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NOTE. A fraction, strictly speaking, is less than a unit; hence, if the numerator is equal to, or greater than, the denominator, it expresses a unit or more than a unit, and is therefore called an improper fraction.
A mixed number is a whole number with a fraction; as, 7, 58.
A simple or single fraction has but one numerator and one denominator, and may be either proper or improper; as,,.
A compound fraction is a fraction of a fraction, connected by the word of; as, of § of §.
A complex fraction is a fraction having a fraction or a mixed number for its numerator or denominator, or both; as 27 81 71 '97' 11'91
ART. 132. When we divided 479956 by 6 (Art. 49, Ex. 12), we had a remainder of 4, which could not be divided by 6, and therefore we wrote it over the divisor with a line between them. This is an expression for division without performing the operation, and is called a fraction; the number above the line being the numerator, and the one below the denominator. Hence, Fractions originate from division; the numerator answers to the dividend, and the denominator to the divisor.
ART. 133. From what has preceded, we perceive that the value of a fraction is the quotient arising from the division of the numerator by the denominator, or from the expression of this division. Thus the quotient of or 6 ÷ 2 is 3; and the quotient of or 3 ÷ 4 is 2.
REDUCTION OF VULGAR FRACTIONS.
ART. 134. REDUCTION of Fractions is changing their form or terms without altering their value.
ART. 135. To reduce a fraction to its lowest terms.
Ex. 1. Reduce to its lowest terms.
QUESTIONS. When the numerator is equal to, or greater than, the denominator, is the expression, strictly speaking, a fraction? Art. 132. From what do fractions originate? To what does the numerator answer? To what the denominator? Art. 133. What is the value of a fraction? - Art. 134. What is reduction of fractions?
We first divide both terms of the fraction by 2, a number that will divide them both without a remainder, and obtain 3. We next divide this result by 3, and obtain, which cannot be divided by any number greater than 1, and therefore the fraction is in its lowest terms. The result would have been the same, if we had first divided by 6, the greatest common divisor.
By dividing the numerator and denominator of a fraction by the same number, it is evident we cancel equal factors in both (Art. 113), and diminish them in the same proportion; consequently, their relation to each other is not changed, and the value of the fraction remains the same. Therefore, Dividing the numerator and denominator of a fraction by the same number does not alter the value of the fraction.
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RULE I. Divide the numerator and denominator by any number greater than 1, that will divide them both without a remainder, and thus proceed with the successive results until the operation can be carried no farther. Or,
RULE II. -Divide both the numerator and denominator by their greatest common divisor, and the result will be the fraction in its lowest
NOTE. A fraction is in its lowest terms, when its numerator and denominator are prime to each other. (Art. 118.)
EXAMPLES FOR PRACTICE.
to its lowest terms.
3. Reduce to its lowest terms. 4. Reduce to its lowest terms. 5. Reduce to its lowest terms.
6. Reduce to its lowest terms.
7. Reduce 123 to its lowest terms.
8. Reduce to its lowest terms.
9. Reduce 7 to its lowest terms. 10. What is the lowest expression of 18?
Ans. Ans. 18. Ans. . Ans. ii. Ans. 1.
Art. 136. To reduce a mixed number to an improper fraction.
Ex. 1. In 73 how many fifths?
QUESTIONS. Art. 135. How do you reduce a fraction to its lowest terms? Why does dividing both terms of a fraction by the same number not alter the value of the fraction? Has the same value as &? Why? What is the rule for reducing a fraction to its lowest terms? a fraction is in its lowest terms?
How may you know when
Since there are 5 fifths in 1 whole one, there will be 5 times as many fifths as whole ones; therefore, in 7 there are 35 fifths, and the 3 fifths being added make 38 fifths, which are expressed thus,
RULE. Multiply the whole number by the denominator of the fraction, and to the product add the numerator, and place their sum over the denominator of the fraction.
