the least denomination given, reduce it to the next greater by division of fractions (common or decimal), and add, or annex it to any given number of the same denomination or name. Proceed thus till the required fraction is found, which reduce to its lowest terms. For convenience write the least denomination first, and the others under it in order. EXAMPLES. [Let the answers be found in both common and decimal fractions.] Ex. 2. What part of a cwt. is 2 qr. 10 Ibs. ? 3. What part of a mile is 26 rods 11 ft. ? 4. What part of a hhd. is 15 gal. 3 qts. ? 162 FRACTIONAL COMPOUND NUMBERS. 5. What part of a yard is 3 qr. 3 na. ? 6. What part of a day is 6 h. 30 m. ? 7. What part of an acre is 2 B. 10 rods ? 8. What part of a yard is 3 qr. 3| na. ? Process. 3| na. -f- 4=V- na. -|- 4=1! qr. 3H qr.-r*=tt yds. 9. What part of 3 is 15 s. 6 d. ? Process. First find what part of 1, and then of that. CASE II. Art* 122. Fractional compound numbers reduced to in- tegral compound numbers ; reduction descending. EXAMPLE 1 . Eeduce f to whole numbers. Process. By common fractions. Since y is less than 1, reduce it to shillings. $ X 20 = ^s. = 14$s. ; f s. X12 = yd. = Sfd. ; f d. X4=V-fer. r= If far. Therefore, $=14s. 3d. l^far., Ans. Ex. 2. Keduce .625 to whole numbers. - .625. Process (by decimal fractions'), the same as in reduc- 20 tion descending of whole numbers, except pointing off, 12.500s. ns in multiplication of decimals. 12 Ans. 12s 6.006d. SPECIAL RULE. Eeduce the fractions to kss denomina- tions, and find the value of each in whole numbers. EXAMPLES. How much in whole numbers is (3.) | (10.) .325. (4.) tfton. (5.) ^-mile. (6.) | yard. (7.) f acre. (8.) f hogshead. (9.) ^ year. (11.) .675cwt. (12.) .75 rod. (13.) .6 yard. (14.) .25 sq. mile. (15.) .0025 tun. (16.) .0785 day. CASE III. Art* 123. Reduction of Fractional Compound Numbers to fractions of other denominations. EXAMPLE 1. Beduce ^y to the fraction of a penny. FRACTIONAL COMPOUND NUMBERS. 163 Process. Reduction descending by multiplication of common frac- tions and cancellation. ^u=^ oX 2 OX12=fd. , Ans. Ex. 2. Reduce f oz. to the fraction of a cwt. Process. "Reduction ascending by division of common fractions. OZ ' = - 16 ' ' 7Xl8*Xl Ex. 3. Keduce .00125 ton to pounds. .000125 ton. 20 Process. Reduction descending by multiplicatiou .002500 cwt. of decimals. 100 -4ns. .250000 Ib. Ex. 4. Keduce .5 rod to the fraction of a mile. Proeess. Reduction ascending by division of * r S ' decimals. 8).Q125 fur. Ans. .0015625 mile. EXAMPU3S. 5. What part of a pound is f of an oz. Av. ? 6. What part of a pound is y^j- o f a cwt. ? 7. What part of a pound is .5 of an oz. Troy ? 8. What part of a pound is .0004 of a ton ? 9. What part of a gallon is of a gill ? 10. What part of a gallon is 3-^ of a hhd. ? 11. W r hat part of a gallon is .25 of a pint ? 12. What part of a gallon is .003 of a tun ? 13. What of an hour is ^ of a minute ? 14. What of an hour is -^ of a day ? 15. What of an hour is .8 of a second ? 16. What of an hour is .04 of a day ? 17. What part of a rod is -^1 of a foot ? 18. What part of a rod is ^ of a mile ? 19. What part of a rod is .165 of a yard ? 20. What part of a rod is .003 of a mile ? 164 FEACTTONAL COMPOUND NUMBERS. Art, 124. Addition and Subtraction of Fractional Compound Numbers. EXAMPLE 1. Add f to f shilling. s. d | =12 8 Process.-- fs.= 10 I3a 6d. Ex. 2. From cwt. subtract Ib. f cwt. = 3 qr. 8 Ib. 5^ oz. Process. f Ib. = 12 Ans. 3 qr. 7 Ib. 9 oz. Ex. 3. Add .6 acre and .06 rod. .6 acre. 160 Process .6 ucre = 96 rods, to which add .06 rods, 95,0 rods and the sum will be 96.06 rods. .'06 Ans, 96.06 " RULE. Reduce the fractions to whole numbers, and add or subtract them as other compound numbers. EXAMPLES. Ex. 4. Add f of a day to f of an hour. 5. From of a mile subtract f of a fur. 6. Add .525 of a gal. to .9 of a qt. 7. Add T % Ib. to | oz. 8. From hhd. take of a barrel. 9. From .625 of an hour take 3.5 minutes. Art, 125. Promiscuous Examples in Fractional Com- pound Numbers. EXEECISE I. 1. How much in whole numbers is % ? 2. How much in whole numbers is .375 of a pound (Troy)? 3. What part of a ton is 5 cwt. 2 qr. 15 Ibs. ? PROMISCUOUS EXAMPLES. 165 4. What part of a hundred weight is 3 qr. 12 Ibs. 8 oz. (in decimals). 5. What part of a drachm is T | of a pound (Apo.)? 6. What part of a yard is f of a nail ? 7. What part of a rod is .825 of a foot ? 8. What part of a square rod is .0025 of an acre ? 9. What is the sum of cord and foot ? 10. How much more is .25 of an hour than 5.5 minutes ? EXEBCTSE H. 11. How much in whole numbers is T 6 T of a hogshead ? 12. How much in whole numbers is .625 of a bushel ? 13. What part of a year is 125 days 12 hours ? 14. What part of a degree is 25' 36" (in decimals) ? 15. What part of a farthing is y^ of a shilling ? 16. What part of a pound is f of a pwt. ? 17. What part of an ounce is .84 of a scruple ? 18. What part of a yard is .64 of a nail ? 19. What is the sum of .375 of an acre and 75.5 rods ? 20. How much more is f of a rod than 3^ yards ? EXEKCISE m. 21. How much in whole numbers is ^ of a mile ? 22. How much in whole numbers is .865 of a cord ? 23. What part of a barrel is 21 gal. ? 24. What part of a bushel is 3 pks. 6 qts. (decimal) ? 25. What part of a second is y^j of a minute ? 26. What part of a degree is ^ of a circular minute ? 27. What part of a farthing is .0025 of a shilling ? 28. What part of an ounce is .9 of a pwt. ? 29. What is the sum of % ton and -g- cwt. ? 30. How much less is .26 cwt. than 28.5 Ibs. ? EXEBCISE IV. 31. How much in whole numbers is | of a pound (A.) ? 32. How much in whole numbers is 2.5 of a yard ? 33. What part of a yard is 2 ft. 9 in..? 34. What part of a cord is 21 cu. ft. 576 in. ? 35. What part of a quart is ^ of a gal. ? 166 FRACTIONAL COMPOUND NUMBEES. 36. What part of a peck is f of a pint ? 37. What part of a minute is .006 of an hour ? 33. What part of a circle is . 72 of a degree ? 39. Add f s. 6| d. 2 far. 40. How much more is .3 s. than 2.25 d.? EXEECISE v. 41. How much in whole numbers is of a pound (Troy) ? 42. How much in whole numbers is .0025 of a ton ? 43. What part of an ounce is 4 dr. 2 scr. ? 44. What part of a yard is 1 qr. 3 na. (decimal) ? 45. What part of an inch is T ^ of a yard ? 46. What part of an acre is | of a rood ? 47. What part of a cord is 512 of a foot ? 48. What part of a gill is .0056 of a gall.? 49. What is the sum of .6 of a bushel and .8 of a peck ? 50. How much more is ^ bushel than -g- peck ? Art. 126, Promiscuous Examples in Fractions, Common and Decimal, and Fractional Compound Numbers. EXEBCISE I. 1. How many yards are there in three pieces of cloth, mea- suring as follows: 30|, 37, 38| yards ? 2. From a piece of cloth which contained 33 yds., 16f yds. have been cut off ; how many are left ? 3. How much will 4^ tons of iron cost at $18f a ton ? 4. At $18f a ton, how many tons of iron can be bought for $60? 5. If 4| tons of iron cost $75, what will a ton cost ? 6. How many are 3 and 7 tenths, 44 and 41 hundredths, 73 and 9 thousandths, 12 and 305 thousandths ? 7. Prom 5 hundredths subtract 476 ten thousandths. 8. At $.1875 a yard, what cost 12.25 yds. of muslin ? 9. At $.1875 a yard, how many yds. of muslin can be bought for $5.75 ? PROMISCUOUS EXAMPLES. 167 10. If 19 yds. of muslin cost 5.377, what is the price per yd. ? 11. At $18.875 a 1000, what will 12.500 feet of pine boards cost? 12. At $18.875 a 1000, how many feet of pine boards can be bought for $12.50 ? 13. If 1200 ft. of pine boards cost $20, what is the price per 1000 ft. ? 14. What part of a Ib. Av. is ^ of an oz. ? 15. What part of a nail is ^ of a yard ? 16. What part of a mile are 3 fur. 8 rods ? 17. What part of a bushel is .5 of a peck ? 18. What part of a pound Troy is . 75 of an oz. ? 19. What part of a pound Troy are 8 oz. 8 pwt. (decimal) ? 20. What decimal fraction is equal to % ? 21. What common fraction is equal to .8 ? 22. What whole numbers are equal to T 9 ^ of a day ? 23. What whole numbers are equal to .5625 of a day ? 24. How much are % of a week, 1% days, 5> hours ? 25. At $.625 a bushel, what cost 15 bus. 3 pks. 4 qts. of rye ? 26. At $.625 a bushel, how much rye can be bought for $10? EXERCISE II. 27. How many pounds are 20}, 21K, 22^, 23^, 24| Ibs.? 28. From a cask containing 64 1 gallons of molasses, 30^ gals, have been used ; how many are left ? 29. At $^ a yard, how much ribbon can be bought for $M? 30. At $2^ a y ar( l, n w niuch will 3% yards of ribbon cost ? 31. If 1% yds. of ribbon cost $3^, what is the price per yd. ? 32. How many are 44 and 19 thousandths, 8 and 71 hundred thousandths, 83 and 3327 ten thousandths, 60 and 301 ten thousandths ? 33. From 14 and 15 tenths subtract 7 and 37 thousandths ? 34. At $12 a ton, how much coal can be bought for $5.64 ? 35. At $12 a ton, what costs 3.047 tons of coal ? 36. If 1.047 ton of coal cost $12.564, what is the price per ton? 168 FKACTIONAL COMPOUND NUMBERS. 37. At $3.875 a 100, how many bricks can be bought for $23.25? 38. At $3.875 a 100, how much wiU 3,750 bricks cost ? 39. If 4,575. bricks cost $161.5625, what is the price per 100 ? 40. What part of a pound (Troy) is % of an oz. ? 41. What part of an oz. is yg^^ of a cwt. ? 42. What part of a pound (Troy) are 9 oz. 12 pwt. ? 43. What part of a yard is .8 of a nail ? 44. What part of a bushel are 3 pks. 2 qts. (decimal) ? 45. What part of a peck is .175 of a bushel ? 46. What decimal fraction is equal to % ? 47. What common fraction is equal to .16 ? 48. What whole numbers are equal to ff of a mile ? 49. What whole number is equal to .03515625 of a Ib. Av.? 50. How much are ^ hhd. , \\ gal. , and \ qt. ? 51. At $4.80 a cord, how much wood can be bought for $70.80 ? 52. At $4.80 a cord, what cost 15 c. 96 ft. of wood ? 53. If 13 cords 96 ft. of wood cost $61.875, what is the price per cord ? EXERCISE HI. 54. How many are 28%, f of 18%, 32, % of 18% ? 55. From 60 subtract % of 100. 56. At $}4 a pound, how many pounds of feathers can be bought for $8^ ? 57. At $> a Ib., how much will 15> Ibs. of feathers cost ? 58. If 141^ pounds of feathers cost $7>, what is the price per pound ? 59. How many are 28 and 45-thousandths, 3 and 91-hun- dredths, 80 and 219 ten-thousandths, 17 and 7 tenths ? 60. From 900 and 9-hundredths subtract 99 and 9 thou- sandths. 61. At $10.375 a barrel, how many barrels of flour can be bought for $72% ? 62. At $10.625 a bbl., how much will 10 barrels of flour cost ? PROMISCUOUS EXAMPLES. 169 63 If 9 bbls. of flour cost $96.75, what is the price per bbl. ? 64. At $7.50 a 100, how many cabbages can be bought for $5.625? 65. At $7.50 a 100, how much will 44 cabbages cost ? 66. If 56 cabbages cost $4.20, what is the price per 100 ? 67. What part of a yard is f of a nail ? 68. What part of an inch is -fe of an E. English ? 69. What part of a pound Av. are 9 oz. 2f dr. ? 70. What part of a yard is .5 of a quarter ? 71. What part of an acre is .875 of a square rod ? 72. What part of an acre are 1 rood, 14 rods (decimal) ? 73. What whole numbers are equal to f of a yard ? 74. What whole numbers are equal to .000175 of an acre ? 75. From f of an ounce subtract % of a pennyweight. 76. At $.625 a bushel, how much corn can be bought for $9.00 ? 77. At $.625 a bushel, what cost 15 bu. 3 pks. 4 qts. of corn ? 78. If 15 bu. 3 pks. 4 qts. of corn cost $9.525, what is the price per bushel ? EXERCISE rv. 79. How many yards in six pieces of cloth measuring as fol- lows : 18%, 19*, 2% 20&, 22^, 24^ ? 80. From a piece of silk which contained 31% yards, 8-f yds. have been cut off; how many remain ? 81. At $| a pound, how much tea can be bought for $2.00 ? 82. At $f a pound, how much will 3| Ibs. of tea cost ? 83. If 2^ Ibs. of tea cost $2, what is the price per pound ? 84. Add 4 and 35 ten thousandths, 10 and 35 hundred thou- sandths, 6 and 35 millionths, 100 and 35 ten millionths. 85. From 113 and 5 tenths subtract 8 and 37 thousandths. 86. At $45.625 a barrel, how many barrels of molasses can be bought for $136. 875? 87. At $.375 a gallon, how much will 28.625 gal. of vinegar cost? 8 170 FRACTIONAL COMPOUND NUMBERS. 83. If 25.5 gals, of vinegar cost $8.925, what is the price per gallon ? 89. At $21 a 1000, how much will 150 shingles cost ? 90. If 200 shingles cost $3.50, what is the price per 1000 ? 91. What part of an acre is % of a square rod ? 92. What part of a minute is y^j of a day ? 93. What part of an A. are 2. B. 20 square rods ? 94. What part of a pint is .03125 of a gal. ? 95. What part of a cwt. are 1 qr. 8 Ibs. 10 oz. (decimal)? 96. What part of a nail is .05 of a yard ? 97. What common fraction is equal to .05 ? 98. What decimal fraction is equal to -^ ? 99. What whole numbers are equal to % of a bushel ? 100. What whole numbers are equal to to .07 of a hhd. ? 101. From f of a mile subtract -/y of a fur. 102. At 27.25 an acre, how much land can be bought for $3500 ? 103. At $105. an acre, how much will 26 A. 2 E. 25 rods of land cost ? 104. If 112 A. 3 E. 20 rods of land cost $11287.50, what is the price per acre ? EXEBCISE V. 105. How many dollars are 5, 7^, 4%, 11%, 12%, llf dollars ? 106. From f of 16| subtract 4^. 107. At 4f cts. a pound, how many pounds of lead can be bought for $1.30f ? 108. At 4| cts. a pound, how much will 28)^ Ibs. of lead cost? 109. If 23K Ibs. of lead cost $1.45f , what is the price per Ib. ? 110. How many are 10 and 19 thousandths ; 106 and 3 hun- dredths ; 17 and 16 millionths ; 9 and 9 tenths ; 71 and 63 ten thousandths ? 111. From 51.004 subtract 31 and 8 hundredths. PROMISCUOUS EXAMPLES. 171 112. At $.347 a pound, how much will 9 Ibs. of tea cost ? 113. At $.375 a pound, how many pounds of tea can be bought for $5.25 ? 114. If 11.23 pounds of tea cost $6.75, what is the price per pound ? 115. At $5.625 a 100 feet, how many feet of boards can be bought for $36.5625 ? 116. At $5.625 a 100 feet, how much will 1000 feet of boards cost? 117. What part of a pound is || of a pennyweight ? 118. What part of a grain is -fa of a dram ? 119. What part of a bushel are 1 peck, 5 qts. , 1 pint ? 120. What part of a pint is .025 of a gallon ? 121. What part of a mile are 110.4 rods ? 122. What part of a mile are 5 fur. 20 rods (decimal)? 123. What common fraction is equal to .625 ? 124. What decimal fraction is equal to % ? 125. What whole numbers are equal to f of a mile ? 126. From % of a hhd. subtract % of a barrel. 127. At $3.50 per gal., how much will 37 gals., 2 qts., 1 pt. of wine cost ? 128. If 25 gals., 1 qt., 1 pt. of wine cost $78.6625, what is the price per gallon ? \- EXERCISE YL 129. How many gals, are 8%, 11%, 9|, 6^, 1Q&, 6| gals.? 130. A man having 137% acres, sold 25f , how many acres had he left ? 131. At $26^ an acre, how many acres of land can be bought for $1000 ? 132. At $26^ an acre, what will 33> acres of land cost ? 133. If 33> acres of land cost $1666%, what is the price per acre? 134. What is the sum of 9 and 9 tenths ; 10 and 12 thous- andths ; 100 and 1 hundredths ; 1000 and 1 thousandths ; 10000 and 1 ten thousandths ? 172 FRACTIONAL COMPOUND NUMBERS. 135. From 300. subtract 3 Imndredths. 136. At SI. 75 a bushel, how many bushels of peaches can be bought for $25 ? 137. At $1.75 a bushel, how much wiU 20^ bushels of peaches cost ? 138. If 20} bushels of peaches cost $36.90, what is the price per bushel ? 139. At $7.20 a 1000, what cost 19.625 bricks ? 140. At $7.20 a 1000, how many bricks can be bought for $148.50 ? 141. If 20000 bricks cost $140, what is the price per $1000 ? 142. What part of a barrel is -ffi, of a gill ? 143. What part of a nail is ^ of a yard ? 144. What part of a cubic foot is .0005 of a cord ? 145. What part of an acre are 3 E. 14 sq. rds. 16 sq. yds. 4 sq. ft. and 72 sq. in. ? 146. What part of a cwt. are 3 Ibs. 11 oz. 3.2 dr. (decimal)? 147. What common fraction is equal to .35 ? 148. What decimal fraction is equal to f ? 149. What whole numbers are equal to .375 of an ounce ? 150. What whole number is equal to f of a month ? 151. Add % of a ton and of a cwt. 152. At $4 a yd., how much cost 4 yds. 3 qrs. 1 na. of cloth ? 153. At $5.25 a yard, how much cloth can be bought for $34.125? EXEECISE VH. 154. How many miles are 7>, 3}, 6), 4, 8 T ^, 12^ miles ? 155. Sold a carriage for $175}^, and gained $14% ; what did it cost ? 156. At $4)^ a bushel, how much will 7f bushels of clover seed cost ? 157. At $4} by 2^ ft. ? 253. How much will it cost to pave a street ^ mile long and 2) rods wide, at $12 a square rod ? 254. If a piece of land containing 5.5 acres be divided into building lots 4 rods long and 2.2 rods wide, what would they all be worth at $200 each ? 255. How many shingles will cover the roof of a house 32 ft. long the rafters on each side being 16% ft. long, allowing one " shingle for every 24^ sq. in. ? 256. How much will it cost to dig a ditch around a garden 6.5 rods square, the ditch to be 3.25 feet deep and 2.5 feet wide, atl ct, a cubic foot ? DUODECIMALS. Arti 127. Duodecimals are a species of fractional com- pound numbers sometimes used in measuring lumber, &c. They arise from successive divisions of 1 foot by 12. (Latin, duodecim). 178 DUODECIMALS. TABLE. 12"" (fourths) make 1" (third) = r ^ ft. 12'" (thirds) 1" (second) = T ^ ft. 12" (seconds) 1' (prime, or inch) = ^- ft. 12' (inches) 1 (foot) The marks used to distinguish the different denomina- tions are called Indices* Duodecimals may be added, subtracted, multiplied and divided, like other compound numbers. In multiplication of duodecimals by duodecimals Feet multiplied by feet give square feet. Square feet multipled by feet give cubic feet. Feet multiplied by inches (as 1XA) give sq. inches, ^ sq. ft. Square feet multiplied by inches (as iXiV) gi ye inches, ^ cubic foot. Inches multiplied by inches (as -^XA) give square inches, or seconds, T | ? square foot. Square inches multiplied by inches (as T^XT^) give cubic sq. in. or seconds, T |^ cu. ft. Inches multiplied by seconds (as iVXyl?) gi ye thirds d? 1 ^) sq. foot. Square inches multipled by seconds give thirds (77^) cu. ft. Seconds multiplied by seconds give fourths, &c. EXAMPLE 1. What are the contents of a board 10 feet 6 inches long, and 2 feet 3 inches wide ? ft. in. 10 6 Process -6X3'=^XA=i L 4 V=18''=r 6". Write J_ 6" and carry 1'. Next 10 feet X 3'=10X-&=ff and 2 76 1' makes ft=Zfa=& feet 7 inches. Then 6'X5 ft. , &c. 52 6 55 ft. 1' 6" RULE. Multiply each term in the multiplicand by each term in the multiplier, giving each product an index equal to the indices of both its factors ; then after reducing and carry- ing, as in compound numbers, add the like terms of the pro- ducts. DUODECIMALS. 179 EXAMPLES. (2.) Multiply 8 feet four inches by 3 feet 9 inches. (3.) 9 ft. 6 in. by 2 ft. 8 in. (4.) 12 ft. 10 in. by 4 ft. 3 in. (5.) 10 ft. 8 in. by 5 ft. 2 in. (6.) 15ft. 5 in. by 3 ft. 4 in. (7.) 13ft. Tin. by 6 ft. 5 in. (8.) 14ft. 8 in. by 7 ft. lin. (9.) lift. 9 in. by 9 ft. 3 in. (10.) 16ft. 4 in. by 3 ft. 3 in. 11. How many sqiiare feet are there in a board 14 feet 9 inches long, and 2 feet wide ? 12. How many square feet in a board 16 feet 8 inches long, and 1 foot 10 inches wide ? 13. How many square feet in a door 6 ft. 6 inches long, and 3 feet 4 inches wide ? 14. How many square feet in a floor 18 feet 10 inches long and 15 feet wide ? 15. How many square feet in a piece of molding 20 feet long and 3 inches wide ? 16. How many square feet in 20 boards, each 12 feet long and 9 inches wide ? 17. How many cubic feet in a stick of timber 10 feet long, 1 foot 3 inches wide, and 4 inches thick ? 18. How many cubic feet in a load of wood 8 feet long, 4 feet 6 inches high, and 3 feet 10 inches wide ? 19. How many cubic feet in a block of marble 6 feet 8 inches long, 2 feet 6 inches wide, and 2 feet thick ? 20. How many cubic feet in a wall 21 feet 6 inches long, 6 feet 3 inches high, and 2 feet thick ? 21. What will it cost to plaster a room 24 feet 6 inches long, 15 feet 5 inches wide, and 8 feet 4 inches high, at 36 cents a square yard ? 22. How many bricks 8 inches long, 4 inches wide, and 2 inches thick, will it take to build a wall 72 feet long, 4 feet 6 inches high, and 1 foot thick, supposing the bricks not to be separated by mortar ? 180 ANALYSIS. ANALYSIS. Art. 128. Analysis in Arithmetic is a method of solv- ing questions without formal rules. Rules are derived from analysis. The process consists generally in reason- ing from a given number to 1 of the same kind, and from 1 to the required number. EXAMPLE 1. If 7 pounds of sugar cost $1.12, how much will 42 pounds cost ? Process. If 7 Ibs. of sugar cost $1.12, 1 Ib. will cost | of $1.12, or 16 cts., and 42 Ibs. will cost 42 times 16 cts., or $6.72, Ans. Or since 42 Ibs. is 6 times 7 Ibs., 42 Ibs. will cost 6 times $1.12 (the price of 7 Ibs.), and $1. 12X6=$6. 72, Ans. Ex. 2. If | of a' yard of cloth cost $4, how much will I of a yard cost ? Process. If yard cost ($4) $|, yard will cost $f, and f, or 1 yard, will cost $6. Then yard will cost $|, and I will cost , or $ 4|, Ans. In examples like the last it is better to express each multiplication by writing the multiplier as a factor in the numerator of a fraction, and each division by writing the divisor as a factor in the denomina- tor, then cancel, &c., thus : 2 Ex. 3. Barter. How many loads of wood at $4 will pay for 3 barrels of flour at $8f ? Process. If 1 bbl. of flour costs $8f, 3 bbls. will cost$2o|-, and if $4 will pay for 1 load of wood, $26 will pay for as many loads as $4|r is contained times in $26^, which is 6. Ans., 6 loads. Ex. 4. Aliquot parts or Practice. What cost 5 cwt. 65 Ibs., at 2 5s. 6d. per cwt. ? Process. 50 lbs.=i cwt. 2 5s. 6d. 5 1176 price of 5 cwt. 10 Ibs. = of 50 Ibs. 129 " 50 pounds. 5 Ibs. =i of 10 Ibs. 4 6f 10 " 2 3-,3 q 5 " Ans.. 12 17 Oft ANALYSIS. 181 Ex. 5. A general lost % of his army in battle, % were taken prisoners, deserted, and he had 2600 men left ; how many had he at first ? Process. iXiXi=$fc nd tne remainder =2600. &=200, 18= 12000, the army at first. Analysis may also be applied to questions under rules to be here- after given, such as Proportion, Partnership, Reduction of Currencies, Alligation, &c. EXAMPLES. 6. If 20 barrels of apples cost $50, what will 35 bbls. cost ? 7. If 33 tons of coal cost $198, how much will 11 tons cost? 8. If 15 pounds of butter cost $4.50, how much will 70 Ibs. cost? 9. If 12 pairs of shoes cost $30, how much will 48 prs. cost ? 10. How much will 65 sheep cost if 5 sheep cost $17.50 ? 11. How much are 50 cows worth, if 10 cows' are worth $450? 12. If % of an acre of land cost $66%, how much will 6% acres cost ? 13. How much will f of a ton of hay cost, if f of a ton costs $12? 14. How much will ^- of a cord of wood cost, if ^- costs $1.12^ ? 15. If f of a pound of tea costs $^, how much will f of a pound cost ? 16. If f of a yard of cloth costs $4.80, how much will of a yard cost ? 17. If | of a cord of wood cost $1.10, how much will f of a cord cost ? 18. How much will ^ of a ton of plaster cost if f of a ton ' 9 19. How much will f of an acre of land be worth if ^ of an acre is worth $30 ? 20. How much will if of a chain 30 feet long cost if of a like chain 36 feet long is worth $24 ? 182 ANALYSIS. 21. How many eggs at 20 cents a dozen must be given for 9 pounds of butter at 30 cents a pound ? 22. How many pounds of lard, at 15 cents a pound, will pay for 14 pounds of sugar, at 12) cents a pound ? 23. How many bushels of oats, at 37^ cents a bushel, will pay for 5 yards of cloth, at $4.50 ? 24. How many yards of calico, at 18% cents a yard, can be bought for 10 pounds of butter, at 31)^ cents ? 25. At $87.50 an acre, what cost 8 acres 110 rods ? 26. At $6.75 a ton, what cost 7 tons 12 cwt. 60 Ibs. of hay ? 27. At $5.37^ a yard, what cost 3 yards 3 quarters 3 nails of cloth ? 28. At 2 11s. 6d. a bushel, what cost 5 bushels, 1 peck, 4 quarts of timothy seed ? 29. At 10 12s. 6d. an acre, what cost 9 acres 60 rods ? 30. At 2 8s. 6^d. a cwt., what cost 7 cwt. 30 pounds of flour? 31. A regiment of soldiers was diminished ^ by sickness, captured by the enemy ; ^ killed and missing, and then con- sisted of 250 men ; how many did it number at first ? 