Analysis. At 100 dollars apiece, it is evident that the 30 pianos would cost 100 X 303000 dollars; and at 500 dollars apiece they would cost 5 times as much as at 100 dollars each, or 5 times 3000 dollars 15000 dollars.

Hence, we use the following

RULE.

OPERATION.

500 Multiplicand 30 Multiplier.

15000 Product.

Multiply the SIGNIFICANT figures together, and to the produc annex as many ciphers as there are at the right hand of both the factors.

EXAMPLES FOR PRACTICE.

2. What would be the weight of 300 bales of cotton averaging 550 pounds per bale?

OPERATION.

550

300

Ans. 165000 pounds.

Instead of writing units under units, tens under tens, etc., It is the usual practice, when ciphers are at the right of the multiplier, to place the multiplier under the multiplicand, as indicated in the foregoing "operation."

DIVISION.

ARTICLE 57. Ex. 1. James has 12 oranges, which he wishes to divide between his 3 brothers; how many oranges can he give to each?

OPERATION.

12 oranges.

3

9 1st remainder.

3

6 2d remainder.

3

Analysis. Here we are required to divide 12 oranges into 3 equal parts. If James gives each brother one orange, it will require 3 oranges, and 9 oranges would be left. If now he gives each of them another orange, it will require 3 oranges, and 6 oranges would be left. If he gives them one apiece a third time, it is evident there would be only 3 oranges left. If he gives them one apiece a fourth time, there would be none left. Hence, it is plain that he can give each of his brothers 4 oranges. Ans. 4 oranges.

3 3d remainder.

3

0 4th remainder.

In this example we see that 12 contains 3 four times, for if we subtract 3 from 12 four times, nothing is left. A number, therefore, may be divided into equal parts by subtraction.

We might have found the answer in a shorter way by saying that each brother would have as many oranges as 3 is contained times in 12; and 3 is contained in 12 four times. How do we know this? Because we know that 3 times 4 are 12. Ilence, to find how many times one number is contained in a second, we must merely find what number multiplied by the first will produce the second.

2. 5 is contained in 20 how many times? Ans. 4 times How do we know this? Because 4 times 5 are 20.

3. 3 is contained in 18 how many times? Ans. 6 times Why? Because 6 times 3 are 18.

Analysis. At 100 dollars apiece, it is evident that the 30 pianos would cost 100 X 30 =3000 dollars; and at 500 dollars apiece they would cost 5 times as much as at 100 dollars each, or 5 times 3000 dollars 15000 dollars.

Hence, we use the following

RULE.

OPERATION.

500 Multiplicand 30 Multiplier.

15000 Product.

Multiply the SIGNIFICANT figures together, and to the produc annex as many ciphers as there are at the right hand of both the factors.

EXAMPLES FOR PRACTICE.

2. What would be the weight of 300 bales of cotton averaging 550 pounds per bale?

OPERATION.

550

300

Ans. 165000 pounds.

Instead of writing units under units, tens under tens, etc., it is the usual practice, when ciphers are at the right of the multiplier, to place the multiplier under the multiplicand, as indicated in the foregoing "operation."

3 Multiply 570 by 80.

4 Multiply 75600 by 320. 5 Multiply 4030 by 5600.

Ans. 45600.

Ans. 24192000.

Ans. 22568000.

DIVISION.

ARTICLE 57. Ex. 1. James has 12 oranges, which he wishes to divide between his 3 brothers; how many oranges can he

give to each ?

12

3

OPERATION.

oranges.

9 1st remainder.

3

6 2d remainder.

3

Analysis. Here we are required to divide 12 oranges into 3 equal parts. If James gives each brother one orange, it will require 3 oranges, and 9 oranges would be left. If now he gives each of them another orange, it will require 3 oranges, and 6 oranges would be left. If he gives them one apiece a third time, it is evident there would be only 3 oranges left. If he gives them one apiece a fourth time, there would be none left. Hence, it is plain that he can brothers 4 oranges.

3 3d remainder.

3

0 4th remainder.

give each of his Ans. 4 oranges.

In this example we see that 12 contains 3 four times, for if we subtract 3 from 12 four times, nothing is left. A number, therefore, may be divided into equal parts by subtraction.

We might have found the answer in a shorter way by saying that each brother would have as many oranges as 3 is contained times in 12; and 3 is contained in 12 four times. How do we know this? Because we know that 3 times 4 are 12. Ilence, to find how many times one number is contained in a second, we must merely find what number multiplied by the first will produce the second.

2. 5 is contained in 20 how many times? Ans. 4 times How do we know this? Because 4 times 5 are 20.

3. 3 is contained in 18 how many times? Ans. 6 times Why? Because 6 times 3 are 18.

4. How many times is 7 contained in 21? Ans.

Why?

5. How often does 4 go into 24? Ans.

Why?

6. If 3 lemons cost 15 cents, what is the cost of 1 lemon? 7. If 5 boys eat 30 peaches, how many will that be for each one?

8. If 5 marbles be put in a ring, how many rings can you make out of 40 marbles?

The foregoing examples are all solved by Division. Hence, ART. 58. DIVISION is the process of finding how many times one number is contained in another.

ART. 59. The DIVIDEND is the number to be divided.

The DIVISOR is the number by which we divide.

The QUOTIENT is the number which shows how many times the divisor is contained in the dividend.

ART. 60. The REMAINDER is that which is sometimes left after dividing. The remainder being a part of the dividend, is always of the same kind or denomination as the dividend. When there is no remainder, the division is said to be exact.

ART. 61. The SIGN OF DIVISION is a short horizontal line with a dot above and a dot below,; it signifies that the number before it is to be divided by the number after it; thus 124 denotes that 12 is to be divided by 4.

Division is frequently indicated by a line, with the dividend above it and the divisor below it; thus, denotes that 15 is to be divided by 5. In practice, Division is usually indicated by placing the divisor on the left of the dividend, with a curved line between: thus, 6)24 indicates that 24 is to be divided by 6.

ART. 62. Division is the opposite of Multiplication. In Multiplication we usually have two factors to find their product. In Division we have the product and one of the factors to find the other factor.