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Constant Product.

$39. The Product of two numbers remains the same, when the multiplicand and multiplier are taken the one for the other. Thus 3 times 15 gives the same product as 15 times 3.

For, 3 times 15 must be 15 times as many as 3 times 1, which is 3; that is, 3 times 15 is just as many as 15 times 3.

Prove that 4 times 23 is equal to 23 times 4. Prove that 5 times 17 is equal to 17 times 5. Prove that 6 times 39 is equal to 39 times 6.

Sign of Multiplication.

$ 40. The sign X, called into, placed between two numbers, signifies that they are to be multiplied together.

Thus 9X4, 9 into 4, signifies 9 times 4, or 4 times 9.

[blocks in formation]

Product of Concrete Numbers.

§ 41. The product of a concrete number taken any num ber of times, is a similar concrete number.

Thus 3 times 5 cents is 15 cents.

What is 3 times 9 pounds? 4 times 7 yards? 3 times 8 dollars? 6 times 9 miles? 7 times 10 acres? 8 times 11 furlongs?

$42. A concrete number cannot be taken concretely as a multiplier; for, as a multiplier, a number can express nothing but repetitions of the multiplicand.

For example, 3 hats at 5 dollars apiece would amount to 3 times 5 dollars, which is 15 dollars; the multiplier 3 expressing only repetitions of 5 dollars. We multiply by 3,-not by 3 hats.

What would 4 slates amount to at 10 cents apiece? What would 5 books amount to at 8 cents apiece? What would 6 pounds of butter amount to at 12 cents a pound? What would 7 cords of wood amount to at 3 dollars a cord? What would 8 pair of boots amount to at 5 dollars a pair? What would 9 tons of hay amount to at 8 dollars

a ton?

If a man can walk 4 miles an hour, how far could he walk in 10 hours? If a yard of cloth sells for 6 dollars, what should be paid for 11 yards at the same rate? Allowing 12 men to do a piece of work in 5 days, in what time ought one man to do the same?

RULE V.

$43. To Multiply by a Number not exceeding 12; or by such Number with Os annexed.

1. Proceeding from right to left, multiply each figure of the multiplicand, and under each set its product, when less than 10.

2. When the product is 10 or more, set down its right hand figure, and add the left figure or figures to the next product. Set down the whole of the last product.

3. Ciphers in the right of either or both of the factors, are omitted in multiplying; but as many Os must be placed in the right of the product.

4. When the multiplier is 10, or 100, &c., the product will be formed by merely annexing to the multiplicand as many Os as there are Os in the right of the Multiplier.

EXAMPLES.

1. To Multiply 5070 by 3; that is, to find 3 times 5070.

50 70
3

15210

We place a 0 in the right of the product, for the 0 in the right of the multiplicand; and then say,

3 times 7 is 21; setting down the 1, we say 3 times 0 is 0. and the 2, from 21, added; makes 2; 3 times 5 is 15.

The required product is 15210.

2. To Multiply 243 by 30; or to find 30 times 243.

243
30
7290

Having placed a 0 in the right of the product, for the 0 in the right of the multiplier, we say

3 times 3 is 9; 3 times 4 is 12; 3 times 2 is 6, and 1 makes 7.

3. To multiply 359 by 100, we merely annex two Os to the multiplicand, and thus find the product 35900.

The adding of the left hand figure in the product of any figure, to the next product, is carrying one for every ten, as in Addition.

In the first example, we might have begun by saying 3 times 0 is 0, and then said 3 times 7 is 21, &c. But this amounts to the same thing as merely placing a 0) on the right, and beginning the multiplication with 3 times 7.

In the second example, 243 multiplied by 3 produces 729. But since the true multiplier is 30, which is 10 times 3, the true product must be 10 times 729; and this tenfold value is assigned to 729 by the 0 on the right, which removes each of its figures one place farther toward the left from units (§ 16).

We thus find that for each 0 omitted in the right of either factor, a 0 must be placed in the right of the product.

The Operation Proved.

$44. Multiplication may be verified or proved, by multiplying the multiplicand by the multiplier minus 1, and adding the multiplicand to the product thus obtained; the sum must be equal to the product found with the entire multiplier.

EXERCISES.

1. Mary bought two books, at 31 cents apiece. How many cents did she pay for them!

The 2 books cost twice 31 cents; or 31 cents+31 cents (§ 38).

Ans. 62 cents.

2. A farmer sold 3 horses at 125 dollars apiece. What sum did he receive for them! Ans. 375 dollars.

3. What would be the value of 20 shares of road stock, at 95 dollars for each share! Ans. 1900 dollars. 4. What would be the weight of 30 bales of cotton, allowing 450 pounds to each bale! Ans. 13500 pounds. in 10 barrels of flour, Ans. 1960 pounds. in a day, how many Ans. 1220 miles.

5. How many pounds of flour are there there being 196 pounds in each barrel! 6. If a steamboat can run 305 miles miles could it run in 4 days?