NOTE.-1. Any whole number may be expressed in the form of a fraction, by taking the number itself for a numerator, and a unit for the denominator; thus, 5 may be written §.
2. To reduce a whole number to a fraction of the same value, having a giren denominator, we multiply the whole number by the given denominator, and make the product the numerator; thus 5, reduced to a fraction, having 3 for a denominator, becomes 15.
EXAMPLES FOR PRACTICE.
2. In 83 dollars how many sevenths?
9. Change 4311 to an improper fraction. 10. What improper fraction will express 279? 11. Change 111 to an improper fraction? 12. Change 125 to an improper fraction. 13. Change 25 to an improper fraction, having nominator.
14. Reduce 75 to ninths.
15. Change 343 to the form of a fraction. 16. Reduce 84 to fifteenths.
Ans. 52. Ans. 13. Ans. 103.
how many one hundred and seventeenths?
6 for a deAns. 159. Ans. 6 675. Ans. 343.
ART. 137. To reduce improper fractions to whole or mixed numbers.
QUESTIONS. Art. 136. What is the rule for reducing a mixed number to an improper fraction? Give the reason. How can a whole number be expressed in the form of a fraction? How do you reduce a whole number to a fraction of the same value, having a given denominator?
Ex. 1. How many dollars in 37 dollars?
This question may be analyzed by saying, As 16 16) 37 (2% sixteenths make one dollar, there will be as many dollars in 37 sixteenths as 37 contains 16, which is 2 times, = $25.
RULE. -Divide the numerator by the denominator, and, if there be a remainder, place it over the denominator at the right hand of the whole number.
EXAMPLES FOR PRACTICE.
2. Reduce to a whole number.
7. Reduce 37 to a whole number.
9. Reduce 43 to a mixed number.
10. Reduce 1848 to a mixed number.
Ans. 12. Ans. 10. Ans. 10
ART. 138. To reduce a compound fraction to a simple fraction.
Ans. 188. Ans. 1429.
Ex. 1. Reduce of to a simple fraction.
To show the reason of the operation, this question may be analyzed by saying, that, if i'r of an apple be divided into 5 equal parts, one of these parts of an apple; and, if of be, it is evident that will be 7 times as much. 7 times
is ; and if be, of will be 4 times as much, 4 times
Or, by multiplying the denominator of by 5, the denominator of, it is evident we obtain of, since the parts into which the number or thing is divided are 5 times as many, and consequently only as large, as before. Again, since of fr=5, of will be 4 times as much; and 4 times 8. This process will be seen to be precisely like the operation.
QUESTIONS. Art. 137. What is the rule for reducing improper fractions to whole or mixed numbers? Give a reason for the rule. Art. 138. How do you reduce a compound fraction to a simple one? operation.
Give the reason for the
Ex. 2. Reduce of of of § of to a simple fraction.
RULE. -1. Multiply all the numerators together for a new numerator, and all the denominators for a new denominator, and then reduce the fraction to its lowest terms.
2. If there are factors in the numerator similar to those in the denominator, cancel them in the operation.
NOTE. All whole and mixed numbers in the compound fraction must be reduced to improper fractions, before multiplying the numerators and denominators together.
EXAMPLES FOR PRACTICE.
3. What is of
4. What is
1. Ans. 5g'g. Ans. 176.
of % of 4?
of 2 of 2 of 7 to a simple fraction. Ans. 23.
7. Required the value of of of 11 of 13 of 52.
Ans.. of § of & to a simple fraction. Ans. of of 41 to a simple fraction. Ans. . to a simple fraction. Ans. . of 15 of 9g to a whole number.
Since some of the numerators and denominators to be multiplied together are alike, we may cancel these common factors, according to the rules of cancellation.
8. Reduce of of
10. Reduce of of
12. Reduce of off of 83 of to a simple fraction. Ans. 1.
QUESTIONS.-When there are common factors in the numerator and denominator, how may the operation be shortened? What is the rule? What must be done with all whole and mixed numbers in the compound fraction?