32. A young man spent -| of his property in 3 years, ^ of it the next 2 years, and then he had $600 left ; how much had he at first ? 33. Paid $2,100 for ^j of a vessel, what was the whole vessel worth? 34. If 18 Ibs. of cheese cost $2.70, what will a cheese weighing 72 Ibs. cost ? 35. If f of hogshead of sugar cost $34, what will f of a hogshead cost ? 36. If of a gallon of alcohol cost $3f , what will T \ of a gallon cost ? 37. How many bushels of corn at 90 cents a bushel, will pay for 63 Ibs. of beef at 14 cents a pound ? 38. At $5.38 a cwt., how much will 60 Ibs. of flour cost ? 39. If a barrel of ale cost 7 14s. 4d., what will 24 gallons 1 quart cost ? ANALYSIS. 183 40. There is a town in which the men are farmers, me- chanics, | laborers, and the rest 26 without employment ; how many men are there in the town ? 41. How many pounds of pork at 12^ cents a pound, will pay for 7 days labor, at $1.121 a day ? 42. If 30 pounds of coffee cost $7.50, what will 9 Ibs cost. 43. If % of a cord of wood is worth $3, how much is j~| of a cord worth ? 44. How much will of a pound of soap cost, if f of a pound cost 12 cents ? 45. At $94 an acre, what will 50 square rods cost ? 46. How much will 29 gals, of vinegar cost at $10}^ a barrel ? 47. A man left his elder son % of his property, the younger YD and the elder son had $1000 more than the younger ; what was the whole property? 48. If % of a ship is worth $56,000, what is ^ of it worth ? 49. What is \ of a bushel of clover seed worth, if f of a bushel is worth $2f ? 50. If a man earn $42 in 12 days, how much will he earn in 50 days ? 51. How many days' labor at $.87> a day will pay for 7 bushels of buckwheat at 75 cents a bushel ? 52. At $21. 31 1 a cwt., how much will 86 Ibs. of honey cost ? 53. If a year's labor is worth 100, how much will it be for 7 months and 20 days ? 54. A benevolent lady gave % of her income to the Bible Society, and f of it to the poor, reserving only $600 for herself, what was her income '? 55. If | of a ton of hay cost f of $9%, how much will ^ of a ton cost ? 56. How many barrels of apples at $2)^ a barrel, will pay for 4 bbls. of flour at $9^? 57. What will ^ of a yard of velvet cost, if f of a yaid cost $1.08 ? 184 PERCENTAGE. 58. What will 15 yards of calico cost, if 9 yds. cost $1.62 ? 59. At $4.50 a yard, what will 3 yds. 1 qr. 3 na. cost ? 60. A lady gave )^ of her property to her son, y z to her daughter, and the rest, amounting to $1,200, to benevolent objects, what was the amount of her property ? PERCENTAGE, Art. 129. Percentage is calculating numbers by hun- dredths, or parts of a hundred. Per cent, (derived from the Latin words per centum,, meaning by the hundred) is used in expressing hun- dredths, or parts of a hundred ; thus, 5 per cent, is 5 hundredths, or five for every hundred (dollars, pounds, &c. ) ; 6 per cent, is 6 hundredths. The sign % is often used for per cent. Art. 130. In Percentage three things are chiefly con- sidered. The Principal, the number on which percentage is calculated. The Rate per cent., the number of hundredths. The Percentage, the number which the principal pro- duces at a given rate. Any two of these being known, the other maybe found. The term Principal thus used includes, but is not limited to money at interest. The rate per cent, is expressed by a fraction, usually a decimal fraction, thus: 1 per cent, is written 01 = y^ 5 per cent, is written .05 = y^ or ^ 10 per cent, is written 10 = ^ 25 per cent, is written 25 = 4 PERCENTAGE. 185 100 per cent, is -written 1.00 = the whole. y 2 per cent, is written 005 or .00>. % per cent, is written 0025 or .00^. Some rates per cent, cannot be exactly expressed by decimals ; as ^ per cent, must be written .00^ ; 33^ per cent. .33^. Write the following rates per cent. 3 per cent.; 6, 4, 12, 7, 8, 15, 9, 20, #, %, %, J, %, 2^, 8X, 18%, 3f, %, 37^, i 33^, %, %, 6#, 12^, 10|, 75, 110, 125, 137%. CASE I. Art t 131. To find the percentage. MENTAL EXERCISES. How much is 3 per cent, of $5 ? Process. Since 3 % is .03, 3 % of $5. is .03 times $5, which is .15 cents. Therefore 3 % of $5. is 15 cents. How much is 4 % of $1 ? $10 ? $12 ? $8 ? 50 cents ? How much is 5 % of 4 pounds ? 20 Ibs. ? 50 Ibs. ? 100 Ibs. 9 1 If a miller take 4% toll for grinding. wheat, how much will it be for grinding 100 bushels ? 200 ? 50 ? 25 ? EXAMPLES FOB THE SLATE. EXAMPLE 1. What is 6^ % of 568 pounds ? 568 .0625 Process. 6 % = .0625, therefore 6 % of 568 ooJn Ibs. = 568 X-0625 = 35. 5000 = 35 Ibs. 1:1 3 6 3408 ^ns735T5000 Ibs. Or <*K=i$ = n * i of 568=351. RULE. Multiply the principal by the rate per cent. Percentage = Principal X Rate. 2. What is 3 % of $8750 ? 6^? 7%? 3. What is 5 % of 364 gallons ? 7 %"? 12 % ? 4. What is 6% of 576 pounds? 8% ? 16%? 5. What is 7 % of 368 bushels ? 9 % ? 4)^ % ? 6. What is 8K % of 261 gallons ? f % ? 16% % ? 186 PERCENTAGE. 7. A farmer has 320 bushels of wheat, 25 % more of oats, and 12)^ % less of corn ; how many bushels of oats has he ? Of corn ? 8. A regiment consisted of 840 soldiers, of whom 16% were killed and missing ; how many were left ? 9. A merchant having $6400 capital, gained 18% % ; how much did he gain ? 10. Another merchant, having $7200 capital, lost 12} % of it ; how much did he lose ? 11. A young man, having $1240, spent 6^ % of it for clothes and board, 10 % of it for a horse, and 12^ % of it in travel- ing ; how much had he left ? 12. A grocer bought 450 pounds of coffee, and found that 10 % of it was damaged ; how much of it was good ? 13. A flock of 175 sheep increased 20 %, how large was it afterwards ? CASE H. Arti 132. To find what per cent, one number is of another. EXAMPLE 14. What per cent, of $10 is $2.50 ? Process. Since $2.50 is a certain per cent, of jn\2 59 $10, the same per cent, of SI. is A- of $2.50, sf which is $.25=25%. Therefore $2.50 is 25% Ans., .25 = 25 %. of $10. RULE. Divide the number which is the percentage by the other number. Eate = Percentage -f- Principal. 15. What % of $75. is $5 ? $10 ? $20 ? $25 ? 16. What % of $87.50 is $5.25 ? $7.87^ ? 17. What % of $60. is $6 ? $9 ? $12 ? 18. What % of $56. is $7 ? $14 ? $21 ? 19. What % of $150 is $9 ? $15 ? $50 ? 20. What % of $200 is $8 ? $16 ? $24 ? $50 ? 21. What % of $1000 is $50 ? $60 ? $75 ? 22. A farmer raised 500 bushels of wheat and kept 50 bushels for family use ; what per cent, of it did he keep ? PROMISCUOUS EXAMPLES. 187 23. A regiment consisting of 900 soldiers lost 75 of them in a battle ; what per cent, was it ? 24. A young man having $350, has spent 850 of it ; what per cent, of it has he spent ? 25. A grocer bought 560 gallons of molasses, and found that 56 gallons had leaked out ; what was the percentage ? 26. A miller took 15 bushels of corn for grinding 300 bush. ; what per cent, was it ? CASE IIL Art , 133, To find the principal when a certain per cent, of it is known. EXAMPLE 27. A man gave $50 to benevolent objects, which was 10 % of his income ; what was his income ? Process. Since $50 is 10 % of his income, his income $ cts. was as many dollars as .10 is contained times iu 50, . 10)50 . 00 which is 500. Therefore his income was $500. .Ans. $500. Or, since $50 is ^ of his income, the whole of it is $50X10=4500. RULE. Divide the given percentage by the rote per cent. Principal percentage -f- rate. 28. $5 is 6 % of how many dollars ? 29. 12 pound is 10 % of how many pounds ? 30. 15 gallons is 25 % of how many gallons ? 31. 37% bushels is 18% % of how many bushels ? 32. $75 is 5 % of how many dollars ? 33. A man pays an income tax of 4 % amounting to $84 ; what is his income ? 34. A debtor, whose property is worth $4,500 is able to pay only 75 % of what he owes ; how much does he owe ? 35. A man wishes to leave his daughter an income of $1,000 a year ; what sum must he invest for her at 7 %. 36. A miller has taken 1 barrel of flour for toll at 2^ %\ how many barrels of flour has he ground ? 37. A regiment of soldiers lost 60 men in a battle which was 10 % of their whole number ; how many belonged to the regi- ment before the battle ? 188 PROMISCUOUS EXAMPLES. CASE IV. Art. 134* To find the principal, when being increased or diminished a certain per cent., the sum or remainder is known. EXAMPLE 38. A man, whose property has increased 50 % is now worth $15,000 ; what was he worth before ? Process. Since his property has increased 50 % $1.50)15000.00 it was formerly as many dollars as it is now times >*^ % less than he expects to sell next year ; how much did he sell last year and how much does he expect to sell next year ? 47. A farmer having 150 tons of hay expects to sell 90 tons ; what per cent, is that of the whole ? 48. A farmer has sold 75 tons of hay which is 60 % of all he had ; how much did he have ? 49. A liquor dealer bought a hogshead of rum and mixed 33i/j % of water with it ; how much did it make ? 50. At another time he mixed 75 gals, of water with 225 gals. of brandy ; what per cent, of the mixture was water ? 51. At another time he filled up a cask containing ale with 20^ gals, of water which was 37)^ % of what the cask would contain ; how much did the cask hold ? 52. At another time he mixed some wine with 15 % of water and then had 57^ gallons ; how much wine was there ? 53. In a certain town the population is 2750, and 4 % are col- ored ; how many colored persons live in the town. 54. In another town there are 2040 whites and 360 blacks ; what per cent of the whole population are black ? 55. In another town 8% % of the population are colored, of whom there are 136 ; what is the whole population ? 56. In another town the population has increased in 10 years 127) %, and is now 2275 ; what was it ten years ago ? 57. The expenses of a family are 18% % greater this year than the last, when they amounted to $963. 75 ; how much are they this year ? 58. A debtor whose property is worth $6,750, is able to pay only 67^ % of what he owes ; how much can he pay ? 59. A man worth $10,000 has invested $2,520 in government bonds ; what per cent of his property is thus invested ? 60. A gentleman traveling, having spent 75 % of his money found that he had $75 left ; how much had he at first ? 190 APPLICATIONS OF PERCENTAGE. Applications of Percentage. Art. 136. Percentage is applicable to Commission and Brokerage, Stocks and Gold at a Premium, Insurance, Profit and Loss, Interest, Discount, Taxes, Duties, Part- nership, Bankruptcy, Exchange, &c. Some of these are so much like percentage that they scarcely need to be separately treated except in a few particulars. Art. 137. Commission is the percentage paid to a commission merchant or agent doing business for another. It is calculated the same as percentage. A consignment consists of goods sent to a person to sell on commission. The gross proceeds are the whole amount of the sales. The net proceeds are what is left after de- ducting the expenses. Art. 138, An Account Of Sales is a written statement of goods sold on commission, with the prices, gross and net proceeds, &c. ; as SALES OF PRODUCE CONSIGNED BY THOS. FAY & Co., DETEOIT. 1867. Sold to Produce. Price. Aug 1 Rogers & Son Flour, 20 bbls. . $11. 00 $ cts. 220 00 " 8 11 25 C. Jones & Co T. Agnew & Co. ... Wheat, 300 bu... 2.50 Corn, 500 bu... 90 750.00 450.00 Charges. $1420.00 Freight on 20 bbls @ 50 cts. $10.00 800 bu @10ctP- 80.00 Cartage and Storage 12.50 Commission on $1240 @ 2)4 % 31. 95 124 . 45 Net proceeds TTT~ $1295.55 SMITH, MYGATT A- CO. New York, Aug. 31, 1867. BEOKEEAGE STOCKS. 191 Art. 139. Brokerage is the percentage paid to brokers. It is sometimes called discount. A. Broker is one who exchanges or loans money, buys and sells stocks, also goods not in his own possession. Art. 140. Stocks are money or property invested in. Banks, incorporated or chartered Companies, Bonds, &c. They are divided into shares, usually of $100 each, for which Certificates or Scrip is issued, liable to be bought or sold. When stocks sell for what they originally cost, they are at par ; when they sell for more, they are above par, at a premium or advance ; and below par, or at a discount, when they sell for less. The premium or discount is a certain percentage on the par value, to be added or subtracted from it, in find- ing the market value. The market value of any number of shares is found by multiplying it by the market value of a single share. Stocks are quoted : at par, 100 ; at a premium of 1 % y 101; 2J& 102|; 18|, 118| ; at a discount of 5 %, 95 ; 12*&87*; 25^,75, &c. Stockholders are the owners of stock. A Dividend is what is paid to stockholders as their part of the profit or gain. Bonds are securities for money loaned, bearing inter- est, issued by Corporations or Governments. The United States have issued the following bonds : U. S. 5's, paying 5 % interest in gold, and payable in 1871 and 1874. U. S. 6's, paying 6 % interest in gold, and payable in 1867.. 1868, and 1881. U. S. 5-20's, paying 6 % interest in gold, and payable in 5 to 20 years. 192 APPLICATIONS OF PERCENTAGE. U. S. 10-40's, paying 5 % interest in gold, and payable in 10-40 years. U. S. 7-30's, paying 7^, or 7.30% interest, in currency, and payable in three years from their date. Art. 141. Gold 9 at a premium, is bought and sold the same as stocks. Art. 142. Insurance is security against loss. Fire Insurance is security against loss by fire ; Marine Insurance, against loss on the ocean, &c. Insurance, also, secures a certain allowance in case of accident, sickness, or death. The last is called Life Insurance. The Policy is the written contract. The Premium is a certain percentage on the amount insured. Art. 143. Promiscuous Examples in Commission, Brokerage, Slocks, Gold, and Insurance. [These examples are to be done the same as others in Percentage.] The amount bought or sold, collected, invested or in- sured, or the par value of stocks and gold, is the principal ; to be found, if required, by Case III. in Percentage ; or Case IV. when the percentage is to be deducted from the given sum, or has been deducted from the required sum. The per cent, is the rate ; to be found, if required, by Case n. in Percentage. The commission, brokerage, dividend, premium, or dis- count, is the percentage ; to be found, if required, by Case I. in Percentage. Ex. 1. What is the commission for selling goods amounting to $2500, at 3^ per cent. ? * Process, the same as in Percentage, Case L PKOMISCUOUS EXAMPLES. 193 2. A commission merchant received $87.50 for selling goods amounting to $2500 ; what per cent, was his commission ? Process. Percentage, Case II. 3. A commission merchant received $87.50 for selling goods at 3- per cent. ; what was the amount he sold ? Process. Percentage, Case HI. 4. A commission merchant received $2587.50 for the pur- chase of goods, after deducting 3-^ per cent, commission ; what was the amount of the goods he purchased ? Process. Percentage, Case IV.; commission to be deducted. 5. A commission merchant, after deducting 3 % commis- sion from the whole sum he had received, had a balance of $2500 for the purchase of goods ; what was the whole sum he received ? Process. Percentage, Case IV. ; commission already deducted. 6. A broker in New York exchanged $1500, uncurrent money, at ^ per cent, discount ; how much was his brokerage ? 7. A broker has $5012^ to invest in bank stock, after de- ducting \ % for brokerage ; how much is to be invested ? 8. A merchant gave a broker $1000 uncurrent money, and received from him $990 current money ; what per cent, was the brokerage ? 9. A merchant gave a broker $10 for exchanging some un- current money, at % % ; what was the amount ? 10. A broker, after deducting % % for brokerage, paid back $1985 current money ; how much uncurrent money had he received ? 11. What is the value of 12 shares of railroad stock, at a premium of 5 % ? 12. What is the value of the same at 5 % discount ? 13. How much stock, at 5 % premium, can be bought for $5250 ? 14. How much stock, at 5% discount, can be bought for $4750? 194 APPLICATIONS OF PERCENTAGE. 15. When gold is 125, how much is $500 in gold worth in paper currency? 16. When gold is 125, how much is $500 in paper currency worth in gold ? 17. What must be paid for insuring a house valued at $2500, at 1} % premium ? 18. Paid $31.25 for insuring a house valued at $2500 ; what per cent, was the premium ? 19. Paid $37.00 for insuring a house, at 1 % premium; what was the amount of the insurance ? 20. For how much must a house, valued at $2500, be in- sured, at \% %, to cover both its loss by fire and the cost of the insurance ? 21. What is the commission for selling goods amounting to $6500, at 3 % ? 22. What is the brokerage on $1500 uncurrent money, at #x? 23. An agent has received $1230 for the purchase of goods, after deducting 2)^ % commission ; what will be the amount of the goods, and what his commission ? 24. How much money at 1 % discount will pay a note for $1500 ? 25. What is the value of 10 shares of bank stock at a pre- mium of 12 % ? 26. How many shares of railroad stock 10 % below par will pay a debt of $1800 ? 27. What must be paid for the insurance on a store and goods valued at $18000, at 3 per cent ? 28. How much current money should a broker give in ex- change for $560, at y z per cent, discount ? 29. What will be the amount of a bill of goods to be pur- chased with the balance of $5616, received by remittance, after deducting 4 % commission ? 30. How much are 15 shares in an Insurance Company worth at 6^ % advance ? 31. What is the premium on an insurance amounting to $5620, at PROMISCUOUS EXAMPLES. 195 32. When gold is 118%, how much in currency should be received for $60 interest on a Government bond, payable in gold? 33. An agent in Chicago has bought grain for a flouring mill in New York amounting to $3500 ; what is his commis- sion, at 3^ per cent. ? 34. What should a broker give in exchange for $640 uncur- rent money, at 3% % discount ? 35. A Southern merchant has remitted to his agent in Phil- adelphia $2370}^, for the purchase of goods, after deducting his commission, at 3^ % ; what will be the amount of the goods to be purchased ? 36. What should a broker deduct from the amount of a draft for $850, taken in exchange for currency, at % % dis. ? 37. What are 18 shares of bank stock worth at 16% % adv. ? 38. What is the premium on the insurance of a house val- ued at $2800, at 4> % ? 39. When gold is 125, how much of it will pay a debt of $600? 40. A gentleman wishes a horse-dealer to purchase for him a span of horses worth $900, and is willing to allow him 3% % commission ; what will the horses cost him ? 41. A gentleman gave a horse-dealer $806 for a span of horses and his commission at 4 % ; what was paid for the horses alone ? 42. What is a draft for $480 worth at % % discount ? 43. What are 6 shares in a steamboat company worth at a premium of 100 per cent. ? 44. When gold is 1.27%, how much currency will pay the duties (payable in gold) on a bill of imported goods amount- ing to $3750.50 ? 45. What must be paid for insuring a steamboat valued at $75000, at 6> per cent. ? 46. The insurance of a steamship, valued at $250000, costs $10000 ; what per cent, is the premium ? 47. A merchant in San Francisco has remitted to his agent im Hew York $4888 in gold for the purchase of goods, after 196 PROFIT AND LOSS. deducting 4 % commission ; gold at the time being 125 ; what is the commission ? 48. What are 10 shares of the Corn Exchange Bank worth at a premium of 16% % ? 49. Bought a house lot for $10000 when gold was at par ; what ought it to be worth, gold being $130 ? 50. A merchant had paid 3}^ % for insurance on his stock of goods at $15000, for 10 years. After that time they were destroyed by fire ; what did he gain by having them insured ? PROFIT AND LOSS. Art. 144. Profit (or Gain) and Loss are usually esti- mated at a certain percentage on the cost (principal.) The profit or loss, at any rate per cent, is found by Case I. in Percentage. The per cent, of profit or loss by Case II. The cost by Case III. or IV. The profit or loss = cost X P er cent. The per cent, of profit or loss == the profit or loss -r- cost. -I . the profit or loss -r- per cent, or The COSt = J %. gain the selling price -r- 1 or per cent, or The profit or loss is the difference between the cost and selling price. The selling price is the cost with the profit added or the loss subtracted. When the rate per cent, is an aliquot part of 100, it is often more convenient to express it in that form ; as 25 = J EXAMPLES. 1. When cloth costs $4 and is sold at 25 % profit, what is the profit and selling price ? EXAMPLES. 197 Process as in Percentage, Case I. S4X-25, or = $1.00 profit, which added to the cost is *5.00 the selling price. 2. When the cost of cloth is $4 a yard and the selling price $5.00, what per cent, is the profit ? Process. 55 84= SI. (gain) then as in Percentage, Case IE. $14- $4 = .25, or 25%. 3. When cloth is sold at 25 % profit and the profit is Si a yard, what did it cost ? Process as in Percentage, Case m. Sl.00-j-.25 or = S4.00 (the cost. ) 4. When cloth is sold for $5.00 a yard and 25% is thus gained, what did it cost ? Process as in Percentage, Case IV. $5. 00 -j- $1.25 = $4.00 the cost. The prices in the following examples may be considered as the prices per yard, pound, &c., &c. 5. Cost 10 cents ; profit 20 % ; selling price is what ? 6. Cost 15 cents ; selling price 18 ; gain per cent, is what ? 7. Profit 5 cents ; per cent. 20 ; cost is what ? 8. Selling price 20 cents ; loss 25 % ; cost is what ? 9. Cost $1. ; selling price Si. 25 ; gain per cent, is what ? 10. Profit 20 cents ; per cent. 15 ; cost is what ? 11. Selling price 35 cents ; profit 16% % ; cost is what ? 12. Cost 32 cents ; profit 12) % ; selling price is what ? 13. Cost $3; profit 50 cents ; per cent, is what ? 14. Cost $300; loss $75 ; per cent, is what ? 15. Selling price $75; loss 25 % ; cost is what ? 16. Profit $625; per cent. 12} ; cost is what ? 17. Cost $150; profit 16% % ; profit is what ? 18. Cost $5.20 ; profit 10 % ; selling price is what ? 19. Selling price $3612> ; loss % % ; cost is what ? 20. Cost $875; selling price $1050 ; per cent, is what ? 21. Cost $1000 ; profit $70; gain per cent, is what ? 22. Cost $1000 ; loss 7 % ; selling price is what ? 23. Loss $120; per cent. 6 ; cost is what ? 24. Profit $300; per cent. 6 ; cost is what ? 25. Selling price 6^ cents ; cost 6 cents ; per cent, is what ? 198 INTEREST. 26. Selling price $5.25 ; cost $6; loss per cent, is what ? 27. Cost $2000 ; profit 18% % ; selling price is what ? 28. Bought a horse for $175 and sold him for $210 ; what was the profit per cent. ? 29. Sold a house for $2850, 5 % less than cost ; what did it cost? 30. Bought a piece of land for $1500 and sold at an ad- vance of 33}^ % ; what was the selling price ? 31. Gained in trade $1600, which 16% % of the capital en> ployed ; what was the capital ? 32. A wool merchant bought 36000 Ibs. of wool at 56 cents a pound, the expenses on it were $108, and he sold it for 62 y 2 cts. a pound ; how much per cent, did he gain ? 33. If 16 cwt. 31 Ibs. of sugar cost $203.875, for what must it be sold per pound to gain 24 % ? 34. A grocer sold tea for $.75 a pound, and lost 12 % % ; what did it cost ? 35. A grocer sold tea for $1 a pound, and gained 33 V^ % ; what did it cost ? 36. A grocer gained 12} cts. a pound on tea, which was 25 % ; what did the tea cost per pound ? INTEREST. Art. 145. Interest is a certain percentage paid for the use of money for a specified time. The principal is the money lent, or loaned. The rate is the per cent, paid annually or per annum. The amount is the principal with the interest added to it. Simple interest is the interest on the principal only. Compound interest is on the principal and interest al- ready due. When only the word interest, is used, simple interest is meant. Legal interest is at the rate fixed by law. In all the United States it is 6 per cent, except New York, Michigan, INTEREST. 