112 REDUCTION OF VULGAR FRACTIONS. RULE. Divide the numerator by the denominator, the quotient will be the whole number, and the remainder, if any, will be the numerator to the given denominator. EXAMPLES. 1. Reduce 343 to its equivalent, whole, or mixed number. 8)293 365 Ans. 2. Reduce '143 to its equivalent, whole, or mixed number. Ans. 1274 3. Reduce to its equivalent whole number. Ans. 9. CASE 5. To reduce a compound fraction to an equivalent simple one. RULE. Multiply all the numerators continually together for a new numerator, and all the denominators for a new denominator, and they will form the simple fraction required.) If part of the compound fraction be a whole or mixed number, it must be reduced to an improper fraction, by Case 2d or 3d. If the denominator of any number of a compound fraction be equal to the numerator of another number thereof, these equal numerators and denominators may be expunged, and the other numbers continually multiplied (as by the rule) will produce the fractions required in lower terms. EXAMPLES. 1. Reduce foff of $ off to a single fraction. 1X2 X3x4 =}. the ans. 2 3 X 4 X 5 Or, by expunging the equal numerators and denominators, it will give } as before. Thus, 1x2x3x -} Ans. 2. Reduce & off of of it'to a single fraction. Ans. it 3. Reduce 5 of 4 of 4 of 124 to a single fraction. Ans. 71=131 4. Reduce 14 of 17 of to a single fraction. CASE 6. To reduce fractions of different denominations to equiva. lent fractions having a common denominator. Ans. . RULE.. Multiply each numerator into all the denominators, except its own, for a new numerator, and all the denominators into each other continually, for a common denominator. EXAMPLES. 1. Reduce , j, and {, to equivalent fractions, having a common denominator. 1x5x8=40, the new numerator for 5 do. do. 4x5x8=160, the common denominator. Therefore the new equivalent fractions are 4*, , and 186, the answer. 2. Reduce }, }, }, b, and }, to fractions having a common denominator. Ans. 179, 193, 194, tist. 3. Reduce i, šof 5, 78, and is, to a common denominator. Ans. 936 1872, 183, 1872, 1817 4. Reduce 1}, & of 1), ta, and , to a common denomi. nator. Ans. 1440, 14148, 4122, 12%. CASE 7. To reduce any given fractions to others, which shall have the least common denominator. RULE. Find the least common multiple (already taught) of all the denominators of the given fractions, and it will be the common denominator required. Divide the common denominator by the denominator of each fraction, and multiply the quotient by the numerator, and the product will be the numerator of the fraction required. EXAMPLES. 1. Reduce }, 4, and }, to fractions having the least common denominator possible. K2 114 REDUCTION OF VULGAR FRACTIONS. 4)3 4 8 4* 3*2=24, the least common de. 3 1 2 nominator 243x1=8, the first numerator; 24;2x3=18, the 2d numerator; 24:8X7=21, the 3d numerator. So the required fractions are 4, 4, 44. 2. Reduce }, $, 4, and , to a fraction having the least common denominator. Ans. , , , CASE 8. To reduce a fraction of one denomination to the fraction of another, but greater, retaining the same value. RULE. Multiply the given denominator by the parts of the de. noininations between it and that denomination you would reduce it to; reduce this compound fraction to a single one, by Case 5, and you will have a fraction of the required denomination, equal in value to the given fraction. EXAMPLES. 1. Reduce 4 of a cent to the fraction of a dollar. 4 Xito=760=1 dollar. 2. Reduce of a penny to the fraction of a pound. Ans. rio 3. Reduce of a pennyweight to the fraction of a pound troy. Ans. Toolb. 4. Reduce of an ounce to the fraction of a pound avoirdupois. Ans. zölb. RULE. Multiply the given numerator by the parts of the denominations between it and that denomination you would reduce it to, for a new numerator; then reduce the compound to a single one. EXAMPLES. 1. Reduce sts of a dollar to the fraction of a cent. TEX?qo=4ct. Ans. 2. Reduce ado of a pound to the fraction of a penny. Ans. id. 3. Reduce of a lb. avoirdupois, to the fraction of an ounce. 4. Reduce ito of a cwt. to the fraction of a lb. avoirdupois. Ans. 14lb. CASE 10. To find the value of a fraction in the known parts of the integer, as of coin, weight, measure, &c. RULE. Multiply the numerator by the parts in the next inferior denomination, and divide the product by the denominator, and if anything remain, multiply it by the next inferior de. nomination, and divide by the denominator as before, and so on, as far as necessary; and the quotients placed after each other, in their order, will be the answer required. In examples of federal money, annex as many ciphers to the numerator as may be necessary. If a remainder occur when at its lowest denomination, place it as a numerator over the denominator, for a fraction of the lowest denomination. EXAMPLES. cts. Ans. 62cts. 5m. or 625m. $1=100 5 8)500 62,5m. 2. What is the value of 4 of a pound? Ans. 14s. 3d. 14. 3. What is the value of of a lb. avoirdupois ? Ans. 12oz. 12 dr. 4. What is the value of of a mile? Ans. 6fur. 26p. 11ft. 5. What is the value of of a day? Ans. 16h. 36m. 55 55. 6. What is the value of 4 of an acre ? Ans. 3r. 17 p. CASE 11. To reduce any given quantity to the fraction of any larger denomination of the same kind. RULE. Reduce the given quantity to the lowest term mentioned, for a numerator; then reduce the integral part to the same term, for a denominator ; which will be the fraction required. 116 REDUCTION OF VULGAR FRACTIONS. Note 1.-If there be a fraction given with the said quantity, it must be farther reduced to the denominative parts thereof, adding thereto the numerator. NOTE 2.-This case is the reverse of the former, therefore proves it. EXAMPLES. 1. Reduce 62cts. 5m. to the fraction of a dollar. 62cts. and 5m.=625m. =fd. Ans. 1 dollar =1000m. 2. Reduce 14s. 3d. 1qr. to the fraction of a pound. Ans. 428=4£. 3. Reduce 12oz. 124 dr. to the fraction of a lb. avoirdupois. Ans. lb. 4. Reduce 6fur. 26p. 11ft. to the fraction of a mile. Ans. (m. 5. Reduce 16h. 36m. 55 s. to the fraction of a day. Ans. is day 6. Reduce 3r. 174p. to the fraction of an acre. Ans. , acre. CASE 12. To reduce a given fraction to another equivalent one, having a given numerator. RULE. As the numerator of the given fraction EXAMPLES. Ans. 46 1. Reduce & to a fraction of the same value, the numerator of which shall be 15. As 3': 15 :: 4 : 20 denominator 4. Ans. 2. Reduce s to a fraction of the same value, having its numerator 42. 3. Reduce to an equivalent fraction, the numerator of which shall be 27. Ans. 27 30% 4. If to be reduced to an equivalent fraction, having its numerator 36, what will that fraction be? Ans. 36 115 |