2. Divide 3515 by 42. Ans. 83f§. OPERATION. rinDING TH1 TRUE REMAINDER. 2) 35 1 5 4 X 3 X 2 — 24, 1st Product. 3) 1 757, 1, 1st Rem. 2x2= 4, 2d Product. 7 ) 5 8 5, 2, 2d Rem. 1, 1st Remainder. 8 3, 4, 3d Rem. 29, true Remainder. Or, (4 X 3 X 2) -f- (2 X 2) + 1 = 29, true Rem. Using as divisors 2, 3, and 7, the factors of 42, we obtain for remainders, 1, 2, and 4. The first remainder, 1, is a unit of the given dividend, since it is a part of it (Art. 71). The second remainder, 2, is units of the quotient 1757, whose units are 2 times as great as those of the given dividend. The third remainder, 4, is units of the quotient 585, whose units are 3 times as great as those of the quotient 1757, of which the units are 2 times as great as those of the given dividend. Now, these remainders must be all of the same units as the given dividend, to constitute the whole or true remainder. We therefore multiply the third remainder by 3 and 2, the divisors used in producing the quotient of which it is a part; and the second remainder by 2, the divisor used in producing the quotient of which it is a part; and the products with the first remainder added together give 29, the whole or true remainder sought. Hence, as shown by these illustrations, when the divisor is a composite number, we may Divide the dividend by one of the factors, and the quotient thus found by another, and thus proceed till each of the factors has been made a divisor. The last quotient will be the quotient required. If there be remainders, multiply each remainder, except the first, by all the divisors preceding the one which produced it; and the first remainder being added to the sum of the products, the amount will be the true remainder. Note. — There will be but one product to add to the first remainder when there are only two divisors and two remainders. Examples. 3. Divide 7704 by 24 = 4 X 6. Ans. 321. 4. Divide 8317 by 27 = 3 X 9. 5. Divide 3116 by 81 = 9 X 9. 6. Divide 61387 by 121 = 11 X H- Ans. 507T^'T. 7. Divide 19917 by 144 = 12 X 12. Ans. 138^. 8. Divide 91746 by 336 = 6 X 7 X 8. Ans. 273^. 9. At 45 dollars an acre, a farm of how many acres can be bought for 5464 dollars? Ans. 121^f acres. 79. When the divisor contains one or more ciphers at the right hand. Ex. 1. If 10 men receive 792 dollars for a job of work, what will be each man's share of it? Ans. 79T20- dollars. Operation. Xo multiply by 10, we annex one 1 I 0 ) 7 9 I 2 cipher, which removes the figures one Quotient"^, 2 Rem. P!fe t0, the}f ■ ?nd, tTM/^*1c£? ^ ' value denoted tenfold (Art. 65). Or thus: 7 9 I 2. Now, it is obvious, that, if we reverse the process, and cut off the righthand figure by a line, we remove the remaining figures one place to the right, and consequently diminish the value denoted by each the same as dividing by 10. The figures on the left of the line are the quotient, and the one on the right is the remainder, which may be written over the divisor and annexed to the quotient. Hence each man's share is 79^. 2. How many years will it take a man at a yearly salary of 700 dollars to earn 3664 dollars? Ans. 5f$$ years. Operation. The divisor, 700, mav • 1I00)36I64 be resolved into the factors „ x~o"7 ~7T. ,, . T, 7 and 100. We first divide 7 ) 3 6, 6 4,1st Kern. by the factor 100i by cut 5, 1, 2d Rem. tinS off two figures at the 'right, and get 36 for the Or thus: 7I00)36I64 quotient, and 64 for a re mainder. We then divide 5, 16 4. the quotient, 36, by the other factor, 7, and obtain 5 for a quotient and 1 for a remainder. The last remainder, 1, being multiplied by the divisor, 100, and 64, the first remainder, added, we obtain 164 for the true remainder (Art. 77); and for the answer required, 5^5 years. Hence, when the divisor contains one or more ciphers at the right, we may, to perform the division, Cut off the ciphers from the right of the divisor, and the same number of figures from the right of the dividend; and then divide the remaining figures of the dividend by the remaining figures of the divisor. Note. — When, by the operation, there is a remainder, to it must be annexed the figures cut off from the dividend to form the true remainder. Should there be no last remainder, then the significant figures, if any, cut off from the dividend, will form the true remainder. Examples. Quotient. Rem. 3. Divide 123456789 by 10. 12345678 9. 4. Divide 987654300 by 100. 10. The entire annual loss to the United States in consequence of intemperance has been estimated to be about 98,400,000 dollars. How many schools at a yearly expense of 600 dollars would that sum support? Ans. 164,000 schools. 11. The late war with Russia was carried on at a cost of 600,000,000 dollars to Great Britain. Allowing that country to have a population of 28,000,000, what was the cost to each individual? 12. If light moves at the rate of 192,000 miles in a second, how long is it in passing from the sun to the earth, a distance of 95,000,000 of miles. Ans. 494{|f seconds. GENERAL PRINCIPLES AND APPLICATIONS. 