CASE I.

ART. 39. Multiplication of abstract numbers, when the multiplier does not exceed 9. 1. Multiply 846 by 8.

OPERATION.

Thousands.
Hundreds.
Tens.

Units.

EXPLANATION. Write the numbers down, placing units under units; then proceed from right to left; thus, 8 times 6 units are 48 units, or 4 tens and 8 units;-place the 8 units in units' place, and reserve the 4 tens to add to the next product, 8 times 4 tens are 32 tens, and 4 tens added are 36 tens, or 3 hundreds and 6 tens;-place the 6 tens in tens' place and reserve the 3 6768 hundreds to add to the next product, 8 times 8 hundreds are 64 hundreds, and 3 hundreds added are 67 hundreds, or 6 thousands and 7 hundreds, which write down; and we have for the product 6768.

8

Multiplicand,8 4 6
Multiplier,
Product,

PROOF. Multiply the multiplier by the multiplicand; if the product thus obtained, equals the first product the work is presumed to be right.

2. Multiply 348 by 2. 3. Multiply 483 by 3. 4. Multiply 684 by 4. 5. Multiply 6482 by 4. 6. Multiply 14682 by 5. 7. Multiply 18623 by 6. 8. Multiply 38943 by 7. 9. Multiply 28462 by 8. 10. Multiply 8946 by 7. 11. Multiply 7683 by 6. 12. Multiply 9898 by 9. 13. Multiply 6847 by 3. 14. Multiply 94762 by 6. 15. Multiply 88992 by 7. 16. Multiply 33449 by 8. 17. Multiply 884682 by 9. 18. Multiply 99999 by 5. 19. Multiply 897654 by 7. 20. Multiply 123456789 by 8.

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PRACTICAL QUESTIONS.

1. A man sold 195 sheep, at 3 dollars a piece; how much did he receive for them?

2. What cost 184 barrels of flour, at 6 dollars a barrel? 3. What cost 1987 acres of land, at 9 dollars an acre ? 4. What cost 4786 barrels of sugar, at 9 dollars a barrel ?

5. In 1 mile there are 5280 feet; how many feet in 5 miles ?

6. In 1 mile there are 1760 yards; how many yards in 5 miles ?

7. If 9 men can mow a certain meadow in 18 days; in how many days can one man do the same?

8. If 6 masons can build a certain wall in 149 days; in how many days can one mason build the same wall?

9. If 460 bushels of oats will feed 1 horse 11 months; how many bushels will be required to feed 8 horses the same time?

10. Bought 245 cords of wood, at 7 dollars a cord. What did the whole cost?

11. A farmer sold 8 horses, at 253 dollars a piece; how many dollars did he receive for them?

12. A lady bought 189 yards of ribbon, at 6 cents a yard; how much did it all cost her?

13. What cost 1786 boxes of raisins, at 3 dollars a box?

14. If a steamship can go 395 miles in 1 day, how far can she go in 9 days?

15. A merchant bought 2864 hats, at 4 dollars a piece; how much did he pay for them all?

16. A farmer sold 9 fat oxen, at 185 dollars a piece; how much did he receive for them all?

17. In 1 day there are 1440 minutes; how many min utes in 8 days?

18. In 1 day there are 86400 seconds; how many seconds in 5 days?

19. If one man receive 4 dollars a week, how much will an army of 35680 men receive in 6 weeks?

36

MULTIPLICATION.

[CHAP. II.

20. At 2 dollars a day, each; how much will it cost, to board 685 men 7 days?

CASE II.

ART. 40. Multiplication of abstract numbers in general? 1. Multiply 437 by 56.

Hundreds.

Tens.
Units.

437

56

OPERATION. EXPLANATION.-Write the numbers down so that units stand under units, tens under tens, &c. Begin at the right and proceed, thus,-6 times 7 units are 42 units, or 4 tens and 2 units; write the 2 units in units' place, and reserve the 4 tens to add to the next product. 6 times 3 tens are 18 tens, and 4 tens added are 22 tens, or 2 hundreds and 2 tens, &c. We next multiply by the 5 tens. 2622 For convenience we say 5 times 7 are 35, and place 2185 the 5 under the multiplier, 5, that is in tens' place, and reserve the three hundreds to add to the next 24472 product, &c. But instead of 5 times 7, &c., it is, 50 times 7 units = 350 units, or 3 hundreds, 5 tens and 0 units. I therefore placed the 5 tens in tens' place, where you perceive it belongs. We proceed in the same way to explain why we place the right hand figure of the product in the third, or hundreds' place when multiplying by that figure, &c.