199 Wisconsin, Minnesota, S. Carolina, and Georgia, 7 % ; Louisiana, 5 % ; Florida, Alabama, Mississippi, and Texas, 8 % ; Kansas and California, 10 % ; Oregon, 12^ %. EXAMPLE 1. What is the interest of $125.50 for 2 years 5 months and 21 days, at 6 % ? Process. Since the interest of $1.00 for a year is $.06 (6 cts.) the interest of $125.50 for 1 year is (.06 times $125.50 or 125.50 times $.06) $7.53 ; for2 years ($7.53X2) $15.06 ; for 5 months (fa of $7.53) $3.138 and for 21 days (ffc or -fa of $.628 the interest for 1 month or -iV of a year) $.439. Therefore, the interest for the whole time is $18.637. It is sufficiently accurate to count mill or more as 1 mill, and re- ject less fractions. $125.50 .06 7.53 =1 year. 2 15.06 =2 years. ^ year's 2 . 01 =4 months. 1 4 month's . 628 =1 month. month's .314 =15 days. ^15 days .105 =5 days. 15 days .021 =1 day. $18.638 whole time. A $125.50 .06 7T53~ 2 15.06 3.138 .439 AM. $18.637 int. 1 year. " 2 years. ; 5 months. 21 days. whole time. RULE. To find the interest for one year, multiply the prin- cipal by the rate. For two or more years, multiply this product by the num- ber of years. For months take aliquot parts of a year's interest. For days take aliquot parts of a month's interest. The above rule is applicable to any rate ; but at 6 % the following method is preferable : Art, 146. At 6 % the interest of $1 being $.06 a year or 12 months, it is half as many cents as months. Hence the interest of $1 for 1 month $.005 2 months $.01 3 months $.015 6 months $.03 9 months $.045 11 months $.055 200 INTEREST. Also, the interest of $1 for one month (30 days) being $.005, it is 1 mill for every 6 days, or as many mills as days. EXAMPLE 1. By second method. Process. The interes 2 years= 5 montlis= 21 days = whole time= . . . ; of $1. for $.12 $.025 $.0035 Therefore \ $125.50 .1485 62750 100400 50200 12550 $.1485 6)18. 636750 at 6 % 3. 106125 at \% I $21. 742875 at 7% The interest at 6 % divided by 6 is the interest at 1 %, and this multiplied by 7 is the interest at 7 % ; or if multiplied by any other rate it would be the interest at that rate. EULE. Find the interest of $l/or the given time, allowing 6 cents for every year, half as many cents as months and as many mills as there, are times 6 days ; then multiply it by the given number of dollars. EXAMPLES. In the following examples, pupils should use both methods of cal- culating interest till they become familiar with them, then either as they may prefer, or the teacher may direct. There may be a slight difference in the answers obtained by the different methods. In business, interest is more generally computed by tables. What is the interest of $1 for (1.) 1 year 5 months 18 days, at 6% ? also of $125.00 ? (2.) 2 " 4 " 12 " " 6%? " 875.50? (3.) 3 " 7 " 15 " "6,%? " $210.25? (4) 4 " 8 " 20 " " 6%? " $87.625? (5.) 5 " 9 " 21 " "6)%? " $10.12',? (6.) 6 "1 " 3 " "6%? " $256.00? (7.) 7 "0 " 4 " "6%? " $160.50? (8.) 9 " 25 " " 6%? " $62.375? (9.) 10 "10 " 10 " "6%? " $100.00? (10.) 27 " " 6%? " $1000.00? INTEKEST. , 201 What is the interest of (11. ) 145. 00 for 1 year 3 months 6 days, at 7 % ? (12.) $130.00 " 2 " 1 " 3 " 7%? (13.) $234.50 " 3 " 8 "25 " 7 % ? (14.) $15.87K " 4 " 5 " 15 " 7 % ? (15.) $112.5614 " 5 "10 " *9 " 4)X? (16.) $215.00 "3 "11 " 13 " 5%? (17.) $321.18% " 2 " 6 " 17 " 7^? (18.) $400. " 4 " "10 " S%? (19.) $500. " 1 " 4 "22 " 5%? (20.) $650. " 3 " 3 " 3 " (21.) $36.50 " 6 " at (22.) $118.75 " 19 " 7%? (23.) $90.00 " 2 " 1 " 6 " 4>%? (24.) $124.80 " 3 " 5 " 7 " 5%? (25.) $84.30 " 1 " 6 " 12 " 6%? (26.) $100. " 4 " 7 " 18 7%? (27.) $72.70 " 2 " 2 " 2 " 7 % ? (28.) $95.50 " 1 " 1 " 1 " 6%? (29.) $100. " 4 " 8 " 7%? {30.) $1000. " 29 " Yz% e ! 31. What is the amount of $10 for 1 year 6 months at 6 % ? 32. What is the amount of $25 for 6 months 12 days at 7 % ? 33. What is the amount of $50 for 2 years 20 days at 7 % ? 34. What is the amount of $60 for 2 months 15 days at 6 % ? 35. What is the amount of $75 for 1 year 10 days at 7 % ? 36. What is the amount of $100 for 7 months 18 days at 5 %? 37. What is the interest of $120 for 2 years 9 months at 6 % ? 38. What is the amount of $150 for 3 months 3 days at 7 % ? 39. What is the amount of $1 for 100 years at 7% ? 40. What is the amount of $10 for 10 days at 6 % ? 41. What is the interest of $125.50 from July 1, 1858, to Jan. 1, 1859, at6%? 42. What is the amount of $200 from Aug. 1, 1860, to Nov. 16, 1860, at 7% 7 43. What is the amount of $137.50 from May 16, 1865, to January 1, 1866, at 6 % ? 9* 202 INTEREST. 44. What is the interest of $300 from June 1, 1864, to Feb. 10, 1865, at 7 %? 45. What is the amount of $250 from July 10, 1865, to March 1, 1866, at 6,%? 46. What is the amount of $320 from Sept. 15, 1867, to April 1, 1868, at 5%?' 47. What is the interest of $250 from July 15, 1866, to Jan. 1, 1867, at 6%? 48. What is the amount of $560 from January 1, 1867, to June 16, 1867, at7%? 49. What is the amount of $67.50 from July 21, 1866, to Sept. 1, 1866, at 6 % ? 50. What is the amount of $100 from June 15 to July 1, at 5> "? Exact Interest. Art. 147. In the preceding examples 30 days have been allowed to each month, which makes a year to con- sist of 360 instead of 365 days. To find the exact inter- est we must count the exact number of days in each month (see Table of Time, Art. 50,) and consider *each day ^5 part of 1 year. EXAMPLE 51. What is the exact interest of $36 from March 1 to June 9, at6%? $36 n/ Process. The number of days are 31 in March, 30 in April, 31 in May, and 8 in June (31+30+ 31+8,) 100. And $2.16, the interest for a year, divided by 365, gives the interest for 1 day, 365)216. 00($. 59+ which multiplied by 100, gives the interest for 1825 100 days. For convenience we multiply by 100 3350 first, and then divide by 365. 3295 55 52. What is the exact interest of $50 from July 1 to Sept. 16, at 6 % ? 53. What is the exact interest of $1000 from Oct. 10 to Jan. 1, at 7^? PARTIAL PAYMENTS. 203 54. What is the exact interest of $100 from July 1 to Aug. l,at6/V? 55. What is the exact interest of $100 from Feb. 10 to March 10, at 6 % ? if it is leap-year ? Art. 148. Interest of Sterling Money. EXAMPLE 56. What is the interest of 50 10s. 6d. for 1 year 4 months, at 5 per cent. ? 50,525 5 3)2.52625 84208 Process. ~50 10s. 6d.= (Article 121) 50.525, on 3 36833 which the interest 4 m. =% year, found as in Federal Money, is 3.36833, or (Art. 122) 3 7s. 4d. 1.5968far. 1.59680 RULE. Reduce the shillings, &c., to the decimal of pound. Then proceed "as in Federal Money, and reduce the decimals to shillings, &c. What is the interest of Ex. 57. 325 12s. 3d. for 5 years, at 6,%"? 58. 174 10s. 6d. for 3 years 6 months, at 6 % ? 59. 150 16s. 8d. for 4 years 7 months, at 6 % ? 60. 45 10s. for 2 years, at 4 % ? PARTIAL PAYMENTS, Art. 149. A partial payment is the payment of part of a note or debt bearing interest. A Note is a written promise to pay a debt ; as NEW YOEK, Sept. 1, 1867. Thirty days after date I promise to pay J. V. Peck, or order, three hundred and twenty-five -ffc dollars, for value received. SAM'L A. EOGEES. 204 PARTIAL PAYMENTS. Sam'l A. Rogers is the drawer or maker. J. V. Peck is the payee. $325.38 is the face of the note. UNITED STATES RULE. When partial payments have been made, apply the payment^ in the first place, to the discharge of the interest then due. If the payment exceeds the interest, the surplus goes toward discharging the principal, and the subsequent interest is to be computed on the balance of principal remaining due. If the payment is less than the interest, the surplus of the interest must not be taken to augment the principal, but interest continues on the former principal, until the period when the payments taken together exceed the interest due, and then the surplus is to be applied towards discharging the principal, and interest is to be computed on the balance as aforesaid. EXAMPLE 1. HARTFORD, Jan. 1, 1860. For value received, I promise to pay Five Hun- dred Dollars on demand, with interest at 6 per cent. Indorsements : May 1, 1861, $175. Sept. 16, 1862, $25. Jan. 1, 1864, $100. July 25, 1864, $120. What was due Jan. 1, 1865. DATES. Note I860.. 1 1 1st Payment, 1861 . . 5 1 2d Payment, 1862 . . 9 16 3d Payment, 1864. . 1 1 4th Payment, 1864. . 7 25 TIME BETWEEN DATES. Years. Months. Days. 1 4 1 4 15 1 3 15 6 24 5 6 Int. of $1, .08 .0825 .0775 .034 .026 Settlement,.. 1865.. 1 1 Principal $500. Interest to May 1, 1861 40. Amount 540. 1st Payment 175. Balance and new principal May 1, 1861 365. Interest to Sept. 16, 1862 $30. 11 2d Payment (less than interest) 25.00 Surplus interest 5.11 PABTIAL PAYMENTS. 205 Interest of the same principal from Sept. 16, 1862 to Jan. 1, 1864 28.29 33.40 Amount 398.40 3d payment (to be deducted) 100.00 Balance and new principal 298.40 Interest to July 25, 1864 10.146 Amount 308.546 4th.Payment (to be deducted) 120.000 Balance and new principal 188.546 Interest to Jan. 1, 1865 4.902 Amount due Jan. 1, 1865 $193.448 Ex. 2. A note of $450 is dated June 16, 1860. Interest at 7 per cent. Indorsements : Aug. 1, 1861, $20. Jan. 13, 1863, $220. "What was due May 16, 1864 ? Ex. 3. A note of $300 is dated July 1, 1861. Interest at 6 per cent. Indorsements : Jan. 1, 183, $15. July 1, 1865, $150. What was due May 1, 1866 ? Ex. 4. A note of $620 is dated Sept. 9, 1863. Interest 5 per cent. Indorsements : Dec. 21, 1863, $75. Sept. 8, 1864, $200. June 20, 1865, $20. What was due Sept. 20, 1865. Ex. 5. A note of $750 is dated May 1, 1862. Int'rst 6 per cent. Indorsements : Nov. 16, 1863, $50. Sept. 1, 1864, $175. What was due May 1, 1865 ? Ex. 6. A note of $500 is dated Sept. 7, 1860. Interest 7 per cent. Indorsements : Jan. 1, 1861, $100. June 19, 1861, $10. Jan. 1, 1862, $200. What was due Sept 1, 1862 ? Ex. 7. $600. NEW HAVEN, Jan. 1, 1863. On demand I promise to pay D P & Co. , or order, six hundred dollars, for value received, with interest. P W . Indorsements : Jan. 1, 1864, $100. July 1, $10. What was due Jan. 1, 1865 ? 206 PARTIAL PAYMENTS. Ex. 8. $1*00. NEW YOBK, Dec. 25, 1863. On demand, we promise to pay G M . & Co. , or order, fifteen hundred dollars, i'or value received, with interest. P W & Co. Indorsements : Jan. 7, 1865, $200. July 7, 1865, $25. What was due Aug. 1, 1865 ? MERCANTILE RULE. Art. 150. Find the amount of the principal for one year, and from it subtract the amount of each payment during the year, to the end ofti; the remainder mil be a new principal, with which proceed as before. If the time of settlement is less than a year, find the amount of the principal for such a portion of a year, and subtract from it the amount of the payments to the same date. EXAMPLE 9. A note of $1000 is dated Jan. 1, 1864 interest 6 per cent. Indorsements : March 12, 1864, $200. October 25, $350. April 18,1865, $100. What was due June 18, 1865 ? Principal $1000. Interest for 1 year 60. Amount 1060. 1st Payment (March 10) $200.00 Interest till Jan. 1, 1865 9.60 2d Payment (Oct. 25) 350.00 Interest till Jan. 1, 1865 3.85 563.45 Balance for new principal 496.55 Interest till settlement 13.82 Amount 510.37 3d Payment (April 18, 1865) 100.00 Interest tiU settlement 1.00 101.00 Balance due June 18, 1865 $409.37 Ex. 10. A note of $1000 is dated Jan. 1, 1866. Interest 6 per cent. CONNECTICUT RULE. 207 Indorsements : March 1, $100. May 25, $100. Sept. 1, $100. July 1, 1867, $500. What was due Sept. 1, 1867 ? Ex. 11. A note of $500 is dated July 1, 1865. Interest 7 per cent. Indorsements : Jan. 1, 1866, $100. April 1, $100. May 16, $100. What was due July 1, 1866 ? Ex. 12. A note of $800 is dated May 1, 1866. Interest G per cent. Indorsements: Sept. 1, $200. Jan. 1, 1867, $200. March 1, $200. What was due May 1, 1867 ? CONNECTICUT RULE. Art. 151. The Connecticut rule is the same as the Mer- cantile when a payment is made in less than a year; other- wise it is the same as the U. S. Rule. Find by the Connecticut Rule what was due in Example 1, page 204. Process. The same as by U. S. Bule till the third payment has been deducted, leaving Balance for a new principal $298.40 Interest for 1 year 17.90 Amount (Jan. 1, 1865) 316.30 4th Payment $120.00 Interest to Jan. 1, 1865 3.12 123.12 Amount due Jan. 1, 1865 $193.18 By the same rule find the amounts due in Examples 2, 3, 4, &c. If the answers are very nearly the same as>by the U. S. Rule they may be considered correct. RATE. Art. 152. To find the rate when the principal, interest, or amount and time are given. 208 INTEREST. EXAMPLE 1. At what rate will $200 yield $30 interest in 2 years 6 months ? $200 .01 Process. At 1% the interest of $200 for 6 mos. )2.00 1 year. 2 years 6 months is $5 . Therefore if the in- 2 terest is $30 the rate must be as many times IjToO 2 yrs. 1 % as $5 is contained times in $30, or 6 %. I.'OO 8 mos. "5^0 30.00 Ans. ' 6% RULE. Divide the given interest by the interest of the principal at 1 %. Ex. 2. At what rate will $320 yield $72.80 in 3 yrs. 3 mos.? TIME. Artt 153* To find tine time when the principal, interest, or amount and rate are given. Ex. 3. In what time will $200 yield $33 interest, at 6 % ? Process. The interest of $200 for 1 year, at 6%, $ $ is $12. Therefore if the interest is $30, the time 1J ) d(J must be as many years as $12 is contained times in Ans. 2 5 years. $30, or 24 years. RULE. Divide the given interest by the interest of the prin- cipal for 1 year. Ex. 4. In what time will $320 yield $72.80, at 7 per cent.? PEINCIPAL. Art. 154. To find the principal when the interest or amount, time, and rate are given. Ex. 5. What principal, at 6 %, will yield $30 interest, in 2 years 6 mos. ? Process. $1 at 6 % will yield in 2 years 6 months $.15 .15^30 00 interest. Therefore $30 interest will require as many dollars as $.15 is contained times in $30, or $200. COMPOUND INTEREST. 209 RULE. Divide the given interest or amount by the interest or amount of $l,/br the given time. Ex. 6. What principal at 7% will yield $72.80 interest in 3 years 3 months ? EXAMPLES. 7. At what rate will $500 yield $34 interest in 1 year 1 month 18 days ? 8. At what rate will $300 amount to $366 in 3 yrs. 8 mo. ? 9. In what time will $560 yield $106.40 at 8 per cent. ? 10. What principal will yield $192 interest in 4 yrs. 3 mos. 6 days, at 6 per cent. ? 11. At what rate will $1200 yield $3 interest in 15 days ? 12. In what time will $360 yield $10.50 interest at 5 % ? 13. What principal will yield $37.50 interest in 4 years 2 months, at 6 % ? 14. In what time will $360 amount to $360.66, at 6 % ? 15. At what rate will $350 yield $101.50 interest in 7 years 3 months ? 16. What principal will yield $9 interest in 1 year 2 mos. 12 days, at6%? 17. At what rate will $500 yield $62.50 interest in 2 years 1 month ? 18. In what time will $65 yield $2.60 interest at 6 % ? 19. What principal will amount to $1245 in 3 years and 6 months, at 7 %? 20. At what rate will $1000 amount to $1150, in 2 years and 6 months ? COMPOUND INTEREST. Art. 155. Compound Interest is interest on the prin- cipal and interest already due. EXAMPLE 1. What is the compound interest of $500 for 3 years 8 months, at 6 % ? 210 COMPOUND INTEREST. $500 .06 30.00 Interest 1st year. 500 530 Amount " " .06 31.80 Interest 2d " 530 561.80 Amount " " 33.7080 Interest 3d 561.80 595.508 Amount .03 17.86524 Interest 6 months. 595.508 613.37324 Amount 500 $113.37324 Compound int'st for 3 y'rs 6 mos. RULE. Find the interest for a year or the time till it is due, and add it to the principal for a new principal ; on which find the interest as before. Proceed thus till the last interest is due, and from the amount subtract the first principal. EXAMPLES. What is the compound interest of (2.) $200 for 3 years, at 7% ? (3.) $300 for 4 years, at 6 % ? (4) $400 for 5 years, at 5 % ? (5.) $200 for 2 years, at Q% (payable semi-ammally ?) (6.) $100 for 1 year, at 6% (payable quarterly ?) 7. What is the amount of 700 for 3 years, 9 months, and 24 days, at 7 %, compound interest ? 8. What is the amount of $740, at 6 %, compound interest, (semi-annually,) from Dec. 20, 1866, to Nov. 2, 1869 ? 9. What is the compound interest of $1000 for 2 years, 8 months, 15 days, at 6 %? 10. What is the amount of $500 for 2 years at 8 %, com- pound interest, payable quarterly ? DISCOUNT. 211 TABLE, Shoioing the amount of $1, or 1, at 3, 4, 5, 6, and 7 per cent com- pound interest, for any nwmber of years from \ to 10. Years 3 per cent. 4 per cent. 5 per cent. 6 per cent. 7 per cent. 1 1.030,000 1.040,000 1.050,000 1.060,000 1.07,000 2 1.060,900 1.081,600 1.102,500 1.123,600 1.14,4:;0 3 .092,727 1.124,861 1.157,625 1.191,016 1.22,504 4 .125,509 1.169,859 1.215,506 1.262,477 1.31,079 5 .159,274 1.216,653 1.276,282 1.338,226 1.40,255 6 1.194,052 1.265,319 1.340,090 1.418,519 1.50,073 7 .229,874 1.315,932 1.407,100 1.503,630 1.60,578 8 1.266,770 1.368,569 1.477,455 1.593,848 1.71,818 9 1.304,773 1.423,312 1.551,328 1.689,479 1.83,845 10 1.343,916 1.480,244 1.628,895 1.790,848 1.96,715 Multiply the amount of $1, by the given number of dollars. Find the answers to the above examples by the table. The semi-annual interest of $1 is the same as the annual interest at the rate per cent. DISCOUNT. Art. 156. Discount is a certain percentage deducted for the payment of money a specified time before it is due. The present worth of a sum, or debt, payable at some future time without interest, is the sum which put at in- terest till it becomes due, will amount to the given sum. EXAMPLE 1. What is the present worth of $500, payable in one year at 6 per cent. ? 212 DISCOUNT. 1.06)500(471.698 424 760 24-2 Process. At 6 % the present worth of $1.06 due in 1 year, is $1.00. Therefore the present worth of $500 is as many dollars as it contains times $1.06, or $471.698+. $500-471.698= 740 $28.302 discount. 636 1040 954 ~860 848 RULE. Divide the given sum by the amount of $1 for the given time, at the given rate ; the quotient will be the present worth. (Percentage, Case IV.) To find the discount, subtract the present worth from the given sum or debt. EXAMPLES. What is the 2. Present worth of $130, due in 5 years, at 6 %? 3. Discount of $115, due in 2 years 6 months, at 6 %1 4. Present worth of $334, due in 1 year 1 month 18 days, at 6^? 5. Discount of $666.40, due in 2 years 4 months 15 days, 6. Present worth of $942, due in 4 years 3 months 6 days, at 6^? 7. Present worth of $534.04, chie in 3 years 5 months 18 days, at 5 %? 8. Discount of $366, due in 3 years 8 months, at 6 %? 9. Present worth of 273.75, due in 1 year 7 months, at 6%? 10. What was the discount of $263.04, due April 27, 1859, but paid Feb. 15, 1858, at 8 % ? 11. What is the value May 10, 1863, of a debt of $200, due Aug. 28, 1865, at 7 %? 12. How much must be paid July 3, for a debt of $142.45, due Nov. 27, at 9 %? BANK DISCOUNT. 218 13. How much should be deducted from a debt of $170.50, due April 19, 1869, if paid Jan. 9, 1867, at 6 %* 14. What is the discount, Feb. 5, 1862, on a note of $407.088, payable Aug. 20, 1864, at 7 %? 15. How much should be paid, May 18, 1867, on a note of $5783.09X, payable Sept. 25, 1870, at 8 %? BANK DISCOUNT, Art, 157. Bank Discount is a certain percentage paid to banks, or bankers, for the use of money paid on notes before they are due. It is the same as simple interest paid in advance. Bank discount is greater than true discount, because it is computed on the amount or face of the note, which includes the interest with the money lent, instead of only the principal. In computing bank discount three days of grace are al- lowed, in addition to the specified time. EXAMPLE 1. What is the bank discount of a note of $150, payable in 90 days, at 6 % ? $ 150. .0155 Process. The interest of $1 for 93 days is $.0155, ==pr and of $150, 150X-0155=$2.325. ? 150 Ans. S2.3250 RULE. Find the interest for three days more than the specified time. If the note bears interest, find the interest on the amount that will be due on it at maturity. The discount subtracted from the given sum gives the present worth. 214 EXAMPLES. EXAMPLES. What is the bank discount 2. On a note of $300 for 6 months, at 6 % ? 3. On a note of $450 for 4 months, at 5 % ? 4. On a note of $500 for 3 months, at 7 % ? 5. On a note of $750 for 9 months, at 5 % % ? 6. On a note of $1000 for 3 months, at 7 % ? What is the present worth 7. Of a note of $120 for 4 months, at 7 % ? 8. Of a note of $360 for 30 days, at 6 % ? 9. Of a note of $340 for 6 months, at 8 % ? 10. Of a note of $480 for 1 month, at 6 % ? 11. Of a note of $1950 for 2 months, at 6 % ? Art. 158. To find for what amount a note must be given that it may be worth a given sum when discounted. EXAMPLE 12. For what amount due in 60 days must a note be given that its present worth may be $500, at 6 % ? Process. The bank discount of $1 for 63 days is $.0105, and the present worth is $.9895. Therefore $500 is the present worth of as many dollars for the same time as $.9895 is contained tunes in $500. $500 -i- $.9895 = $505.30. RULE. Divide the required sum by the present worth of II 13. For what amount must a note payable in 90 days be given that its present worth may be $300, at 6 % discount ? 14. For what amount must a note payable in 10 months be given that its present worth may be $500, at 7 % discount ? 15. For what amount must a note payable July 1 be given that it may be worth Jan. 1, the same year, $730, at 6 % dis. ? 16. A man wishes to procure from a bank $1000. At 5 % discount what will be the amount for which he must give his note, payable in six months ? Promiscuous Examples in True and Bank Discount. If bank discount is not specified, discount means true discount. 1. What is the discount of $100 for six months at 6 % ? TAXES. 215 2. What is the bank discount of the same ? 3. What is the present worth of the same at true discount ? bank discount ? 4. A debt of $300 is due Oct. 1 ; if paid June 1, the same year, how much should be paid ? 5. Bought $100 worth of goods on G months' credit ; how much should be deducted for cash, at 7 % discount ? 6. A note of $3000, payable in 60 clays, was discounted at a bank at 6 % ; how much was received for it ? 7. A speculator wished to procure from a bank $10,000 for 4 mos. For what amount should he have given a note for it at 1% discount? 8. A merchant bought goods amounting to $3000 on 6 mos. credit, but was allowed 5 % of the amount for cash ; money being worth 7 %, how much did he gain by paying cash ? 9. A merchant bought 75 barrels of flour for $500 and sold it for $640, receiving for it a note payable in 8 months, which he had discounted at 6 %, bank discount ; how much did he gain? 10. A drover wishes to procure from a bank $2000 for 2 mos. 15 days ; for what amount must he give his note, at 7 % dis. ? TAXES. Artt 159. A Tax is money required by law to be paid for the support of the government and its institutions, or public improvements. A poll tax is a certain sum on male citizens, called polls. An income tax is a certain percentage on incomes. Taxable property is either Personal Property or Real Estate. Real Estate is that which is not movable; as land, houses, &c. Personal Property is such as is movable, as money, notes, furniture, &c. 216 TAXES. Taxes are commonly a certain percentage on property, of all which an inventory, or list of articles, is first made. EXAMPLE 1. In a certain town $5150 is to be raised by tax. The number of polls is 300, each taxed 50 cents. The real estate is valued at $800,000, personal property $200, 000. What is the rate per cent, on $1, and what is the amount of A's tax, who pays for 5 pells, and whose real estate is valued at $4000 and personal property $1500 ? Process. -300 polls X 50 cents = $150 poll tax ; $5050 $150 = $5000 property tax ; $5000 -f- 1000000 (800000 -f 200000) =.005 mills the per cent, or tax on $1. (Percentage, Case II.) A's property, $5500 (4000 + 1500) X -005 = $27.50 A's property tax, to which add his poll tax (5 polls X 50 cents) $2.50. Ans. $30.00. RULE. Subtract the poll tax from the whole tax ; find the per cent, of the remainder on all the property, and then each man's property tax, to which add his poll tax. After finding the tax on one dollar, it will be convenient in prac- tice to make a tax table, as follows : Tax on |1.=$.005 2.= .01 3.= .015 4.= .02 5.= .025 6.= .03 7.= .035 TABLE. Tax on $8 .=$.04 9.= .045 10.= .05 20.= .10 30.= .15 40.= .20 50.= .25 Tax on $60. =$.30 70.= .35 80.= .40 90.= .45 100.= .50 1000. =5. 