How do you know? Ans. Because 8, the divisor, is one factor, and therefore 12 must be the other; for 12 X 8—96. 10. Divide 72 by 6. 72÷6-12 Ans. Into what factors is the dividend separated in this example? Ans. 6 and 12. Why not 8 and 9? They are also factors of 72.-—Ans. Because neither is like the divisor.

Let the teacher propose similar inquiries in regard to the following exercises:

11. Divide 84 by 12. 12. Divide 108 by 9. 13. Divide 121 by 11. 14. Divide 132 by 12.

Let the student read the following forms of implying division, and write others similar:

OBS. 3.-For the reading of the following forms, see Art. 48.

16÷8 2

Art. 43.-Illustration of general principles.-What effect upon the quotient has multiplying the dividend? What effect upon the quotient has multi16×2÷8=4 plying the divisor ?

What would be the effect of multiplying 16÷8×2=1 both divisor and dividend? Illustrate.

What effect upon the quotient has dividing

16÷2÷8=1 the dividend?

What effect upon the quotient has dividing

16÷8÷2=4 the divisor?

What effect upon the quotient has dividing both the dividend and divisor? Illustrate. Art. 44. From the foregoing illustrations, the following principles are manifest. The larger the dividend, with a given divisor, the larger the quotient; and the less the dividend, with a given divisor, the less the quotient. Therefore, To multiply the dividend is the same as to multiply the quotient, and to divide the dividend is the same as to divide the quotient. To divide the divisor is the same as to multiply the dividend, and to multiply t› divisor is the same as to divide the dividend.

MULTIPLICATION AND DIVISION, BY CANCELLING. Art. 47.--The operation of questions, involving Multiplication and Division, may be greatly abridged by the following

RULE.

I. Draw a perpendicular line, and place dividends and numbers to be multiplied for dividends, on the right, and divisors on the left hand.

OBS. 1.-The perpendicular line is the same as the curve line in Division, separating divisors from dividends.

II. If there be two equal numbers on each side of the line, cross them out, and omit them in the operation.

Thus Multiply 8 by 9, and divide by 8.

Operation. As 8 is found on both sides of the line, cross 18 them both, and 9, remaining on the right, is the $19 Ans. answer.

The principle upon which this RULE proceeds is that of cancelling, or rejecting equal factors from dividends and divisors. Thus, taking the above example, 8 and 9 are the factors of 72: 8×9=72. The quotient of 72 divided by 8, is 9, one of its factors; the other factor, 8, equal to the divisor, is rejected.

III. If a number on one side of the line will divide a number on the other side, without a remainder, erase both numbers, and substitute for the larger the number of times it contains the smaller. Multiply the remainders together, on the right, for a dividend, and the remainders on the left, for a divisor.

Thus: Multiply 6 by 3, and divide by 2.

16 3

23

In this example the divisor, 2, is not the same as either figure of the ridend, but it is a factor 9 Ans. of one of them, 2×3=6. We may, therefore, cross 2 and 6, since the divisor, 2, cancels one of the factors of 6, the dividend, and write 3, the other factor, against 6 as the quotient. The remainders on the right multiply together, 3×3=9, and 18÷29, the answer, as before.

When there is no remainder on either side of the line, and the numbers are all cancelled, the answer is 1: that is, the right-hand side contains the left-hand, once.

OBS. 2.-A stroke drawn through any number denotes its being cancelled; and any number which takes its place may be set alongside of it.

3. Multiply 8 by 5, and divide the product by 3; multiply the quotient by 18, and divide the product by 9; multiply again by 9, and divide the product by 6; multiply the quo

QUESTIONS.-1. What is the rule for Multiplication and Division by cancelling? 2. First, second, and third steps? 3. Is the answer affected by striking out equals on each side of the line? 4. Why not? 5. What is done with remainders? 6 When there is no number left on either side of the line, what is the answer?

tient by 24, and divide the product by 12; multiply the quotient by 2, and divide the product by 4.

Having stated the question, according to the foregoing RULE, we proceed to cancel, or cross equals on each side of the perpendicular line. In the first place, 9 is found on each side of the line. We therefore cross them both; for 9 is contained in 9 once, and multiplying any number by 1 does not alter its value. Secondly: 3 and 6, on the left hand of the line, multiplied together make 18-equal to 18, on the right hand of the line, which may be crossed out. Again: 4 and 12, on the left, multiplied together, are 48, equal to the numbers 2 and 24 on the right multiplied together, and may be crossed out. The numbers now are all cancelled, except the 5 and 8, on the right, which, multiplied together, give 40, the answer.

4. A boy gathered 16 nuts under each of 4 trees, and divided them equally between himself and 7 schoolmates. How many did each receive?

Operation.
|16 2
84×2=8

In this example, it is evident, that had the boy gathered but 16 nuts, there would have been but 2 apiece; but as he gathered the same number under each tree, the 16 must be multiplied by 4; and as there were 8 to share them, the product of 16 multiplied by 4 must be divided by 8.

8 Ans.

5. Multiply 20 by 5, and divide by 6; multiply by 7 and divide by 14; multiply this again by 6, and divide by 10, and multiply by 12. Ans. 60.

6. Multiply 120 by 40, divide by 400, multiply by 20, divide by 30, multiply by 250, divide by 50, multiply by 300, divide by 500, and give the answer. Ans. 24.

QUESTION.—7. Give the reason for placing 16 and 4, in Example 4, on the right of the line, and 8 on the left.

SUPPLEMENT

TO THE FOUR FUNDAMENTAL RULES OF ARITHMETIC, VIZ:

ADDITION, SUBTRACTION, MULTIPLICATION, AND

DIVISION.

EXERCISES.

1. A man purchased a farm for 6720 dollars; sold it for 199 dollars more than he gave. For how much did he sell it? Ans. 6919 dollars.

2. Suppose a tree broken by the wind 39 feet from the ground, and the part broken off to be 56 feet in length. How high was the tree ? Ans. 95 feet.

3. A merchant having 784 bushels of salt, sold 99 bushels. How many had he left? Ans. 685 bushels. 4. A man left his estate, valued at 8956 dollars, to his wife and daughters, giving his wife 4688 dollars. How much did the daughters receive? Ans. 4268 dollars.

5. Sir Isaac Newton was born in the year 1642, and died in the year 1727. What was his age? Ans. 85 years.

6. The greater of two numbers is 624; their difference is 89. What is the less number? Ans. 535.

7. What will 58 yards of broadcloth cost, at 4 dollars per yard? Ans. 232 dollars. 8. Bought 122 bushels of wheat, at 2 dollars a bushel; 8 oxen for 27 dollars each; 4 cows, 16 dollars each, and a wagon for 60 dollars. How much was paid for the whole, and how much more for the wheat and oxen than for the cows and wagon? 584.

{

Ans.336.

9. The factors of a certain number are the difference between 1632 and 1700, and between 94 and 5 dozen. What is that number? Ans. 2312.

10. How many barrels of flour may be bought for 6721 dollars, at 13 dollars per barrel ? Ans. 517 barrels. 11. Paid 57600 cents for eggs, paying at the rate of 12 cents a dozen. How many dozen did I buy?

Ans. 4800 dozen.

12. What will 168 firkins of butter cost, at 29 dollars a firkin? Ans. 4872 dollars.