The process of omitting the needless factors is called finding the least common denominator of several fractions. It consists in resolving the several denominators into their prime factors (42, Def. 5). Hence, To find the least common denominator of two or more fractions,-RULE. Write the several denominators on a line from left to right, and divide them successively by the primes, 2, 3, 5, 7, 11, 13, 17, 19, &c., written on the left as divisors, which will divide one or more of them without a remainder, writing their several quotients and undivided numbers on a line below, until the last quotients terminate in unity. The product of all these several divisors will be the least common denominator. NOTE 3.-Or, we may discontinue the division when no number greater than 1 will divide two of the numbers without a remainder, and the continued product of the last quotients and the divisors will be the least common denominator sought. 1. Find the least common denominator of the several fractions, 2, 4, o' 3 3 , and, and their new numerators. The foregoing is an illustration of the operation of the rule, to which the pupil is referred, and he will find the least common denominator to be 120. New Numerators. To find the new numerators of the several fractions,— RULE. Divide the common denominator by the denominator of one of the fractions, multiply the quotient by its numerator, and write the product over the common denominator for its new numerator. So with each of the other fractions. Thus, ; 120-2=60x1=120 3 ; 120÷4=30×3=90, &c. 2. Reduce, and 13 to equivalent fractions having the least common denominator. Ans. 24, 14, 14. 24 13 3. Reduce, 23, and of to equivalent fractions having the least common denominator. Ans. 72 936 405 360, 360 360.360° 4. What is the greatest common measure, or divisor, of the numbers, 4, 5, 8, and 10 (46, Note 2.) ? 5. What is the least common multiple of 45, 117, and 18? Ans. 1170. 54. Addition and Subtraction of Common Fractions. NOTE 1.-Fractions having different denominators, before being added or subtracted, must be reduced to a common denominator; then their numerators can be added or subtracted in the same way as whole numbers. 8 6 1. Reduce, to fractions having the least common denominator, and add them together. NOTE 2.-In writing fractions for addition and subtraction which have a common denominator, the numerators may be written in a line, connected by the appropriate signs, one line extended under them all, and the denominator written under this line but once. Thus, in the last example, 2+3+3=12+9+20=1=117 24 Amount. 2. There are 3 pieces of cloth, one containing 73 yards, another 13 yards, and the other 157 yards; how many yards in the 3 pieces? 72 = 718 135 = 1320 24 157 = 1521 Ans. 37 yards. SOLUTION.-Before adding, reduce the fractional parts to their least common denominator; this being done, add together all the 24ths, viz., 18+20+21=53=211. We write down the fraction ¦¦ under the other fractions, and reserve the 2 integers to be carried to the amount of the other integers, making in the whole 3711 3. There was a piece of cloth containing 343 yards, from which were taken 123 yards; how much was there left? 34 = 34, 123 = 1210 9 241 SOLUTION.—We cannot take from 24 we therefore borrow 1 integer=24+=23, 33 and 33-18=17; then, 1+12=13 integers, Ans. 211 yards. and 34-13 Ans. 2117 yards. 24 16 We have, then, for the addition and subtraction of fractions, this general RULE. Add or subtract their numerators as the question may require, when they have a common denominator; otherwise, they must first be reduced to a common denominator. EXAMPLES. 4. What is the amount of 9, 43, and 12? Ans. 17. 5. A man bought a ticket, and sold of it; what part of the ticket had he left? 5 6. Add together, §, 1, 7, 1, and 1. ΤΟ 7. What is the difference between 14 and 1633 ? 8. From 11 take 2. 9. From 3 take . 10. From 147 take 484. 11. Add together 112, 3113, and 1000. 12. Add together 14, 11, 43,, and . 13. From take 1. From 7 take 2. Ans. 118. Remainder, 3. Rem. 23. Rem. 988. Ans. 1424}}. 14. What is the difference between and }? and? and ? and ? and? and ? 15. How much is 1-? 1-? 1-3? 1-? 2-3? 2-4? 21-? 3-? 1000-? 55. Multiplication of Common Fractions. Multiplication of fractions does not always imply increase. Multiplication of Fractions is taking such part of the multiplicand as is expressed by the multiplier; thus, 8×2 is taking of 8 three times: = 6. 