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OPERATION.

34. Multiply forty-seven thousand and thirteen by eighty thousand eight hundred and seven. Ans. 3798979491. Art. 41.

A COMPOSITE number is one produced by multiplying together two or more numbers greater than unity or one; thus, 12 is a composite number, since it is the product of 3 x 4; and also 24 is a composite number, since it is the product of 2 X 3 X 4.

A FACTOR of any number is a name given to one of two or more numbers, which, being multiplied together, produce that number ; thus, 3 and 4 are factors of 12, since 3 x 4 = 12.

Art. 42. To multiply by a composite number.

Ex. 1. A merchant bought fifteen pieces of broadcloth, at 96 dollars per piece; how much did he pay for the whole ?

Ans. 1440 dollars. The factors of 15 are 3 and

5. Now, if we multiply the 96 dolls., price of 1 piece.

price of one piece by the factor 3

3, we get the cost of 3 pieces ; 2 8 8 dolls., price of 3 pieces.

and then, by multiplying the 5

cost of 3 pieces by the factor

5, it is evident we obtain the 1440 dolls., price of 15 pieces.

cost of 15, the number of pieces

bought, since 15 is equal to 5 times 3. Hence we adopt the following

RULE. Multiply the multiplicand by one of the factors of the multiplier, and the product thus obtained by another, and so on until each of the factors has been used as a multiplier. The last product will be the answer.

NOTE. — The product of any number of factors is the same in whatever order they are multiplied. Thus, 3 x 4 = 12, and 4 X 3 = 12.

EXAMPLES FOR PRACTICE. 2. Multiply 30613 by 25 = 5 x 5.

Ans, 765325. 3. Multiply 1469 by 84 = 7 X 12.

Ans. 123396. 4. Multiply 7546 by 81, using its factors. Ans. 611226. 5. Multiply 7901 by 125, using its factors. Ans. 987625.

6. In 1 mile there are 63360 inches; how many inches in 45 miles ?

Ans. 2851200 inches. 7. If in one year there are 8766 hours, how many hours in 72

Ans. 631152 hours.

years ?

QUESTIONS. — Art. 41. What is a composite number? What is a factor of any number?

Art. 42. What are the factors of 15? How do we multiply by a composite number ? Repeat the rule. In what order must the factori of a composite number be multiplied ?

OPERATION.

8. If sound moves 1142 feet in a second, how far will it move in one minute ?

Ans. 68520 feet. ART. 43. When the multiplier is 1 with one or more ciphers annexed to it, as 10, 100, &c.

Ex. 1. In 1 day there are 24 hours ; how many hours in 10 days? In 100 days?

Answers. 240, 2400.

The removal of a figure one Multiplicand 24 24

place to the left increases its Multiplier

10 100 value ten times. (Art. 7.) If, Product 240 2400

then, we annex a cipher to the

right of 24, the multiplicand, Or thus, 240, 2400.

each figure is removed one place to the left, and its value is increased ten times, or multiplied by 10. If two ciphers are annexed, each figure is removed two places to the left, and its value is increased 100 times, or multiplied by 100 ; every additional cipher increasing the value ten times. Hence the following

RULE. · Annex to the multiplicand as many ciphers as has the multiplier. The number thus formed will be the product required.

EXAMPLES FOR PRACTICE. 2. Multiply 2356 by 10.

Ans. 23560. 3. Multiply 5873 by 100.

Ans. 587300. 4. Multiply 7964 by 1000.

Ans. 7964000. 5. Multiply 98725 by 100000. Ans. 9872500000.

ART. 44. When there are ciphers on the right hand of the multiplier or multiplicand, or both.

Ex. 1. What will 600 acres of land cost at 20 dollars per acre ?

Ans 12,000 dollars.

The multiplicand may be resolved into Multiplicand 600

the factors 6 and 100, and the multiplier Multiplier 20 into the factors 2 and 10. Now, it is evi

dent (Art. 42), if these several factors be Product 12000 multiplied together, they will produce the same product as the original factors 600 and 20. Thus 6 X 2=12, and 12 X 100= 1200, and 1200 X 10= 12000, the same result as in the operation. Hence the following

OPERATION.