00 &c. By this table A's property tax in the above example is on $5000 $25.00 500 2.50 $27.50 property tax. Ex. 2. The tax of a certain town is $4500. The number of polls is 500, each taxed $1. The real estate is valued at $600,000, and personal property $200,000. What per cent, is the tax, and what is B's tax, whose property is valued at $3500, and who pays for two polls ? When district schools are supported by families in proportion to DUTIES. 217 the attendance, divide the whole expense by the whole number of days' attendance, and multiply the quotient by the days' attendance from each family. DUTIES. Art. 160. Duties are taxes on goods imported or exported. A Port Of Entry is a place where duties are collected. A Custom-House is an office where duties are collected. Ad valorem duty is a certain percentage on the cost of the goods. Specific duty is a certain price per weight or measure. An invoice is a list of goods, with the prices. Tare is an allowance for weight of boxes, casks, &c., containing the goods. Draft is an allowance of weight for waste. Leakage is an allowance for the waste of liquors in casks or barrels ; breakage, for the same in bottles. Gross weight is the weight of goods, including the packages. Net weight is the weight after deducting the tare, &c. In all these allowances reject fractions less than |, and add 1 for 3 or more. EXAMPLE 1. What is the duty, at 30 per cent., on 100 gals, of oil, invoiced at 75 cts. a gallon, allowing 2 % for leakage ? Process. 100 X 02 = 2. gallons leakage. 1002 = 98 net. 98 X -75 cts. = $73.50 net value. $73. 50 X 30 = $22. 05 duty. Ex. 2. What is the duty on 10 bbls. of sugar, each weighing 215 Ibs. gross, at 2 cts. a pound draft 2 Ibs. each ; tare 12 > per cent. ? 218 EXCHANGE. Process. 215 2 = 213 pounds. 213 X 10 = 2130 2130 X - 12 i= 266 Ibs. tare. 2130 266 = 1864 Ibs. net. 1864 X 2 cts - =$37.28 duty. KULE. Deduct all allowances, then to find the ad valorem duty multiply the cost by the given rate per cent.; to find the specific duty multiply the net weight or quantity by the duty on ONE of the same. Ex. 3. What is the duty, at 18 per cent., on 200 bags of coffee, each weighing 150 Ibs. , invoiced at 12)^ cts. a pound draft 2 Ibs. each ; tare 3 per cent. ? Ex. 4. What is the duty, at 33 > per cent., on 36 pieces of silk, each containing 45 yards, invoiced at $2 a yard ? Ex. 5. What is the duty, at 10 cts. a pound, on 34 chests of tea, each weighing 118 Ibs. draft on each 1 Ib. ; tare 8 per ct. ? Ex. 6. What is the duty, at 40 per cent., on 500 yards of satin, at 11.62)^ ? Ex. 7. What is the duty, at 25 cts. a gallon, on 18 casks of wine, each containing 68 gals., allowing 2 per cent, leakage ? Ex. 8. What is the duty, at 12% cts. a pound, on 12 boxes of tobacco, each weighing 130 Ibs. ; draft 1 pound on each box ; tare 6 per cent. ? Ex. 9. What is the duty, at 30 per ct. , on 9 cases cf broad- cloth, each case containing 20 pieces, and each piece 36 yards, at $4 a yard ? EXCHANGE. Art. 161. Exchange is a means employed by persons in one place of making payments in another, such as drafts, or bills of Exchange. Exchange in the same country is Domestic; from one country to another, Foreign. FORMS. 219 FORM OF A DRAFT. ^ (^/Lav-en, C/ct. /<5, & '__ / y (/ vaute tecewea. and cnaiae / 4a?ne to tne account /5OO E. Clark is the Drawer. S. Staples is the Drawee. Ezra Jones is the Payee. S. Staples accepts, or promises to pay the above, by writing " Accepted " oyer his name, on the back. If Ezra Jones indorses it, by writing his name on the back, any person who has it is entitled to the amount. Drafts, or Bills of Exchange, like Stocks, may be at par, at premium, or discount, and their value found in the same way. If they are not payable at sight they are subject to bank discount, the same as notes. EXAMPLES OF DOMESTIC EXCHANGE. 1. What is -the cost of the following draft, at a premium of ., Q^aa. SO, S&&7. ' cet/i/ed, and cnabae to tne account o/ / / 220 EXCHANGE ON ENGLAND. Process. $1 costs $1.0025. Therefore $300 cost $1. 0025 X 300 =s= $300.75, Ans. 2. What is the cost of the following draft, at 2)^ % dis- count ? a. so, to and t <7 /k vauie kecewea, and cnaiae to 6ne account Process. Present worth of $1, by bank discount=$.9936; $.9936 .025 $.9686 the cost $1 of the draft, $.9686X.480=$464.928, Ans. 3. A merchant in St. Louis wishes to remit a draft for $1000 to New York ; what will it cost, exchange being at 2}^ per cent, premium ? 4. A merchant in New Orleans wishes to remit to Philadel- phia a draft for $1500, at thirty days' sight ; what will it cost, exchange being at 4 per cent, premium at sight. Drafts are usually drawn at 30, 60, or 90 days' sight, and at a cer- tain per cent, premium or discount, including allowance for the time. EXCHANGE ON ENGLAND. Art. 162. Exchange on England is always at a pre- mium in this country, because in making it, a pound sterling is valued at $444f, instead of its true value, $4.84 PEOMISCUOUS EXAMPLES. 221 Ex. 1. BILL OF EXCHANGE ON ENGLAND. and wwia o/ Aame aafa and S nfoatd,/ /iau to ' / / wn What is the cost of the above bill, when exchange is premium ? Process. 1=S4| (old value) 300X^=$1333.33i $1 at premium=$1.105. $1333. 33><1- 105^$M73.33i, Ans. Art. 163. To Find {he Cost of a Bill of Exchange on England. EULE. Reduce the pounds and decimal of a pound, at the old value, ( $4fJ to dollars, &c.; which multiply by the cost Ex. 2. A gentleman, about to visit England, wishes to buy a bill exchange for $2000 ; what will be the amount of it at 9} premium? Process. $1.095 = cost of $1. $1.095 X 44 = $4.86| the cost of 1. $2000 ~ 4. 86| = 410.959 + = 410 19s. 2d. + Art. 164, To find the amount of a Bill on England which can be bought with U. S. Money. RULE Divide the given number of dollars by the cost of 1. 222 PARTNERSHIP. EXAMPLES IN EXCHANGE ON ENGLAND. 3. What will a bill of exchange on London, amounting to 500, cost in New York, exchange being 9% per cent. prem. ? 4. A gentleman wishes to buy a bill of exchange on Liver- pool with $2500 ; what will be its amount at 12 % % pre- mium ? 5. An importer wishes to pay for goods ordered from Man- chester, England, amounting to 1000; what will a bill of exchange cost at 9^ per cent, premium ? 6. An importer wishes to remit to Leeds, England, $3000 ; what will be the amount of a bill of exchange, at 10 per cent, premium. ? Exchanges on other foreign countries are made by reduction of currencies. Promiscuous Examples in Exchange. 1. What is the cost of a draft on Memphis for $3600, at 1% per cent, premium ? 2. What is the amount of a draft which costs $1012.50, at 1 per cent, premium ? 3. What will a draft on New York for $600 cost, at 2 per cent, discount? 4. What will be the cost of a bill of exchange on Liverpool, England, of 150 10s., at 9 per cent, premium ? 5. What will be the amount of a bill of exchange on Lon- don, bought in Boston for $3000, at 9^ per cent, premium ? 6. What will be the cost of a draft for $750, on Hartford, at ^ per cent, discount ? 7. What is the amount of a draft bought for $1250, at a premium of 2^ per cent. ? EXAMPLES. 223 PARTNERSHIP. Art. 165. Partnership is a company of two or more persons in the same business. They are called a firm, or house, and each member a partner. The capital, or stock, is the money or property em- ployed in their business, of which the profit or loss is a certain part or percentage. In Bankruptcy the creditors are the same as partners in business, and the property of the bankrupt as profit or loss. CASE I. Art. 166. To find each partner's share of the gain or loss when their capital has been used the same length of time. EXAMPLE. Peck, Staples & Clark are partners in business. P.'s capital is $4800 ; S.'s $2400, and C.'s $1800. The whole gain is $3000 ; what is each one's share ? Process.-P.'s capital, $4800 \ The whole gain is IW= i. or S 's " $2400 33 3 P er cent - of the wllole ca P~ r ' " SfftMl -I ital - Therefore each partner's f^ share is i, or 33^% of his capi- Whole capital, &9000 tal Therefore, $4800 X i or -33^= $1600 P.'s gain, ) $2400' " $800 S.'s " [ Ans. $1800 " S 600 C.'s " ) RULE. Take such a part of each partner's capital as the whole gain or loss is of the whole capital"; or Multiply each partner's capital by the gain or loss per cent. EXAMPLES. 2. Messrs. Staples & Clark are in partnership ; S. 's capital is $3000, C.'s $2000; they have gained $1500; what is each one's share ? 3. Messrs. Howland & Son shipped a cargo of goods, amounting to $20,000, % of which belonged to H. The profits were $4000 ; what was each one's share ? 4. The capital of a Eailroad Company is $2,000,000, the 224 PROMISCUOUS EXAMPLES. annual earnings are $400,000, the expenses $240,000. I own 25 shares ; what is my annual dividend ? 5. A factory owned by three men was damaged by fire to the amount of $3000 more than the insurance. S. owned , E. f , and W. the remainder ; what was each man's share of the loss ? 6. A merchant having failed in business, owed J. Strong $825, T. Williams $700, S. Vernon $1175. He can pay the three only $900 ; what is each one's share ? CASE H. Art. 167. To find each partner's share of the gain or loss, when their capital has been used unequal portions of time. EXAMPLE 7. Mead, Rogers & Smith have furnished capital as f oUows : Mead, $5000 for 2 months ; Eogers, $4000 for 4 months, and Smith $3000 for 3 months. They have gained $3500 ; what is each one's share ? Mead's capital, $5000 X 2 = S 10000 for 1 month. Kogers' " $4000 X = $16000 Smith's " $3000X3=$ 9000 The whole " = $35000 " The whole gain is ^ftftfty = -, L - of the sum of the products of each one's capital multiplied by the time it was used ; therefore each partner's share is -fa, or 10 per cent, of his capital, multiplied by its time. Mead's share is A of $10000 = $1000 ) Bogers' share is -^ of $16000= $1600 } Ans. Smith's share is ^ of $ 9000 = $ 900 ) KULE. Multiply each partner's capital by the time it was used, and treating the product as his capital, proceed as in Case I. EXAMPLES. 8. Messrs. Hoyt & Lane formed a partnership as drovers ; H. furnished $3200 for 2 months, and L. $2000 for 4 months. They gained $1350 ; what was each one's share ? 9. Two carpenters, Adams & Nelson, contracted to build a PROMISCUOUS EXAMPLES. 225 house for $600 ; A. furnished 9 men for 100 days, and Nelson 12 men for 75 days ; what was each one's share ? 10. Three men performed a piece of work in 28 days alto- gether, for which they were paid $42. A. received $15, B. $12, and C. the remainder ; how many days did each work ? 11. Messrs. Todd & Howe engaged in business with a capi- tal of $3600. T.'s capital was in the business 4 months, and his share of the profits was $160 ; B. 's capital was in the busi- ness 6 months, and he received as his share $192 ; how much capital did each furnish ? Promiscuous Examples in Partnership. 1. Messrs. Tappan, Edwards & Kimballwere partners. T. furnished $800 capital, E. $700, K. $1300 ; they lost $560 ; what was the loss of each ? 2. A man bequeathed $8400 to his three sons, in proportion to their ages, 10, 14, and 18 years ; how much did each re- ceive ? 3. Messrs. Hawley, King & Stebbins were partners. H. furnished $10,000 capital for 15 months, K. $12,000 for a year, and S. $15,000 for 9 months. They gained $9652.50 ; what was each man's share ? 4. Messrs. Benedict & Coe were partners three years. B. furnished $6150 capital, C. $8100 ; B.'s share of the gain was $1250 ; what was the whole gain ? 5. C. Jenkins commenced business with a capital of $2500 ; after 6 months he took into partnership A. Heed, with $3503 capital ; 3 years afterwards their joint capital was doubled ; how much had each gained ? 6. I agreed to pay three men $72 for delivering to families, at different distances, 36 tons of coal. A drew 9 tons 6 miles. B 12 tons 4 miles, C thf jggmainder, 2 miles ; how much is due to each ? 7. Mesdz&Scott & Taylor were partners three years ; S. fur- nished % as much capital as T. ; they gained $5000 ; what was each one's share ? 226 PERCENTAGE. 8. Three drovers hired a pasture for $36. A had 24 head of cattle, and paid $12 ; B paid $16, and C the remainder, as their portions ; how many cattle did B and C each have in the pasture ? 9. Three drovers had 500 sheep each, for which they hired a pasture and paid $56, each agreeing to pay in proportion to the number of weeks his sheep were in the pasture. A paid $14 for 2 weeks, B paid $17^, and C $24^ ; how long were B and C's sheep in the pasture ? 10. A bankrupt owes $18,000, and his property is worth $3600 ; how much will a creditor receive, whom he owes $1500? 11. Messrs. Paige, Cassidy & Warren are partners. P. 's capital, $20,000, has been in the business 3 years ; C.'s capi- tal, $15,000, 2 years 6 months ; W.'s capital, $12,000, 1 year 9 months ; they have gained $2962)^ ; what is each man's share ? Art, 168. Promiscuous Examples in the various applications of Percentage. EXEECISE I. 1. At 2j^ % commission, how much will an agent receive for selling goods amounting to $920 ? 2. A commission merchant has received $624 for the pur- chase of goods after deducting 4 % commission ; what amount must he expend ? 3. At 3 % commission, what amount of goods must be sold in a year to realize an income of $2400 ? 4. What is the premium on an insurance policy of $4500, at iy*%t 5. For what must a house valued at $5000, be insured at 2 %, to cover the cost of insurance ? 6. What is the value of 15 shares of bank stock, at 3 % be- low par ? 7. What will a broker receive for exchanging $578 in bank notes, at 1) % discount ? PROMISCUOUS EXAMPLES. 227 8. What will a broker give for $720 in bank-notes at % % discount ? 9. What is the interest of $1500 for 4 years 3 months 6 days, 10. What is the amount of $360 from June 14, 1863, to Sept. 28, 1865, at 7 %? 11. A note of $300 was dated Jan. 1, 1863. Interest 6 %. Indorsed July 1, 1863, $109. Jan. 1, 1864, $100. What was due July 1, 1864 ? 12. A man has paid in 3 years, $341.75 interest, at 5%; what was the principal ? 13. A man pays $800 rent for a house valued at $10,000 ; what per cent, interest does he pay ? 14. What is the compound interest of $600 for 2 years, 6 months, at 6 % ? 15. What is the present worth of $399.60, due in 1 year 10 months, at 6 % discount ? 16. What is the bank discount on $750, payable in one month ? 17. What is the present worth of a note for $360, payable in 1 month, at 6 % bank discount ? 18. For what sum must a note at 6 % for 90 days, be given at a bank to obtain $393.80 ? 19. A merchant sold goods amounting to $300, and gained 20 %; what did he gain ? 20. If silk cost $1.80 a yard, for what must it fee sold to gain 25^? 21. A man paid $75 for a wagon, and sold it for $100 ; what per cent, did he gain ? 22. A man sold a wagon for $112, and gained 40 %, what did it cost ? 23. A town is taxed $6250 on its property valued at $1, 200, 000, and there are 500 polls taxed 50 cts. each ; what per cent is the tax on the property ? 24. What is the duty, at 20 % ad valorem, on 80 bales of imported wool, each weighing 400 Ibs. , invoiced at 25 cts. per pound ; tare 5 %1 228 PERCENTAGE. 25. Me Kae & Oakley shipped goods in partnership. Me Kae furnished $5000 ; Oakley, $3000 ; they gained $2320 ; what was each one's share ? 26. When gold is 130, what is $100 in currency worth ? 27. What will be the cost of a draft at sight on St. Louis for $1000, at 1 % premium ? 28. What must be paid for a bill of exchange on Liverpool, Eng., for 400, at 9 % premium ? EXERCISE H. 29. A man having $8,000 lost 12) % of it ; how much had he left ? 30. A steamboat is insured for $30,000 at 2) %\ premium ? 31. What will 20 shares of the Central Kailroad cost at % advance ? 32. A bank has failed whose circulation is $75,000, and is able to pay only 66% %\ what amount can it pay ? 33. What is the duty on 840 bags of coffee, each weighing 120 Ibs. at 3 cts. a pound ; tare 3 per cent. ? 34. The expenses of a school district are for teacher's salary $600 ; fuel, &c., $37. The whole attendance has been 5600 days ; what must a man pay for 80 days' attendance ? 35. Bought 195 cords of wood at $4.12)^ a cord, and sold it for $4.87> a cord ; what was the gain per cent.? 36. A gentleman sold a carriage for $150, which was 75 % less than it cost ; what did it cost ? 37. If I pay $960 for wheat, for what must I sell it to gain 25 per cent. ? 38. What is the interest of $691.20 from March 5, 1862, to Sept. 20, 1864, at 7 per cent. ? , 39. How much money at 6 % interest will yield a semi-an- nual income of $650 ? 40. What is the interest of $350, at 6 % t for 1 year 6 mos. 12 days ? Also, the discount and bank discount of the same for the same time ? PEOMISCUOUS EXAMPLES. 229 41. $87. 25 NEWBUEGH, N. Y., Aug. 14, 1857. Six months after date I promise to pay H. S. Barnes Eighty- seven ^^ dollars, value received, with interest. EOBT. LIVINGSTON. Indorsed, Dec. 8, 1858, $5.00. May 20, 1859, $45.50. How much was due August 26, 1859 ? 42. What is the compound interest of $200 for 3 years at 7 per cent. ? 43. What is the value of a draft on New Orleans for $1500, at 3 % discount ? 44. When gold is 125, how much of it is equal to $500 in currency ? how much currency is equal to $500 in gold ? 45. For what amount will $2427.77| purchase a bill of ex- change on Liverpool, at 9^ per cent. ? 46. Messrs. Taylor & Olmsted are partners in business. T. furnished $5000 capital for 6 months, O. $4000 for 5 months. They have gained $2500 ; what is each one's share ? EXERCISE TTT. 47. A merchant having shipped flour amounting to $1260, paid 2^ % for insurance ; what was the whole cost ? 48. An auctioneer sold goods amounting to $956.20, at 2) per cent, commission ; how much did he receive ? 49. A merchant paid a broker $3090 for a draft on Charles- ton, allowing him 3 % brokerage ; what was the amount of the draft ? 50. What is the amount due on a note of $391, dated Sept. 7, 1864, and payable Feb. 1, 1865, at 6 per cent. ? 51. $1000. UTICA, May 1, 1862. For value received I promise to pay H. Low & Co., or order, One Thousand Dollars, with interest at 7 %. Indorsed : Nov. 1, 1862, $150. March 13, 1863, $200. What was due Sept. 19, 1863 ? 52. A man at his death bequeathed $8000 to an asylum, to be paid after it amounted to $10000, at 1 % ; how long was the time? 230 PERCENTAGE. 53. What is the interest of 113 10s. for 1 year 6 months, at 6 per cent. ? 54. What is the compound interest of $250 for 3 years, at 6 per cent.? 55. What is the present worth of $500.76 payable in 8 inos., at 6 per cent, per annum ? 56. What is the present worth of the same by bank dis- count ? 57. What is the difference between the discount and the interest of $1000 for 1 year, at 6 per cent. ? 58. For what amount must a note for 90 days be given at a bank to obtain $600, at 7 % ? 59. Shipped 5 loads of furniture, worth $1200, to Mobile, and paid 2)^ % for insurance ; what was the premium ? 60. For what amount must the same furniture have been insured to have covered the whole loss if the ship had sunk ? 61. If a man pays $90 premium at 2)^ % for insurance, for what amount is he insured ? 62. A gentleman bought a harness for $60, and afterwards sold it for 12 % less ; how much did he lose ? 63. Bought a firkin of butter for $26.63 ; for how much must it be sold to gain 20 % ? 64. A man bought a farm at $85 an acre, and afterwards sold it for $100 an acre ; what per cent, did he gain ? 65. A gentleman sold his watch for $72, which was 28 per cent, less than it cost ; what did it cost ? 66. What is the duty, at 2 cents a pound, on 10 boxes of sugar, each weighing 125 Ibs., allowing 6 Ibs. a box for tare ? 67. What is the duty on a cargo of coffee, invoiced at $3560, at 30 per cent. ? 68. What is the amount of tax due from J. Carpenter, whose property is assessed at $1800, and who pays for three polls, the whole tax of the town being $3600 ; the number of polls 602, paying 75 cents each, and the amount of property assessed is $250,000 ? 69. How many shares of the Erie Eailroad stock can be bought for $7500, when it is 37)^ % below par ? PROMISCUOUS EXAMPLES. 231 70. When gold is 129)^, how much of it must be paid for $750 in currency ? How much currency, also, is equal to $750 in gold ? 71. What is the value in Memphis of a draft on New York for $3200, &i2%% premium ? 72. What must be paid for a bill of exchange on England for 300 10s., at 9^ per cent, premium ? 73. A bankrupt's debts amount to $30,000 ; the property in his possession $18,000 ; how much can be paid a creditor whom he owes $1251.37}^ ? 74. Messrs. Halsey & Coe are partners in business; H.'s capital is $5000, and has been employed 3 years ; C. 's capital is $4500, but was not paid in till 4 months after H. 's ; they have gained $5400 ; what is each one's share ? EXERCISE IV. 75. The population of a city has increased 250 per cent, in 20 years, and now contains 175,000 ; what was its popula- tion 20 years ago ? 76. A lawyer charges 4 % commission for collecting a debt of $625.25 ; how much will he receive ? 77. A country merchant has forwarded $3681 to his agent in the city for the purchase of goods, after deducting 2^ % com- mission ; how much will be the commission ? 78. What is the premium for insuring a mill valued at $4620, at 1% per cent.? 79. For what sum must a house, valued at $3800, be insured, at 2 % to cover the property and premium ? 80. What is the value of 16 shares in a Manufacturing Com- pany, at 12) % premium ? 81. What must be paid for a draft for $1275.25, at % % premium ? 82. What is the amount of $1475.28 from March 29, 1866, to July 4, 1867, at 7 per cent. ? 83. $200. NEW HAVEN, July 18, 1861. Three months after date I promise to pay J. Atwater, or order, Two Hundred Dollars, with interest, at 6 % f value re- ceived. B. CLAEK. 232 PERCENTAGE. Indorsed : Jan. 18, 1864, $20. July 1, 1867, $60. What was due Nov. 7, 1867 ? 84. A man wished to lea.ve his daughter an income of $600 a year ; what principal would be required at 7 % interest ? 85. What is the amount, at compound interest, of $500 for 2 years, at 5 % ? 86. What is the present worth of $800, due in 4 years and 8 months, at 6 % ? 87. What is the discount of $800, due in 10 years, at 6 % ? 88. What is the bank discount of $800, due in 10 years, at 89. For what sum must a note, payable in 60 days, be given to obtain from a bank $800, at 6 % discount ? 90. Bought a span of horses for $400, and sold them at 25% profit ; for what were they sold ? 91. Bought a span of horses for $400, and sold them for $450 ; what per cent, was gained ? 92. Sold a span of horses for $400, which was 25 % less than they cost ; what did they cost ? 93. Sold a span of horses for $400, which was 25 % more than they cost ; what did they cost ? 94. A man whose property was valued at $6240, and who paid for three polls at $1 each, was taxed $34. 20; what per cent, tax did he pay ? 95. What is the duty, at 12 %, on 20 boxes of tobacco, each weighing 250 Ibs. , and costing 20 cents a pound ; tare 6^ % ? 96. Three men bought a farm of 300 acres, and agreed to divide it according to the amount each one could pay. A paid $12,000, B $10,000, C $8,000; how many acres did each one have ? 97. When gold is 135, what is the value of $270 in currency ? How much currency must be given for $270 in gold ? 98. For what amount can a bill of exchange on Liverpool, England, be purchased with $720, at 8 % premium ? EXEECISE v. 99. If a city, with a population of 120,000, should increase 10 % a year, what would be its population after 10 years ? PROMISCUOUS EXAMPLES. 233 100. An auctioneer having sold some goods, retained ?..08 and paid the owner $3492 ; what per cent, commission did he charge ? 101. A broker received $2412 for the purchase of stocks, after deducting % per cent, brokerage ; how many $100 shares did he buy ? 102. What is the premium for the insurance of a factory, valued at $5400, at 2% per cent.? 103. For what must a house, valued at $1987.50, be insured, a ^ % %> to cover both the house and premium ? 104. How many shares of bank stock, at 30 % premium, can be bought for $1300 ? 