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15. Multiply 32100421 by 65; by 85. A. 4815063150. 16. Multiply 32100421 by 27; by 33. A. 1926025260

17. Write down one thousand, multiply it by 25, add five thousand to the product, subtract twenty-nine thousand nine hundred and ninety-nine from the amount, and see if the remainder be 1. 1 XII. WHEN THE MULTIPLIER IS 10, 100, 1000, &c.

Q. How many are 10 times 5? Now, if we annex a cipher to the 5, thus, 50, it will produce the same effect: why is this?

A. Because, by annexing a cipher to 5, the 5 is removed to the tens' place; hence the value is increased 10 times.

Q. What effect would two ciphers have, or three ciphers, &c.?

1. Two ciphers would remove any figure two places towards the left, and of course increase it 100 times, and so on for 1000, &c.

RULE, Q. What, then, appears to be the rule ?

A. Annex to the multiplicand all the ciphers in the multiplier.

Exercises for the Slate. 1. What will 36 bushels of rye cost, at 100 cents a bushel ? A. 3600 cents.

2. What will 100 bushels of salt cost, at 87 cents a bushel ? What will 1000 bushels ? What will 10000 bushels ? What will 10 bushels ? A. 966570 cents.

3. Multiply 8978 by 10; by 100; by 1000; by 10000; by 100000; by 1000000. A. 9975545580. XIII. WHEN THERE ARE CIPHERS AT THE RIGHT

HAND OF EITHER OR BOTH THE FACTORS. RULE. Q. From the illustrations given, 1 XII., how does it appear that we can multiply?

A. Multiply without the ciphers first, and annex them to the product.

Exercises for the Slate. (1.)

Q. How do you get the 3 Multiplicand, 63500

ciphers in the product? 50 A. There are 2 at the right of

the multiplicand, and 1 at the 3175000 | right of the multiplier, making 3.

* The Key contains the entire work of the six following exansples; and similar aid is afforded the teacher in other parts of this treatise, when the pro cess of solution is long and tedious. 2. Multiply 02123000 by 130000.

A. 8075990000000. 3. Multiply 35432000 vy 256000.

A. 9070592000000. 4. Multiply 6789354270000 by 685300.

A. 4652744481231000000 5. Multiply 78954398765 by 7235000.

A. 571235075064775000, 6. Multiply 123456789 by 123450000.

A. 15240740602050000. 7. Multiply 1234567890 by 1234500000.

1. 1524148134278400000.

1 XIV. WHEN THE MULTIPLIER IS A COMPOSITE

NUMBER.
Q. How many are 5 times 8? 7 times 9 ? 12 times 12 ?
Q. What are these products, 40, 63, 144, called ?
A. Composite Numbers.

Q. What are the multiplying numbers, 5 and 8,7 and 9, 12 and 12, called ?

A. The Component Parts.

Q. What are the coinponent parts nf 36 ? Of 72 ? Of 100? or 27 ? Or 15 ? Of 35 ? Of 132 ? Of 144 ?

Q. What, then, is the product of any two numbers called!
A. A Composite Number.

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1. What will 14 barrels of flour cost, at 8 dollars a barrel ?

Q. What does multiplying
OPERATION

8 dollars by 7 barrels give ? 8 dollars.

A. The price of 7 barrels at

8 dollars a barrel, making 56 ho barrels.

dollars.

Q. How much more will 14 56 dollars.

barrels cost than 7 barrels ?
2 times 7 are 14. A. 2 times as much as 7,

9

that is, 2 times 56, making 112 112 dollars, Ans.

dollars.
RULE. Q. How, then, would you begin to multiply?
A. By one of the component parts first.
Q. What would you multiply this product by?
A. By the other component part.

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of:

More Exercises for the State. 2. What will 36 hundred weight of sugar cost, at 29 dollars a hundred ? A. 1044 dollars. 3. Multiply 3065428 by 35.

A. 107289980. 4. Multiply 4078945 by 96.

A. 391578720. 5. Multiply 18934 by 108.

A. 2044872. 6. Multiply 45678 by 144,

A. 6577632.

SIMPLE DIVISION.

1 XV. 1. If you divide 12 apples equally between two boys, how many will each have ? How many times 2 in 12, then? Why? A. Because 2 times 6 are 12?

