PROBLEM II. To reduce fractions of different denominators to a common denominator. When fractions have their denominators alike, they are said to have a common denominator, and may then be added or subtracted as easily as whole numbers. Thus, į and s are ģ, &c. But when they have different denominators we must reduce them to a common denominator before we can add or subtract them. Thus, reduce 1 and } to a common denominator ; that is, find how many parts and į must be made into, so that the parts shall be of equal size. Now if we divide into 3 equal parts, it will take 12 such parts to make a whole 1 ; and if we divide } into 4 equal parts, it will take 12 such parts to make a whole 1. Hence the common denominator is 12, and ļof 12 is i'i, and } of 12 is iż. RULE I. Multiply each numerator into all the denominators except its own, for a new numerator; then multiply all the denominators together for a common denominator, and place it under each new numerator, and it will form the fraction required. EXAMPLES. 1. Reduce ?, ?, and to equivalent fractions having a .common denominator. 2 3 5 3 4 9. 15 12 6 72 common denominator. 2. Reduce and to a common denominator. Ans. 200, 40, 3. Reduce g, and 1 to a common denominator. 96. Ans. 135, 135 117 288, 288, 288, 288 4. Reduce }, }, ž, and to a common denominator. Ans. 144. 192 252 240 5. Reduce , 1, § and is to a common denominator. Ans. 2052, 513. 1710 2106 Compound fractions must be reduced to simple fractions before finding the common denominator ; and mixed num bers reduced to an improper fraction ; or, the fractional part of mixed numbers may be reduced to a common denominator, and annexed to the whole numbers. 6. Reduce } of and off to a common denominator. Ans. 83, 1982 7. Reduce 127, ã and to a common denominator. Ans. 12365, 266, 216. 8. Reduce 18, and } of of to a common denominator. (Reduce the 18 to an improper fraction first, then reduce the compound fraction to a simple fraction.) Ans. 9409. 120 512 : 512 9. Reduce 103, 123, and of is to a common denominator. Ans. 3465, 4158. 180 The common denominator of fractions is multiple, or such a number as can be divided by all the denominators of the given fractions without a remainder ; and multiplying all the denominators continually together produces a common multiple of those factors. But it will not always produce their least common multiple; and as the operations are more easily performed by having the fractions always in their lowest terms, and as their least cominon multiple is their least common denominator, the following Rule is much preferable. 330) 330, 330 a common RULE II. Find the least common multiple of all the denominators (Problem 2, page 166,) which will be the common denominator of the given fractions. Then divide the common denominator by the denominator of each fraction, and multiply the quotient by the numerator, which will give the new numerator of each fraction; and the new numerator written over the common denominator, will express the fractions in their lowest terms. EXAMPLES. 1. Reduce }, \, and ę, to a common denominator. Operation. The denominator 2)1 4 2 being 12ths, it is ev ident that the nu1 2 1 merator must be in 3X2 X2=12 common denominator. proportion, and to increase this, we take }, , and of the 12ths, as below 12:3X1=4 new numerator, written over the 12=i. 12-4X3=9 new numerator, written over the 12= 12-6x5=10 new numerator, written over the 12=1 2. Reduce , , and 45, to their least common denominator. Ans. 1: 13's 3. Reduce }, 35, and ja, to their least common denominator. 4. Reduced, s, and } of , to their least common denominator. 5. Reduce }, 1o, 25, and 30, to a common deriominator by Rule I. then reduce them by Rule II. Ans. by Rule 1, 235006 22 500 22 560 325 70 3000 5250 Rule II, 30 30 to 30 Ans. 106 Ans. 12 $5 a PROBLEM III. To reduce fractions of higher denominations into those of lower denominations RULE. Multiply the numerator of the given fraction, by the common parts of its own integer, and under the product, write the denominator ; or make a compound fraction, by comparing the given fraction with all the denominations between it and the denomination you would reduce it to; then reduce the compound fraction to a simple one. EXAMPLES. 1. Reduce zo of a pound to the fraction of a penny. Operation. Or, by comparing the given 1 fraction with the several de. nominations, and making a 20 compound fraction, it will stand 12 thus, zło of of 4, then i numerator 240 ***¥=, and 318= then 32= Ans. of a penny, the answer as be fore. 2. Reduce 18 of a pound to the fraction of a shilling. Thus, T&X 20 ==s. Ans. 3. Reduce 12go of a pound, to the fraction of a farthing. Ans. 4. Reduce Tào of a hogshead, to the fraction of a gallon. Ans. įgal. X20 5. Reduce 1680 of a guinea, to the fraction of a penny. Ans. d. 6. Reduce 1920 of a pound Troy, to the fraction of a pwt. Ans. žpwt. 9. Reduce mo of a mile, to the fraction of a rod. Ans. | rod. 8. Reduce 10% of a cwt. tơ the fraction of a pound. Ans. 1 lb. 9. Reduce como of a day; to the fraction of a minute. Ans. &m. 10. Reduce of a guinea, to the fraction of a pound. Compounded thus, of of brill=f Ans. PROBLEM Iỹ. To reduce fractions of lower denominations, into those of higher denominations. RULE. Multiply the denominator of the given fraction, by the common parts of an integer of the required fraction, for a new denominator, over which write the numerator; of a compound fraction by comparing the given fraction with the denominations between it and the one you would reduce it to, then reduce this compound fraction to a simple one. 48 EXAMPLĖS. 1. Reduce of a penny, to the fraction of a pound. Operation. denominator 4 Or by comparing 12 the given fraction with the several de nominations and ma. 20 king a compound fracnew denominator 960 tion, it will stand numerater 3 1 ihen thusdenominator 960=320€ Ans. of of brdo and po£ an swer as before. 2. Reduce of a shilling to the fraction of a pound. Ans. To 3. Reduce of a farthing to the fraction of a pound. Ans. 1280 4. Reduce } gallon to the fraction of a hogshead. Ans. 126 5. Reduce of a penny, to the fraction of a guinea. Ans. do 6. Reduce of a pwt. to the fraction of a pound Troy. Ans. 7. What part of a mile, is i of a rod ? Ans. 3320 8. Reduce 1g of a pound, to the fraction of a cwt. Ans. 10964 9. Reduce of a minute, to the fraction of a day. Ans. zodo 10. Reduce of a £ to the fraction of a guinea. Compounded thus, of 2 of 3=80 Ans. 4. 1920* ADDITION OF VULGAR FRACTIONS: RULE. Reduce compound fractions to single ones, and all of them to their least common denominator (Rule 2, page 169) then the sum of the numerators written over the common denominator, will be the sum of the fractions required. EXAMPLES. Add together 123,95, and soft Thus, į of }= then &, and , reduced to a common denominator, by Rule 2; Problem II. are equal to 120, 120, 1o 63, and the sum of the numerators 90+ 100+ 63=253, which written over the common denominator, will be , which reduced to a mixed number, is equal to 212., then the whole numbers 12+ 9+ 2133=2312 Ans. 2. What is the whole amount of 7} yards, 135 yards, and 8} yards ? Ans. 29 20 yds. 3. Add together , , and as: Ans. 13. 4. Find the sum of , , and á. Ans. 15. 5. Add together , , , and Ans. 37. 6. Find the sum of 18, and 295. Ans. 48%. Note. In adding mixed numbers that are compounded with other fractions, reduce them first to improper fractions, and all of them to a common denominator, by Rule 2, page 196, then add as before. |