7. There being 1760 yards in a mile, what number of yards is there in 5 miles ?

8. A farmer sold 100 acres of land, at What sum did he receive for the land?

Ans. 8800 yards. 43 dollars per acre Ans. 4300 dollars. how many feet are Ans. 5280 feet.

9. There being 660 feet in a furlong, there in a mile, which is 8 furlongs? 10. A merchant bought 60 sacks of coffee, at 13 dollars a sack. What did the whole amount to?

Ans. 780 dollars.

11. A butcher bought 7 fat oxen, at an average of 32 dollars a head. What sum did he pay for them? Ans. 224 dollars. 12. Find the sum that should be paid for 9 horses at 130 dollars apiece, and 10 cows at 21 dollars apiece.

Ans. 1380 dollars. 13. Bought at one time 5 coils of rope, each containing 139 yards; and at another, 7 coils, each containing 150 yards. What was the whole number of yards bought?

Ans. 1745 yards. 14. A farmer bought 5 horses, at 95 dollars each; 12 mules, at 60 dollars each; and 40 head of cattle, at 27 dollars a head. What did he pay for all the stock purchased?

Ans. 2275 dollars.

15. Sold to A 7 pieces of cotton, to B 11 pieces, and to C 12 pieces, each piece containing 31 yards. What was the number of pieces, and what the number of yards that was sold? Ans. 30 pieces; and 930 yards. 16. A merchant bought 218 beaver hats, at 4 dollars apiece. What did the whole amount to?

The hats amounted to 213 times 4 dollars; but 4 times 213 is the same number as 213 times 4. (§ 39). Ans. 852 dollars.

17. Bought 175 cords of wood, at 3 dollars per cord. What did the whole amount to? Ans. 525 dollars. 18. A planter sold 325 bales of cotton, at the average sum of 30 dollars per bale. What did the sale amount to?

Ans. 9750 dollars. 19. If a steam ship can run at the rate of 300 miles per day, how far would she run in 31 days. Ans. 9300 miles. 20. A drover sold 573 head of cattle, at the average price of 20 dollars a head. What did the sale amount to?

Ans. 11460 dollars. 21. A farmer filled, at one time, 83 sacks of corn; and at another time 112 sacks;-each sack containing 5 bushels. What quantity of corn was put into all the sacks?

Ans. 975 bushels.

22. Bought of A, 137 acres of land; of B, 89 acres; and of C, 384 acres;—at the average price of 40 dollars per acre. What quantity of land was bought, and what sum was paid for it! Ans. 610 acres; and 24400 dollars. 23. Shipped to New Orleans, at one time, 120 barrels of apples; at another, 75 barrels; at another, 100 barrels; and at another, 9 barrels;-each barrel containing 3 bushels. Required the number of barrels, and the number of bushels shipped. Ans. 304 barrels; and 912 bushels.

24. A merchant bought 275 yards of cloth, at 6 dollars per yard, of which he has sold 133 yards, at 10 dollars per yard. What would he gain on the whole, by selling the remainder at 9 dollars per yard? Ans. 958 dollars. 25. If 10 masons can build a wall, in 34 days; in how many days ought one mason to build the same wall!

It would take 1 mason 10 times as long as it would 10 masons, to build the wall; that is, 10 times 34 days. Ans. 340 days. 26. How long ought one man to subsist on a stock of provisions which would support 7 men for 29 days?

Ans. 203 days. 27. If 9 men could mow a certain meadow, in 19 days; in how many days ought one man to mow the meadow ?

Ans. 171 days.

28. If 115 bushels of oats will feed one horse for 3 months; what quantity of oats would feed 12 horses, the same time? Ans. 1380 bushels.

29. Allowing 20 pieces of artillery to demolish a fortress in 48 hours, in what time ought one piece to demolish the fortress? Ans. 960 hours. 30. A lent B 8 yoke of working oxen, for 13 days. How long ought B to lend A 1 yoke, to requite the favor?

Ans. 104 days. 31. C exchanges silk, at 1 dollar per yard, with D, for broadcloth, at 7 dollars per yard. How many yards of silk should be given for 125 yards of broadcloth ? Ans. 875 yards.

32. Two merchants barter as follows: A gives 75 yards of cloth, at 9 dollars per yard, to B, for 109 beaver hats, at 7 dollars apiece; the deficiency on either side being made good in money. Which of them receives money, and how much?

Ans. B receives 88 dollars.

33. A gave B 3 horses, and 30 head of cattle, for 50 barrels of flour and 200 barrels of corn. A sold the flour at 6 dollars, and the corn at 3 dollars a barrel; while B sold his horses at 120 dollars, and his cattle at 25 dollars, each? Which of the two gained by the trade, and how much?

Ans. B gained 210 dollars.

Ans. 247387300.
Ans. 615360200.
Ans. 848208000.
Ans. 963702000.
Ans. 2118000000.
Ans. 957003300.

34. Find the Product 10X24738730. 35. Find the Product 200×3076801. 36. Find the Product 1200X706840. 37. Find the Product 90X10707800. 38. Find the Product 300×7060000. 39. Find the Product 1100X870003. 40. Find the Product 12000×70000. Ans. 840000000.

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