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204. A FRACTION is an expression denoting one or more equal parts of a unit.
205. A fractional unit is one of the equal parts into which the whole thing or integral unit has been divided. Thus halves, thirds, &c., being equal parts of integral units or whole things, are fractional units.
206. The unit of a fraction is the unit or whole thing from which its fractional parts have been derived.
207. A COMMON Fraction is expressed by two numbers one above the other, with a line between them.
208. The number below the line is called the denominator. It shows into how many parts the whole number has been divided. It gives name to the fraction and value to the fractional unit. Thus, in the expression 4, the denominator is 7, indicating that the unit of the fraction has been divided into 7 equal parts, and that the value of the fractional unit is one seventh.
The number above the line is called the numerator. It shows how many parts have been taken, or numbers the fractional units expressed by the fraction. Thus, in the expression %, the numerator is 2, indicating that the fractional unit, which is one seventh, has been taken 2 times.
209. The terms of a fraction are its numerator and denomi. nator. Thus, the terms of the fraction are the numerator 2 and the denominator 3.
210. A proper fraction is one whose numerator is less than the denominator ; as ,, 8.
211. An improper fraction is one whose numerator is equal to, or greater than, the denominator ; as 7, 11, 12
212. A mixed number is a whole number with a fraction ; as 31, 163, 90%.
213. A simple or single fraction has but one numerator and one denominator. It may be either proper or improper; as di ti, is.
214. A compound fraction is a fraction of a fraction, or two or more fractions connected by the word of; as of } of jo, of of 7.
A complex fraction is a fraction having a fraction or a mixed number for its numerator or denominator, or both; as 1 7 35 21 7' 44' 13' 4
216. A fraction is an expression of division , the numer. ator answering to the dividend, and the denominator to the divisor: (Art. 67); and the value of a fraction is the quotient arising from the division of the numerator by the denominator (Art. 80). Thus, in the fraction 43, the numerator 15 is the dividend, the denominator 7 is the divisor, and the value expressed 24, or the quotient arising from the division of the 15
by the 7.
217. Since a fraction is an expression of division, it follows,
1. That, if the numerator be multiplied, or the denominator be divided, by any number, the fraction is multiplied by the same number (Art. 81).
2. That; if the numerator be divided, or the denominator multiplied, by any number, the fraction is divided by the same number (Art. 82).
3. That, if the numerator and denominator be both multiplied, or both divided, by the same number, the fraction will not be changed in value (Art. 83).
REDUCTION OF COMMON FRACTIONS.
218. Reduction of fractions is the process of changing their form of expression without altering their value.
219. A fraction is in its lowest terms, when its numerator and denominator are prime to each other (Art. 166).
220. To reduce a fraction to its lowest terms. Ex. 1. Reduce is to its lowest terms.
Ans. 1. By dividing both terms of the fraction by 4) 1 =
4, a factor common to them both, it is re4) = $ s Ans. duced to it. Dividing both terms of us by
4, a factor common to them both, it is re
duced to s. Now, as 1 and 3 are prime to each other, the fraction } is in its lowest terms.
The same result is often more readily ob
tained by dividing the terms of the fraction 16 ) to
} 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. 217).
RULE. Divide the numerator and denominator by any number greater than 1 that will divide 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.
Ans. i: Ans. Ans. .
2. Reduced to its lowest terms.
Ans. 2. 8. Reduce 64. to its lowest terms. 9. Reduce $15 to its lowest terms. 10. Reduce 814 to its lowest terms.
TITO 11. Reduce G15 to its lowest terms. 12. Reduce 1816to its lowest terms. 13. Reduce to its lowest terms.
221. To reduce an improper fraction to an equivalent whole or mixed number. Ex. 1. How many yards in 11- of a yard ?
Ans. 615 Since 19 nineteenths make one yard,
it is evident there will be as many 19 ) 117 ( 62 Ans. yards in 117 nineteenths as 19 is con114
tained times in 117, which is 6 times.
Therefore, bi yards is the answer re 3
quired. 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.
2. Reduce 167 to a mixed number.
Ans. 11 15. 3. Reduce 1613.4 to a mixed number.
Ans. 14116 4. Reduce 1374 to a mixed number. 5. Reduce iti to a mixed number.
Ans. 3181 6. Change 10,00 to a mixed number.
Ans. 1115 7. Change 4123 to a mixed number
Ans. 9145 8. Change 125 to a whole number.
Ans. 125. 9. Change 37 to a whole number.
222. To reduce a whole or mixed number to an improper fraction.
Ex. 1. Reduce 19 to a fraction whose denominator shall be 7.
Since there are 7 sevenths in 1 19 x 7 = 133. whole one, 19 whole ones = 133 133 sevenths
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 A, and read 4 ones.
3. Reduce 15 to fourths.
3. Change 5 to a fraction whose denominator shall be 17.
Ans. is. 9. Reduce 988 to an improper fraction.
Ans. 9594. 10. Reduce 116&+ to an improper fraction.
Ans. 797. 11. 71834 equal how many ninety-sevenths Ans. 6.9691. 12. Reduce 100198 to an improper fraction. Ans. 218%. 13. Reduce 7 an improper fraction. 14. Reduce 19 to a fraction whose denominator shall be 13.
Ans. 443 15. 1164 yards equal how many fourths of a yard?
Ans. 465 fourths. 223. To reduce a compound fraction to a simple fraction. Ex. 1. Reduce of } to a simple fraction.
By multiplying the denominator of z by
4, the denominator of a, it is evident, we i XX= 41, Ans.
obtain 4 of 1 = 3a, since the parts into
which the number is divided are 4 times as many, and consequently only , as large as before ; and since of x = id; l of į will be 3 times ja
2 RULE. Multiply all the numerators together for a new numerator, and all the denominators for a new 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 denomi. nator, they may be cancelled in the operation.
2. Reduce f of of 13 of 3 to a simple fraction. Ans. It
It , Ans.
11 3. What is şof á of } of 11 ?
Ans. 1725 = 4. What is of i} of } of i's . ? 5. Reduce 4 of of of t} to a simple fraction.
. 6. What is the value of of of 1 of 21 ?
Ans. 75% = 23