105. When gold is 130, how much of it can be bought for $650 in currency ? How much currency is equal to $650 in gold? 106. What is the interest of $720.50 from Jan. 16, 1865, to July 31, 1865, at 7 % ? 107. What is the discount, Jan. 16, 1865, of $720.50, due July 31, 1865 ? 108. What is the bank discount, Jan. 16, 1865, of $720.50, due July 31, 1865 ? 109. $500. NORWALK, July 25, 1865. On demand, for value received, I promise Wm. Kellogg, or order, Five Hundred Dollars, with interest. J. CARTER. Indorsed : Dec. 19, 1866, $62. Aug. 4, 1867, $48. What was due Oct. 8, 1867 ? 110. What is the compound interest of $400, for 2 years 8 months, at 5 % ? 111. If cloth cost $5.50 a yard, for what must it be sold to gain 20 per cent. ? 112. If cloth cost $5.50 a yard, and is sold for $3, what per cent, is gained ? 113. If cloth is sold for $6, and the gain is 25 %, what did it cost ? 114. What is the duty on cutlery invoiced at $2400, at 30 %t 115. The tax levied on a town is $3440, the number of polls 234 PERCENTAGE. is 1000, at $1 each ; the inventory of property is $320, 000 ; what is a man's tax whose property amounts to $2400, and who pays for 2 polls ? 116. What must be given for a bill of exchange on London for 600 12s., at $% % premium ? 117. A bankrupt owes $17,000, and is able to pay only 9000 ; how much will a man lose whom he owes $680 ? 118. Messrs. Carter, Bo we & Foster are partners in business. C. furnishes ^ the capital, E. , and F. the rest ; they have gained $3300 : what is each one's share ? EXERCISE VI. 119. A merchant having a capital of $6480, lost 25 % of it ; how much had he left ? 120. A farmer having purchased some land, spent 10 % of the price in improvements, and then found that the whole cost was $8800 ; what did he pay for the land ? 121. A merchant having lost 10 % of his capital, had $1800 left ; how much had he at first ? 122. A merchant having $1800 capital, lost $300 ; what per cent, was his loss ? 123. Bought 100 tons of iron for $3865, and sold it at 3% % less than cost ; what was the selling price ? 124. Bought 500 bushels of wheat for $1300, and sold it for $1560 ; what per cent, was the gain ? 125. Sold 500 bushels of corn, and gained $100, which was 20 % of the cost ; what was paid for it ? 126. Borrowed, Jan. 1, in New York, $1250, and returned it with interest the following Sept. 15 ; what was the amount re- turned ? 127. What is the interest of $340 from July 20, 1860, to July 2, 1861, at 7 per cent.? 128. What is the interest of 760 5s. 6d., at 6 %, for 2 years 4 months ? 129. What is the amount of $850 for 5 years 6 months 3 days, at 6 % ? PROMISCUOUS EXAMPLES. 235 130. $500. HAKTFOKD, Jan. 16, 1855. On demand, I promise to pay A. Terry, or order, Five Hun- dred Dollars, for value received, with interest. J. E. DAY. Indorsed .-April 1, 1856, $50. July 16, 1857, $400. Sept. 1, 1858, $60. What was due Nov. 16, 1858 ? 131. A gentleman has a son 15 years old, and he wishes to invest for him such a sum as will amount to $10,000 when he is of age ; what sum must be invested, at 7 per cent. ? 132. A young man has received a legacy which yields a semi-annual income of $750, at 6 per cent. ; what is the amount of the legacy ? 133. A man left his son $6000, possession to be given after it amounts to $9000, at 6 per cent. ; how long must the son wait for it ? 134. A young man has a legacy of $622.75, to be paid in 3} years, but he wishes to have it immediately ; what is it worth at 5 % discount ? 135. What is the present value of a note for $900, payable in 6 months, at 6 per cent, bank discount ? 136. What is the present, value of a note for $900, payable in 6 months, with interest, at 6 %, provided the discount at a bank is 7 per cent. ? 137. For what amount must a note, payable in 60 days, be given to obtain from a bank $400, at 6 % ? 138. An auctioneer receives $112.50 for selling goods, at 2% commission ; what was the amount sold ? 139. A commission merchant has received $1656 for the pur- chase of goods, after deducting 3}^ % commission ; what is the amount of the goods to be purchased ? 140. What is the value of 50 shares of the Illinois Central Eailroad, at a premium of 9 per cent. ? 141. How many shares of stock, 10,% below par, can be bought for $9000 ? 142. When gold is 131, how much of it can be bought for $360 in currency ? How much currency is equal to $360 in gold? 236 EQUATION OF PAYMENTS. 143. A factory having been insured 7 years for $21,000, at 1% %> was destroyed by fire ? what was the actual loss to the Insurance Company ? 144. A certain town is taxed $2328, and pays 3 % for col- lecting it ; the property is valued at $419,568, and there are 300 polls taxed $1 each ; what is H. Scott's tax, whose prop- erty is valued at $3100 ? 145. For what amount can a bill of exchange on England be bought for $3633,635, at 9 % premium ? EQUATION OF PAYMENTS. Art, 169, Equation of Payments is finding the time when several payments, due at different times, may be made at once, or the balance of an account may be paid, without loss to either debtor or creditor. The time for such payment is called equated time. An account current is a record of what one person is debtor or creditor to another. CASE I. Art, 170, When the payments are due at different times to find the equated time. EXAMPLE 1. I owe J. Brush & Co. $400, payable in 3 mos., $300 in 4 mos., and $200 in 6 months ; when should I pay the whole debt at once ? Process. $400 for 3 months = $1200 for 1 month. $300 for 4 =$1200 $200 for 6 " = $1200 $900 for =$3600 Now $3600 for 1 month = $900 for as many { $900) $3600 months as it contains times $900. j Ans. 4 inos. When the times of different payments are reckoned from different dates, any date may be assumed from which to reckon them and the equated time. It is more convenient to assume the date of the ear- liest payment. EXAMPLES. 237 Ex. 2. What is the equated time of the following account ? T. A. SCOTT, Dr. To STJYDAM & JACKSON : 1860. Aug. 10. Mdse. (3 mos. credit) $300 " Sept. 15, " (6 mos.) 1861. Jan. 1, Cash Process. These different sums will be due 1860. Nov. 10, $300 X days = $0 1861. Mar. 15, $400X124 " =$49600 for 1 day. " Jan. 1, $500 X 51 " = $25500 $1200 X " =)$75100 Ans. 63 days. RULE. Multiply each payment by its time, and divide the sum of the products by the sum of the payments. This rule is according to bank discount. If the date is required, reckon the equated time from the given or assumed date. Count i day or more as 1 day. Cash payments have no products when the date is given, but must be included in the sum of the payments. EXAMPLES. 3. A merchant buys goods, pays $200 at the time, and agrees to pay $200 in 4 months, and $200 in 8 months ; what is the equated time of payment ? 4. A merchant bought goods on 4 months' credit, as follows : April 10, $200 ; May 15, $160 ; June 1, $440 ; what is the equated time of payment ? 5. A man borrowed $500, and agreed to pay $100 in 2 mos., $200 in 4 mos., and the balance in 6 mos. ; what is the equated time ? 6. The following bills of goods were bought on six months' credit : Aug. 5, $62.50 ; Aug. 11, $24.50 ; Sept. 10, $32.60 ; Oct. 15, $50. If a note payable in 6 months was given for the whole amount, when ought it to have been dated ? 7. A man bought a farm for $8000, and agreed to pay $2000 at the time, and the rest in three annual payments ; what is the equated time of payment ? 238 EQUATION OF PAYMENTS. 8. The following bills of goods have been bought on 60 days credit ; May 1, $150 ; June 16, $200 ; July 20, $320 ; Sept. 1, $300 ; what is the equated time of payment ? 9. A man owes $1600, % of which is now due ; % of it in 4 months ; % in 6 months, and the rest in 8 months ; what is the equated time of payment ? 10. A merchant has bought goods as follows : Aug. 10, $160 on 60 days' credit ; Sept. 1, $250, 90 days' credit; Oct. 12, $300, 60 days ; Nov. 10, $200, 90 days ; what is the equated time of payment ? 11. A merchant has sold goods amounting to $1200, of which he received $300 at the time ; $300 was to be paid in 3 months ; $300 in 6 months, and the remainder in 9 months ; what is the equated time of payment ? 12. Sold goods as follows : Sept. 5, $500 on 2 months' credit ; Oct. 10, $400, 3 months ; Nov. 15, $600, 3 months ; Dec. 1, $500, 1 month ; what is the equated time of payment ? CASE H. Art. 171* To find ike equated time of the balance of a debt, when partial payments have been made before it is due. Ex. 13. J. Howe owed me $500 payable in 6 mos., but at the end of 3 months he paid me $100, and at the end of another month $200 ; how long may the balance remain unpaid ? Process. $100 for 3 months before due = $300 for 1 month. $200 for 2 months before due = $400 $300 for " " " =$700 Therefore the balance ($500 $300) $200 may remain unpaid as many months as it is contained times in $700. $200) $700 Ans. 85 mos. EULE. Multiply each partial payment by the time it was made before it was due, and divide the sum of the products by the balance unpaid. EXAMPLES. 14. J. King promised to pay $800 in 10 months. At the end of 4 months he paid $200, and after 3 more months $100 ; how long may he wait before paying the balance ? EXAMPLES. 239 15. A merchant owed $1000, payable in 6 months. Atrthe end of two months he paid $200, and at the end of three more months $300 ; how long may he leave the balance unpaid ? 16. A man gave his note for $1200, payable in 6 months ; at the end of the first, third and fifth months he paid $100 each time ; how long may he keep the balance ? 17. A man bought a house for $2400, payable in 2 years, but at the end of 1 year he paid -| of it ; how long after the whole was due may he wait before paying the balance ? CASE HI. Art. 172. To find, the equated time of an account cur- rent. EXAMPLE 18. DR. J. H. MEQLEB. CB. 1867. 1867. April 1 To Mdse. $700 July 11 By Cash $200 " 16 200 Aug. 1 < : 300 May 11 it 100 Sept. 21 100 June 16 Cash 400 Oct. 1 Mdse. 200 Process. First find the equated time of each side. Dr. $700X0 days = 200X15 " =$3000 for Id. 100X41 " = 4100 " 400X77 " =30800 " 1400 )37900 " From April 1, 27 days. Dr. April 28, $1400. Or. $200X0 days = 300X20" =$6000 for Id. 100X72 " = 7200 " 200X82 " = 16400 " $800 ) $29600 " From July 11, 37 Cr. Aug. 17, $800. Difference between April 28and August 17 = 111 days. If the account is balanced April 28, credit is given for $800, 111 days before it is paid, which would be a loss to the creditor. There- fore the balance, ($1400 $800) $000, is due as many days before April 28 as it is contained times in ($800X111 days) $88800, or 148 days. Hence the balance was due Dec. 2, 1866. RULE. Find the equated time of each side ; multiply the amount of the smaller side by the number of days between the two dates of equated time, and divide the product by the bal- ance ; the quotient will be the number of days to be ADDED to 240 EQUATION OF PAYMENTS. the equated time of the larger side when its amount becomes duei;A.ST, but SUBTRACTED from it when it becomes due first. The cash value, or true balance, at any time of settlement, is found by adding the interest up to the time of settlement, when the balance is due beforehand, and subtracting it when due afterwards. EXAMPLES. 19. What must be the date of a note to balance the follow- ing account ? DR. J. WILSON. CB. 1867. 1867. May 1 To Mdse. $900 April 1 By Cash $200 " 16 700 " 16 400 June 1 Mdse. 400 May 15 Draft 500 20. What must be the date of a note to balance the follow- ing account ? DB. G. S. WOOD. CB. 1866. 1866. March 1 To Mdse. (4m.) $1000 June 16 By Cash $500 April 10 " (3m.) 800 July 10 400 June 11 600 Draft (10 d.) 600 CASE IV. Art. 173. To find the true balance of an account bearing interest when the time of settlement is given. EXAMPLE 21. What is the true balance of the following ac- count, at the time of settlement, August 20, 1868, allowing 60 days' credit to each charge, and 7 per cent, interest ? DB. THOS. GOODWIN. CB. 1868. 1 1868. Jan. 2 To Mdse. $200 1 Feb. 20 By Mdse. $100 April 20 400 May 10 300 REDUCTION OF CURRENCIES. 241 Dr. Due Mar. 2, $200. Int. till Aug. 20, AM'T. (172 days). . . . 6.60 $206.61 Due June 20 $400 Int. till Aug. 20, (61 days) 4.66 404.66 $611.27 404.69 Or. Due April 20, $102. Int. till Aug. 20, AMT. (121 days) . . . . $2.36 $102.32 Due July 10 300 Int. till Aug. 20, (41 days) 2.37 302.37 $404.69 True balance, Aug. 20 . . $206.58 EULE. Find the interest on each entry up to the time of settlement. Add the several amounts on each side, and the difference between the sums will be the true balance. EXAMPLE 22. DK. HOWE & STEBBINS. OB. 1863. 1863. July 1 To Mdse. $250 July 21 By Consignm't $130 Aug. 13 200 Oct. 1 Draft 160 Sept. 24 550 10 n 210 What is the balance due Jan. 1, 1864, allowing 6% interest? REDUCTION OP CURRENCIES, Art, 174, Reduction of Currencies is changing one kind of Money into another without altering its value. State Currencies. Sterling Money was formerly the currency of this country before its separation from Eng- land, and is still used to some extent. But its value is not the same as in England ; nor the same in all the States, because its paper bills depreciated more in some than in others. Hence arose the following currencies : N. E, States, Virginia, Kentucky, Tennessee, N. E. Currency in 11 242 SEDUCTION OF CURRENCIES. ( New York, ) N. Y. Currency in -< Ohio, V $1 = 8s. = . ( N. Carolina, ) f Pennsylvania, Penn. Currency in g^SST [ Maryland, , = 7s. 6d.=f. Maryland, Geor. Currency in j g^afdina ' $ 1=4s - 8d - = Hence, also, in New England Currency. .1=$^=$3. 33^ ; ls.=16f cts. New York Currency l=$f =$2.50 ; Is. =12^ " Pennsylvania Currency.. !=$ =$2.66| j Is.^l3| ' Georgia Currency l=$^=$4.28f ; ls.=22| " Art. 175. To reduce U. S. Money to State Currencies. EXAMPLE 1. What is the value of $836.50 in Pennsylvania currency ? $836.50 Process. $1 (7s. 6d.) 7s., or | ; there- fore $836.50:=7i times as many shillings, or | times as many pounds. 20)6273. 75 i Ans. 313.13s. 9.00d. RULE. Multiply the given sum by the value of $1 in the required currency. EXAMPLES. 2. Reduce $315.4375 to New York currency. 3. Reduce $490. 38 to New England currency. 4. Reduce $325.00 to Pennsylvania currency. 5. Reduce $245.00 to Georgia currency. Art. 176. To reduce State Currencies to U. S. Money. EXAMPLE 6. What is the value of 312 18s. 9d. ? 312 18s.9d.=.75s. Process. 7s .=$1, therefore 312 20 18s. 9d. reduced to shillings, and di- 7^6258 75 videdby7S=$834.50. EXAMPLES. 243 EULE. Divide the given sum reduced to shillings, by the number of shillings in $1 ; or reduced to pounds, by the frac- tion of a pound, equal to $1. EXAMPLES. 7. Reduce 90 6s. New York currency to U. S. money. 8. Eeduce 120 7s. 6d. Pennsylvania currency to U. S. money. 9. Eeduce 35 14e$. 9d. New England currency to U. S. money. 10. Eeduce 57 12s. 6d. Georgia currency to U. S. money. Art. 177. TABLE OF THE PRINCIPAL FOREIGN COINS AND THEIR VALUE IN U. S. MONEY. Austria, Florin of Canada, Pound Ster.. China Tael $0.48^ 4.00 1 48 Italy, Dollar of Eome, ' ' Ducat of Naples, " Lira of Sardinia $1.05 .80 ISA Denmark, Dol., (sp.). England, Pound Ster. * * Crown 1.05 4.84 1 06 ' ' Lira of Tuscany, " Livre of Genoa, Mexico, Doubloon *- w io .16 .ISA 15 60 .18 A Portugal, Milrea .... 1.12 ** LlVT6 18 Prussia Florin 22 ^ Germany Florin 40 Eussia Euble ** 74 75 Eix Dollar, India Pagoda .69 1 94 Spain, Eeal Plate Switzerland, Livre . . . .10 27 " Eupee 44^o To reduce Sterling Money of England to U. S. Money, multiply pounds and decimal of a pound by $4.84. To reduce U. S. Money to pounds sterling divide by 4. 84 . Eeduce the decimal to shillings, &c. EXAMPLES. 11. Eeduce 1500 francs to U. S. Money. 12. Eeduce 3000 livres of Switzerland to U. S. Money. 13. Eeduce 140 15s. 9d. to U. S. Money. 14. Eeduce 100 dollars (specie) of Denmark, to U. S. Money. 15. Eeduce 500 rupees of India to U. S. Money. 16. Eeduce 400 rubles of Eussia to U. S. Money. 17. Eeduce $705.6115 to Pounds Sterling, &c. BATIO. Promiscuous Examples, 1. Beduce 89 18s. N. York Currency to U. S. Money. 2. Beduce $4S8.3S to New England Currency. 3. Beduce 36 9d. N. England Currency to U. S. Money. 4. Beduce $314.43% to New York Currency. 5. Beduce 500 francs to U. S. Money. 6. Beduce $810 to Swiss livres. 7. Beduce 60 15s. Pennsylvania Currency to U. S. Money. 8. Beduce $417.25 to Pennsylvania Currency. 9. Beduce 400 India rupees to U. S. Money. 10. Beduce 114 16s. Georgia Currency to U. S. Money. 11. Beduce $954.12^ to Georgia Currency. RATIO. Artt 178. Ratio denotes the magnitude of one number compared with another of the same kind. Arithmetical ratio denotes the difference between two numbers ; geometrical ratio the number of times one contains the other. The word ratio, used alone, means the latter. Ratio equals the quotient of one number divided by another, and is usually expressed by ( : ) written be- tween them ; thus the ratio of 12 to 3 is written 12 : 3. The two numbers are called the terms of the ratio ; written together they are called a couplet, of which the first is called the antecedent, and the second the conse- quent. Ratio respects only things of the same name, or which may be re- duced to the same ; thus the ratio of 12 inches, or 1 foot to 6 inches is 2, but there is no ratio .between 12 inches or 1 foot and 6 cents. Art. 179. -A Compound ratio is the product of two or more simple ratios ; thus the ratio of 2:6. and 3 : 9. make the compound ratio 2 X 3 : 6 X 9 > or 6 : ^ PROPORTION. 245 EXERCISES. Express the ratios of 4 to 8, 3 to 9, 5 to 15, 7 to 14. 12 to 24. What is the ratio of 7:2:12 6:24 40:10 5:25 $4 : $12 10 Ib. : 30 Ib. 12 gal. : 3 gal. 8 rods : 32 rods 50 Ibs. : 1 cwt. 3 qrs. : 4 yds. 24 qts. : 3 gals. 1 : 2s. 6d. A ratio may be reduced to its lowest terms by dividing both terms by their greatest common divisor ; thus, 5 : 25 = 1 : 5. PROPORTION. Art. 180. Proportion is an equality of ratios. The ratio of 4 : 2 = 12 : 6 ; hence 4:2 is in proportion as 12: 6. Proportion is usually expressed by (::) four dots placed between two ratios ; thus, 6 : 3 : : 24 : 12 ; which is read 6 is to 3 as 24 to 12. The first and last terms are called the extremes ; the second and third the means. When the consequent of the first ratio is the same as the antecedent of the next, it is called a mean proportion- al ; thus, 8 : 4 : : 4 : 2 ; 4 is a mean proportional. Art. 181. Proportion is often the most convenient method of solving arithmetical questions, when the re- quired number is evidently as many times greater or less than another of the same kind, as is expressed by the ratio of two other numbers on which they respect- ively depend ; thus, Prices depending on quantities ; Times, &c., depending on distances, &c., are evidently in proportion. 1 Ib. of coffee : 10 Ibs. of coffee :: $0.45 : $4.50 ; 12 men: 4 men:: 6 days: 2 days (in doing a certain work.) 246 PEOPOBTION. The proportion is called direct when a greater number of one kind requires a greater number of another kind, or less requires less ; but indirect or inverse when a greater number requires a less, or a less requires a greater ; thus the first proportion above is direct, because the greater quantity (10 Ibs.) requires the greater price (34 50 ;) but the other is inverse, because the greater number of men (12) requires the less number of days (2,) to do a certain work. In every proportion the product of the extremes is equal to the product of the means ; thus, 3 : 4 : : 6 : 8, 3X8 = 4X6. Hence, The product of the means divided by either extreme gives the other ; and The product of the extremes divided by either mean gives the other. EXAMPLE I. If. 5 yards of linen cost $6.00, what will 20 yards of linen cost ? Process. This is an example in proportion, because it is evident that the prices must be in proportion to the given numbers of yards. Since, too, the price of 20 yards is unknown, it is usually considered as the fourth term of the proportion, which is to be found, and there- fore the corresponding number or given price will be the third term. It is evident, also, that 20 yards will cost more than 5 yards ; there- fore, 20 yards must be the second term, and 5 yards the first term, in order to make the ratios equal, or a proportion. Hence, 5 yards : 20 : : $6 : Ans. Beducing the first ratio to its lowest terms, 1 yard : 4 yards : : $6 : Ans. = $6 X*_ $24 BULK. Consider the answer the fourth term, and make the given number of the same name or kind the third term ; then, if the answer will evidently be greater than the third term, make the greater of the other two numbers the second term ; but, if less, make the smaller number the second term. The remaining number will be the first term. Reduce the left hand ratio to its lowest terms, or cancel any factor common to the first lerm and either the second or third. Then multiply the second and third terms together, and di- vide by the first. EXAMPLE& 247 Compound Numbers must be reduced to the lowest denomination mentioned, and the first and second terms must be of the same name as well as kind. EXAMPLES. 2. If 5 yards of cloth cost $35, how much will 20 yds. cost ? 3. If 12 tons of coal cost $72, how much will 3 tons cost ? 4. If 7 Ibs. of coffee cost $2.33)^, how much will 4 Ibs. cost ? 5. If 3 bbls. of flour cost $22.50, how much will 50 barrels cost? 6. If 44 Ibs. of tea cost $33.00, how much wiU 11 Ibs. cost ? 7. If 10 acres of land produce 250 bushels of corn, how much will 45 acres produce ? 8. If a man travel 300 miles in 12 days, how far can he travel in 4 days ? 9. If 1 Ib. 6 oz. of silver is worth $12.50, what are 3 oz. 10 pwts. worth? 10. If 124 men can build a mill-dam in 60 days, in how many days could 248 men build it ? 11. If a man can walk 3% miles in 1 hour, how long will it take him to walk 12 miles 120 rods ? 12. How many yards of cambric, % yd. wide, will it take to line 20 yards of cloth, 1% yds. wide ? 13. If a quantity of provisions will last 315 men 56 days, how many days will the same last 45 men ? 14. If a quantity of provisions will last 316 men 56 days, how many men will it feed 14 days ? 15. If % of a yard of cloth cost $ 1 \, what will 3% yds. cost ? Art. 182. Proportion may be used in solving ques- tions under many of the preceding rules. 16. Percentage. What is 7 % of $256 ? By Proportion, $100 : $256 : : 7 % : $17.92, Ans. 17. What per cent, of $400 is $24 ? $400: $100:: $24:6,%, Ans. 18. Problems in Interest How long must $200 be at 6 % in- terest to gain $36 ? Interest 1 year is 12, therefore $12 : $36 : 1 year : 3 yrs. , Ans. 248 PROPORTION. 19. If the interest of $600 for 1 year 8 mos. is $60, what is the rate per cent.? Interest at 1 % is $10, therefore 10 : 60 : : 1 : 6 %, Ans. 20. What principal at 6 per cent, will yield $4.52 interest in 1 year 4 months be ? Interest of $1 is 8 cents, therefore .08 : 4.52 : : $1 : $56.50, Ans. 21. Discount. What is the present worth and discount of $306, due in 4 mos., at 6 % discount ? The present worth of $1.02 is $1.00, therefore $1.02 : 100 : : $306 : $300, Ans. The discount of $1.02 is $.02, therefore $1.02 : $.02 : : $306: $6, Ans. 22. Profit and Loss. Sold a horse for $150, and gained 25 % ; what did the horse cost ? $125 : $100 : : $150 : $120, Ans. 23. Partnership. Messrs. Platt, Wood & Torrey are in part- nership ; P.'s capital is $5000, W.'s $4000, T.'s $3000; they have gained $3600 ; what is each one's share ? The whole capital $12,000 : $5000 : : $3600 : $1500, P.'s share, &c. Solve the f ollowing questions by proportion . 24. What is 6% of $750? 25. What per cent, of $500 is $60 ? 26. What is $650 in currency worth when gold is 130 ? 27. What is the interest of $150 for 4 years 2 mos., at 6 % ? 28. What principal, at 6 %, will gain $60 in 4 years ? 29. What is the present worth of $1350, due in 5 years 10 mos., at 6%? 30. What was the cost of a yard of cloth, which being sold for $4.25 occasioned a loss of 20 per cent. ? 31. A bankrupt's debts amount to $9600, and he has proper- ty amounting to $7500 ; how much can he pay a creditor whom he owes $1600 ? 32. Messrs. Bay & Stearns are in partnership ; B. 's capital is $3000, S.'s capital $2000 ; they have gained $2500 ; what is each one's share ? 