equal parts the unit is divided, and the numerator how many of the parts are taken.

Hence, also, we may conclude that,

PROPOSITION I. If the numerator of a fraction be multiplied by any number, the denominator remaining the same, the value of the fraction will be multiplied as many times as there are units in the multiplier. Hence,

To multiply a fraction by a whole number, we simply multiply the numerator by the number.

119. If three apples be each divided into 6 equal parts, there will be 18 parts in all, and these parts will be expressed by the fraction 18. If it were required to express but one-third of the parts, we should take in the numerator but one-third of 18; that is, the fraction would express one-third of 18. If it were required to express one-sixth of the parts, we should take one-sixth of 18, and would be the required fraction.

In each case the fraction 18 has been divided as many times as there were units in the divisor. Hence,

PROPOSITION II. If the numerator of a fraction be divided by any number, the denominator remaining un

118. If an apple be divided in six equal parts, how do you express one of those parts? Two of them? Three of them? Four of them? Five of them? Repeat the proposition. How do you multiply a fraction by a whole number?

changed, the value of the fraction will be divided as many times as there are units in the divisor. Hence,

A fraction may be divided by a whole number, by dividing its numerator.

EXAMPLES.

1. Divide by 2, by 7, by 14.

4

Ans., ts, s.

2. Divide 112 by 56, by 28, by 14, by 7. 3. Divide 100 by 25, by 8, by 16, by 4.

Ans.

Ans.

120. Let us again suppose the apple to be divided into 6 equal parts. If now each part be divided into 2 equal parts, there will be 12 parts of the apple, and consequently each part will be but half as large as before.

Three parts in the first case will be expressed by 8, and in the second by. But since the value of each part in the second is only half the value of each part in the first fraction, it follows that,

3. = one-half of .

If we suppose the apple to be divided into 18 equal parts, three of the parts will be expressed by, and since the parts are but one-third as large as in the first case, we have

= one-third of

and since the same may be said of all fractions, we have

119. If 3 apples be each divided into 6 equal parts, how many parts in all? If 4 apples be so divided, how many parts in all? If 5 apples be so divided, how many parts? How many parts in 6 apples? In 7? In 8? In 9? In 10? What expresses all the parts of the three apples? What expresses one-half of them? One-third of them? One-sixth of them? One-ninth of them? One-eighteenth of them? What expresses all the parts of four apples? One-half of them? One-third of them? One-fourth of them? One-sixth of them? One-eighth of them? One-twelfth of them? One twenty-fourth of them? Put similar questions for 5 apples, 6 apples, &c. Repeat the proposition. How may a fraction be divided by a whole number? 120. If a unit be divided in 6 equal parts and then into 12 equal parts, how does one of the last parts compare with one of the first? If the second division be into 18 parts, how do the parts compare? If into 24? What part of 24 is 6? If the second division be into 30 parts, how do they compare? If into 36 parts? Repeat the proposition. How may a fraction be divided by a whole number?

PROPOSITION III. If the denominator of a fraction be multiplied by any number, the numerator remaining the same, the value of the fraction will be divided as many times as there are units in the multiplier.

Hence, A fraction may be divided by any number, by multiplying the denominator by that number.

121. If we suppose the apple to be divided into 3 equal parts instead of 6, each part will be twice as large as before, and three of the parts will be expressed by instead of. But this is the same as dividing the denom. inator 6 by 2; and since the same is true of all fractions, we have

PROPOSITION IV. If the denominator of a fraction be divided by any number, the numerator remaining the same, the value of the fraction will be multiplied as many times as there are units in the divisor. Hence,

A fraction may be multiplied by a whole number, by dividing the denominator by that number.

EXAMPLES.

1. Multiply by 2, by 4.

2. Multiply

by 2, 4, 8, 16, 32.

Ans.

16

Ans. 18, 16, 18, 18, 16.

3. Multiply by 2, 4, 6, 8, 12, 16, 24, 48.

Ans.

&c.

121. If we divide 1 apple into three parts, and another into six, how much greater will the parts of the first be than those of the second? Are the parts larger as you decrease the denominator? If you divide the denominator by 2, how do you affect the parts? If you divide by 3? By 4? By 5? By 6? By 7? By 8? Repeat the proposition. How may a fraction be multiplied by a whole number?

4. Multiply 2 by 2, 4, 6, 12, 21, 42.

19 19 Ans.

12, 11, 1, &c., &c. Ans. 5, 5, 5. 151 151 151 20

5. Multiply 151 by 5, 10, 20. 122. It appears from Prop. I., that if the numerator of a fraction be multiplied by any number, the value of the fraction will be multiplied as many times as there are units in the multiplier. It also appears from Prop. III., that if the denominator of a fraction be multiplied by any number, the value of the fraction will be divided as many times as there are units in the multiplier.

Therefore, when the numerator and denominator of a fraction are both multiplied by the same number, the increase from multiplying the numerator will be just equal to the decrease from multiplying the denominator: hence we have,

PROPOSITION V. If both terms of a fraction be multiplied by the same number, the value of the fraction will remain unchanged.

EXAMPLES.

1. Multiply the numerator and denominator of by 7: this gives Ans. 35. 2. Multiply the numerator and denominator of by 3, by 4, by 5, by 6, by 9, by 12, by 15, by 20.

3. Multiply each term of 12 by 7, by 8, by 12, by 14, by 15, by 17, by 45.

123. It appears from Prop. II., that if the numerator of a fraction be divided by any number, the value of the

122. If the numerator of a fraction be multiplied by a number, how many times is the fraction increased? If the denominator be multiplied by the same number, how many times is the fraction diminished? If then the numerator and denominator be both multiplied at the same time, is the value changed? Why not? Repeat the proposition.

123. If the numerator of a fraction be divided by a number, how many times will the value of the fraction be diminished? If the denominator be divided by the same number, how many times will the value of the fraction be increased? If they are both divided by the same number, will the value of the fraction be changed? Why not? Repeat the proposition.

fraction will be divided as many times as there are units in the divisor. It also appears from Prop. IV., that if the denominator of a fraction be divided by any number, the value of the fraction will be multiplied as many times as there are units in the divisor. Therefore, when the numerator and denominator of a fraction are divided by the same number, the decrease from dividing the numerator will be just equal to the increase from dividing the denominator: hence we have,

PROPOSITION VI. If both terms of a fraction be divided by the same number, the value of the fraction will remain unchanged.

EXAMPLES.

1. Divide both terms of the fraction by 4: this gives

Ans. 2.

3. Divide each term of the fraction by 2, by 4,

by 8, by 16, by 32.

32 128

4. Divide each term of the fraction 60 4, by 5, by 6, by 10, by 12, by 15, by 20,

GREATEST COMMON DIVISOR.

by 2, by 3, by by 30, by 60.

124. Any number greater than unity that will divide two or more numbers without a remainder, is called their common divisor: and the greatest number that will so divide them, is called their GREATEST COMMON DIVISOR.

Before explaining the manner of finding this divisor, it is necessary to explain some principles on which the method depends.

One number is said to be a multiple of another when it contains that other an exact number of times. Thus, 24 is a multiple of 6, because 24 contains 6 an exact

124. What is a common divisor? What is the greatest common divisor of two or more numbers? When is one number said to be a multiple of another? What is the first principle? What is the second? What is the third? Give the rule for finding the greatest common divisor. How do you find the greatest common divisor of more than two numbers?