PROOF by the excess of 9's.-Find the excess of 9's in each FACTOR. Then if the excess of 9's in the product of these excesses, equals the excess of 9's in the product of the two factors, the work is right. Take for illustration the preceding example :

EXPLANATION. Commence at the left of the first factor, (the altiplicand,) and add, thus, 4+3+7 are 14, or 1 nine and 5

units, the excess of 9's in that factor. Add the second factor, (the multiplier,) 5+6 are 11, or 1 nine and 2 units, the excess of 9's in that factor. The product of these excesses is 10, or 1 nine and 1 unit, the excess of 9's in the product of the excesses. Add the total product, 2+4+4+7+2 are 19, or 2 nines and 1, the excess of 9's in the product of the factors. This excess equals the required excess; hence the work is right.

2. Multiply 4624 by 35. 3. Multiply 3846 by 39. 4. Multiply 8462 by 47. 5. Multiply 7846 by 147. 6. Multiply 3976 by 183. 7. Multiply 2243 by 144. 8. Multiply 2882 by 414. 9. Multiply 1414 by 323. 10. Multiply 2463 by 382. 11. Multiply 8632 by 132. 12. Multiply 4862 by 897. 13. Multiply 9876 by 678. 14. Multiply 4567 by 7654. 15. Multiply 1234 by 4321. 16. Multiply 8362 by 8496. 17. Multiply 146832 by 8376. 18. Multiply 36847 by 8324. 19. Multiply 1384697 by 476324.

20. Multiply 897654321 by 123456789.

PRACTICAL QUESTIONS.

1. If 15 men can build a certain wall in 235 days, how long will it take 1 man to do it?

2. If 45 men can accomplish a certain piece of work in 360 days by working 8 hours a day, how many days will it take one man to do the same by working 4 hours a day?

3. If 360 bushels of oats will last 185 horses 3 days, how long will it last 1 horse?

4. A drover bought 685 oxen, at 104 dollars a piece; what was the cost of all of them ?

5. A merchant bought 25 pieces of broadcloth, each piece containing 48 yards, at 9 dollars a yard. How much did he pay for the whole ?

6. If a steamship can sail 18 miles in 1 hour, how far can she sail in 34 days of 24 hours each?

7. A speculator bought 8968 acres of land, at 195 dollars an acre. How much did the whole cost him? 8. In 1 furlong there are 660 feet; how many feet in 8 furlongs, (1 mile)?

9 How many pounds of flour are there in 395 barrels; there being 196 pounds in each barrel?

10. What is the value of 346 shares of railroad stock, at 125 dollars a share?

11. How many pages are there in 5896 books, there being 394 pages in each book?

12. A speculator bought 302 cattle, and 293 times as many sheep; how many sheep did he buy?

13. If a garrison of men consume 987 pounds of beef in 1 day; how many pounds will a garrison containing twice as many men consume in 365 days?

14. Farmer A has 245 acres, sowed with wheat, which produces 32 bushels to the acre. Farmer B has 360 acres, sowed with wheat, which produces 25 bushels to the acre. What quantity of wheat was raised by A and B respectively?

15. A speculator bought 146 head of oxen; 230 head of cows; and 69 head of calves. He made a profit of 16 dollars a head on the oxen; 12 on the cows; and 5 on the calves. How much did he gain on the oxen, cows, and calves respectively?

16. A merchant bought 12 boxes of linen, each con'taining 25 pieces, and each piece containing 36 yards, at 65 cents a yard. How many pieces, and how many yards did he buy, and how much did it all cost him?

17. A has 395 acres of land, worth 27 dollars an acre; and B has 493 acres, worth 19 dollars an acre. What is

the value of each of their farms?