Rule.-Multiply the integral part of the mixed number by the denominator of the fractional part; to the product add the numerator; and write the result over the denominator.

11. In 221 bushels, how many elevenths of a bushel?

Case IV.

An Improper Fraction to an Integer or Mixed Number.

110. 1. Reduce 1 to an integer or mixed number.

OPERATION.

1753, Ans.

Since there are 3 thirds in 1 unit, there will be in 17 thirds as many units as 3 is contained times in 17, or 53. Therefore, y is equal to 5.

Rule. - Divide the numerator by the denominator.

Explain the operation of reducing an improper fraction to an in

teger or mixed number. Recite the Rule.

10. In 1g of a dollar how many dollars?

11. What is the value of 8224 miles?

Ans. $33.

Ans. 514 miles.

COMMON DENOMINATOR.

111. Fractions have a Common Denominator when they have the same number for a denominator.

1. Reduce and to fractions having a common denominator.

OPERATION.

2

2×7

14

5

5 × 7

35

Ans.

3

3 × 5

15

7

7 × 5

35

Since multiplying both the numerator and denominator of a fraction by the same number does not change its value (Art. 105), we multiply both terms of the fraction by 7, the denomi

nator of the second fraction, and have = 1; and both terms of the fraction by 5, the denominator of the first fraction, and have. Therefore, and are equal to and

When have fractions a Common Denominator? Explain the manner of finding the common denominator by the first operation.

Rule.-Multiply both terms of each fraction by the denominators of the other fractions. Or,

Multiply both terms of one or more of the frac tions by such a number as will make all the denominators alike.

If there are integers or mixed numbers with the given fractions, they must first be reduced to improper fractions.

Examples.

Reduce to equivalent fractions having a common de nominator: :

112.

ADDITION.

Fractions may be added by means of a common denominator. Thus,

We can add halves and halves, thirds and thirds, fourths and fourths, etc.; but we can not add directly halves and thirds, thirds and fourths, etc., any more than we can add other things of different kinds, as dollars and days.

1. Find the sum of 1, §, and §.

Recite the Rule. By what means may fractions be added?

The sum of 1 fifth, 3 fifths, and 4 fifths, is 8 fifths, or ‡, which, reduced, gives 18..

Since numbers must be of the same name or kind, in order to be added, we reduce the given fractions to equivalent fractions having a common denominator, and have 15 twentieths, and 16 twentieths, which, added, give 31 twentieths, 11, or, by reduction, 111. Therefore, the sum of and is 11.

Rule.-Reduce the fractions, if necessary, to equivalent fractions having a common denominator, and write the sum of the numerators over the common denominator.

113. When there are mixed numbers, the fractions and integral parts may be added separately; and then their sums may be added.

13. What is the sum of 31, 61, and 23?

OPERATION.

¿ + ¿ + } = } + 1+}=} = 1};

3+6+211; 11 + 13 = 12}, Ans.

What is the Rule? How may the addition be performed when there are mixed numbers?

14. What is the sum of 53, 63, and 41? 15. What is the sum of 17, 22, and 3? 16. Required the sum of 5, 6, and 73.

Ans. 174.

Ans. 231.

Ans. 1885.

17. Bought a handkerchief for § of a dollar, a vest for 2 dollars, a pair of gloves for of a dollar, and a hat for 41 dollars; how much did the whole cost? Ans. $71. 18. A merchant has sold from a piece of cloth, 12, 43, 84, and 2 yards; how many yards is this in all?

Ans. 2711. 19. What is the sum of $91, $121, $103, and $73 ? 20. I have bought three lots of coal, weighing respectively, 13,, and 9 tons; how much is there in all? Ans. 11 tons.

21. A farmer sold sheep for $627, cattle for $102, and a horse for $125; how much did he receive for all?

SUBTRACTION.

114. One fraction may be subtracted from another, by means of a common denominator. Thus,

We can subtract halves from halves, thirds from thirds, fourths from fourths, etc.; but we can not subtract directly thirds from halves, fourths from thirds, etc., any more than we can subtract days from dollars, which are of different names or kinds.

1. Find the difference between 17 and 1.

is 3 eighteenths, or, which, reduced, gives . Therefore, the difference required is 1.

How may an integer be expressed in a fractional form? (Art. 144.) How may one fraction be subtracted from another?