10. What is the greatest common measure of 48 and 192?

Ans. 48.
Ans. 1.

12. What is the greatest common measure of 35, 42, 63 ?

Ans. 7.

Art. 63.—To reduce a complex fraction to a simple fraction,

To multiply numerator is the same as to divide denominator, 2 ? 2×4 (Art. 44;) therefore, ; and to multiply denominator

If the numerator be whole or mixed numbers, reduce them to improper fractions.

Then multiply the numerator of each fraction by the denominator of the other; the product will be the fraction required.

Art. 64.-To change a simple fraction to a complex. 1. Change to a complex fraction.

If the numerator or denominator, or both, be a composite numher, separate them into factors, and transfer one or more from numerator to denominator, and from denominator to numerator, observing that a factor transferred, becomes a divisor.

2. Change to a complex fraction.

Ans.

OBS.-The answer depends pon the factor transferred.

or

3. Change to a complex fraction. 4. Change 15 to a complex fraction.

Art. 65.—The following, if made familiar, will aid the scholar in cancelling.

1. Any number ending with an even number, or cipher, is divisible, or can be divided, by 2.

2. Any number ending with 5, or 0, is divisible by 5.

3. If the right-hand place of any number be 0, the whole is divisible by 10; if there be two ciphers, it is divisible by 100; if 3 ciphers, by 1000, and so on, which is only cutting off those ciphers.

4. If the two right-hand figures of any numbers be divisible by 4, the whole is divisible by 4; and if the three right-hand figures be divisible by 8, the whole is divisible by 8, and so on.

5. If the sum of the digits in any number be divisible by 3 or by 9, the whole is divisible by 3 or 9.

6. If the right-hand digit be even, and the sum of all the digits be divisible by 6, then the whole will be divisible by 6.

7. A number is divisible by 11, when the sum of the 1st, 3d, 5th, etc., or all the odd places, is equal to the sum of the 2d, 4th, 6th, etc., or of all the even places of digits.

8. If a number cannot be divided by some quantity less than itself, that number is a prime, and cannot be divided by any number whatever.

Art, 66.-To multiply and divide fractions by whole numbers, whole numbers by fractions, fractions by fractions.

RULE.

Draw a perpendicular line, and write numerators, in all cases, as you would a whole number standing in the place of the fraction; viz., the numerators of fractions to be multiplied or divided on the right of the line, and their denominators on the left. The question thus stated, equals on each side of the line may be crossed, as cancelling each other. (See Art. 47.) When no two numbers remain, one on each side of the line, capable of being divided by any one figure, multiply the figures on the right of the line, for a numerator, or dividend, and those on the left, for a denominator, or divisor, and the result will be the answer in the lowest terms of the fraction.

Multiplication of Fractions by Whole Numbers.

Art. 67.—1. If a man receive 1 of a dollar for 1 day's work, what will he receive for 2 days' work?

It is evident, if a man receive of a dollar for 1 day's work, that he would receive, for 2 days' work, twice as much, or 2= Multiplying the numerator by 2, the denominator remaining the same, we have twice the number of parts, while the value of each part remains the same. Dividing the denominator by 2, the numerator remaining the same, we have the same number of parts, while the value of each part is twice as great. Hence, to multiply the numerator of a fraction is the same, in effect, as to divide the denominator. If the numerator of be multiplied by 2, it becomes 1. If the denominator be divided by 2, it becomes 11. Therefore, to multiply a fraction by a whole number, we have the following

1

RULE.

Multiply the numerator, or divide the denominator, and the result will be the answer required.

2. If a pound of lead cost of a dollar, how much must 16 pounds cost?

Operation.
16|1

16

1 Ans.

It is evident, if one pound cost 1 dollar divided by 16, that 16 pounds would cost 16 dollars divided by 16, equal to 1 dollar. Therefore-A fraction is multiplied into a quantity equal to its denominator, by cancelling or removing the denominator.

