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When a () appears in the multiplier, we need not multiply by it, since each of the products is 0.; but when we multiply by the next figure to the left, we must observe to set the first figure of the product directly under it's multiplier :

CASE I.

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25. When the multiplier does not exceed 12.

RULE I. Set down the multiplicand and under it set-the multiplier, so that units of the same order shall fall under éach other, and draw a line beneath.

II. Multiply every figure of the multiplicand by the multiplier, setting down and carrying as in addition:

EXAMPLES.
(1.)
(2.)
(3.)

(4.)
867901 278904 678741 3021945
1
2
3

4 867901

12087780

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13. A merchant sold 104 yards of cotton sheeting at 9 cents a yard: what did he receive for it?

14. A farmer sold 309 sheep at four dollars apiece. how much did he receive ?

15. Mrs. Simpkins purchased 149 yards of table linen at two dollars a yard : how much did she pay for it?

24. When a 0 is found in the multiplier need you multiply by it? When you multiply by the next figure to the left, where do you place the first figure of the product?

25. When the multiplier does not exceed 12, how do you set it down? How do you multiply by it?

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CASE II.

26. When the multiplier exceeds 12.

RULE. I. Set down the multiplier under the multiplicand, so that units of the same order shall fall under each other, and draw

ne beneath.

II. Begin with the right hand figure, and multiply all the figures of the multiplicand by each figure of the multiplier, setting down and carrying to the next product as in addition: obseroing also to write the first figure of each product di. rectly under its multiplier.

III. Add up the several products and their sum will be the product sought.

Note 1. There are three numbers in every multiplica. tion. First, the multiplicand : second, the multiplier : and third, the product.

Note 2. In multiplication either of the factors may be used as the multiplier without altering the product.

Let it be required to multiply 5 by 3.
Place as many 'ones - in a

OPERATION. horizontal row as there are

5 units in the multiplicand, and make as many rows as there

1 1 1 1 are units in the multiplier: the 31 1 1 1 1 product will then be equal to

1 1 1.1 1 one row taken as many times as there are rows; that is, to the whole number of ones : viz., 15. But if we consider the number of ones in a vertical row to be the multiplicand, and the number of vertical rows the multiplier, the product will then be equal to a vertical row taken as many times as there are vertical rows; that is, it will be equal to the whole number of ones : viz., 15. Hence,

26. When the multiplier exceeds 12, how do you set it down? How do you multiply by it? How do you add up? How many numbers are there in every multiplication ? Name them? Is the product of two numbers altered by changing the multiplicand into the multiplier ? Is 7 multiplied by 8 the same as 8 multiplied by 7? If we write 6 horizontal lines with 8 units in each, it is evident that the product of 8x6=48, is the number of units in all the lines.

But let us first connect the lines in sets of two each, as on the right; the number in each set will then be ex. pressed by 8 X 2=16. But there are 3 sets ; hence, the number of units in all the sets is 16 X3=48.

If we divide all the lines into sets of 3 each, as on the left, the number of units in each set will be equal to 8X3=24, and there being two sets, the whole number of units will be expressed by 24 X2=48.

Hence, when the multiplier is a composite number, we have the following

RULE.

Multiply by each of the factors in succession, and the last product will be the entire product sought.

EXAMPLES.

1. Multiply 327 by 12. The factors of 12, are 2 and 6, or they are 3 and 4, or they are 3, 2 and 2: for, 2x6=12, 3X4=12, and 3 X2 X2=12. 327 327

327
6
3

3
1962
981

981
2
4

2 Product, 3924

3924

1962

2 Product, 3924

2. Multiply 5709 by 48; the factors being 8 and 6, or 16 and 3.

Ans. 3. Multiply 342516 by 56.

Ans. 19180896. 4. Multiply 209402 by 72.

Ans. 5. Multiply 937387 by 54.

Ans. 50618898. 6. Multiply 91738 by 81.

Ans. 7. Multiply 3842 by 144.

Ans. 553248.

18. Multiply 6795634 by 918546.

Ans. 6242102428164. 19. Multiply 86972 by 1208.

Ans. 20. Multiply 1055054 by 570. Ans. 601380780. 21. Multiply 538362 by 9258. Ans. 4984155396. 22. Multiply 50406 by 8050. Ans. 405768300. 23. Multiply 523972 by 15276,

Ans. 24. Multiply 760184 by- 16150. Ans. 12276971600. 25. Multiply 1055070 by 31456. Ans. 33188281920. 26. Multiply 91874163 by 27498765.

Ans. 2526426017908695.

CASE III.

28. A composite number is one that may be produced by the multiplication of two or more numbers, which numbers are called the components or factors. Thus, 2x3=6, Here 6 is the composite number,

. and 2 and 3 are the factors or components.

The number 16=8X2: here 16 is a composite number, and 8 and 2 are the factors; and since 4X4=16, we may also regard 4 and 4 as factors or components of 16. Sixteen has three factors : for 2x2x4=16. It also has four : for 2x2x2x2 316.

EXAMPLES ILLUSTRATING PRINCIPLES. Let it be required to multiply 8 by the composite number 6, in which the factors are 2 and 3.

8

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28. What is a composite number? Is 6 a composite number? What are its components or factors? What are the factors of the composite number 16 ? What are the factors of the composite number 12? How do you multiply when the multiplier is a composito number?

If we write 6 horizontal lines with 8 units in each, it is evident that the product of 8x6=48, is the number of units in all the lines.

But let us first connect the lines in sets of two each, as on the right; the number in each set will then be ex. pressed by 8x2=16. But there are 3 sets; hence, the number of units in all the sets is 16 X3=48.

If we divide all the lines into sets of 3 each, as on the left, the number of units in each set will be equal to 8x3=24, and there being two sets, the whole number of units will be expressed by, 24 X 2=48.

Hence, when the multiplier is a composite number, we have the following

RULE.

Multiply by each of the factors in succession, and the last product will be the entire product sought.

EXAMPLES.

1. Multiply 327 by 12.

The factors of 12, are 2 and 6, or they are 3 and 4, or they are 3, 2 and 2: for, 2x6=12, 3x4=12, and 3X2 X2=12. 327 327

327
6
3

3
1962
981

981
2
4

2 Product, 3924

3924

1962

2 Product, 3924

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2. Multiply 5709 by 48; the factors being 8 and 6, or 16 and 3.

Ans. 3. Multiply 342516 by 56.

Ans. 19180896. 4. Multiply 209402 by 72.

Ans. 5. Multiply 937387 by 54.

Ans. 50618898. 6. Multiply 91738 by 81.

Ans. 7. Multiply 3842 by 144.

Ans. 553248.

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