ART. 137.-To change Improper Fractions to Whole or Mixed Numbers.

ORAL EXERCISES.

1. How many units in 13?

Solution. Since there are

in 1 unit, 3 is equal to as many

units as is contained times in 12, which is 23 times. Proof: 13 ÷

ART. 138. Rule for changing an Improper Fraction to a Whole or Mixed Number.-Divide the numerator of the fraction by the denominator. If there be a remainder, write the denominator under it and annex the resulting fraction in its simplest form to the quotient.

ART. 139.-To change a Fraction or an Integer to an Equivalent Fraction having a Given Denominator.

ORAL EXERCISES.

1. How many eighths are there in ?

Solution. Since there are

in 1, in of 1 there are of, or .

6. How many sixteenths in one eighth? fourth? In one half?

7. How many eighths in ? In ? In ?

8. Change to fourths.

616 T

In one

Solution. It is readily seen that the denominator must be multiplied by 2 in order to make fourths, and since the value of a fraction is not changed by multiplying its terms by the same number, we multiply the terms of by 2; hence changed to fourths, is 4.

9. Change to ninths; to twelfths; to fifteenths. 10. Change to twentieths; to thirtieths; to fortieths 11. Change to twenty-firsts; to twenty-eighths.

12. Change to fortieths; to hundredths; to two hundred twentieths.

ART. 140.-Rule for changing a Fraction to an Equivalent Fraction having a Given Denominator.-Divide the given denominator by the denominator of the fraction, and multiply both its terms by the quotient thus obtained.

NOTE.-An integer is changed to an equivalent fraction having a given denominator by multiplying it by the denominator required and writing the product over the given denominator.

ART. 141.—To change Fractions to Equivalent Fractions having a Common or a Least Common Denominator.

WRITTEN EXERCISES.

1. Change and to equivalent fractions having a common denominator.

Process.

Analysis. The common denominator must be a common multiple of the denominators given. A multiple of 3 and 7 is their product, 21. Changing to 21sts by multiplying both terms by 7, we obtain the equivalent fraction; changing ‡ to 21sts by multiplying both terms by 3, we obtain the equivalent fraction.

ART. 142.-Rule for Changing Fractions to Equivalent Fractions having a Common Denominator-Multiply both terms of each fraction by all the denominators except its

own.

Change to equivalent fractions having a common denominator:

ART. 143. Since it is necessary to change fractions to equivalent fractions having a common denominator in order to add or subtract them, it is convenient to have the common denominator as small as possible. Hence, we change the fractions to equivalent fractions with the least common denominator.

WRITTEN EXERCISES.

1. Change, 8, 8, to equivalent fractions with the least common denominator.

Process.

2 × 15 = 15

3 × 13 = 38

8 × 18 = 88

Analysis. The least common denominator of 3, 3, 5, is the least common multiple of 4, 5 and 6, or 60.

To change each fraction to 60ths, we divide 60 by each denominator in turn, and multiply both terms of each fraction by the quotient thus obtained.

ART. 144.-Rule for finding the Least Common Denominator. Find the least common multiple of the denominators, divide this by each denominator in turn, and multiply both terms of each fraction by this quotient.

Change the following to equivalent fractions having the least common denominator :

Addition of Fractions.

ORAL EXERCISES.

ART. 145.-1. Harry had of a dollar and his mother gave him † of a dollar: how much, then, had Harry?

2. Mary paid of a dollar for a ribbon and of a dollar for a book: how much did she pay for both?

3. James gave of a dollar for a handkerchief and of a dollar for a knife: how much did he spend ?

4. If a duck costs of a dollar and a chicken % of a dollar, how much do both cost?

5. Kate gave of her money to her brother, to her sister, and to her cousin : how much did she give away?

ART. 146.—Principle. Only like fractions can be added.

6. George paid of a dollar for an arithmetic and of a dollar for a copy book: how much did he pay for both?

Solution. He paid the sum of 1⁄2 of a dollar and of a dollar. and = £; £ + &= he paid of a dollar.

=

:

7. One lot contains of an acre, and another of an acre: how many acres in both lots?

8. Mr. A. gave his son of a dollar and his daughter of a dollar how much did he give to both?

9. John bought of a dozen of eggs at one store and of a dozen at another store: how many did he buy? 10. Thomas rode how far did he go?

11. Mary received from her father, and did she receive?

of a mile and walked of a mile:

of a dollar from her mother,

from her brother: how much