33L For what amount must a note for sixty days be given, to obtain from a bank $1800, at 6 % discount ? COMPOUND PROPORTION. 249 34. What is the interest of $225 for 2 years 7 mos., at 7 per cent.? COMPOUND PROPORTION. Art. 183. Compound Proportion is the equality of a compound and a simple ratio. It consists of two or more simple proportions. EXAMPLE 1. If 7 men can cut 42 acres of wheat in 3 days, working 10 hours a day, how many acres can 14 men cut in 4 days, working 9 hours a day ? Process. Since the answer will be acres, 42 acres will be the third term. The other numbers, two of the same kind make the first and second terms of as many simple proportions, each couplet to be ar- ranged as if the answer 'depended upon it alone ; thus, % 523 H : U : : 42 $ X 3 X*0)2 X .* $ : 4 by cancellation. 5 *0: 03 5) (42X^X3)504 Ans. lOOf acres. RULE. Make that number which is of the same name as the answer the third term. Arrange* the other numbers in pairs of the same kind, as the first and second terms of as many simple proportions as there are pairs, and each couplet as if the answer depended on it alone. Cancel as in simple proportion, or reduce the ratios and divide the product of all the second and third terms by the product of att the first terms. EXAMPLES. 2. If 6 men can dig a ditch 36 rods long in 8 days, how many men will it require to dig a ditch 72 rods long in 4 days? 3. If 90 Ibs. of beef will supply 12 men 20 days, how long will 144 pounds last 36 men ? 11* 250 CONJOINED PROPORTION. 4. If 6 horses eat 48 busliels of oats in 12 days, how many horses will eat 96 bushels in 8 days ? 5. If ^ mediate exchanges between other cou&teafefik.- '^SMs is done by Conjoined Proportion. EXAMPLES. 2. If $9 in the United States are equal to 12 rubles at St. Petersburg, and 8 rubles in St. Petersburg are equal to 15 florins in Frankfort, and 9 florins in Frankfort are equal to 20 francs in Paris, how many dollars in the United States are equal to 75 francs in Paris ? 3. If 25 yards of cloth are worth 22 barrels of flour, 11 bbls. of flour are worth 150 pounds of wool, and 15 Ibs. of wool are worth 18 Ibs. of butter, how many pounds of butter will pay for 10 yards of cloth ? ALLIGATION. Art. 186. Alligation is finding the prices or quantities of mixtures. It is Medial or Alternate. The word alligation means connecting together, and is used in arithmetic because the prices of mixtures are connected one with another. Art. 187. Alligation Medial is finding the price of a mixture when the quantity and price of each ingredient are given. EXAMPLE 1. A farmer has mixed 20 bushels of corn, at 75 cents, 30 bu. barley at 60 cts., 40 bu. oats at 50, and 10 bu. rye at $1 ; what is a bushel of the mixture worth ? Process. 20 bushels corn X $. 75 $15. 00 30 barley X 60= 18.00 40 " oats X 50= 20.00 _10 rye Xl-00= 10.00 100 bu. of mixture = 53.00 1 =53 cts., Ans. RULE. Divide the price of the whole mixture by the whole quantity. Ex. 2. A grocer mixed 8 Ibs. of tea at 75 cents, 12 Ibs. at 60 6 16 252 ALLIGATION. cts., 15 Ibs. at 50 cts., and 20 Ibs. at 40 cts. ; what was a pound of the mixture worth ? Art, 188, Alligation Alternate is finding the quanti- ties of different ingredients whose prices are known, to form a mixture worth a certain price. Ex. 3. A grocer wishes to make a mixture of tea worth 56 cts. He has one kind at 40 ots. a pound, another 50 cts., another 60 cts., and another 75 cts. ; how much of each may he take ? Process. Since the gain and loss f40 19 must be equal, we connect a less price j 50 4 than that of the mixture with one 56 j greater. On every pound, at 40 cents, there is gained 16 cts., and on every [75 pound at 75 cts. there is lost 19 cents. Therefore, since the gain is to the loss The whole mixtore=40 Ibs. as 16 to 19, the quantities must be as 19 to 16. For the same reason the other ingredients at 50 and 60 must be as 4 to 6. Hence the mixture must consist of 19 Ibs. at 40 cents, 4 Ibs. at 50 cts., 6 Ibs. at 60 cts., and 16 Ibs. at 75 cts. These relative quantities are found by writing the difference be- tween the price of the mixture and that of each ingredient opposite the one with which it is connected. Art, 189, The quantity of one of the ingredients is some- times given. In the above example, let the quantity at 75 cts. be 8 Ibs. Then since 8 is only 4 of 16, only of the other quantities must be taken to form the mixture. Art, 190, Again, the quantity of the whole mixture may be given. In the same example, let the quantity of the mixture be 100 Ibs. in- stead of 40, as found. Then, since 100 =2^ times 40, the quantity of each ingredient as first found must be multiplied by 2. EULE. Write the prices under one another, and the mean price on the left. Connect each price less than that of tJ.-e. mixture with one greater, and each greater with one less. Write the difference between the price of the mixture and that of each ingredient opposite the price with which the latter is connected. The relative quantity of each ingredient will thus be found opposite its price. ALLIGATION. 253 If one of ike quantities, or the whole quantity, is given, and is greater or less than that found, all the others must be in- creased or diminished in the same proportion. EXAMPLES. Ex. 4. A grocer mixed different kinds of sugar at 10, 13 and 16 cents a pound, so that he could sell the mixture for 12 cts. a pound ; how much of each kind did he take ? 5. A farmer mixed 10 bushels of wheat at $1.40, with rye at 96 cents, corn at 72 cts., and oats at 60 cts., so that he could sell the mixture at 76 cts. a bushel ; how much of each did he take? 6. How much water must be mixed with wine, at 90 cents a gallon, so that there may be 100 gallons worth 60 cts. a gal. ? 7. A grocer has different kinds of sugar at 12, 11, 9, and 8 oents a pound ; how may he *nix them so that he can sell the mixture at 10 cents a pound ? 8. How many bushels of corn at $1, and oats at 60 cents a bushel, must be mixed with 20 bushels of rye at $1.30, to make the mixture worth 82 cents ? 9. A wine merchant mixed different kinds of wine at 62)^, 87% and 112} cents a gallcn, with water, so that the mixture was worth 75 cts. ; how much of each did he take ? 10. A grocer having sugar at 8, 16 and 24 cents a pound, made a mixture of 240 Ibs. worth 20 cts. a pound ; how many pounds of each did he take ? INVOLUTION. Art. 191. Involution is multiplying a number by itself. The product is called its power. 3X3 = 9; 3X3X3 = 27; 9 and 27 are powers of 3. The different powers are distinguished as the first, second, third, &c. ; and are expressed by small figures 254 INVOLUTION. above the numbers at the right, called an index (sing.) indices (plu.,) the index of the first power being usually omitted. The first power of 3 is written 3 The second power of 3 is written 3 2 = 3 X 3 The third power of 3 is written 3 3 = 3X3X3 The second power is usually called the Square; the third power the Cube. KULE. To involve a number, or find any power of it, mul- tiply the number by itself one less times than the name of the power denotes. There is one less multiplication than the times the number is used as a factor. In 3X3=9, there is one multiplication, but 3 is used twice as a factor. Powers already found, when multiplied together, produce the power denoted by the sum of their indices ; 3 2 X^ 3 or 9X27 = 3 6 or 243. EXAMPLES. What are the squares or second powers of 1, 3, 5, 7, 9, 11, 13, 25 ? What are the cubes or third powers of 2, 4, 6, 8, 10, 12 ? What are the squares, cubes, and fourth powers of 14, 20, 27, 36 ? What are the squares and cubes of J, |, f , f, -^, .5, 1.2, .05, 1.02? EVOLUTION, Art. 192. Evolution, the opposite of Involution, is finding a number which, multiplied into itself, will pro- duce the given number. The number found is called the Root of the other, and is one of its equal factors : 9 = 3X3; 27 = 3X3X3; 3 is the root of 9 or 27. EVOLUTION. 255 The root of a square is called the square root ; of a cube, the cube root ; of a fourth power, the fourth root, &c. The Radical Sign, +/ is used to express roots. When written alone before a number it expresses the square root ; with the figure 3 over it, the cube root, &c. ^/9 expresses the square root of 9 = 3. 4/27 expresses the cube root of 27 = 3. expresses the fourth root of 81 = 3. Square Root. Art. 193. The square root of a number, is another number, which multiplied by itself once will produce the given number or square. The square of 1= 1 therefore the -v/I 1 " " 2= 4= " T?> , -25 ? EXAMPLES. What is the square root of (2 ) 9801 (8 ) . 12321 (3.).. 4489 (9.).. ...8.1225 6561 (10 ) . . . . 40401 (5.).. 8649 (11 ).. ... .06 9 5 (6.;.. ..7225 (12.).. . 56644 (13.) 531441 (14.) 5499025 (15.) 36372961 (16.), (17.), (18.) >m .0045369 Art, 194, Applications of the Square Root. The areas of circles, squares, and all similar figures, are to each other as the squares of their corresponding dimensions. A circle whose diameter is 4 feet, is to a circle whose diameter is 2 feet, as 4* to 2 2 , or 16 to 4, four times greater. 258 EVOLUTION. A Triangle is a figure bounded by three straight lines, The difference in the direction of two lines which meet, is called an angle ; and if one line is perpendicular to the other, the angle is a right angle. A Bight-angled Triangle is a triangle that has a right angle ; as A B C. The Ilypotlieniise of a right-angled tri- angle is the side opposite the right angle ; as if A C. The Base is the horizontal line ; as B C. The other is the Perpendicular ; as B A. It is demonstrated in geometry, that the square of the hypothenuse is equal to the sum of the squares of the other two sides. This may be illustrated by the opposite figure. The small squares are all square inches or feet, &c. ; and those on the hy- pothenuse are equal to those on the other two sides : 25=16+9. BULB FOR FINDING THE HYPOTHENUSE. Add the squares of the base and perpendicular and extract the square root of the sum. BULE FOR FINDING EITHER THE BASE OR PERPENDICULAR. Subtract from the square of the hypothenuse the square of the other given side, and extract the square root of the remainder. EXAMPLES. 19. How long must a ladder be, to reach to the top of a house 40 feet high, when its foot is placed 30 feet from the house ? 20. A ladder 50 feet long, and having its foot 30 feet from a house, just reaches the top ; what is the height of the house ? 21. A ladder 50 feet long just reaches the top of a house 40 feet high ; how far is its foot from the house ? SQUARE KOOT. 259 22. A room is 16 feet long and 12 feet wide ; what is the diagonal distance between its opposite corners ? 23. Two ships, one sailing directly north and the other di- rectly west, are 100 miles apart, both having sailed the same distance from the same place ; how far have they sailed ? Art. 195. The side of a square equal to any given surface^ is found by extracting the square root of the surface. Kectangles, whose length is a certain number of times greater than their breadth, may be divided into that number of squares, with sides equal to the breadth of the rectangles. 24. It takes 25 square yards of carpeting to cover a room ; how many feet square is it ? 25. A rectangular field containing 20 acres is twice as long as it is wide ; what is its length and breadth ? Art. 196. The areas of similar figures are in proportion to the squares of their similar sides or dimensions. 26. A man having a grass plot 16 feet square, wishes to make it 4 times larger ; how many feet long must each side be ? 27. A man having water conducted from a spring to his house by a lead pipe )^-inch in diameter wishes to increase the quantity four times ; how large a pipe must he use ? The square root of the product of two numbers is a mean proportional between them. What is the mean proportional between (28.) 9 and 25 (29.) 4 and 16 (30.) Sand 18 (31.) 3 and 27 (32.) 1 and 49 (33.) 5 and 20 (34.) 2 and 8 (35.) 4 and 9 (36.) 5 and 125 Promiscuous Examples in Square Root, 37. A man about to build a house 32 feet square, wishes to have the peak 12 feet higher than the plate beams ; how long must the rafters be ? 38. A man having 1764 peach trees, wishes to set them out 260 EVOLUTION. in a square field, so that it shall be exactly filled with the trees ; how many must be in row each way ? 39. A man having 1152 apple trees, wishes to set them in rows twice as long one way as the other ; how many must be in a row each way ? 40. A farmer having a ditch 3 fee.t deep and 2 feet wide* wishes to make it twice as large in the same proportion ; how deep and wide must it be ? Cube Root. Art. 197. The cube root of a number, is another num- ber, which multiplied by itself twice, will produce the given number, or cube. The cube of 1 is 1 therefore the l/\ is 1 " " 2 " 8 " 4/8 " 2 " 3 " 27 " 4/27 " 3 4 64 " 4/64 " 4 " " 5 " 125 " 4/125 " 5 " " 6 " 216 " s/216 " 6 7 343 a/343 7 " " 8 " 512 " 4/512 " 8 " " 9 " 729 " 4/729 " 9 " " 10 " 1000 " v'lOOO 10 To find the cube root of any large number, we ob- serve First That in cubing a number, each figure is cubed, and multi- plied three times into the squares of each of the others, also its square is multiplied three times into each of them. Secondly That the cube of any number contains as many periods of three figures each, beginning at the right, as there are figures in the root, except the left-hand period may have only one or two figures. Thirdly That the cube of a unit's figure is contained in the cor- responding unit's period ; of a ten's figure in the ten's period, &c. ; also, that the cube of the left hand figure is the greatest cube in its corresponding period. CUBE ROOT. 261 The cube of any number from 1 to 9 is from 1 to '729 and has 1 period. 10 to 99 is from 1000 to '970 299 and has 2 periods. 100 to 999 is from 100000 to 997'002 999 and has 3 periods. Therefore the cube root of any number from 1 to '729 is from 1 to 9, 1 figure. 1000 to 970'299 is from 10 to 99, 2 figures. 1000000 to 997'002'999 is from 100 to 999, 3 figures. Or we may consider any cubic number as representing the contents of a cube, in cubic inches, feet &c. The cube of 36 is 36 (36)2 36 "210 6X6 108 30X6 19QR nr 6X30 36 30X30 7776 46656 6X6 X6 = 63 = 216 30X6 X6 = 30 X6 2 = 1080 6X30X6 = 30 X6 2 = 1080 30X30X6 =(30) 2 = 5400 6X6 X30= 30 X6 2 = 1080 30X6 X30=(30 2 X6 = 5400 2x6 = 5400 2 == 27000 46656 30X30X30=(30 (36) To find the cube root of 46656, we reverse this process. Dividing the number into periods of 30 2 X3 =2700 three figures, except the left-hand period, 30 X6X3= 54 46'656(36 we ascertain that the root has two figures. 6 2 = 36 27 The greatest cube in the left-hand period 3275 )r9656 being 27 thousands, its corresponding 19656 root is 3 tens, which we write on the 262 EVOLUTION. right, and subtract its cube from the period. The remainder with the next period annexed is 19656. This must contain three times the square of the root (30, already found) multiplied by the next root figure (to be found, ) three times the root (found, ) multiplied by the square of that figure, and its cube ; therefore 19656-^-2700, allowing for an in- crease of the divisor, equals (6) the next root figure. Three times this figure multiplied by the former part of the root, we add to the divisor, also its square that in multiplying, its square thus multiplied and its cube may be included in the product. The divisor, thus in- creased, is ( 2700 -f 540 -|- 36 ) 3276 which multiplied by (6) the last root figure equals 19656 the lust remainder. Therefore the cube root of 46656 is exactly 36. EULE FOR EXTRACTING THE CUBE BOOT. Separate the given number into periods of three figures each, beginning at the right of whole numbers, and the left of decimals. Find the greatest cube in the left-hand period, and placing its root at the right, subtract it from that period, and bring down the next for a dividend. Write three times the square of the root already found, with a cipher annexed, for a trial divisor, and, allowing for its increase, find how many times it is contained in the dividend, and annex the quotient to the root. Add three times the pro- duct of the last root figure with a cipher annexed, and the former part of the root, to the trial divisor, also the square of the last root figure. Multiply the completed divisor by the last figure in the root, subtract the product from the dividend and bring down the next period. Proceed thus, till the figures in the root are equal in number to the periods. Treat fractions as in square root. EXAMPLES. What is the cube root of Ex. (1.) 205379 (5.) 80.763 (2.) 614125 (6.) 29.503629 (3.) 41421736 (7.) 146363.183 (4.) 162771336 (8.) 122615327232 (9.) t& (10.) ^ (11.) y*W (12-) 5JM (13.) CUBE EOOT. 263 Artt 198. The contents of all similar solids are to each other as the cubes of their corresponding dimensions. 14. What is the side of a cube containing 1728 cubic feet ? 15. What is the side of a cube equal to a block 36 in. long, 8 in. wide, 6 in. high ? 16. If a ball 3 inches in diameter weighs 24 Ibs., what will be weight of a similar ball 6 inches in diameter ? 17. A man wishes to have a cubical cistern made, which will hold 25hogsh'ds; what must be its depth, &c., allowing 231 cubic inches to a gallon ? 18. A stack of hay 16 feet high, is worth $50 ; what is the value of a stack of similar shape, 20 feet high ? 19. A farmer wishes to have a cubical box made that will hold 100 bushels of grain; allowing 2150.4 cu. in. to a bushel, what must be its depth, &c. ? 20. A cistern 15 feet deep, holds 1410.048 gals.; how deep must a cistern of similar shape be to hold half the quantity ? Promiscuous Examples in Square and Cube Root, 1. The pole of a circular tent 96 feet in diameter is 36 feet high; what must be the length of a rope that will reach from the top of the pole to the circumference of the tent ? 2. A speculator has bought 1000 acres of western land, to be laid out in a square ; what must be the length of its sides ? 3. A man wishes to have a cubical ice-house under ground, that will contain 4096 solid feet of ice ; what must be its inside dimensions ? 4. A gentleman wishes to have a new house built with a foundation twice as large, but in the same proportions, as the old one, which is 40 feet long and 30 feet wide ; what must be its length and width ? 5. What is the mean proportional between 45 and 96 ? 6. If a cylindrical cistern 6 feet in diameter will hold 30 hogs- 264 EVOLUTION. heads of water, how many gallons will a similar cistern hold, whose diameter is 12 feet ? 7. A general has an army of 5184 men ; how many must he place in rank and file to form them into a square ? 8. If a pipe 2> 2 ' in. diameter discharges 10 hogsheads of water in an hour, what must be the diameter of another pipe, to discharge 40 hhds. in the same time ? 9. What is the side of a cubical box that will hold just one bushel (1250. 4 cu. in.)? 10. A certain square, containing 20736 sq. feet, is paved with stones a foot square ; how many are there in a row across one of its sides ? 11. If a piece of silver 3 inches in length, is worth $150 ; what is a similar piece worth which is 6 inches long ? 12. The area of a circle 80 feet in diameter, is 5026.56 sq. feet ; what is the area of a circle whose diameter is 60 feet ? 13. A sphere 30 in. in diameter contains 8 cu. ft. 313.2 cu. in. ; what is the solidity of a sphere 50 feet in diameter ? PROGRESSION. Art. 199. Progression is a series of numbers increas- ing or decreasing uniformly. There are two kinds, Arith- metical and Geometrical. The Terms are the numbers forming th series ; the first and last of which are called the Extremes^ the others, the Means. When the terms increase the series is Ascending ; when they decrease, Descending. Arithmetical Progression. Art. 200. Arithmetical Progression is a series of numbers increasing or decreasing by a common difference / as, 2, 4, 6, 8, 10, 12, 9, 6, 3. ARITHMETICAL PEOGBESSION. 265 In arithmetical progression it is required to find the first term (a), the last term (I), the common difference (d), the number of terms (w), or the sum of the series (s), of which three must be given. In the series ascending, 3, 5, 7, 9, 11, 13, 15, descending, 15, 13, 11, 9, 7, 5, 3, if the common difference (2) be multiplied by the number representing any term, less 1, and added to or subtracted from the first term, it will give the term sought. Again, half the sum of the first and last terms is the average of all the terms, which, multiplied by the number of terms, equals the series. Hence the following rules : Art, 201. To find the last term, the first, the common difference, and the number of terms being given. Multiply the common difference by the number of terms less 1, and (in an ascending series) add the product to the first term, or subtract it (in a descending series). l = adx(n-l) Art, 202, To find the common difference, the extremes and number of terms being given. Divide the difference of the extremes by the number of terms less 1. I a or a I n-l Art, 203, To find the number of terms, the extremes and common difference being given. Divide the difference of the extremes by the common difference, and add one to the quotient. I a or a I -- -3 + 1 Art, 204. To find the sum of the series, the extremes and number of terms being given. Multiply half the sum of the extremes by the number of terms. 12 266 GEOMETRICAL PROGRESSION. a + I s = x n Many more cases might be added. EXAMPLES. 1. What is the last term of a series whose first term is 2, common difference 3, and the number of terms 25 ? 2. What is the common difference in a series whose first term is 2, last term 200, number of terms 10? 3. What is the number of terms of a series, whose extremes are 2 and 32 ; common .difference 3 ? 4. What is the sum of a series whose extremes are 3 and 23 ; the number of terms 11 ? 5. I agree to give a man for work 3 cents the first hour, V cts. for the second, 11 cts. for the third, &c., for 10 hours; what will he receive for the last hour ? 6. A man had nine children at equal intervals, the oldest 35, and the youngest 3 years old ; what was the difference in their 7. A man traveled on foot the first day 5 miles, and the last 45, increasing his journey each day 4 miles ; how many days did he travel? 8. How many times does a clock strike in twelve hours ? Geometrical Progression. Art. 205, Geometrical Progression is a series of num- bers increasing by a common multiplier, or decreasing by a common divisor. In geometrical progression it is required to find the first term (a), the last term (), the ratio (r), the number of terms (n), and the sum of the series (s). In the series ascending, 3, 6, 12, 24, 48, 96, or descending, 96, 48, 24, 12, 6, 3, if the ratio (2 or ) be raised to a power one less than the number of any term, and multiplied by the first term, the product will be the other term. Again, if we multiply the ascending series by the ratio, we shall have another series, 6, 12, 24, 48, 96, 192, twice as great as the other, 3, 6, 12, 24, 48, 96, . The difference between them MENSURATION. 267 (192 3, the other terms being alike in both series), 189, is therefore the sum of the given series. Hence the following rules: Art. 206. To find the last term, the first term, the ratio, and number of terms being given : Multiply the first term by that power of the ratio whose index is one less than the number of terms. l=a x r n - 1 Art, 207. To find the sum of the series, the extremes and number of terms being given : Multiply the last term by the ratio, and divide the differ- ence between the product and first term by the difference between the ratio and 1. The last term is frequently to be first found. (iXr) a ~ r 1 or 1 r Many more cases might be added. EXAMPLES. 1. What is the 12th term of a series whose first term is 3, and the ratio 2 ? 2. "What is the sum of a series whose first term is 2 ; ratio 3 ; and number of terms 10? 