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, and are 3, &c. But when they have different denominators we must reduce them to a common denominator before we can add or subtract them. Thus, reduce and to a common denominator; that is, find how many parts and must be made into, so that the parts shall be of equal 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 of 12 is, and of 12 is 12. size. 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. 135 4. Reduce, 3, 7, and 8 to a common denominator. 5. Reduce,, Ans. 144, 183, 283, 248 288 288° and 13 to a common denominator. Ans. 2052, 513 1710 2106 3075' 3078' 3078' 3078 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 of to a common denominator. 192 48 Ans. 18, 192. 72 7. Reduce 123, and to a common denominator. Ans. 1235, 360' 360° 200 216 8. Reduce 183, and of of to a common denominator. (Reduce the 183 to an improper fraction first, then reduce the compound fraction to a simple fraction.) 9. Reduce 101, 123, and of nator. Ans. 9408 120 512 512' to a common denomiAns. 3465 4158 180 330 330 330 The common denominator of fractions is a common 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 common multiple is their least common denominator, the following Rule is much preferable. 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, 2, and 3, to a common denominator. increase this, we take 1, 2, and § of the 12ths, as below 12÷3×1=4 new numerator, written over the 12= 12÷4x3=9 new numerator, written over the 12=12÷6×5=10 new numerator, written over the 12= 2. Reduce,, and, to their least common denomi nator. 6 Ans. 18131s. 3. Reduce , and, to their least common denomi nator. 3 35' Ans. 106 45 70707 4. Reduce 2, §, and of %, to their least common denominator. $55 Ans. 147 5. Reduce, 10, 15, 3 2, and 7 to a common denominator 30 by Rule I. then reduce them by Rule II. Ans. by Rule I, 22500 22500 22500 22500* 4500 6750 3000 5250 Rule II, 3 30 30 30. PROBLEM III. To reduce fractions of higher denominations into those of lower denominations RULE. 6 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 of a pound to the fraction of a penny. Operation. X 20 20 12 numerator 240 then 2. Reduce 3. Reduce 12 4. Reduce of a pound to the fraction of a shilling. Thus,×20=60s. Ans. of a pound, to the fraction of a farthing. Ans. 2. of a hogshead, to the fraction of a gallon. Ans. gal. 5. Reduce 1880 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. Zpwt. 7. Reduce 3520 of a mile, to the fraction of a rod. 8. Reduce 104 9. Reduce Ans. rod. of a cwt. to the fraction of a pound. Ans. 18lb. 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 28 of PROBLEM IV. 113=# Ans. 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; or make 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. EXAMPLES. 1. Reduce of a penny, to the fraction of a pound. Operation. denominator 4 12 48 20 new denominator 960 Or by comparing the given fraction with the several denominations and ma king a compound fraction, it will stand thus 2 of and of 20=980 320 an swer as before. 2. Reduce of a shilling to the fraction of a pound. 3. Reduce of a farthing to the fraction of a pound. Ans. 180 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. 840 6. Reduce of a pwt. to the fraction of a pound Troy. 7. What part of a mile, is 7 Ans. 1920* Ans. 3520 of a rod? 9 Ans. 1964 9. Reduce of a minute, to the fraction of a day. Ans. 5040 10. Reduce of a £ to the fraction of a guinea. Compounded thus, of 20 of Ans. 4. 80 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. 4' EXAMPLES. 90 100 63 Add together 123, 95, and 7 of 3. Thus, 7 of 3=1 then 2, 5, and 21, reduced to a common denominator, by Rule 2, Problem II. are equal to 1926, 128, 1926, and the sum of the numerators 90+ 100+ 63-253, which written over the common denominator, will be 253, which reduced to a mixed number, is equal to 213, then the whole numbers 12+9+22=23120 Ans. 2. What is the whole amount of 73 yards, 135 yards, and 8 yards? 3. Add together 4, 3, and 4. Find the sum of 3, 3, and §. 5. Add together 3, 4, 5, and 3. 6. Find the sum of 183, and 2998. 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. |