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15. Multiply 32100421 by 65; by 85.

16. Multiply 3211001421 by 27; by 33.

A 4815063150

A. 1926725260,

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.

¶ 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 !

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

RULL. 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. 8600 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? 4. 966570 cents.

3. Multiply 8978 by 10; by 100; by 1000%; by 10000; by 100000; by 1000000. A. 9075545580.

XM. When there are Ciphers at the Right HAND OF EITHER OR BOTH The FactorS.

Rut.z. Q. From the illustrations given, ¶ Xlk, how does it appeas that we can multiply?

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

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The Key contains the entire work of the six following examples; and similar aid is afforded the teacher in other parts of this treatiss, when the preBess of solution is long and tedious.

2. Multiply 62123000 by 130000.
3. Multiply 35432000 by 256000.
4. Multiply 6789354270000 by 685300.

A. 8075990000000.

A. 9070592000000.

A. 4652744481231000000.

5. Multiply 78954398765 by 7235000.

6. Multiply 123456789 by 123450000.

A. 571235075064775000.

7. Multiply 1234567890 by 1234560000.

A. 15240740602050000.

A. 1524143134278403000.

¶XIV. When the Multiplier is a Composite NUMBER.

A

Q. How many are 5 times 8? 7 times 9? 12 times 121

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 alled?

A. The Component Parts.

Q. What are the component parts of 361 Of 721 Of 100! "OF #71 Of 151 Of 351 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 four cost, at 8 dollars a barrel ?

OPERATION.

8 dollars.

7 barrels.

56 dollars.

2 times 7 are 14.

112 dollars, Ans.

Q. What does multiplying 8 dollars by 7 barrels give? A. The price of 7 barrels at 8 dollars a barrel, making 56 dollars.

Q. flow much more will 14 barrels cost than 7 barrels?

A. 2 times as much as 7, that is, 2 times 56, making 112 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.

More Exercises for the Slate.

2. What will 36 hundred weight of sugar cost, at 29 dolla

hundred? A. 1044 dollars.

3. Multiply 3065428 by 35. 4. Multiply 4078945 by 96. Multiply 18934 by 108. 6. Multiply 45678 by 144.

A. 107289980.

A. 391575720.

A. 2044872.

A. 6577632.

SIMPLE DIVISION.

1 XV. 1. If you divide 12 apples equally between twe boys, how many will each have? How many times 2 in 13 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? 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, 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 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 10 cents?

8. How much is one half (1) of 4" Of 8? Of 16? Of 20? Of 24? Of 20? Of 100? Of 200?

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

10. How much is one third (1) of 6? Of 24? Of 30 Of 36?

11. How much is one half (1) ɔf 8? One third (1) of 24.7 One fourth (†) of 16? One fifth (†) of 35? One sixth (4) One seventh (4) of 35? One eighth (4) of 562 Que ninth () of 108? One twelfth (T) of 144 ?

of 24?

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

Q. What is this method of finding how many times one number Contained in another, or of dividing a number into equal parts, called 7 A. Division.

Q. What is this method of finding how many times one number contained in another of only one name, or denomination, called 1 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 éperation 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 407 and why

Q. From this example, what does Division appear to be the oppas site of?

A. Multiplication.

James, having 12 Stanges, was desirous of dividing them equally among his little sisters, and, in order to do this, he handed them at first one apiece; how many had he left f

Q. When he handed Chem another apiece, how many had he left ?

When he handed

them one more apiece,

How many had he left?

One to each makes

1st time he had

12 oranges.

4

8 left.

One to each makes

4

4 left.

One to each makes

4

U left.

2d time he had

3d time he had

Q. From these illustrations; how does it appear that a number marg

e divided into equal parts?

A. By Subtraction

E Q. How many times did James give to each of his sisters an orange spiece 1

Q. How many times did you subtract ?

A. Three times.

How many times 4 in 12 ?

By this we see that the quotient represents the number of sub tractions: now, if the quotient were 4000, how many times would it be secessary 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.

XVI. Q. What is SHORT DIVI9108! 4. When the divisor is 12, or less.

1. How many oranges, at 3 cents apiece, may be bought for €57 cents?

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Q. How do you ob tain the 2 (hundreds) in the quotient?

4.I begin on the left of the dividend, and say, 3, the divisor, is contained in 6 (hup

dreds) 2 (hundreds) times, that is, 200 times, writing the (hundreds) down under the 6 (hundreds).

Q. How do you get the I (ten)?

A. 3 in 5 (tens) I 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 aking 27.

Q. How do you proceed to get the 9, then?

A. 3 in 27, 9 times.

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Q. How many times 6 in 30, and why?

Q. How, then, would you proceed to prove the foregoing example?

A. I would multiply 219 (the quo tient) by 3 (the divisor), inaking 657 (the dividead)-therefore right

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