QUESTIONS. Art. 43. What is the effect of removing a figure one place to the left ? What is the effect of annexing a cipher to any figure or number? Two ciphers ? &c. What is the rule when the multiplier is 1 with ciphers annexed ? — Art 44. How do you arrange the figures for multiplication, when there are ciphers on the right hand of either the multiplier or multiplicand, or both ? Why does multiplying the significant figures and annexing the ciphers produce the true product ?

RULE. Write the significant figures of the multiplier under those of the multiplicand, and multiply them together. To their product annex as many ciphers as there are on the right of both multiplicand and multiplier.

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4. Multiply 8010700 by 9000909.

Ans. 72103581726300. 5. Multiply 700110000 by 700110000

Ans. 490154012100000000. 6. Multiply 4070607 by 7007000.

Ans. 28522743249000. 7. Multiply 4110000 by 1017010.

Ans. 4179911100000. 8. Multiply twenty-nine millions two thousand nine hundred ind nine by four hundred and four thousand.

Ans. 11717175236000. 9. Multiply eighty-seven millions by eight hundred thousand seven hundred.

Ans. 69660900000000. 10. Multiply one million one thousand one hundred by nine hundred nine thousand and ninety. Ans. 910089999000.

11. Multiply forty-nine millions and forty-nine by four hundred and ninety thousand.

Ans 24010024010000. 12. Multiply two hundred millions two hundred by two millions two thousand and two. Ans. 400400800400400.

13. Multiply four millions forty thousand four hundred by three hundred three thousand.

Ans. 1224241200000. 14. Multiply three hundred thousand thirty by forty-seven thousand seventy,

Ans. 14122412100. 15. Multiply fifteen millions one hundred by two thousand two hundred.

Ans. 18000220000. 16. Multiply one billion twenty thousand by one thousand one hundred.

Ans. 1100022000000.

QUESTION. - What is the rule ?

Ś V. DIVISION.

MENTAL EXERCISES. ART. 45. WHEN it is required to find how many times one number contains another, the process is called Division.

Ex. 1. A boy has 32 cents, which he wishes to give to 8 of his companions, to each an equal number; how many must each receive ?

ILLUSTRATION. — It is evident that each boy must receive as many cents as the number 8 is contained times in 32. We therefore inquire what number 8 must be multiplied by to make 32. By trial, we find that 4 is the number; because 4 times 8 make 32. Hence 8 is contained in 32 4 times, and the boys receive 4 cents apiece.

The following table should be studied by the learner to aid him in solving questions in division :

DIVISION TABLE.

2 in 2 1 time
2 in 4 2 times
2 in 6 3 times
2 in 8 4 times
2 in 10 5 times
2 in 12 6 times
2 in 14 7 times
2 in 16 8 times
2 in 18 9 times
2 in 20 10 times
2 in 22 11 times
2 in 24 12 times

3 in 3 1 time
3 in 6 2 times
3 in 9 3 times
3 in 12 4 times
3 in 15 5 times
3 in 18 6 times
3 in 21 7 times
3 in 24 8 times
3 in 27 9 times
3 in 30 10 times
3 in 33 11 times
3 in 36 12 times

4 in 4 1 time
4 in 8 2 times
4 in 12 3 times
4 in 16 4 times
4 in 20 5 times
4 in 24 6 times
4 in 28 7 times
4 in 32 8 times
4 in 36 9 times
4 in 40 10 times
4 in 44 11 times
4 in 48 12 times

5 in 5 1 time 5 in 10 2 times 5 in 15 3 times 5 in 20 4 times 5 in 25 5 times 5 in 30

6 times 5 in 35

7 times 5 in 40 8 times 5 in 45 9 times 5 in 50 10 times 5 in 55 11 times 5 in 60 12 times

6 in 6 1 time
6 in 12 2 times
6 in 18 3 times
6 in 24 4 times
6 in 30 5 times
6 in

