4. Divide 27 × 35 × 52 by 18 x 7 x 13.

5. Divide 92 × 87 × 57 × 69 by 23 × 23 × 19 × 29.

6. Divide 140 × 169 × 510 by 39 × 68.

7. How many baskets of eggs, each containing 12 dozens, at 15 cents a dozen, will pay for 8 bolts of cloth, each containing 24 yards, at 30 cents a yard?

128. ANALYSIS.

1. If 63 books cost $126, what will 125 books cost?

FORM.

$126 × 125
63

ANALYSIS. Since the question asks for the cost of certain articles, I begin with $126, writing it above a short horizontal line. If 63 books cost $126, each book will cost one sixty-third of $126, which is expressed by writing 63 below the line as a divisor. 125 books will cost 125 times this number of dollars, which is expressed by writing 125 above the line as a factor of the dividend. Cancelling the common factors and completing the work, the result is $250.

2. If 15 men can do a piece of work in 7 days, in how many days can 21 men do the same work?

3. If 24 men dig a ditch in 18 days, how many would be required to dig the same ditch in 27 days?

4. If 11 tons of hay can be made from 5 acres, at the same rate, how many tons can be made from 65 acres?

5. If 12 acres of land raise 720 bushels of corn, how many acres would be needed to raise 1,800 bushels at the same rate?

6. If 26 horses eat a certain quantity of grain in 39 days, how many days would it last 338 horses?

7. If a certain quantity of grain last 46 horses 34 days, how many horses would eat the same amount in 391 days?

8. 64 men can do a piece of work in 57 days, working 9 hours a day; in how many days can 38 do the same work, working 8 hours a day?

9. If 42 men do a piece of work in 18 days, working 10 hours a day, how many men can do the same work in 90 days, working 7 hours a day?

10. If 91 men can do a certain amount of work in 54 days, working 9 hours a day, how many hours a day must 162 men work to perform the same labor in 39 days?

11. The interest on what amount of money at 8 per cent, for 87 days, equals the interest on $2,500 at 7 per cent for 261 days?

12. At what rate will the interest on $3,200 for 92 days equal the interest on $4,800 for 46 days, at 6 per cent?

13. For how many days must $965 be loaned in order that the interest on it at 5 per cent shall equal the interest on $2,123 for 125 days, at 11 per cent?

14. How many men working 12 hours a day will be needed to dig a ditch 1,500 ft. long, 8 ft. wide, and 6 ft. deep, in 250 days, if 36 men in 180 days of 9 hours each can dig a ditch 1,080 ft. long, 9 ft. wide, and 12 ft. deep, the work to be uniformly difficult?

For further problems see Simple and Compound Proportion.

SECTION VI.

FRACTIONS.

129. DEFINITIONS.

1. A fractional number, or, more briefly, a fraction, is a number that is composed of one or more fractional units. Illustration. 3, 12.

2. A fractional unit is one of the equal parts into which a whole has been separated. Illustration. 3, 12.

3. The numerator of a fraction is the number of fractional units in the fractional number. In the fractions 3, §, the numerators are 2 and 5.

4. The denominator of a fraction is the number that shows the size of the fractional unit. This it does by showing the number of equal parts into which the whole has been separated. Illustration. In the fractions,

The numerator and de

5 and 12 are the denominators. nominator are the terms of the fraction.

5. Fractions are classified, with respect to their value, into proper and improper.

6. A proper fraction is a fraction whose value is less than one. Its numerator is less than its denominator. Illustration. 3, 13.

5

7. An improper fraction is a fraction whose value is equal to or greater than one. Its numerator is equal to or greater than its denominator. Illustration. §. 4.

8. Fractions are classified, with respect to their form, into simple, complex, and compound.

9. A simple fraction is a fraction whose terms are integers. Illustration. §,‡.

10. A complex fraction is a fraction which contains a

fraction in one or both of its terms.

11. A compound fraction consists of two or more simple fractions joined by of. Illustration. 3 of 1.

12. A mixed number is composed of an integer and a fraction. Illustration. 4ğ.

13. Tell whether each of the following is proper or improper, and give the definition in each case.

14. Tell whether each of the following is simple, complex, or compound, and give the definition in each case.

15. A fraction is in its lowest terms when the numerator and denominator are prime to each other.

130. PRINCIPLES.

I. Multiplying the numerator of a fraction by an integer multiplies the fraction by the integer.

Since the number of fractional units in the fractional number is multiplied by the integer, while their size is unchanged, the fraction is multiplied by the integer. Illustrate.

II. Dividing the numerator of a fraction by an integer divides the fraction by the integer.

Since the number of fractional units in the fractional number is divided by the integer, while their size is unchanged, the fraction is divided by the integer. Illustrate.

III. Multiplying the denominator of a fraction by an integer divides the fraction by the integer.

If the number of equal parts into which a unit has been separated be doubled, each part will be one half as large as before. If the denominator of a fraction be multiplied by any integer, the unit will be divided into as many times the number of parts that it was before as the integer is times The resulting fractional units will be the same part of the former fractional units that one is of the integer. Since the number of fractional units is unchanged, the fraction must be divided by the integer. Illustrate.

one.

IV. Dividing the denominator of a fraction by an integer multiplies the fraction by the integer.

If the number of equal parts into which a unit has been separated be divided by an integer, the resulting fractional units will be as many times the former fractional units as the integer is times one. Since the numerator is unchanged, the fraction is multiplied by the integer. Illustrate.

V. Multiplying both terms of a fraction by the same number does not change its value.

Illustration.

There are 4 times as many frac

tional units in as in, but each is only as large.

VI. Dividing both terms of a fraction by the same number does not change its value.

Illustration.

=

There are only as many fractional units in as in, but each is 4 times as large.

131. REDUCTION.

Reduction of fractions is the process of changing their denomination without changing their value.

Review Reduction, page 6.

132. Illustrative Example. In $5 there are how many quarters?

ANALYSIS. Since in $1 there are 4 quarters, in $5 there are 5 fours of quarters, which are 20 quarters.