uct, the remainder is 5159 pounds, which, being less than the divisor, shows that the quotient figure is correct. Therefore, by the conditions of the question, 6 stacks can be made, and there will remain 5159 pounds of hay unused, which might have been taken from the whole quantity without affecting the number of stacks of 8743 lb. each.

4. In a pasture there are 191 sheep, which their owner wishes to divide equally among 26 farmers. How many can he give to each farmer?

In performing the necessary division in this example, observe that as the divisor, 26, is nearer 3 tens than 2 tens, we shall be more likely to obtain the true quotient by taking 3 as the trial divisor, than by taking 2. 3 being contained 6 times with a remainder in 19, we try 6 for a quotient. But 26 parts of 6 each, give 156 for a product, which, subtracted from the dividend, leaves a remainder of 35. As the remainder is greater than the divisor, the trial quotient must be too small. We must, therefore, try a larger number for a quotient. Inspecting the remainder we see that it can contain the divisor but once, with a remainder of 9. Therefore, the dividend will contain the divisor one more than 6 times, which will be 7 times, and there will be a remainder of 9; that is, each man could take 7 sheep, and there would be 9 left that could not be distributed without giving to one man more than to another.

The following examples are to be performed in a similar

manner.

5. How many rooms can be carpeted by 237 yd. of carpeting, if each room can be carpeted by 24 yd.?

6. How many gold watches, at $87 apiece, can be bought for $613?

7. How many dollars will 448 oranges cost, if 56 oranges cost 1 dollar?

8. How many square feet are there in 1327 inches?

9. How many cubic feet in 12346 cubic inches?

15. If 495 men consume 2367 lb. of meat in 1 week, how many pounds will 1 man consume?

16. If 384 kegs will contain 3456 gallons, how many gallons will 1 keg contain?

17. If 7427 barrels of flour can be bought for 69359 dollars, for how many dollars can 1 barrel be bought?

18. If 37 lb. of rice cost $1.48, how much will 1 lb. cost? 19. If 67 yd. of cotton cloth cost $6.03, how much will 1 yd. cost?

G. 1. A miller has 9233 lb. of flour, which he wishes to put into bags, each containing 28 lb. How many bags can he fill? Solution. If 28 lb. will fill 1 bag, 9233 lb. will fill as many bags as there are parts of 28 lb. each in 9233 lb., to ascertain which we must divide 9233 by 28.

The following represents a convenient way of writing the figures used in performing the work :

Ans. 329 bags, with 21 bushels remaining unused, or 32931 bags.

Explanation. The divisor being nearer 3 tens than 2 tens, 3 will be a more convenient trial divisor than 2. Since 3 is contained 3 times in 9, we infer that 28 is contained 3 hundred times in 92 hundreds. Multiplying 28 by 3, placing the product under the hundreds of the dividend to mark its denomination, (which is equivalent to multiplying 28 by 300,) we have 84 hundreds, which, being less than the part of the dividend we consider, shows that the trial quotient figure is not too large. Subtracting this product from the hundreds of the dividend, we find a remainder of 8 hundreds, which being less than 2800 shows that the trial quotient figure is the true one. 8 hundreds =80 tens, to which adding the 3 tens of the dividend, we have 83 tens to be divided by 28. Since the trial divisor, 3, is contained twice in 8, we infer that 28 is contained 2 tens times in 83 tens. 2 tens, or 20, times 28 56 tens, which, being

=

tens

=

less than 83 tens, shows that the trial quotient figure is not too large. 56 tens from 83 tens 27 tens, which, being less than 28 tens, shows that the trial quotient figure is the true one. 27 270 units, to which adding the 3 units in the units' place of the dividend, we have 273 to be divided by 28. Since the trial divisor, 3, is contained 9 times in 27, we infer that 28 is contained 9 times in 273. 9 times 28 = 252, which being less than 273 shows that the trial quotient is not too large. 252 from 27321, which, being less than 28, shows that the trial quotient figure of the units is the true one. The entire quotient is, therefore, 329, and the remainder is 21, thus showing that the miller could fill 329 bags, and have 21 lb. of flour left; or, that he could fill 32934 bags.

The answer might have been obtained in another form by reducing the remainder, 21 units, to lower denominations and continuing the division as follows:

21 units 210 tenths. Since 3 is contained 7 times in 21, we infer that 28 is contained 7 tenths times in 210 tenths. 7 tenths times 28: 196 tenths, which, being less than 210 tenths, shows that the trial quotient figure is not too large. Subtracting as we have hitherto done gives a remainder of 14 tenths, which is equal to 140 hundredths. The trial divisor 3 is contained almost 5 times in 14, and as it is larger than the tens of the true divisor, it is probable that the true divisor will be contained .05 times in 140 hundredths. But .05 times 28= 140 hundredths. The quotient is, therefore, 329.75. This form might have been obtained by reducing to a decimal fraction.

