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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 treatise, when the so cess of solution is long and tedious. 2. Multiply 62123000 by 130000.

A. 8075990000000. 3. Multiply 35432000 by 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 1234560000.

A. 1524148134278400000

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 component parts of 36 ? Of 72? Of 100? Of 7 ? Of 15? Of 35? Or 132 Of 144 ?

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

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 7 barrels.

8 dollars a barrel, making 56

dollars. 56 dollars.

Q. How much more will 14

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

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.

More Exercises for the Slate. 2. Wtat will 36 hundred weight of sugar cost, at 29 dollars o nuadrid ? 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

ught 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 ma' y pounds can you buy for 45 cents? For 54 cents? For 1us cents ?

8. How much is one half (1) of 4? Of 8? Os 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 (3) of 6? Of 24? Of 30 ? Of 36?

11. How much is one half (3) of 8? One third (b) of 21? One fourth () of 16? One fifth (5) of 35? One sixth (+) of 24? One seventh (+) of 35? One eighth (b) of 569 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 info 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 re mainder be?

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

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

A. Multiplication.
Q. James, having 12

12 oranges oranges, was desirous of dividing them equally One to each makes 4 among his 4 little sisters, and, in order to do this, 1st time he had 8 left. he handed them at first one apiece; how many

One to each makes 4 had he left 1 Q. When he handed

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

Q. When he handed them one more apicce,

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

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

A. By Subtraction.

Aprece?

Q. How many times did James g eto each of his sisters an orange
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. I XVI. Q. What is Sport Division ? A. When the divisor is 12, or less. 1. How many oranges, at 3 cents apiece, may be bought for 657 cents ? OPERATION.

1 Q. How do you ob

tain the 2 (hundreds) in Dividend.

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 (hundreds).

Q. How do you get the 1 (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 unite 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 ? Divisor,

3

A. I would multiply 219 (the quo

tient) by 3 (the divisor), making 657 Dividend, 657 (the dividend)-- therefore right.

From the illustrations now given, we derive the following

RULE.
Q. At which hand of the dividend do you place the divisor ?
A. At the left.
Q. How inany figures do you lake first ?
A. Enough to contain the divisor once, or more
Q. What do you set down underneath?
A. The quotient.
Q. If there should be a remainder, how would you proceed ?

A. I join or carry it to the next figure of the dividend, as so many tens.

Q: For example, suppose 3 remain, and the next figure be 8, how would you say?

A. I would say, 3 (to carry) to 8, makes 38.

Q. How do you proceed if the divisor be not contained in the next figure of the dividend ?

A. Write a cipher in the quotient, and join this figure to the figure next to it, as so many tens.

PROOF. Q. Which terms do you multiply together to pirme the operation ? A. The divisor and quotient. Q. What is to be done with the remainder, if there be any? A. Add it to this product. Q. What must the amount be like! A. The dividend.

More Exercises for the Slate. 2. Rufus divided 42 oranges e prally between his two little brothers; how many had they apiece?' 21.

3. If 3 bushels of apples cost 36 ants, how much is that a bushel ? A. 120 cents.

4. How many months are there in 452 weks, there being 4 weeks in each month ? A. 113 months.

7. A man, having 416 dollars, laid them alı ou il cider, at 4 dollars a barrel; how much cider did he buy? A. 104 barrels.

6. A man bought 6 oxen for 318 dollars; how much did he pay a head? A. 53 dollars.

4. How much flour, at 7 dollars a barrel, can be bought for 1512 dollars ? A. 216 barrels.

8. At 8 cents apiece, how many oranges will 8896 cents buy. A. 1112 oranges.

9. At 10 dollars a barrel, how many barrels of flour may be bought for 1720 dollars: A. 172 barrels.

tä. 12 men by contract are to receive 1500 dollars for a job

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