7 in 7 1 time
7 in 14 2 times
7 in

21

3 times 7 in 28 4 times 7 in 35 5 times 7 in 42 6 times 7 in 49 7 times 7 in 56 8 times 7 in 63 9 times 7 in 70 10 times 7 in 77 11 times 7 in 84 12 times

8 in 8 1 time
8 in 16 2 times
8 in 24

3 times
8 in 32 4 times
8 in 40 5 times
8 in 48 6 times
8 in 56 7 times
8 in 64 8 times
8 in 72 9 times
8 in 80 10 times
8 in 88 11 times
8 in 96 12 times

9 in 9 1 time 9 in 18 2 times 9 in 27 3 times 9 in 36 4 times 9 in 45 5 times 9 in 54 6 times 9 in 63 7 times 9 in 72 8 times 9 in 81 9 times 9 in 90 10 tiines 9 in 99 11 times 9 in 108 12 times

36 6 times 6 in 42 7 times 6 in 48 8 times 6 in 54 9 times 6 in 60 10 times 6 in 66 11 times 6 in 72 12 times

10 in 110 11 times
10 in 120 12 times

11 in 88 8 times
11 in 99 9 times
11 in 110 10 times
11 in 121 11 times
11 in 132 12 times

10 in 10 1 time
10 in 20 2 times
10 in 30 3 times
10 in 40 4 times
10 in 50 5 times
10 in 60 6 times
10 in 70 7 times
10 in 80 8 times
10 in 90 9 times
10 in 100 10 times

11 in 11 1 time
11 in 22 2 times
11 in 33 3 times
11 in 44 4 times
11 in 55 5 times
11 in 66 6 times
11 in 77 7 times

12 in 48 4 times
12 in 60 5 times
12 in 72 6 times
12 in 84 7 times
12 in 96 8 times
12 in 108 9 times
12 in 120 10 times
12 in 132 11 times
12 in 144 12 times

12 in 12 1 time
12 in 24 2 times
12 in 36 3 times

2. A farmer received 8 dollars for 2 sheep; what was the price of each?

ILLUSTRATION. — It is evident, since he received 8 dollars for 2 sheep, for 1 sheep he must receive as many dollars as 2 is contained times in 8. 2 is contained in 8 4 times, because 4 times 2 are 8; hence 4 dollars was the price of each sheep.

3. A man gave 15 dollars for 3 barrels of flour; what was the cost of each barrel ?

4. A lady divided 20 oranges among her 5 daughters; how many did each receive ?

5. If 4 casks of lime cost 12 dollars, what costs 1 cask ?

6. A laborer earned 48 shillings in 6 days; what did he receive per day?

7. A man can perform a certain piece of labor in 30 days ; how long will it take five men to do the same ?

8. When 72 dollars are paid for 8 acres of land, what costs 1 acre? What cost 3 acres ?

9. If 21 pounds of flour can be obtained for 3 dollars, how much can be obtained for 1 dollar ? How much for 8 dollars ? How much for 9 dollars ?

10. Gave 56 cents for 8 pounds of raisins ; what costs 1 pound? What cost 7 pounds ?

11. If a man walk 24 miles in 6 hours, how far will he walk in 1 hour ? How far in 10 hours ?

12. Paid 56 dollars for 7 hundred weight of sugar; what costs 1 hundred weight ? What cost 10 hundred weight?

13. If 5 horses will eat a load of hay in 1 week, how long would it last 1 horse ? .

14. In 20, how many times 2? How many times 4? How many times 5 ? How many times 10?

15. In 24, how many times 3? How many times 4 ? How many times 6 ? How many times 8 ?

16. How many times 7 in 21 ? In 28? In 56? In 35 ? In 14? In 63? In 77 ? In 70? In 84 ?

17. How many times 6 in 12? In 36? In 18? In 54 ? In 60 ? In 42 ? In 48? In 72? In 66 ?

18. How many times 9 in 27? In 45? In 63? In 81? In 99 ? In 108?

19. How many times 11 in 22? In 55? In 77? In 88? In 110 ? In 132 ?

20. How many times 12 in 36? In 60? In 72 ? In 84? In 120? In 144 ?

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