3. A man gave his son $100 when he was 20 years old, and promised to give him $200 the next year, and to double the sum every year for ten years ; how much did he give his son when he was 30 years old ? and how much in the ten years? MENSURATION. Art. 208. Mensuration is finding the length of lines, and the contents of surfaces and solids. The rules for Mensuration being derived from Geometry, can not be explained by Arithmetic. Mensuration has already been applied to squares, cubes, &c. It may also be applied to other surfaces and solids, as Parallelograms* A parallelogram is any four-sided 268 MEN8UKATION. figure whose opposite sides are equal and parallel. They include squares and rect- angles. The Base is the line (AB) on which the figure appears to stand. The Altitude is the length of a perpendicular line (CD) from the base to the opposite side. Art, 209. To find the area of a parallelogram. RULE. Multiply the base by the altitude. Triangles. (Art. 194.) The altitude of a triangle is the length of the per- pendicular drawn to the base, or the base produced from the opposite angle. B Art, 210. To find the area of a triangle. RULE. Multiply the base by half the altitude. Or, when the three sides are given, from half the sum subtract each side separately ; multiply together the re- mainders and half the sum, and extract the square root of the product. Circles, (See Art. 51.) To find the circumference. RULE. Multiply the diameter by 3.14159. Art, 211, To find the diameter. RULE. Multiply the circumference by .3183. Art. 212. To find the area. RULE. Multiply the square of the diameter by .7854 ; or the square of the circumference by .07954. Cylinders. A cylinder is a body whose diame- ter is uniform, and whose ends are parallel circles. Art, 213. To find the surface of a cylinder. RULE. Multiply the circumference by its length or height, and add to the product the area of the two ends. (Art. 212.) MENSUKATTON. 269 Art. 214, To find the solidity. RULE. Multiply the area of either end by the length or height. The areas and solidity of prisms may be found by the same rules. A Prism is a solid whose sides are rect- angles, and whose ends are similar and equal. A Pyramid is a solid having triangular sides meeting at a point at its top called its vertex. A Cone is a solid having a circular base and tapering to a point at its top called its vertex. A Frustum of a pyramid or cone is the part which remains after the top is cut off by a plane parallel to the base. Art, 2 15, To find the lateral surface of a pyramid or cone : KULE. Multiply the perimeter or circumference of the base by one-half its slant height. To find the solidity of a pyramid or cone : RULE. Multiply the area of its base by one-third its height. Art* 216, To find the surface of a frustum : RULE. Add the perimeters or circumference of the two ends together, and multiply the sum by one-half the slant height ; to the product add the areas of the ends. To find the solidity of a frustum : RULE. Add the areas of the ends to the square root of their product, and multiply the sum by one-third the height. Spheres, A sphere is a body every part of whose surface is equally distant from the center. Art, 217, To find the surface of a sphere : RULE. Multiply the square of the diameter by 3.14159. 270 MENSURATION. Art. 218. To find the solidity of a sphere. RULE. Multiply the cube of the diameter by .5236. Art, 219, To find the curved surface of a cone. RULE. Multiply the circumference of the base by half the slant height. The slant height differs from the perpendicular height, as the hypot- enuse of a right-angled triangle from its perpendicular. Art. 220 To find the contents of a cask in gallons. RULE. Add two-thirds the difference between the head and bung diameters to the head diameter / or, .6 if the staves are little curved, to find the mean diameter then multiply the product of the square of the mean diameter into the length by .0034. EXAMPLES. 1. One side of a triangular field is 16 rods long, and a straight line at right angles with that side from the opposite corner is 20 rods ; how many acres are there in the field? 2. The sides of a triangular field are respectively 20, 24, and 32 rods long ; what are the contents of the field ? 3. In laying out a grass-plot, I fastened a rope, 36 feet long, at one end, and with the other end, extended the whole length of the rope, I described a circle ; what are the contents of the plot? 4. How many square feet of sheet-iron will it take to make a stove-pipe 18 feet long and 6 inches in diameter ? 5. How many square feet of tin will cover the ball of a spire 3 feet in diameter? 6. How many gallons of oil will a cylindrical tank hold, that is V feet deep, and 4 feet in diameter, allowing 231 cubic inches to a gallon ? 7. How many cubic feet of gas will fill a balloon 25 feet in diameter ? 8. How many acres are there in a field which can be divided MENSURATION. 271 into two triangles, one having a side 32 rods long, and distant in & straight line, at right angles to it, 24 rods from the opposite corner; the other having its sides respectively, 36, 28, and 20 rods long? 9. The water-wheel of a mill is 24 feet in diameter ; what is its circumference ? 10. A circular pond is 300 feet in circumference ; how much land does it cover ? 11. The diameter of the earth is 8,000 miles (nearly) ; suppos- ing it to be a perfect sphere, how many square miles are there on its surface ? 12. How many sheets of tin, 18 inches long and 1 foot wide, will cover a conical spire, 40 feet high and 18 feet in diameter at its base, allowing for waste in cutting ? 13. How many cubic feet in a stick of timber. 40 feet long, 3 feet in diameter at the larger end, and 2 feet 6 inches at the smaller ? 14. How many gallons in a cask whose bung diameter is 18 inches, head diameter 12 inches, and length 24 inches ? 15. How many square feet of tin will be required to make an oil-can, in the form of a frustum of a cone, 12 inches in diameter at the bottom, and 4 inches at the top, the slant height being 2 feet, allowing ^ for waste in cutting ? PROMISCUOUS EXAMPLES. UNITED STATES MONEY AND COMPOUND NUMBERS. 1. A lady purchased a shawl for $16, 2 pairs of gloves at $1.37^ a pair, and 14 yds, of silk ; she gave the merchant three $20 bills and received back $16.05 ; what was the price of the silk per yard ? 2. A silversmith had 10 Ibs. 3^ oz. of silver, and made it into spoons ; these he sold for $!> apiece, receiving for them all I 272 PEOMISCUOUS EXAMPLES. $42% ; how many spoons did he make, and what was the weight of each ? 3. Beduce 3 m. 4 fur. 21 rods, 3 yds. 6 in. to inches and prove the operation. 4. How many shingles will it take to cover the roof of a house 42 feet long, and 30 feet wide, whose rafters are 18 feet long, allowing each shingle to be 6 inches wide and to lie 6 inches to the weather ? 5. What will it cost to build a brick house 36 feet long, 32 feet wide, and 24 feet, high (on an average,) the walls 1 foot thick, each brick being 8 inches long, 4 inches wide and 2 inches thick, at $6^ per 1000 bricks ? 6. A man was born at 8)4 o'clock A. M., Sept. 5, 1835, and died at 6 o'clock P. M., April 21, 1863 ; what was his age ? 7. The longitude of New York is 74 1', and that of Cincin- nati, 84 24' ; what is the difference of time ? 8. The difference of time between Washington and St. Petersburg is 7 hours 9 minutes ; what is the difference of longitude ? 9. A farmer exchanged 3 pairs of oxen at $112 a pair, and 7 cows at $42 each, for 126 sheep ; what was the price of the sheep per head ? 10. Bought of J. Ayers 7 yds of cloth at $6 a yard, 3 bbls. of flour at $9 barrel, and a cheese for $3 ; he agreed to take in payment 6 cords of wood at $7 a cord, and 18 bushels of corn at $1; how much cash must I pay him to balance the account ? 11. In measuring a portion of a railroad, one man made the distance 43 m. 7 fur. 31 rds. 1 yd. l^ft. ; and another 42 m. 1 fur. 39 rods, 5 ft. 7 in. ; what was the difference in the meas- urements ? 12. In measuring a field, one man made it contain 4 acres, 3 roods, 18 square rods, 6 feet, 64 square inches ; but another 4 acres, 2 roods, 39 square rods, 8 yards, 100 square inches ; what was the difference in the measurements ? 13. A certain railroad is 56) miles long ; in making it, it COMPOUND NUMBERS. 273 was divided equally between 6 contractors ; how much of it did each make ? 14. The difference in longitude between London and Boston is 71 8'; what is the difference of time ? 15. The difference of time between Jerusalem and Balti- more is 7 h. 24 m. 36 sec. ; the long, of Baltimore is 76 37' west; what is the long, of Jerusalem ? 16. How many blocks of marble 6 inches square, will pave two halls of a hotel, crossing each other in the centre ; one 32 ft. long, and 12 ft. wide, the other 64 ft. long and 8 feet wide ? 17. In a town 5 miles square, how many farms can there be containing 150 acres each ? 18. A man has agreed to deliver 48 cords of wood ; the wood is 4 feet long, and he makes the pile 5 feet high ; how long must he make it ? 19. A man has $1623 which is 4 times as much as he had last year, wanting $121 ; his brother had last year 3 times as much as he had, and $10.50 more, but has since lost $1000 ; how much has his brother left ? 20. A merchant having $1000, paid $510 for dry goods and the remainder for 20 barrels of molasses ; how much was the molasses per barrel ? 21. How many suits of clothes, each requiring 4 yds. 2 qrs. , can be made from 639 yards ? 22. How many bricks will be required to pave a sidewalk 80 yds. long and 3 yds. wide, each brick being 8 in. long, and 4 in. wide ? 23. How many yards of muslin, 3 qrs. wide, will line 9 yds, 1 qr. 2 na. of merino cloth, 1 yd. 1 qr. wide ? 24. How much will it cost at 2s. 6d. a square yard to plaster a room 24 feet long, 18 feet wide, and 12 feet high, there being two doors 7 ft. high, and 3 ft. 6 in. wide ; also two win- dows 5 feet high, and 3 feet 4 in. wide, and a mop-board 8 inches wide. 12* 274 PROMISCUOUS EXAMPLES. 25. A man wishes to divide a field of 4 acres into 7 building lots of equal size ; how much will each lot contain ? 26. The longitude of New York is 74 1'; a sea captain, sailing thence, finds that his watch has lost 2 hours ; what is his longitude? 27. The difference of longitude between New York and New Orleans is 15 4'. What is the difference of time? 28. A lady shopping in New York, bought 14 yds. of silk at 15s. 6d. a yard; 5 yds. of linen at 7s. 4d.; 2 yds. of thread-lace at 18s., and a pair of gloves for 10s. 9d. She gave in payment a $100 bill ; how much money should she have been paid back ? 29. Purchased 24 A. 1 B. 27 rods at $360 an acre. I sold the same for building lots at $3.15 a square rod ; what did I gain? 30. A grocer bought 5 barrels of cider at $4 a barrel, and after making it into vinegar, sold it at 10 cts. a quart ; how much did he gain? COMMON AND DECIMAL FRACTIONS. 1. A farmer had of his sheep in one pasture, in another, in another, and the remainder, 46, in a fourth ; how many sheep had he ? 2. A merchant bought 6| cords of wood at $5 a cord ; and paid for it in cloth at $4 a yard ; how many yards did it require ? 3. Bought 640 sheep at $2| a head, and afterward 270 at $2f ; sold f of them at $3^; | of the remainder were killed by dogs, and what still remained I sold for $3 a head ; how much did I gain ? 4. What will 75 yds. 1-J qrs. of silk cost, at .375 a yard? 5. A butcher had 351 Ibs. of beef; he sold f of it, corned of the remainder, and used what was then left in his family ; what was the value of that which his family consumed at 17^ cts. a pound ? 6. What part of of a solid yrd., is of a yard solid ? 7. How much can a man earn in f of a year at $1 a day ? COMMON AND DECIMAL FRACTIONS. 275 8. A piece of oil-cloth 3 yds. square is worth $30. What is 3 square yards of it worth ? 9. A farmer sold 4 fields containing respectively, 6^, 8^, 5-|, and 7^ acres, at $100 an acre ; what did he receive for them ? 10. A hogshead of molasses contained 96 gallons; -f% of it lias been sold ; of the remainder has been used ; how much now remains ? 11. Two men were 82 miles apart, one of them traveled -fo of this distance, and the other ^ of the remainder ; - how far were they then apart? 12. A laborer hired to a farmer for a year for $313, but he was sick of the working time, and was absent of the remain- ing time; how much wages was due to him at the end of the year? 13. A man had a field 36^ rods long, and 24f rods wide ; he sold $ of it for $112.40 ; what is of the remainder worth at the same rate ? 14. A merchant sold $ of a hogshead of sugar weighing 879 Ibs. for $96. What is f of the remainder worth at the same rate? 15. Bought a piece of merino cloth containing 48f yards, and having cut off of it, sold f of the remainder at $! a yard, and what still remained at 87-| cents a yard ; what was the amount sold? 16. A man having 28 tons of coal, sold f of it at $10f a ton, and the remainder at $9f . What amount was received for the coal? 17. A man bought 42^ tons of hay, at $11^ a ton, and sold \ of it at $11|, and the remainder at $13 a ton ; how much did he gain ? 18. A grocer bought a cask of vinegar containing 43f gals, for *15^, and has sold 19 gals. 2 qts. 1^ pts. at cost; for what must he sell the remainder in order to gain as much as the whole cost? 19. What cost .778125 ton of buckwheat flour, a 2d. at pound? 20. What cost 1.8875 acre of land at $1 a rod ? 276 PROMISCUOUS EXAMPLES. 21. What cost 112 hhds. 3.35 gals, of molasses, at 3 8s. 9d. a hogshead ? 22. If .3125 yard of cloth cost $ ; what is the price per yard? 23. If .0625 bbl. of flour cost $^-; what is the price pel- barrel ? 24. A man having 185f acres, sold ^ of it, at another time | of it; what is the value of the remainder at $115f an acre? 25. A man deposited his money in 4 banks, in one f , in another J, in the third , and in the fourth the remainder, which was $48 more than -jL of the whole ; how much money did he deposit? 26. Bought f of a box of starch, sold of it for $4f ; what was the whole box worth at the same price ? 27. A man gave of his money for a horse ; of the remainder for a wagon ; | of what then remained for a saddle; he then had $24.25 left; how much had he at first? 28. A man bequeathed his property to his five children : to the first % of it ; to the second ; to the third ; to the fourth | ; and the remainder to the fifth, who had $665 less than the fourth ; what was the amount of the property ? 29. A person being asked the time of day, said : The time past noon is f of the time from now to midnight. 30. Two men, A and B, were playing cards for money ; f of A's money was equal to f of B's ; but B lost $42, and then had only -j^ times f as much as A then had ; how much had each ? 81. A man gave $1750 to two benevolent societies, and gave one $150 more than the ^ of what he gave the other; what were the amounts given ? 32. A man bequeathed $5,420 to his son and daughter, so that $240 more than of what his daughter had was equal to f of what his son had; how much had each? 33. A pole 102 feet high stands near a house, and of the part above the house equals the other part ; how much higher is the pole than the house ? 34. James said to Henry, 2 years added to of my age equals COMMON AND DECIMAL FRACTIONS. 277 of yours, and the sum of our ages is 37 years ; what was the age of each ? 35. A person being asked his age, replied, that f of it, f of it, of it, and 7 more, would be twice his age ; what was his age ? 36. The difference between my neighbor's property and my own is $1000 ; f of mine equals f of his ; but of his is 1^ times ^ of mine ; what is the property of each ? 37. I wish to make 3 boxes, each 5| feet long, 3^ feet wide, and 2^ feet high ; how many square feet of boards 1-| inches thick will they require ? 38. A man has a garden 10 rods long and 7 rods wide ; what will it cost to dig a ditch around it 3 feet wide and 4 feet deep, at 3 cents a cubic foot? 39. A man traveled 28f miles the first day, 33|| miles the second, and 29^ T miles the third ; how far did he travel in the three days ? 40. The distance from Boston to Worcester is 40 miles ; a man havjng traveled -fo of the distance, and afterward -ft of it ; how far was he then from Boston? 41. Bought 23| bushels of corn at $lf a bushel, and sold f of it for $1-|, and the remainder at $1-| ; what was the whole gain ? 42. Bought f of a flouring mill for $1,837.50, and having sold 7 of my share, I gave of the remainder to cancel a mortgage on it, and what still remained I gave the miller for ^ a year's wages ; what was the amount of his wages a year ? 43. Bought a horse, carriage, and harness for $350. The har- ness cost -fy as much as the horse, and the horse f as much as the carriage ; what did each cost? 44. Bought I of a ton of plaster, and sold -fa of it for $4.50 ; what was the price per ton? 45. One of my horses usually travels 6 miles in of an hour, and the other 7 miles in ^ of an hour ; how much longer will it take one to travel 20 miles than the other ? 46. I have a garden 13f rods long, and f as wide, surrounded by a fence 6 feet high. Next to the fence is a border 1 rod wide, for shrubbery and fruit-trees ; then a gravel walk 8 feet 278 PROMISCUOUS EXAMPLES. wide, the rest is for cultivation; how much is there to be cultivated ? 47. How many pieces of paper, 9-| yards long and 20 inches wide, will it take to paper a room 22-^ feet long, 15^ feet wide, and Hi feet high? 48. A man invested - of his property in his business, of the remainder in stocks, ^ of what still remained, $454, in a farm ; what part of his property was thus invested, and what was the amount of the whole? 49. A man invested \ of his property in a farm, and of the remainder he spent in building a house ; the farm cost $1000 more than the house ; what was the amount of his property ? 50. After spending f of my money, and J of what remained, I had $62.50 left ; what sum had I at first? 51. A cask was full of vinegar ; after drawing from it 8 gal- lons it was \ full ; how many gallons did it hold ? 52. If f of the time past noon is f of the time to midnight, what time is it? 53. If a man draw 350 loads of bricks, and 1,500 bricks at each load ; how much will he receive at the rate of 87^ cents a thousand ? 54. Sold 3000 feet of boards, at $9.50 a thousand, and 700 lath at 50 cents a hundred, what was the amount ? 55. Bought 160 sheep at $2 a head, and 215 at $1.87 a head, sold of them at $2|, and the remainder at $2 ; did I gain or lose, and how much ? 56. Bought 364 pounds of sugar, at 16f cents a pound ; if the price per pound had been 3 cents less, how many pounds could have been bought for the same money ? 57. My farm is f meadow, ^ orchard, and the remainder, 20 acres, more than of the whole, is timber; how large is my farm? 58. The income of a farm, consisting of 125A. 1R. 7$ rods, was 202 11s. 3fd. ; how much was it an acre? 59. How much wheat will 32 A. 1R. 10 rods yield, at the rate of 25 bu. 3 pks. 1 qt., per acre ? PERCENTAGE AND ITS APPLICATIONS. 279 60. I have two small farms, of the acres in one, added to f of the acres in the other, make 90 acres ; and of the first is $ of f of the second ; how much larger is the first farm than the second ? PERCENTAGE AND ITS APPLICATIONS. 1. Sold a quantity of grain for $251.50, for which I received a note dated Oct. 1, 1862, payable in 6 months ; what is the value of the note Dec. 27 ? 2. I have forwarded to my agent in St. Louis, $5000 for the purchase of flour ; after deducting ty% commission, what will be the cost of the flour? 3. What must be the face of a note payable in 4 months, for which I may receive from a bank $600 at 6% discount ? 4. A woolen factory was insured for $37,500 at 2-|% ; after 2 years it was burnt; what was the loss to the company ? 5. What is the duty at 30% ad valorem, on 26 barrels of sugar, each weighing 225 pounds; tare 15%, and the sugar costing 16 cents a pound ? 6. Bought 25 barrels of flour at $10 a barrel, and sold it im- mediately at $11.96 a barrel, on 8 months' credit, what per cent, did I gain, allowing 6% interest ? 7. Sold 34 tons of coal at $8 a ton, for which I received a note payable in 90 days, and had it discounted at a bank. I then found that I had lost 10% on the coal ; what did it cost? 8. If I buy cloth at $7.50 a yard on 9 months' credit, for what must I sell it for cash, to gain 12% ? 9. Messrs. Mead, Gage & Storrs, of Chicago, made a consign- ment of flour to New York ; M. furnished $1,400 ; G. $600, and S. 200 barrels of flour. They gained $270, of which S. had $90 ; at what was his flour valued per barrel, what was M.'s and G.'s share of fche profits? 10. Three men A, B, and 0, hired a pasture for $72. A put in 3 horses for 6 weeks, B put 3 pairs of oxen for 5 weeks, and put in 12 cows for 4 weeks. It was agreed that 5 cows should 280 PROMISCUOUS EXAMPLES. be considered equal to 3 oxen, and 4 oxen to 3 horses ; what was each one's share of the expense ? 11. Messrs. Dudley & Swift contracted to build a section of a railroad for $26,000 a mile ; D. furnished 60 men, and S. 40 horses and carts, with boys to drive them. It was agreed that 3 men be considered equal to 2 horses and their drivers. Swift also was to be allowed $100 a mile for overseeing the work. After completing 5| miles, what was each one's share ? 12. A merchant in Philadelphia wishes to remit to .Liverpool 1000 ; what will a bill of exchange for this amount cost him at 9K % premium ? 18. When gold is 135, which is the better investment, U. S. 5-20's at 103, or bank stock at 108, paying a semi-annual divi- dend of 4% ? 14. Bought goods amounting to $256.50, and having kept them 6 months, sold them so that I gained $% ; for what were they sold? 15. Borrowed of my neighbor $450 for 6 months ; I after- ward lent him $300, long enough to compensate him ; how long did he keep it ? 16. A certain town is taxed $3022.75 ; the taxable property amounts to $146,637.50; there are 150 polls, each taxed 60 cts. ; what per cent, is the tax, and what is a man's tax who pays for two polls and whose property is valued at $1837.50? 17. What is due on the following note, at *I% interest, Jan. 1, 1868? $500. SING SING, K Y., Oct. 10, 1862. On demand, I promise to pay S. Wilbur, or order, five hundred dollars, value received, with interest. H. VAN WYCK. Indorsed, Jan. 1, 1863, $60. April 1, 1865, $75. June 15, 1864, $150. Jan. 1, 1866, $100. 18. Sold a lot of lumber for $500, and gained Vb\% ; what did it cost ? 19. Clark & Smith are partners; 0. put in $2000, and they PERCENTAGE AND ITS APPLICATIONS. 281 have gained $203, of which Smith's share is $87 ; what was his capital ? 20. Lent $176 for 1 year 6 months; it then amounted to $195.14 ; what was the per cent.? 21. Sold several shares in an Oil Company for $6875, which was 40 per cent, less than they cost; what did they cost ? 22. Bought goods to the amount of $1200, sold them for $1344 ; what was the gain per cent. ? 23. Paid $10,000 for a cargo of cotton, and sold it for $15,000, but invested it in stock, which I sold at 15 per cent, less than it cost ; what was the net gain ? 24. Sold 36 bushels of corn for $29.70, and lost \*l\%\ what per cent, should I have gained if I had sold it for $40^ ? 25. Bought 30 yards of cloth at 5% less than the first cost, and sold it at 5% more than the first cost ; I gained $15 ; what was the first cost per yard ? 26. A bank discounted a note payable in 60 days, at 6% dis- count ; and gave for it $2.52 less than the face of the note ; what was the amount of the note ? 27. Bought a house for $3000 ; rented it at $350 ; paid $% for insurance, for taxes If/^, for repairs $106 ; what per cent, did the investment yield ? 28. A man sold his farm for $2500, which was 16f% less than he paid for it; .he then bought another, and sold it for 16%" more than he paid for it ; he thus gained as much as he had lost ; what did he pay for each farm ? 29. A merchant bought 120 yards of cloth, at $4 a yard, on 6 months' credit, and sold it immediately for $500, money being at 6%; what did he gain? 30. The tax in a certain town is \\% besides each poll $1. One man's tax is $127, including 2 polls ; what is the amount of his property? 31. A collector received $36 for his services, at %\%\ what was the amount he collected ? 