3. If the multiplier exceeds 12, multiply by each of its figures in turn, writing the first figure of each partial product under the figure used in multiplying. Then add the partial products.

4. Prove by multiplying multiplier by multiplicand.

78. NAUGHT IN THE MULTIPLIER.-When 0 occurs in the multiplier, bring it down, and go on multiplying by the next figure, all in the same line.

EXAMPLE.-Multiply 7967 by 4005. First multiply by 5. Bring down the two naughts, each in its own column. Then multiply by 4, setting the product in the same line; its first figure is thus brought under the 4. Finally, add the partial products.

7967

4005

39835

3186800

Ans. 31907835

EXAMPLES FOR PRACTICE.

1. Multiply four thousand two hundred and ninety-three, by eight thousand and seventy-six.

Ans. 34670268. 2. Multiply fifty-seven thousand and three, by seventy-five thousand and four. Ans. 4275453012.

3. The factors of a certain product are 51, 4, 6, and 108. Required the product. Ans. 132192. 4. How much money must be divided among 2065 persons, that each may have $908?

5. A drover who had 967 head of cattle, bought 92 more, and then sold the whole for $63 apiece. How much did he receive? 6. How many books in three rooms, furnished with four bookcases apiece, each case containing 108 books?

7. What cost 825 horses at $145 apiece?
8. What is the product of 9263 and 7603?

9. What is the value of 68753 × 10408?

10. Multiply MDLXV. by VIX.

11. Multiply LCCXLI. by XVII.

Ans. 7839085.

Ans. 502761687.

78. When 0 occurs in the multiplier, how must we proceed? Illustrate this with

the given example.

79. NAUGHTS AT THE RIGHT.-We learned in § 25 that every naught placed after a number increases its value ten times. Hence, to multiply by 10, 100, 1000, &c., annex as many naughts as are in the multiplier.

76 × 100 7600

76 × 10000 760000

76 × 10 760
=
80. EXAMPLE.-Multiply 4200 by 40.

4200 40

168 000

Multiplying in the usual way, we find the product to be 168000. The result is the same as if we had multiplied the significant figures together, and annexed the naughts at the right of both factors. Hence,

4200 40

Ans. 168000

When there are naughts at the right of either factor or both, multiply the other figures, and annex to their product as many naughts as are at the right of both factors.

EXAMPLES FOR PRACTICE.

13. One gold eagle is worth $20; how many dollars are 6500 eagles worth?

14. If 100 pounds make one hundred-weight, how many pounds in seventy-eight hundred-weight?

15. Light travels 192000 miles in a second; how many miles will it travel in 60 seconds?

16. What will 140 miles of railroad cost, at an average of $42900 a mile?

79. What is the effect of annexing a naught to a number? How, then, may we multiply by 10, 100, 1000, &c. ?-80. Multiply 4200 by 40 in the usual way. What other way of obtaining the result is there? When there are naughts at the right of either factor or both, what is the shortest mode of proceeding?

17. If an army consume 840 barrels of provisions in one day, how many will it consume in 1 year, or 365 days?

18. If sound travels 1120 feet in a second, how far will it travel in 20 seconds?

19. A farmer has 6 orchards, each containing 10 piles of apples. In each pile are 1200 apples; how many apples in all? 20. Multiply 640 by 10; by 50; by 940: add the products. 21. How much more is 6800 × 140 than 970 × 850 ?

81. MULTIPLYING BY A COMPOSITE NUMBER.—A Composite Number is the product of two or more factors greater than 1. 16 is a composite number, being equal

to 8 × 2.

When the multiplier is a composite number, we may either multiply by the whole at once, or by its factors in turn. The result will be the same. Multiplication by a composite number may, therefore, be proved by multiplying by its factors.

EXAMPLE.-Multiply 93 by 24.

What are the factors of 33-that is, what two numbers multiplied together produce 33 ? What are the factors of 108? 72? 44? 21? 132? 35? 99? 42? 54? 49 ?

121?

In examples 1-6, first multiply by the whole multiplier; then prove the result by multiplying by its factors.

1. What cost 63 firkins of butter, at $16 apiece?

81. What is a Composite Number? How may we multiply by a composite number? How may multiplication by a composite number be proved?

2. What is the weight of 84 barrels of flour, averaging 196 pounds each?

3. How many hours in 365 days, there being twenty-four hours in one day?

4. If a person travels 96 miles a day, for 108 days, how far does he travel in all?

5. What will 27 miles of plank road cost, at $4200 a mile? 6. How many bushels of apples does an orchard of 107 trees yield, if each tree produces 12 bushels?

7. What will 13 square miles of land cost, at $17 an acre, there being 640 acres in one square mile? Ans. $141440.

8. How many miles will a locomotive go in 7 days of 24 hours each, if it moves 29 miles an hour? Ans. 4872 miles.

9. The earth moves in its orbit 68000 miles an hour. How far will it move in 365 days of 24 hours each?

10. In an orchard of 219 apple-trees, the average yield of each tree was 3 barrels of fruit, worth $3 a barrel. What was the whole yield worth? 11. A man owing $8213, gives in payment 5 horses valued at $175 each, 29 cows at $38 each, and $765 in cash.

remains unpaid?

Ans. $1971.

How much

Ans. $5471.

12. A ship, after sailing 23 hours east at the rate of 8 miles an hour, is driven west by a storm 10 miles an hour for 5 hours. At the end of the 28 hours, how far is she from where she started?

13. In one year there are 31556929 seconds. one trillion exceed the seconds in 1864 years?

14. A man bought two farms; one of 87 acre; the other of 101 acres, at $37 an acre. how much was still due?

Ans. 134 miles.
How much does

acres, at $54 an He paid $8140; Ans. $295.

15. If I give 3 horses, each worth $150, and 13 cows, each worth $56, for 50 acres of land, valued at $19 an acre, do I gain or lose, and how much? Ans. Lose $228.

16. If a man travels 47 miles a day for 5 days, and then goes 54 miles a day for four days, how many more miles will he have to go, to complete a journey of 600 miles?

CHAPTER VII.

DIVISION.

82. Two pints make a quart; how many quarts in 8 pints?

If 2 pints make 1 quart, in 8 pints there are as many quarts as 2 is.. contained times in 8. Here we are required to find how many times 2 is contained in 8. This process is called Division.

83. Division is the process of finding how many times one number is contained in another.

DIVISION TABLE.

Any number is contained in 0, 0 times.

1 in 1, once; 1 in 2, twice; 1 in 3, 3 times; 1 in 4, 4 times, &c.

82. Solve the given example. What are we here required to do? What is this process called?-83. What is Divizion? How many times is any number contained in 0? How many times is 1 contained in any number? Recite the Table.