9. How do you get number? How much is 10. How do you find

+? 16? 11⁄2? TOO?

of a thing? How do you find
of 24? Of 32? Of 36? Of 45?
of a number? How do you

of a

Of 19?

find ?

of 72?

11. How much is † of 40? { of 42? † of 84?

Ans. One fourth of 12 is 3; and

What part of 5 is 3? What part of 7 is 1? What part of 7 is 6?

14. How many half-pence in 9 pence?

15. How many quarters of beef in 12 oxen?

16. If I cut 10 oranges into sixths, how many pieces have I? 17. A vessel containing 48 passengers was wrecked. of the passengers escaped. How many escaped, and how many perished?

18. If a pound of coffee costs 40 cents, what will half a pound cost? of a pound? of a pound?

19. A boy having 60 marbles lost of them, gave away, and kept the rest. How many did he lose, give away, and keep? 20. If of a ton of coal costs $3, what will a ton cost? Half

a ton?

140. Fractions may be reduced, added, subtracted, multiplied, and divided.

Reduction of Fractions.

141. Reducing a fraction is changing its form without changing its value.

140. What operations may be performed on fractions ?-141. What is meant by leducing a fraction?

142. CASE I.—To reduce a fraction to its lowest terms. A fraction is in its lowest terms when its numerator and denominator have no common divisor greater than 1. EXAMPLE.-Reduce 45 to its lowest terms.

Dividing both numerator and denominator by the same number does not alter the value of the fraction (§ 137). We therefore divide by their common factors in succession. Dividing by 5, we Dividing the terms

45) 75 (1 45

30) 45 (1

30

15) 30 (2

15|4= Ans.

30

get

of this fraction by 3, we
get. This is the answer,

315 = & Ans.

since its terms have no common divisor but 1.

In stead of dividing as above, we might have found the greatest common divisor (§ 123), and divided by it at once. This is the best method, when the numbers are large.

RULE.-Divide numerator and denominator successively by every factor common to both. Or, divide them at once by their greatest common divisor.

EXAMPLES FOR PRACTICE.

Reduce the following fractions to their lowest terms:

142. What is the first case of reduction of fractions? When is a fraction in its lowest terms? Solve the given example, explaining the steps. What other method is shown? Recite the rule for reducing a fraction to its lowest terms.

Ans. 1.

31. 1889.

Ans. 14.

Ans. f.

Ans.

143. CASE II.-To reduce an improper fraction to a whole or mixed number.

A fraction indicates division. The numerator is the dividend, the denominator is the divisor. To find the quotient, that is the value of the fraction, we have only to divide, as indicated.

EXAMPLE 1.-Reduce 2 to a whole or mixed number.

279 3 Ans.

EXAMPLE 2.-Reduce 30 to a whole or mixed number. 30 ÷ 933 = 31 Ans.

RULE.-Divide the numerator by the denominator.

If there is a remainder, the answer is a mixed number; if not, a whole number. If the answer is a mixed number, the fractional part must be reduced to its lowest terms.

Reduce these fractions to whole or mixed numbers :

143. What is the second case of reduction of fractions? What does a fraction indicate? With what do the numerator and denominator correspond? How may we find the quotient,—that is, the value of the fraction? Give the rule for reducing an improper fraction to a whole or mixed number. Give examples.

Ans. 85711.

41. 4875.

Ans. 571.

144. CASE III.-To reduce a mixed number to an im

proper fraction.

EXAMPLE.-Reduce 93 to an improper frac

tion.

The denominator of the fraction being 5, we reduce to fifths. In 1 there are 5 fifths, and in 9 nine times 5 fifths, or 45 fifths. 45 fifths and 3 fifths make 48 fifths.

Proof. 48 485: = 93

9

5

45 fifths.

3 fifths.

Ans. 48.

48 fifths.

RULE.-1. Multiply the whole number by the denomi nator of the fraction, add in the numerator, and set their sum over the denominator.

2. Prove by reducing the improper fraction obtained back to a mixed number.

145. To reduce a whole number to an improper fraction with a given denominator, the process is the same, except that there is no numerator to add in. Multiply the whole number by the given denominator, and set the product over the denominator.

EXAMPLE.-Reduce 9 to fifths.

9 × 5 45 Ans. 45.

EXAMPLES FOR PRACTICE.

Reduce the following to improper fractions; prove each:

17. Reduce 13 to a fraction with 7 for its denominator.

10000

Ans. 21.

18. How many 89ths in 746? In 29? In 450 ?
19. Reduce 26 to fortieths. To fiftieths. To sixtieths.
20. How many quarters of beef in 1225 oxen?

21. Reduce 387 to nineteenths. To eighty-fifths.

144. What is the third case of reduction of fractions? Solve and prove the given example. Recite the rule for reducing a mixed number to an improper fraction.145. How does the operation differ, when a whole number is to be reduced to an improper fraction? Recite the rule. Give an example.

146. CASE IV.-To reduce a fraction to higher terms. A fraction is reduced to lower terms (§ 142) by division, to higher terms by multiplication.

EXAMPLE.-Reduce

to twenty-fourths.

Multiplying both numerator and denominator by the same number does not alter the value of the fraction. We therefore multiply both terms by such a number as will change fourths to twenty-fourths-that is, 6 (because 24 ÷ 4 6). Ans. 1.

RULE.-1. Divide the given denominator by the denominator of the fraction, and multiply both terms by the quotient.

2. Prove by reducing the fraction back to its lowest

terms.

Mixed numbers must first be reduced to improper fractions.

147. A fraction can thus be reduced only to such higher terms as are multiples of the original terms. Thus, can be reduced to eighths, twelfths, sixteenths, &c., but not to fifths or sixths.

1. Reduce

EXAMPLES

FOR PRACTICE.

to seventieths.

Ans. 18.

2. Reduce the following to 36ths:-;;; t

24

29

3. Reduce to 288ths :- -7; 2; H; 48; 4; 122;
4. Reduce 14 to twenty-seconds.

5. How many 840ths in 17?

6. How many 360ths in ? In 411? In 2?
7. How many seventy-seconds in? In 2?
8. Reduce the following to 2400ths :—

78; 1.

Ans. 314.

Ans. 1575.

In #?

In 47?

391

, 1200.

840

148. CASE V.-To reduce two or more fractions to others having a common (that is, the same) denominator. EXAMPLE.-Reduce 2, 1, and §, to fractions that have a common denominator.

146. What is the fourth case of reduction of fractions? How is a fraction reduced to lower terms? How, to higher terms? Reduce & to twenty-fourths, explaining the steps. Recite the rule. What must first be done with mixed numbers ?147. To what higher terms alone can a fraction thus be reduced?-148. What is the fifth case of reduction of fractions? Work out and explain the given example.