32. A man bought a house, and after spending 10% of the 282 PBOMI6CTJOTTS EXAMPLES. price in repairs, found that the whole cost was $4400 ; what was the price of the house ? 33. Bought a house for $4000 ; paid for repairing it $1500, it remained unoccupied 3 months, when I sold it for $6000, for which I received a note payable in 90 days, after one month I had the note discounted at a bank at *1%\ what per cent, did I gain by the purchase ? 34. Lent my neighbor $900 from Jan. 1 to Sept. 1 ; then borrowed of him $1150 from Sept. 1 to March 1 ; what is the balance of interest at 6%, and to whom due ? 35. What investment at 5% will yield a semi-annual income of $250 ? 36. I have bought a bill of goods amounting to $420, on 60 days' credit. Is it better for me to pay cash at 2^% discount, or receive Q% interest for the money till the time of credit expires? 37. In order to pay the above bill is it better for me to obtain the money from a bank at 6% discount, or take the goods on credit ? 38. Exchanged 100 shares of Erie stock at 40% below par, for stock in a Gold Mining Company at 125 ; how many shares did I receive ? 39. Gained $175 by buying 50 shares of bank stock, at 5% advance, and selling them again : for how much a share were they sold? 40. A man has $4000, invested in U. S. 5-20's ; what income will it yield when gold is 130 ? 41. Bought 1% bonds at 103, amounting to $7210; what an- nual income will they yield ? 42. How much must be invested in 6% bonds at 90, to yield a semi-annual income of $500 ? 43. If I buy 6% bonds at 90, amounting to $6000, what % will the investment yield ? 44. Which is the better investment, D". S. 7-30's at 102, or 6% State bonds at 98 ? 45. What must be the price of gold that U. S. 5% bonds at 95 may yield 6% interest ? PERCENTAGE AND ITS APPLICATIONS. 283 46. A bankrupt settled with his creditors by paying them 70 cents on a dollar ; one o'f them received $507.50 ; what was the amount of his claim ? 47. What will be the premium at l^% for insuring a vessel to cover both its value $10,000 and the premium? 48. At the age of 40 a man insured his life for $5000, at the rate of $36 a 1000, not to be paid after 20 years. "What will his family gain or lose if he die at 45, 50, 55, 60, or afterward, money being worth 6% ? 49. How much must be paid in currency for duties on 25 bar=" rels of sugar, each weighing 224 Ibs., tare 12,%, and the duty 5 cts. a pound in gold, when gold is 130? 50. A gentleman in St. Louis owns 50 shares of the Corn Exchange Bank in New York, which has declared a semi-annual dividend of 5%", a draft for which he sells at \% premium ; what does he receive for it ? 51. A merchant in New York owes a debt in Liverpool of 250. When gold is 130, and exchange 9^, is it better to buy a bill of exchange, or remit U. S. bonds at 95, which can be sold in Liverpool at 60 ? 52. A man left $10,000 to be divided between his two sons, 16 and 18 years old, so that at 6% interest they should each have the same amount when 21 years old ; what did he leave for each? 53. A man owed $287.70; he paid at one time 40% of the debt ; at another time 25% of what he then owed ; and after- ward 12-^%" of what he still owed; how much of the debt remained to be paid ? 54. A merchant having $5000, lost | of it in speculation, and ^ of the remainder in bad debts ; what per cent, of the whole did he lose ? 55. A man spent $487.50 in traveling, which was 15% of his income ; what was his income ? 56. A man exchanged 14 shares of bank stock, at T% premium, for 25 shares of railroad stock, at 12^% discount, and agreed to pay the difference in cash ; how much did he pay ? 284 PROMISCUOUS EXAMPLES. MISCELLANEOUS RULES. [These include various rules not used in the last promiscuous examples.] 1. I have 3 rooms, respectively 12, 16, and 20 feet wide, which I wish to cover with oil-cloth that will exactly fit all of them without cutting off the width ; how wide must the oil-cloth be ? 2. I have 3 pieces of oil-cloth, respectively 3, 6, and 9 feet wide ; what must be the width of a room that either of them will exactly fit without cutting off the width ? 3. What will it cost to gild a ball 10 in. in diameter, at $10.80 a square foot ? 4. What must be the height of a pole which, being broken 30 feet from the top, struck the ground 18 feet from the bottom? 5. Jan. 1. I owe J. Bush $325 due in 4 months ; $362.50 due in 8 months ; and $250 due in 12 months ; at what date should I give my note to settle the account ? 6. What is the equated time of payment of the following bill : 1867. Walter Hickok to H. Walcot. June 1. Mdse, $225 " 12. " (4mos.) 250 Aug. 16. Cash, 125 7. What is the equated time of settling the following account : Dr. A. Knapp. Or. 1867. June 1. Mdse, . . . $200 " 16. " (3 mos.) 400 Oct. 20. Cash, ... 175 1867. July 4. Cash, . . . $200 Aug. 20. Mdse, . ... 76 Sept. 20. " ... 250 8. J. Smith has a horse worth $250, but in trading, values it at $280 ; W. Read's horse cost $300 ; at what should he value it in trading with Smith ? 9. If 24 men can build a wall 33f feet long, 5f ft. high, and 3 ft. thick, in 126 days, by working 9h. 20min. each day; how many hours a day must 217 men work to build a wall 23^ ft. long, 3f ft. high, and 2 ft. thick, in 3 days ? 10. Bought 4 tubs of lard, each weighing 50 Ibs., at 13 cts. a pound; 10 tubs of 40 Ibs. each, at 10 cts. ; 24 tubs, 25 Ibs, each, MISCELLANEOUS RULES. 285 at 7 cts. ; sold the whole at an average of 9^ cts. a pound ; how much was the gain? 11. A butcher bought lambs worth $2, $2-, $3, and $4 a head, for which he gave, on an average, $2^ ; how many at each price did he buy ? 12. A butcher bought 12 calves at $6 a head ; how many must he buy at $9 and $15 a head that he may sell them all at $12 a head without loss ? 13. A butcher bought 85 sheep at an average price of $lf a head ; for some he paid $1, for some $1, for some $2, and for others 2^ ; how many at each price did he buy ? 14. The diameter of a circle is 10 feet ; what will be the diame- ter of another circle twice the area of the first ? 15. A farmer wishes to make a bin which will contain 250 bushels of grain ; its width to be twice its depth, and its length twice its width ; what must be its dimensions ? 16. A boy agreed to work 19 days, for which he was to receive 4 cents the first day, and 3 cents more every day than the preced- ing ; how much did he receive the last day ? 17. A boy bought ten apples ; for the first he agreed to pay 1 mill, for the second 2 mills, and so on ; what did he pay for the last? 18. The circumference of a park is 84 rods; what is its area? 19. A cistern is 6 feet deep and 5^ feet in diameter ; how many hogsheads of water will it contain? 20. "What is the solidity of the largest ball that can be cut out of a cubical block whose sides are five inches square? 21. Two boys are running around a block the larger boy runs around it every 5- minutes, and the smaller boy every 6 minutes ; if they started together how many times must each run around the block before they will be together. 22. How much corn must I take to a mill, that there may be 4 bushels left after taking 4 % from each bushel for toll? 23. A liquor dealer has 60 gallons of brandy, worth $3 a gal- -on, which he wishes to reduce so that he can sell it at $2.40 a gallon ; how much water must he add to it ? 286 MISCELLANEOUS RULES. 24. A thief started from a place at midnight and traveled 8 miles an hour ; the sheriff started in pursuit 3 hours later, and traveled 10 miles an hour; at what time was the thief over- taken ? 25. J. Taylor can mow 4 acres in 3 days, and his son can mow 5 acres in 4 days ; in how many days can they both mow 12|^ acres ? 26. I mixed 16 Ibs. of tea at 75 cents, 20 Ibs. at 87 cents, and 12 Ibs. at $1.25 ; what is a pound of the mixture worth? 27. If 1 Ibs. of tea be worth 8| Ibs. of coffee, 3 Ibs. of coffee be worth 5 Ibs. of sugar, and 3 Ibs. of sugar be worth 40 cents, what is the price of the tea ? 28. What is the equated time of settling the following account : W. JOHNSON. Dr. Or. 1868. 1868. Jan. 1, . Mdse $224 Feb. 1, . Draft (3 days) . 182 " 20, . Cash . . . . 116 Jan. 20, . Cash .... $280 Feb. 6, . Draft (10 days). 132 " 25, . Consm't ... 450 29. A grocer has spices at 9d., Is., 2s., 2s. 6d. ; how must he mix them so that he can sell the mixture at Is. 8d. a pound ? 30. What is the greatest number of hills of corn that can be planted on a square acre, the centers of the hills to be 3^ feet apart ? 31. If 10 barrels of water flow through a pipe 2-| inches in diameter, what must be the diameter of a pipe that will discharge four times as much in the same time ? 32. If a silver ball 3 inches in diameter be worth $270, what is another 6 inches in diameter worth? 33. A man bought 19 yards of linen ; for the first he gave Is., and for the last 1 17s. ; what did the whole cost? 34. What is the area of a circular plot, 11 rods in diameter? 35. What are the solid contents of a column whose average diameter is 18 inches, and whose length is 20 feet? QUESTIONS. Article 1. What is arithmetic? Number? Abstract numbers? Concrete ? [2] What is notation ? How many kinds of notation are in com- mon use ? [3] What is the Roman method? How many letters does it use ? What is the effect of repeating the letters ? of writing a letter before another of greater value ? after it ? [Examples. ] [4] What is the Arabic notation ? How many and what figures does it employ ? Which are called digits ? What is the simple value of a figure? [Examples.] What is its value when written before anoth- er? [Examples.] When written before two others? three, four, etc. ? How do figures increase in value ? Kepeat the French Notation and Numeration table. How is it divided ? Name the periods in order. How may numbers consisting of several figures be written, or what is a rule for Notation? [Instead of learning the rules in the book, it is better that the pupil should thoroughly understand them, and express the ideas in his own words.] [5] What is numeration ? How may large numbers be read ? [6] What are the fundamental rules of arithmetic ? [7] What is addition? What is the number found, or answer called? What is simple addition? [Illustration.] What is the sign of addition ? Give a rule for adding numbers, consisting of one fig- ure or units ; of units, tens, etc. , or several figures. [8] What is subtraction? What is the answer called? the num- ber to be subtracted ? the number from which another is subtracted? What is simple subtraction? [Illustration.] What is the sign of subtraction ? [Illustration. ] Give a rule for subtraction. [9] What is multiplication? What is the number to be multi- plied called ? the number by which another is multiplied ? the an- swer ? What are the factors ? What is simple multiplication ? the sign ? What two numbers in multiplication are properly of the same name ? What kind of a number is the multiplier ? [Illustration. ] [10] Give a rule when the multiplier consists of one figure or a numberless than 12 : [12] when it is greater than 12 : [13] for the 288 QUESTIONS. multiplication of numbers having ciphers on the right ; [14] What is a composite number ? [Illustration. ] Give a rule for multiplying by composite numbers. [15] What is division? What is the number to be divided called? the dividing number? the answer? the number that is some- times left ? What is the sign of division ? In what other way is division sometimes expressed ? What is such an expression called ? With what do the divisor and quotient correspond ? the dividend ? With what must the name of one of the factors in division corres- pond? What is the other factor? [16] How many kinds of division are there ? What is short divi- sion? long division? Give a rule for short division; [17] long division. [18] When there are ciphers on the right hand of the divisor ; [19] when the divisor is a composite number. [20] How is the quotient affected when the dividend is multi- plied by any number ? when the divisor is divided ? when the divi- dend is divided? when the divisor is multiplied? when the divi- dend and divisor are both multiplied or both divided by the same number ? United States Money. [22] What is United States Money ? Of what does it consist? What are its coins in gold? silver? Repeat the table. [23] What are aliquot parts? What part of $1 are 10 cents? 12|? 16$? 20? 25? 33? 37i? 50? 62? 75? 87i? [24] Give a rule for writing United States Money. [25] for read- ing United States Money. [26] for reducing dollars to cents; cents to mills ; dollars to mills ; dollars and cents to cents ; mills to cents ; cents to dollars ; mills to dollars. [27] In what respect is United States Money like simple num- bers? [28] Rule for addition of United States Money ; [29] sub- traction ; [30] multiplication ; [31] division ; [32] when the price of anything is an aliquot part of $1. [34) when the price is per hundred or thousand ? Compound Numbers. [36] What are compound numbers? [Illus- tration. ] What are the general names ? [37] What is English or Sterling Money? Eepeat the table. [38] For what is Troy Weight used? Table. [39] Avoirdupois Weight? Table. [40] Apotheca- ries Weight? Table. [41] How many pounds in a barrel of flour? of beef, pork, or fish? firkin of butter? bushel of wheat? rye or corn? barley? oats? [42] What is Cloth Measure? Table. [43] Long Measure? Table. [44] Surveyor's Measure? Table. QUESTIONS. 289 [45] Square Measure ? Table. What is a square ? How may the contents of a square be found ? What is the difference between 3 square feet and 3 feet square, etc., etc.? What is a rectangle? How may the contents of a rectangle be found? [46] Cubic Mea- sure? Table. What is a cube? How may the contents of a cube be found? How may one of the dimensions be found? [47] Wine Measure? Table. [48] Beer Measure? Table. [49] Dry Measure? Table. [50] Time Measure? Table. [51] Circular Measure? Ta- ble. What is a circle? the circumference? diameter? radius? [52] Miscellaneous Table of units, etc. , paper and books ? Reduction of Compound Numbers. [53] What is reduction of compound numbers ? What two kinds are there ? [54] What is re- duction descending ? ascending ? Give a rule for reduction descend- ing; ascending. [68] Give a rule for addition of Compound Numbers. [69] Sub- traction. [71] Multiplication. [72] Division. [74] Give a rule for finding the difference of time when the dif- ference of longitude is known, [75] for finding the difference of longitude when the difference of time is known. Cancellation. [77] What is cancellation? Why may we cancel? Give a rule for cancellation. Properties of Numbers. [78] What are even numbers? odd? prime ? composite ? [79] Give a rule for resolving composite num- bers into prime factors. [81] What is the greatest common divisor of two or more num- bers? [82] How may it be found? [83] What is the multiple of a number ? the common multiple of two or more numbers ? the least common multiple ? How may it be found ? Rule I., II., III., IV. Fractions. [86] What are fractions? What is meant by , , ? [87] How many kinds of fractions are there? What are common fractions? decimal fractions? [88] How are common fractions written ? What is the number above the line called ? the number below the line ? What are the terms of a fraction ? What does the denominator show? With what does it correspond in division? What does the numerator show ? With what does it correspond in division ? [89] How are common fractions divided ? What is a simple frac- tion? when is it proper? when improper? What is a compound fraction ? complex ? mixed number ? What are like fractions ? un- like ! [90] What is the value of a fraction? [91] How is the value 12 290 QUESTIONS. . of a fraction affected by multiplying the numerator by any number ? dividing the denominator? dividing the numerator? multiplying the denominator? multiplying both the numerator and denominator? dividing both ? [92] What is reduction of fractions? What are their simplest forms? [93] When is a fraction in its lowest terms? How may it be reduced to its lowest tertns ? [94] To what may an improper frac- tion be reduced ? How ? [95] To what may a mixed number be re- duced ? How ? How may a whole number be reduced to a fraction ? [96, 97] How may unlike fractions be reduced to like fractions ? [Rules L, II.] [98] In what is reduction of compound fractions in- cluded. [100] Give a rule for addition of fractions? [101] subtraction? [102, 103, 104] multiplication? [106, 107] division? [108] In what is reduction of complex fractions included ? In what respects ? Decimal Fractions. [109] What are decimal fractions ? How are they distinguished from whole numbers? [110] Repeat the table. [Ill] What is the denominator of a decimal fraction ? What is the effect of prefixing a cipher? why? annexing a cipher? why? Bead the examples. [112] Eule for writing decimals, examples. [113] Eule for addition of decimals ? [114] subtraction ? [115] What is the general principle of multiplication of decimals ? Eule ? Eule for multiplying by 10, 100 etc. ? [116] What is the general principle of division of decimals ? Eule ? Eule for dividing by 10, 100 etc. ? [118] How may common fractions be reduced to decimals? [119] decimal to common fractions ? [120] Fractional Compound lumbers, what arc they ? Examples. General Eule. [121] How may compound whole numbers be re- duced to fractions ? [122] How may fractional compound numbers be reduced to whole numbers ? [123] to other denominations ? [127] Duodecimals, what are they? Whence arise? From what is 'the name derived? How added, subtracted etc. ? What denomi- nation is the product of feet by feet ? square feet by feet ? feet by inches ? square feet by inches ? inches by inches ? square inches by inches? inches by seconds? square inches by seconds? seconds by seconds ? [128] What is Analysis ? General process ? [129] What is Percentage ? From what is the term per cent, de- rived ? [130] What three things are chiefly considered in percentage ? IThatis the principal? rate? percentage? How is the rate expressed ? QUESTIONS. 291 examples. [131] How may the percentage be found ? [132] the rate per cent. ? [133, 134] the principal ? [136] Applications of Percentage. To what is percentage appli- cable ? [137] What is Commission ? A. consignment ? gross pro- ceeds ? net proceeds ? [138] an account of sales ? [139] brokerage ? a broker ? [110] stocks ? common value of a share ? when are stocks at par ? above par, at a premium or an advance? below par or at a discount? stockholders? dividend? bonds? [141] How is gold bought and sold ? [142] What is insurance ? fire insurance? marine insurance? life insurance? a policy? the pre- mium? [144] What is profit and loss ? how estimated? [145] Interest, what is it ? What is the principal when interest is to be found? rate? amount? simple interest? legal interest? the legal interest in the different States ? How is interest found for one year? two or more years ? months? days? [146] Another method ? [147] Exact interest? [148] How is interest on Sterling Money calculated ? [149] Partial Payments, what are they? What is a note ? Write a note in the usual form. Who is the maker or drawer? the payee? What is the face of a note ? How is the amount due on a note found when partial payments have been made ? [150] Merchants' Rule ? [151] Connecticut rule ? [152] How is the rate found from the principal, interest and time ? [153] The time from the principal, interest and rate? [154] The principal from the interest, rate and time ? [155] Compound interest, what is it ? how found? [156] Discount, what is it ? What is the present worth of a sum or debt? How is the present worth found? How is the discount found? [157] Bank discount, what is it ? How does it differ from true discount ? How may it be found ? [158] How may the amount of a note be found, which will be worth a given sum at bank discount? [159] Taxes. What is a tax? poll tax? income tax? What is real estate ? personal property ? an inventory or list ? explain by an ex- ample the process of finding a person's tax. [160] Duties, what are they? What is a port of entry? a custom house? ad valorem duty ? specific duty? an in voice? tare? draft? leakage ? gross weight ? net weight ? how is the duty on goods found ? [161] Exchange, what is it? domestic? foreign? write a draft. Who is the drawer ? drawee ? payee ? How may a draft be accepted ? 292 QUESTIONS. What is an acceptance? How may a draft be indorsed? What is the use of an indorsement? [162] Why is the exchange on England always at a premium ? [165] how may it be found ? how may we find the amount of a bill of ex- change which can be bought with U. S . money ? [165] Partnership, what is it ? a firm or house ? [166] How may each partner's share of profit and loss be found? [167] when the times are unequal? [169] Equation of Payments is what? equated time? an account current? [170] How may the equated time be found? [171] when partial payments have been made? [172] of an account current? [173] of an account bearing interest ? [174] Reduction of Currencies, what is it ? Why have State cur- rencies been used? Wha't is New England Currency? its value? New York Currency? its value? Pennsylvania Currency? its value? Georgia Currency? its value? [175] How may U. S. money be re- duced to State currencies? [176] How may State currencies be re- duced to U. S. money? [178] Ratio, what is it? arithmetical? geometrical? To what is the ratio of two numbers equal? how expressed? What are the terms of a ratio ? What are they both called ? What is the first term caUed? the last? [179] Compound ratio, what is it? [180] Proportion, what is it ? how expressed? what are the ex- tremes ? the means ? what is a mean proportional? [181] a direct pro- portion? an indirect or inverse proportion? How is the product of the extremes compared with that of the means? How may the terms be arranged ? How may the required term be found ? [183] Compound Proportion,"what is it? How may the terms be arranged? How may the required term be found? [184] Conjoined Proportion, what is it ? How may the terms be arranged ? How may the required term be found ? [186] Alligation, what is it? [187] What is alligation medial? How is the price of a mixture found ? [188] What is alligation alter- nate ? How may the quantities of different ingredients be found, that will form a mixture at a certain price ? [191] Involution, what is it? What is a power? How are differ- ent powers distinguished ? What is the second power usually called ? the third ? How are powers found ? QUESTIONS. 293 [192] Evolution) what is it ? What is a root ? a square root '( cube root ? the radical sign ? How are the different roots expressed ? [193] What is the square root of a number? of 1? 4? 9?&c.? How may we find the square root of any number ? [194] How is square root applicable to circles, squares, or other similar figures ? What is a triangle ? a right angle ? a right angled triangle? the hypothenuse? base? perpendicular? How may the hypothenuse be found ? the base or perpendicular ? [195] the side of a square ? [196] the areas of similar figures ? [197] What is the cube root of a number ? of 1? 8 ? 27 ? 64 ? 125, &c. ? How may the cube root of any number be found ? [198] How is cube root applicable to solids ? [199] Progression, what is it? what kinds are there? What are the terms ? the extremes ? the means ? an ascending series ? descend- ing? [200] Arithmetical Progression, what is it? What are to bo found? [201] How may the last term be found? [202] the common difference? [203] the number of terms? [204] the sum of the, [205] Geometrical Progression, what is it? What are to be found? [206] How may the last term be found? [207] the sum of the series? [208] Mensuration, what is it? What is a parallelogram? cylin- der ? prism ? pyramid ? cone ? frustum ? sphere ? How is the sur- face of each found ? How are the solid contents found ? . UNIVERSITY OF CALIFORNIA LIBRARY Due two weeks after date. 30m- 7,' 12 YB I74IS 19 inch condemn sod, il\ comprised m a st>ri< s of bool .Hi of which, much of d. It is !>oand in tw. : ; PART ! nil os, - hv ;t pr I will pupils. PART with Cane