duced to J. Now, as 1 and 3 are prime to each other, the fraction J is in its lowest terms. The same result is often more readily obSecond Operation. tained by dividing the terms of the fraction 1" ) i I = i Ans. by their greatest common divisor, as by the second operation. Since dividing the numerator and denominator of a fraction by the same number, or cancelling equal factors in both, changes only the form of the fraction, while the value expressed remains unchanged (Art. 21 7). Role. — Divide the numerator and denominator by any number greater than 1 that mill diride them both without a remainder, and thus proceed until they are prime to each other. Or, Divide both the numerator and denominator by their •greatest common divisor. Examples. 2. Reduce |f to its lowest terms. Ans. 24r 3. Reduce §§ to its lowest terms. . Ans. £ 4. Reduce |f to its lowest terms. Ans. $ 5. Reduce -j'g'j to its lowest terms. 6. Reduce to its lowest terms. Ans. ^ 7. Reduce |£f to its lowest terms. Ans. 8. Reduce Jvts t0 'ts lowest terms. 9. Reduce §}§ to its lowest terms. Ans. |ff 10. Reduce tvtv t0 'ts lowest terms. Ans. -i8TVB 11. Reduce f^f to its lowest terms. 12. Reduce t80't6s- t0 lts lowest terms. Ans. |g| 13. Reduce to its lowest terms. Ans. -j*^ 221. To reduce an improper fraction to an equivalent whole or mixed number. Ex. 1. How many yards in JTy- of a yard? Ans. 6T^. Since 19 nineteenths make one yard, Operation. it is evident there will be as many 1 9 ) 1 1 7 ( 6T3? Ans. yards in 117 nineteenths as 19 is con114 tained times in 117, which is 6j3s times. Therefore, 6j, yards is the answer required. Rule. — Divide the numerator by the denominator. Note. — Should there a remainder occur, write it over the denominator, and make this fraction a part of the answer. Examples. 2. Reduce -ii6^- to a mixed number. Ans. 11T2S. 3. Reduce JT6iV to a mixed number. Ans. 14T£5. 4. Reduce Jjv- to a mixed number. 5. Reduce f £\ to a mixed number. Ans. 3^|f. 6. Change J-°?oa to a mixed number. Ans. 11J J. 7. Change to a mixed number Ans. 91ff. 8. Change Jfa to a whole number. Ans. 125. 9. Change §f to a whole number. 222i To reduce a whole or mixed number to an improper fraction. Ex. 1. Reduce 19 to a fraction whose denominator shall be 7. Operatiou. Since there are 7 sevenths in 1 19 X 7 =: 133. whole one, 19 whole ones = 133 133 sevenths = Vf&, Ans. sevenths — if>. 2. Reduce 17f to an improper fraction? Ans. OPERATION. 1 7 3 » Since there are 5 fifths in 1 whole one, 0 in 17 whole ones there are 85 fifths, and g rj adding 3 fifths for the fraction, we have 8| as the equivalent of 173. Hence the o 8 8 fifths = Ans. Rule. — Multiply the whole number by the given denominator, and to the product add the numerator of the fractional part, if any; and write the result over the denominator. Note. — A whole number may be expressed in its simplest fractional form, by taking it for a numerator with 1 for a denominator. Thus, 4. may be written 4, and read 4 ones. Examples. 3. Reduce 15 to fourths. Ans. 4. Reduce 161^ to sixteenths. 5. Reduce 171|£ to an improper fraction. Ans. 6. Change 11 to a fractional form. Ans. J^. 7. Change 100 to an improper fraction. 8. Change 5 to a fraction whose denominator shall be 17. Ans. f-f. 9. Reduce 98|f to an improper fraction. Ans. -If-^ 10. Reduce 116gT to an improper fraction. Ans. z£?-x. 11. 718|f equal how many ninety-sevenths r Ans. S-W9-1-. 12. Reduce 100jg§ to an improper fraction. Ans. 2T§§a. 13. Reduce 7 to an improper fraction. 14. Reduce 19 to a fraction whose denominator shall be 13. Ans. 15. 116^ yards equal how many fourths of a yard? Ans. 46.5 fourths. 223. To reduce a compound fraction to a simple fraction. Ex. 1. Reduce £ of £ to a simple fraction. Ans. § By multiplying the denominator of I by Operation. 4^ tne denominator of |, it is evident, we $ X \ — §i , Ans. obtain \ of \ = j^, since the parts into which the number is divided are 4 times as many, and consequently only i as large as before; and since i of s = as' I of s wil l be 3 times 37j = §J • Rule. — Multi-ply all the numerators together for a new numerator, and all the denominators for a ne w denominator. Note 1. — All whole and mixed numbers in the compound fraction must be reduced to improper fractions, before multiplying. Note 2. — When there are factors common to both numerator and denominator, they may be cancelled in the operation. Examples. 