3. If a pound of iron cost of a dollar, how much will 9 pounds cost?

QUESTIONS.-7. What is the rule for the multiplication and divisior of fractions, etc., by cancelling? 8. When the question is stated, what is the method of procedure? 9. When no two numbers are left, one on each side of the line, capable of being divided by any one figure, what is to be done? 10. How do you multiply a fraction by a whole number? 11. Why, in example 2, are 16 and the numerator of the fraction placed on the right of the line?

If 1 pound cost 1 dollar, 9 pounds would cost 1×9=9 dollars; but the cost of 1 pound is 1 dollar divided by 18: therefore, 9 dollars, the cost of 9 pounds, must be divided by 18. By the rule already given, the numerator of the fraction, with 9, its multiplier, is placed on the right of the line, and 18, the divisor, on the left. 9 and 2 are factors of 18 therefore, cross 9 and 18, and write 2, the remaining factor, in the place of 18. The answer, then, is, 1 divided by 2; or, 1. (See Art. 66.) On the principle above stated, A fraction may be multiplied into any factor in its denominator, by cancelling that factor.

6

4. If a pound of lead cost of a dollar, how much will 8 pounds cost?

16

If the cost of 1 pound be of a dollar, 8 pounds will cost 18=1=1 of a dollar. Making the horizontal line, which separates the numerator of the fraction from the denominator, perpendicular, it will be seen that the numerator occupies the place of dividends, (the right of the line,) and the denominator the place of divisors, (the left of the line,) thus:

2 1611

$

2 |1= Ans.

Mul

In the latter mode the question is resolved into this. tiply 1 by 8, and divide by 16; therefore, the numerator of the

fraction and 8, its multiplier, occupy the right of the line, and 16, the divisor, the left. It is to be remembered, that the numerator of a fraction, in all cases, is to be disposed of as a whole number, without regard to its denominator. On whichever side of the line the numerator falls, the denominator must be placed on the opposite side.

5. What will 8 bushels of apples cost, at a dollar per bushel ? Ans. $4. OBS.-This character ($) placed before any number, shows that it is

dollars.

6. If one man can plant of an acre in one day, how much could 12 men plant in the samé time ? Ans. 9 acres. 7. If 1 barrel of fish cost 61 dollars, what will 9 barrels cost? Ans. 561.

OBS.-Mixed numbers must be reduced to improper fractions. 8. If 1 chest of tea cost $254, what will 15 cost?

Ans. $3783

9. If a man can walk 29 miles in 1 day, how far could he walk in 30 days? Ans. 885 miles. 10. What will 600 pounds of cotton cost, if 1 pound cost 91 cents? Ans. $57. OBS.-Dividing any number of cents by 100, reduces them to dollars.

Division of Fractions by Whole Numbers.

Art. 68.-1. If a man receive of a dollar for two days' work, what does he receive per day?

We have seen, that a fraction is multiplied either by multiplying its numerator, or dividing its denominator; then, as Division is the reverse of Multiplication, the reverse of the rule for Multiplication will be the rule for Division. If I divide. the dollar into 4 parts, or quarters, and pay a man 1, or quarter, it is the same as though I should divide it into 8 parts, or half quarters, and pay him, or 2 half quarters Multiplying the denominator by 2, the numerator remaining the same, is dividing the unit into twice as many parts, and consequently the value of each part is diminished by one half. Dividing the numerator by 2, the denominator remaining the same, is taking half as many parts, while the value of each part is the same. Therefore, to divide a fraction by a whole number, we have this

RULE.

Divide the numerator of the fraction by the whole number, when it can be done without a remainder; otherwise, multiply the denominator.

2. If 8 pounds of lead cost of a dollar, what does it cost per pound? Operation. 2/1 8

16|1=Ans.

Were the cost of 8 pounds 1 dollar, the cost of 1 pound would be the quotient of 1 divided by 8. Regarding the numerator as expressing the cost of the lead, without reference to the denominator, we place it on the right, as a dividend, and 8, the number of pounds, on the left, as a divisor; but the cost of 8 pounds is 1 divided by 2; therefore, write 2 also on the left. As no

QUESTIONS.-12. How do you divide a fraction by a whole number? 13. Why, in example 2d, are 8 and 2, the denominator of a fraction, placed on the left? 14. Multi plying the denominator of a fraction, is the same as what?.