By the previous illustrations, it appears that when the divisor is a large number, we proceed as follows:

1st. Find a convenient trial divisor.

2d. With the trial divisor, determine the probable quotient figure of the highest denomination.

3d. Find the product of the divisor by this figure.

4th. Compare this product with the part of the dividend considered, to see that the quotient figure be not too large.

(If it should be found to be too large, a smaller one must be substituted for it, and the product of the divisor by the smaller figure must be substituted for the former product.)

5th. Subtract the product from the part of the dividend considered. 6th. Compare this remainder with the divisor, to see that the quotient figure is not too small.

(The quotient figure will be too small whenever the remainder is larger than the divisor. In that case, the quotient figure, product, and remainder should be removed, and a larger quotient figure should be substituted with its appropriate product and remainder.)

7th. If all the denominations of the dividend have not been considered, reduce the remainder to the next lower denomination, and add to it the figure of that denomination in the dividend.

8th. Proceed with this as before to find the next quotient figure, and so go on till all the denominations of the dividend have been considered, if decimal fractions are used in the quotient, till as many decimal places as are desirable have been obtained.

or,

The position of the decimal point is to be determined in the same manner as when the divisor is a small number.

2. How many square feet are equal to 42912 square inches? 3. How many boxes containing 73 quarts each can be filled from 658764 quarts of meal?

4. How many watches at $48 each can be bought for $3456? 5. If a boy can perform 37 examples in one day, how many days will it take him to perform 8764 examples?

6. If a vessel sails 147 miles per day, how many days will it take her to sail 3579 miles?

7. If 252 pints of cider will fill a barrel, how many barrels can be filled from 876438 pints?

8. How many weights, each weighing 56 lb., will be required to balance 7 loads of iron, each weighing 9478 lb.?

9. James can pump 37 quarts of water per minute, and William can pump 43 quarts of water per minute. How many minutes will it take William to pump as much as James can pump in 473 minutes?

10. What will 1 acre of land cost if 237 acres cost $236526? 11. If 687 bushels of potatoes cost $515.25, what will 236 bushels cost?

12. If 67 yd. of silk cost $75.375, what will 48 yd. cost?

13. Gave 7 loads of hay at $17.48 per load, and 238 bushels of corn at $.67 per bushel, for 32 acres of land. How much money ought I to give for 7 acres of the same land?

14. If g of a vessel is worth $6735.96, what is the whole vessel worth?

15. What will of a vessel cost, if the whole vessel costs $1874.34 ?

16. What is gg of $1976.43?

By performing the subtraction at the same time that we do the multiplication, we can avoid writing our products. Suppose that we wish to find the quotient of 1947.63 divided by 396. We will give the method, leaving the pupil to find the explanation of it. 396 is contained 4 times in 1947. 4 times 6: - : 24,

WRITTEN FIGURES.

396) 1947.63 (4.9182 quotient.

363.6

7.23

3.270

*.1020

.0228 remainder.

4

which subtracted from 27 leaves 3; this we write in the units' place. times 9—36, and 2 added make 38, which subtracted from 44, leaves 6. 4 times 312, and 4 added make 16, which subtracted from 19 leaves 3. The first remainder is, therefore, 363. Changing this to tenths, and adding the 6 tenths, we have 3636 tenths, which divided by 396, gives 9 tenths for a quotient. We multiply and subtract as before, and thus proceed till the division is completed, or till we have as many decimal places in the quotient as we wish.

Perform the division in the following examples in the same

17. What is the quotient or 78643 ÷ 47?

18. What is the quotient of 137648 ÷ 326?

23. What is the value of 764379 cubic inches, expressed in

cubic feet and inches?

24. How many square feet and inches are equal to 954376 sq. in. ?

26. 4743864 = .72 of what number?

27. If the wages of 96 laborers for 1 year are $26470.08, what will be the wages of 1 laborer for the same time?

28. If 875 pairs of boots cost $2737.16, how much will 1 pair cost?

29. At $.68 a pair, how many pairs of slippers can be bought for $97.58?

30. A man failing in business was able to pay $7578.38, which was only 37 per cent of his debts. What was the amount of his debts?

31. Mr. French spent $228.69 for tea, at $.27 per lb. He sold the tea for $.30 per lb., and with the money he received for