30. How many barrels of flour, at $6 a barrel, must be given for 3 pieces of linen, each containing 36 yards, at 25 cents a yard?

31. How many bushels of wheat, at 90 cents a bushel, will pay for 3 barrels of sugar, each containing 200 pounds, at 6 cents per pound?

32. A laborer worked 8 days for 24 bushels of potatoes, worth 40 cents a bushel. What were his daily earnings?

33. How many boxes of tea, each containing 24 pounds, worth 45 cents a pound, must be given for 4 loads of wheat, each containing 54 bushels, worth $.95 a bushel?

34. If 34 bushels of wheat make 8 barrels of flour, how many bushels will be necessary to make 72 barrels ?

35. A grocer sold 18 boxes of soap, each containing 55 pounds, at 10 cents a pound, and received as pay 66 barrels of apples, each containing 3 bushels. What was the price per bushel of the apples?

36. A farmer exchanged 96 bushels of corn, worth $.55 a bushel, for an equal number of bushels of rye, worth $.75 a bushel, and oats worth $.35 a bushel. How many bushels of each did he receive?

37. If 48 men can dig a trench in 25 days, working 9 hours a day, how many days will be required by 20 men to do the same work if they work 10 hours per day?

38. If 12 barrels of pork, each containing 200 pounds, are worth $192, what will 80 pounds cost at the same rate per pound?

39. How many days' work, at $1.25 a day, will pay for 75 bushels of corn, at $.60 cents a bushel ?

FRACTIONS.

116. 1. When anything is divided into two equal parts, what is each part called? Into three equal parts? Into seven equal parts? Into eight equal parts? Into nine equal parts? Into fifteen equal parts?

How many

2. How many halves are there in anything? thirds? How many fifths? How many tenths? many fifteenths? How many twentieths?

How

3. What part of an apple will each boy receive when it

is divided equally among 7 boys? Among 8 boys?

4. How much is one fifth of 10 cents? Of 15 cents? Of 30 cents?

5. How much is one sixth of 12 cents? Of 18 cents? Of 24 cents?

6. How much is one seventh of 14 oranges? Of 21 oranges? Of 28 oranges?

7. In 6 hours James earned 36 cents? How much did

he earn per hour?

8. A school has 4 classes of the same size, and the whole number of pupils in the classes is 40.

in each class?

How many are there

117. One or more of the equal parts of anything is called a Fraction.

Two numbers, written one above the other with a line between them, are used to express a fraction.

118. The number which shows into how many parts a thing has been divided is called the Denominator.

It is written below the line.

Thus, in the fraction, 9 is the denominator. It shows that something has been divided into 9 equal parts.

119. The number which shows how many parts form the fraction is called the Numerator.

It is written above the line.

Thus, in the fraction 3, 7 is the numerator. It shows that the fraction contains 7 of the 9 equal parts.

120. The numerator and denominator together are called the Terms of the Fraction.

121. A fraction whose numerator is less than its denominator is called a Proper Fraction.

Thus, 4, 5, and 15 are proper fractions.

The value of a proper fraction is, therefore, less than 1.

122. A fraction whose numerator equals or exceeds its denominator is called an Improper Fraction.

Thus, 7, 15, and 11 are improper fractions.

The value of an improper fraction is, therefore, 1 or more than 1.

123. A number expressed by an integer and a fraction is called a Mixed Number.

Thus, 35, 83, and 1413 are mixed numbers.

124. The unit which is divided into equal parts is called the Unit of the Fraction.

A fraction whose unit has been divided into any number of equal parts is called a Common Fraction.

A fraction whose unit has been divided into tenths, hundredths, thousandths, etc., is called a Decimal Fraction.

125. One of the equal parts into which a unit has been divided is called a Fractional Unit.

126. A fraction also expresses unexecuted division.

Thus, 10 is equal to 10 ÷ 5; 19 is equal to 19 ÷ 4.

127. Fractions are read by naming first the number of fractional units, and then the kind of them.

Thus, is read three eighths; nine thirty-firsts.

128. Read the following:

31

46

43

26

314 520

305 319 327 81% 113 339 533 713 21 829

Express in figures:

719

224 512

639 518

463

649

1. Seven ninths. Eight elevenths. Four fifteenths.

2. Six nineteenths. Nine fourteenths. Five seventeenths. 3. Three twentieths. Six forty-fifths. Nine eighteenths. 4. Seventeen twentieths. Thirteen forty-sevenths. 5. Twelve thirtieths. Fifty-five eighty-fifths.

6. Seventeen fifty-fifths. Thirty-four ninety-eighths.

129. 1. Interpret the expression 3.

EXPLANATION.

- 3 represents 7 of 9 equal parts of a thing. It also represents one ninth of 7, and 7 divided by 9. It is read seven ninths.

REDUCTION.

130. To reduce fractions to higher terms.

1. In yard, how many fourths are there? How many eighths?

2. In of a foot, how many sixths are there? How many ninths?

3. Since is equal to 4, how may the terms of the fraction be obtained from ? from ? from?

from?

4. Since the terms of the fraction may be obtained from by multiplying them by 4, how may the terms of the fraction be obtained from ?

5. What changes, then, may be made in the terms of a fraction without changing its value?

131. The process of changing the forms of fractions without changing their values is called Reduction of Fractions.

132. A fraction is expressed in higher terms, when its numerator and denominator are expressed by larger numbers.