2. Reduce f of ^ of £| of §§ to a simple fraction. Ans. -fa OPERATION. 1 11 3. What is § of % of | of \h? Ans. X%.V = Iff 4. What is f of^of^-of rV? 5. Reduce ^ of \ of % of \\ to a simple fraction. Ans. XWB. 6. What is the value of of % of \ of 21? Ans. £ft§ = 2TV 7. What is the value of T7T of 151 of 5^ 0f 100? Ans. 5758££. 8. What is $ of \ of H? 9. What is the value of fT of H of of $ 7£? Ans. $1.75. 10. What is the value of f of fT of | J of 3f gallons? Ans. * gal. 11. What part of a ship is ^ of § of f? 12. What is the value of f of & of if of |f of $ 34? Ans. $ 6.75. A COMMON DENOMINATOR. 224■ Fractions have a common denominator when all their denominators are alike. 225. A common denominator of two or more fractions is a common multiple of their denominators; and their least common denominator is the least common multiple of their denominators. 226i To reduce fractions to a common denominator. Ex. 1. Keduce £, T52, and -Jr to other fractions of equal value, having a common denominator. FIRST OPERATION. 7X12X16 = 1344 new numerator. £ = ) 5 X 8 X 1 6= 640" « A = T\vf ["Ans. 1 1 X 8 X 1 2 = 1 05 6" « H = jet ) 8X12X16 = 1536 common denominator. We first multiply the numerator of i by the denominators 12 and 16, and obtain 1344 for a new numerator. We next multiply the numerator of by the denominators 8 and 16, and obtain 640 for a new numerator; and then we multiply the numerator of 15 by the denominators 8 and 12, and obtain 1056 for a new numerator. Finally, we multiply all the denominators together for a common denominator, and write it under the several numerators, as in the operation. By this process, since the numerator and denominator of each fraction are multiplied by the same numbers, their relation to each other is not changed, and the value of the fraction remains the same. (Art. 217.) 16 X 3 = 48, least common multiple, and least common denomina . tor. Having first obtained the least common multiple of all the denominators of the given fractions, we assume this to be their least common denominator. We then take such a part of this number, 48, as is expressed by each of the fractions separately for their respective new numerators. Thus, to get a new numerator for -J, we take ^ of 48, the least common denominator, by div iding it by 8, and multiplying the quotient 6 by 7. We proceed in like manner with each of the fractions, and write the numerators thus obtained over the least common denominator. In this process the value of each fraction remains unchanged, as .both terms are multiplied by the same number. (Art. 217.) The method used in the second operation, it will be perceived, expresses the fractions of the result in lower terms than that used in the first. On this account it is often to be preferred to the other. Rule. — Find the least common multiple of the denominators for the LEAST COMMON denominator. Divide the least common denominator by the denominator of each of the given fractions, and multiply the quotients by their respective numerators, for the new numerators. Or, Multiply each numerator by all the denominators except its own, for the new numerators; and all the denominators together for A Common denominator. Note 1. — Compound fractions must be reduced to simple ones, whole and mixed numbers to improper fractions, before finding a common denominator, and ail to their lowest terms, before finding the least common denominator. Note 2. — Fractions may sometimes be reduced to a common denominator most readily by multiplying both terms of one or more of them by such a number as will make all the denominators alike. Thus 4 and \ may be brought to a common denominator simply by multiplying both terms of the ^ by 2, and changing in that way its form to |. Note 3. — Fractions may often be reduced to lower terms, without destroying their common denominator, by dividing all their numerators and denominators by a common divisor. Examples. Reduce the following fractions to their least common denominator : — 2. Reduce f, f, and Ans. f£, f J. 3$. 3. Reduce A.A,i8.and A- ATM- Tv&. Tv&, T5j*. |