Ex. 69.

1. How many yards of cloth, at $3 a yard, can be bought for 12 tons of hay, at $15 per ton.

2. How many pairs of boots, at $4 a pair, can be bought for 40 pounds of butter, at 40 cents per pound?

3. How many jars of lard of 36 pounds each, at 8 cents per pound, must be given for 16 pieces of cloth containing 24 yards each, at 48 cents a yard?

4. How many coats, at $4 each, can be bought for 32 yards of broadcloth, at $2.50 a yard?

5. A milkman having 30 cows which daily give 8 quarts each, sells the milk at 5 cents per quart. How many pieces of cloth containing 40 yards each, at 12 cents per yard, ought he to receive for the milk of 6 days? 6. A market gardener sold 16 lòts of celery, 120 bunches in each, at 28 cents per bunch; how many 240-pound barrels of sugar, at 8 cents a pound, will the celery pay for?

7. John Peters sold 9 firkins of butter weighing 78 pounds each, at 25 cents per pound; how many pieces of matting having 45 yards in a piece, at 30 cents per yard, should he receive?

8. A workman has received for 15 days' work of 7 hours each, 21 dollars. How much would he receive for 19 days' work of 5 hours each?

9. Thirty workmen have made in 9 days 215 yards of wall. At the same rate, how much would 36 workmen make in 15 days?

10. A telegraph operator transmits 50 words, averaging 4 letters each, in the space of 5 minutes. At the same rate, how many minutes will be required to send a dispatch of 120 words, averaging 5 letters each?

CHAPTER VIII.

103.

COMMON FRACTIONS.

What is the name of one of the parts when a unit

When a unit is divided into twelve equal parts, what is

Express in figures:

1. Three-sevenths.

2. Five-ninths.

3. Seven-eighths.

4. Five-twelfths.

6

5

5. Seven-sixteenths.

6. Five-eighteenths.

7. Four-elevenths.

8. Nine-twentieths.

73

23

Read: £2, A, 1, 7%, 24, 11, 18, 78, 28.

I. Seven of the parts when a unit has been divided into nine equal parts.

II. One-ninth of seven units; for, if seven units be divided into nine equal parts, one of these parts will be seven times as great as one of the parts obtained by dividing one unit into nine equal parts.

III. The quotient of seven divided by nine.

105. In the fraction, the lower figure shows the number of equal parts into which the whole has been divided, and is therefore a divisor; but, since it shows the number of parts into which the whole has been divided, it shows the name of each part, and is therefore called the denominator.

The upper figure shows the number of these parts taken, and is therefore called the numerator.

The figure, then, above the line denotes number, the figure below the line name.

106.

The numerator and denominator are called the terms of a fraction.

107. A proper fraction is one of which the numerator is less than the denominator; as 7.

108. An improper fraction is one of which the numerator equals or exceeds the denominator; as, 17.

When the numerator is greater than the denominator, more than one unit must be regarded as divided into equal parts; thus,

means that three units have been divided each into four equal parts, and that all the parts of two units and one part of the third unit are taken.

109. A mixed number is an expression consisting of a whole number and a fraction; as 4, 5.35. These expressions are read four and three-sevenths, five and thirty-five hundredths.

Every mixed number means that some entire units are taken, and the fraction of another unit.

Select the proper fractions, the improper fractions, and mixed numbers from the following expressions:

†, 21, 41, 8, 94, 125, 7, 18, 17, 68, 15, 187, 1, 1, 51, 1, 8, 194, 17, 68, 25, 183, §.

110. An improper fraction represents a quantity which can also be represented by a whole number or else by a mixed number. Thus, 12 = 25.

For, if we suppose several units to be divided each into seven equal parts, and we take 19 of these parts, 14 (that is, 2 × 7) will make 2 units, and the five remaining parts will be five-sevenths of another unit.

111. To reduce an improper fraction to a whole number,

Divide the numerator by the denominator.

or

mixed

The quotient will be the whole number, and the remainder, if any, will be the numerator of the fractional part, of which the denominator is the same as the denominator of the improper fraction.

112. A whole number or a mixed number represents a quantity which can also be represented by an improper fraction. Thus, $3-$15.

For each dollar contains 4 fourths; therefore 3 dollars contain 3×4 fourths or 12 fourths; which, together with the 3 fourths, make 15 fourths. Hence,

113. To reduce a mixed number to an improper fraction, Multiply the whole number by the denominator of the fraction, and to the product add the numerator; under this sum write the denominator.