196. From the foregoing illustrations may be derived the following

Rule

To change an integer or a mixed number to an improper fraction: Multiply the integer by the denominator of the fraction, and to the product add the numerator; the result will be the numerator of the required fraction.

Oral Exercises.

197. Change the following to 7ths; 8ths; 9ths; 12ths. 8; 3; 5; 2; 4; 7; 6; 9; 11; 10.

Examples for Written Work.

198. Change the following to improper fractions:

What name

What is a fractional unit? A fractional number? is applied to both? Name and define the terms of a fraction. Explain the expression §. How do you change fractions to smaller terms? To larger terms? When is a fraction expressed in its smallest terms? How do you change improper fractions to integers or mixed numbers ? How do you change integers or mixed numbers to fractions?

*First change the fractional part to smallest terms.

ADDITION OF FRACTIONS.

199. Like fractions are like parts of the same or like units.

Thus apple and apple are like fractions, while apple and pear are unlike fractions, and so are apple and

Only like fractions can be added together. a. Add,, and .

b. Add 1, 3, and 1.

apple.

8

d. Add 6,2%, and 2.

e.

Add,, and .

200. Like fractions have the same denominator, which, because it belongs to several fractions, is called a common denominator.

How do you add fractions that have a common denominator?

201. To add fractions not having a common denominator. Illustrative Example. Add and

4 5

To be added, these fractions must be changed to like fractions,

or to fractions having a common denominator.

may be changed to

18ths, 27ths, 36ths, 45ths, 54ths, 63ds, 72ds, etc. may be changed to

24ths,

36ths, 48ths, 60ths, 72ds, etc.

36 is the least multiple (Art. 71) common to 9 and 12, and may be taken for the common denominator. Multiplying both terms of by 4, and both terms of by 3, the resulting fractions are 18 and 1}}, which added equal 3. Ans.

202. Examples for Oral or Written Work.

62.+? 65. 1+1=? 68. +‡=? 71. √+12=?

63. +? 66. +? 69. +1=? 72

=

64.+? 67. +? 70. +? 73.

3+3=?

203. In adding fractions, any common denominator may be used, but the least common denominator is to be preferred. This is always the least multiple common to all the denominators and is called the least common multiple (L. C. M.).

A common multiple of the denominators must contain all the denominators, and hence all their prime factors; the least common multiple must contain only these factors, each occurring as many times as it occurs in any one of the numbers. When these factors cannot be readily seen, the method at the left of the written work below may be used for finding them, and from them obtaining the least common denominator. For another explanation, see Supplement, Art. 8.

105+64+108

108

=

;

168

277

=

168

168

=110%. Ans.

L. C. M.=2×7×4×3=168

=

Here by repeated divisions the factors which are common to two or more of the denominators, and which, therefore, should enter into the common denominator but once, are taken out. The product of these factors with those that are not common must be the least common denominator, which is 168.

To change to 168ths, the denominator 8 is multiplied by 7 × 3, so the numerator 5 must be multiplied by 7 × 3.

In a similar way, is found to equal, and to equal 18. Adding these, the sum is 27 1188 Ans.

=

205. From the preceding examples may be derived the following

Rules.

I. To change fractions to equivalent fractions having the least common denominator:

1. For the common denominator, find the least common multiple of the given denominators.

2. For the new numerators, multiply the numerator of each fraction by the number by which its denominator must be multiplied to produce the common denominator.

II. To add fractions:

1. If they have a common denominator, add their numerators for the numerator of the answer.

2. If they have not a common denominator, change them to equivalent fractions that have a common denominator, and then add their numerators.

Examples for Written Work.

206. Add the following in lines and in columns:

- In performing the following examples, add the integers (See Supplement, Art. 9.)

NOTE. ·and fractions separately.

95. 96.

97.

91. 251+673 +1475. 92. 43+163 +8718. 93. 75+ 9+ 6%. 94. 405+7719+9813.

102. 103. 104. 98. 25 +110 +84. 99. 40+ 1772+137. 100. 12+ 2541+9111. 101. 65+ 841+551⁄21⁄2•

105. Out of a barrel of vinegar were drawn 1 gallons, 3g gallons, and 21 gallons, after which 12 gallons remained. How many gallons did the barrel contain at first?

106. How much silk is there in 4 remnants measuring 12ğ yards, 15 yards, 113 yards, and 91 yards? 107. How much iron railing is needed to fence a court, whose sides are 4 rods, 35 rods, 21 rods, and 74 rods? 108. John weighs 92 and Richard 783 pounds.

pounds, James 65, Charles 855, What is the sum of these weights? 109. From a piece of ribbon measuring 10 yards 3ğ yards, 2 yards, and 3 yards were cut. How much remained?

SUBTRACTION OF FRACTIONS.

Oral Exercises.

207, Illustrative Example.

From of a yard of velvet

of a yard was taken. What part remained? Ans. §. Find the remainders in the following examples:

When the minuend and subtrahend are like fractions, how do you subtract? In mixed numbers when the fraction in the subtrahend is larger than the fraction in the minuend, how do you proceed?

208. Illustrative Example. If Mary had of a yard of satin and has used of a yard, how much has she left?