2. How many oranges, at 8 cents apiece, can you buy for 48 cents ? For 96 cents? How many times 8 in 48? 8 in 96 ? Why?

3. A man bought 8 lemons for 80 cents; how much did he "give apiece? How many times 8 in 80? Why, or proof?

4. How many gallons of brandy, at 3 dollars a gallon, can be bought for 36 dollars ? For 60 dollars ? For 90 dollars ? For 300 dollars? Why?

5. Four boys found a bag containing 48 silver dollars ; how many will they have apiece, if it be divided equally?

6. When oranges are 2 cents apiece, how many will 8 cents buy? Will 16 cents buy? Will 32 cents ? Will 36 cents ? Will 48 cents? Will 100 cents ?

7. If you pay 9 cents for one pound of sugar, how many pounds can you buy for 45 cents? For 54 cents? For 108 cents ?

8. How much is one half (!) of 4? Of 8? Of 16 ? Of 20? Of 24? Of 30? Of 100? Of 200 ?

9. Harry had 16 apples, and gave one half (1) of them to Thomas; how many did Thomas receive ?

10. How much is one third (?) of 6? Of 24? Of 30 ? Of 36?

11. How much is one half (1) of 8? One third () of 24 ? One fourth (1) of 16? One fifth (3) of 35? One sixth (+) of 24? One seventh (7) of 35? One eighth (f) of 56 ? One ninth ($) of 108? One twelfth (12) of 144 ?

12. How many times 4 in 40? 3 in 60? 5 in 100 ? 6 in 1200 ? 8 in 480 ?

Q. What is this method of finding how many times one number is contained in another, or of dividing a number into equal parts, called ?

A. Division.

Q. What is this method of finding how many times one number is contained in another of only one name, or denomination, called ?

A. Simple Division.
Q. What is the number given to divide by called ?
A. The Divisor.
Q. What is the number to be divided called ?
A. The Dividend.

Q. What is the number of times that the divisor is contained in the dividend called ?

A. The Quotient.

Q. What is that which is sometimes left after dividing, or after the operation is performed, called?

A. The Remainder, which must always be less than the Divisor.

Q. Of what name, or denomination, is the remainder ?
A. The same as the Dividend.

Q. If your dividend, for instance, be ounces, what will your remainder be?

A. Ounces.
Q. How many times 4 in 40 ? and why?

2. From this example, what does Division appear to be the opposite of?

A. Multiplication.
Q. James, having 12

12 oranges.
oranges, was desirous of
dividing them equally

One to each makes 4 anong his 4 little sisters, and, in order to do this, 1st time he had 8 left. he handed them at first

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One to each makes one apiece; how many

4 had he left? Q. When he handed

2d time he had 4 left. them another apiece, how One to each makes 4 many had he left?

Q. When he handed thein one more apiece,

3d time he had O left. how many had he left ?

Q. From these illustrations, how does it appear that a number may be divided into equal parts ?

A. By Subtraction.

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Q. How many times did James give to each of his sisters an orange apiece ?

Q. How many times did you subtract !
A. Three times.
Q. How many times 4 in 12 ?

Q. By this we see that the quotient represents the number of subtractions: now, if the quotient were 4000, how many times would it be necessary to take the divisor from the dividend to perform Division by Subtraction?

A. 4000 times.
Q. What, then, is Division a quick way of performing ?
A. Many subtractions.

SHORT DIVISION. 1 XVI. Q. What is Short Division ? A. When the divisor is 12, or less. 1. How many oranges, at 3 cents apiece, may be bought for 657 cents ? OPERATION.

Q. How do you obDividend,

tain the 2 (hundreds) in

the quotient? Divisor, 3 ) 657 cents.

A. I begin on the left

of the dividend, and Quotient, 219 oranges, Ans. say, 3, the divisor, is

contained in 6 (hundreds) 2 (hundreds) times, that is, 200 times, writing the 2 (hundreds) down under the 6 (hunareds),

Q. How do you get the 1 (ten)?
A. 3 in 5 (tens) 1 time, and 2 (tens) left.
Q. What do you do with the 2 which is left?

A. I join, or carry it as 2 tens, that is, 20 units, to the 7 units, making 27.

Q. How do you proceed to get the 9, then?
A. 3 in 27, 9 times.

Q. How many times 6 in 30, and
PROOF.

why? Quotient, 219

Q. How, then, would you proceed

to prove the foregoing example ? 3

A. I would multiply 219 (the qum Dividend,

tient) by 3 (the divisor), making 657 657

(the dividend)—therefore right,

Divisor,

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