49. REDUCE Compound fractions to simple ones; and all the fractions to a common denominator. Then add the nu merators together and place the sum over the common denominator for the answer. When the fractions are large, or numerous, it will be best to reduce them to the least common denominator. Examp. 1. What is the sum of i, j, and †? 1+2+3= 6. Ans. or 11. 2. Required the sum of, and y,? 2+4+5= II. Aus. ‡¦ or lo 3. What is the sum of }, }, and {? The fractions when brought to a cominon denominator will be 29, 15. and : 20+15+12 = 47. Aus.. 4. Required the sum of 9, and } of !! } // == 1; and brought to a common denominator are 7 and ††: 50. When mixed numbers, or mixed numbers and fractions are to be added together, bring the fractions to a common denominator, then set down the integers as in common addition, and the fractions on the right hand : Add the fractions together, and carry the integers (if any) from the sum, to the numbers on the left, which add up as in common addition. Es. 5. What is the sum of 4213, 674, and fi 4211 490 Answer. 1 6. Required the sum of 1000}, 74, and 6 The fractions t, th of when brought to a common denominator are 31. LET the fractions be prepared the same as for Addition: then the difference of the numerators set over the common denominator will give the difference of the proposed fractions. Ex. 1. What is the difference of † and †? The difference of the numerators 1 and 3 is 2; therefore the required difference is for . ↑ and & brought to a common denominator, are }; and }}; therefore }¿~}}=}· Ans. 4. What is the difference of † and H¦! 4 and 1 reduced to a common denominator are!} and {}}}} therefore the fractions are equal, } and reduced to a common denominator are and 52. When the difference of two mixt numbers, or a mixt number and a fraction is required, bring the fractions to a common denominator as before; then place the less number under the greater and take their difference for the answer. But if the lower fraction is greater than the upper one, subtract the nume rator of the former from the sum of the terms of the latter, then set down the difference for the numerator of the remaining frac tion, and carry 1 to be subtracted. 8. Required the difference of 17 and If. The fractions and † reduced to a common denominator are TM and ʼn 9. From 101290 5610 In this example I take from Hof 1. And in the preceding example, 7 is taken from 18 (the sum of the terms of the fraction ), which is the same thing as subtracting → from added to ; for in either case 1 is borrowed, and evidently for the same reason that we borrow 10 in the subtraction of whole numbers when the figure to be subtracted is greater than that above it. 33. The reason why fractions must be brought to a common denominator for the purposes of addition and subtraction, will be evident, if we consider that in order to compare their several values, it is necessary to exhibit them in like parts of the integer. Thus to compare with, if we suppose the integer 1 to be divided into 12 equal parts, will be, and will be; now the values being expressed in 12ths (instead of 3ds and 4ths) it appears that is les than by ¦ falso, that both together make †1. MULTIPLICATION OF VULGAR FRACTIONS. 51. REDUCE mixt numbers to improper fractions; and whole numbers to the form of fractions, by putting 1 for the denominators. Then multiply the numerators together for the numerator, and the denominators together for the denominator of the product. This rule is the same as that for reducing a compound fraction to a simple one; for when the multiplier is a fraction, the product will be a part or parts of the multiplicand: 2 x 3 thus +of+is+or+and+of+is+ or ; and there. 3 x 4 fore the fractions to be multiplied may be set down in the form of a compound fraction, aud the product found in the same inanner as that is reduced to a simple one. Examp. 1. What is the product of and f? Then, 3. What is the continued product of 4, 71, }, and & of product of a fraction and a whole number, multiply by the numerator, and divide by the denominator. VOL. 1. 35. When one factor is a whole, and another a mixt number, or if one is a small fraction, and another a large mixt number, multiply the parts of the latter separately, and add the products together. Ex. 5. Required the product of 6742 by st 7210704+7210704 Ans. 56. And when both factors are mixt numbers, the product may be found by multiplying the parts separately, as in the next example. Ex. 8. Required the product of 5741 by 485!? DIVISION OF VULGAR FRACTIONS. 57. PREPARE the fractions the same as for multiplication; then divide the terms of the dividend by the respective terms of the divisor, if they will exactly divide; but if not, then in vert the divisor and proceed as in multiplication. |