8d In division, the value of the quotient depends upon the relative values of the divisor and dividend. 81 i If the dividend be multiplied, or the divisor divided, by any number, the quotient is multiplied by the same number. Thus, if the dividend be 20 and the divisor 4, the quotient will be 5; but if the dividend be multiplied by any number, as 2, and the divisor remain unchanged, the quotient will be 2 times as large as before, or 10; as (20 X 2) -h 4 = 10; and if the divisor be divided by the 2, and the dividend remain unchanged, the quotient will be, likewise, 2 times as large, or 10; as 20 -i- (4 2) = 10. 82. If the dividend be divided, or the divisor multiplied, by any number, the quotient is divided by the same number. Thus, if the dividend be 32 and the divisor 8, the quotient will be 4. But if the dividend be divided by any number, as 2, and the divisor remain unchanged, the quotient will be only half as large as before, or 2; as (32 -T- 2) 8=2; and if the divisor be multiplied by 2, and the dividend remain unchanged, the quotient will be, likewise, only half as large, or 2; as 32 -i(8 X 2) — 2. 83. If the dividend and divisor be both multiplied, or both divided, by the same number, the quotient will not be changed. Thus, if the dividend be 16 and the divisor 4, the quotient will be 4. Now, if we multiply the dividend and divisor by some number, as 2, their relative values are not changed, and we obtain 32 and 8 respectively, and 32 ,— 8 = 4, the same as the original quotient. Also, if we divide the dividend and quotient by some number, as 2, their relative values are not changed, and we obtain 16 and 2 respectively, and 16 -f- 2 = 8, the same quotient as before. 84. If a factor in any number is rejected or cancelled, the number is divided by that factor. Thus, if 24 is the dividend and 6 the divisor, the quotient will be 4. Now, since the divisor and quotient are the two factors which, being multiplied together, produce the dividend (Art. 72), it follows, if we reject or cancel the factor 6, the remaining 4 is the quotient; and, by the operation, the dividend 24 has been divided by 4. CANCELLATION. 85. Cancellation is the method of abbreviating arithmetical operations by rejecting any factor or factors common to the divisor and dividend. Ex. 1. Sold 19 thousand shingles at 4 dollars a thousand, and received pay in wood at 4 dollars a cord; how many cords of wood was received? Ans. 19 cords. Operation. Having indicated by signs Dividend A- X 1 9 the multiplication and di . ~l = >,^ Quotient, vision required by the ques Uivisor 4 t;on^ tljenj since dividing both dividend and divisor by the same number will not change the quotient (Art. 83), we divide them by tne common factor 4, by cancelling it in both, and obtain 2a for the quotient. 2. Divide the product of 15, 3, 28, and 13, by the product of 7, 30, and 4. OPERATION. Dividend Ii!x3x^ X 13_ 39 Divisor 7 X # 0 X <f _ 2 ~ Quotlent . 2 The product of the 7 and 4 in the divisor equals the 28 in the dividend; we therefore cancel all these numbers. Finding 15 in the dividend to be a factor of 30 in the divisor, we cancel both of the numbers, and use the remaining factor 2 in place of the 30. There now being no factor common to both dividend and divisor uncancelled, we multiply together the remaining factors in the dividend, and divide the product by the remaining factor in the divisor, and obtain the quotient 19 *. Rule. — Cancel the factor or factors common to the dividend and divisor, and then divide the product of the factors remaining in the dividend by the product of those remaining in the divisor. Note. — 1. In arranging the numbers for cancellation, the dividend may be written above the divisor, with a horizontal line between them, as in division (Art. 67); or, as some prefer, the dividend may be written on the right of the divisor, with a vertical line between them. Note. — 2. Cancelling a factor does not leave 0, but the quotient 1, to take its place, since rejecting a factor is the same as dividing by that factor (Art. 84 '. Therefore, for every factor cancelled, either in the dividend or divisor, the factor 1 remains. Examples. 3. Multiply 24 by 16, and divide the product by 12. Ans. 32. 4. Divide 48 by 16, and multiply the quotient by 8. Ans. 24*. 5. Divide the product of 7, 10, 12, and 5, by the product of 18, and 6. 6. If 15 be multiplied by 7, 27, and 40, and the product divided by 54 multiplied by 14, 10, and 2, what will be the result? Ans. 7£. 7. Divide the product of 13, 15, 20, and 5, by the product of 26, 10, 2, and 3. Ans. 12£. 8. Divide the product of 28, 27, 21, 15, and 18, by the product of 7, 54, 7, 3, and 9. 9. How many pounds of butter at 28 cents a pound will be required to pay for 56 pounds of sugar at 11 cents a pound? Ans. 22 pounds. |