1. If 36 dollars be paid for a piece of cloth, what will 3 of it cost? SOLUTION. Reduce the integer to a fractional form, 36 (50, Note 1). Then multiply the two numerators together for a new numerator, and the two denominators together for a new denominator; thus, 36 × 3 = 108 = Ans. 27 dollars. NOTE 1.-In the foregoing example and solution you have two things to learn 1st, When integers are involved in the multiplication of fractions, before operating, reduce them to a fractional form. 2d, As the multiplicand and multiplier are factors of the product (42, Def. 5), it matters not which is taken as multiplier, or which as multiplicand, the integer or the fraction, the result will be the same either way. 2. To multiply one fraction by another.—At of a dollar for 1 bushel of corn, what will of a bushel cost? SOLUTION. Here the multiplicand and multiplier are factors of a product sought; taken together they are a compound fraction, of; hence, we multiply the numerators together and the denominators together, as in the foregoing example; thus, of Ans. of a dollar = NOTE 2.-Here, while passing, the attention of the pupil is called to the fact, that dividing the denominator of a fraction is the same in effect as multiplying the numerator (43, PRIN. I.). Illustration.-Let it be required to multiply by 2; 2= ; so also, 3x2 = 6 = 3, as before. Hence,— 3 There are two ways to multiply a fraction by an integer, or an integer by a fraction, I. Divide the denominator by the integer when it can be done without a remainder, and over the quotient write the numerator. Otherwise, II. Multiply the numerator by the integer, and under the product write the denominator. (43, PRIN. I. and II.) If then it be an improper fraction, reduce it to an integer or a mixed number. 3. How much is of of 2 of 3 ? OPERATION. AN of of A of 6 1 SOLUTION.-Here cancellation will come to our aid in shortening the operation. When applied to fractions, it is simply the reduction of a compound fraction to its lowest terms, by rejecting common factors from the numerators and denominators. Performing the operation as instructed (44), we cancel, in other words, reject, all like factors of the numerators and denominators, and we have remaining, 2×2 1 = Ans. 1. 2 2 2 4. What will be the product of of of of 5 of 221 ? SOLUTION. We reduce the mixed number to an improper frac$ A 6 tion, 1 $ (22) 199 = This being done, we reject 9 the common factors, 4, 6, 5; and the numerator, 3, being twice a factor in the denominator 9, we reject the 3 and one factor of the denominator 9, retaining the other, and we have the remaining factors, 199 Ans. 911. = 56. These principles in the multiplication of fractions, as presented and illustrated in the foregoing examples, are summarily comprehended in the following general RULE. Reduce the given numbers to a fractional form, and cancel common factors. Multiply the numerators together for a new numerator and the denominators together for a new denominator, write the new numerator over the new denominator, and if it be an improper fraction, reduce it to a whole or mixed number. 5. Multiply 53 by 7. EXAMPLES. Product, 401. 6. What will 913 tuns of hay come to, at 17 dollars per tun? Ans. 164 dollars. hour, how far will he in 12 hours? 7. If a man travel 20 miles in 1 travel in 5 hours? in 8 hours? in 3 days, supposing he travel 12 hours each day? Ans. to the last, 77 miles. 8. If a ship be worth 1367 dollars, what is of it worth? Ans. 3037 dollars. 9. What cost of a tun of butter, at 225 dollars per tun? Ans. 190 Ans. 219 dollars. 10. At 63 dollars per barrel for flour, what will rel cost? 13 Ans. 1432 dollars. 12. How much is 13. How much is of 3 of 3 of g? Ans. 180 of 3 of 7 of 3? 15. What is the continual product of 7, 1, of, and 3!? 16. What is the continued product of 3, 2, 5 of 3, 25, and H off of 씀? 245 Ans. 209 17. Reduce of 3 of 4 of 5 of 223 to a simple fraction? Ans. 91 18. A horse consumed of of 8 tuns of hay in one winter; how many tuns did he consume? Ans. 22 tuns. 19. Reduce 20. At 9 yards? of 3 of 4 of 5 of ; of 1 to a simple fraction. Ans. . dollars per yard, what cost 4 yards of cloth? 20 yards? 5 Ans. to the last, 15 dollars. 21. If 2 tuns of hay keep 1 horse through the winter, how much will it take to keep 3 horses the same time? horses? 13 horses? Ans. to the last, 13 tuns. 7 22. A owned of a ticket; B owned of the same; the ticket was so lucky as to draw a prize of 1000 dollars; what was each one's share of the money? Ans. A's share, 600 dollars; B's share, 400 dollars. |