3. In one square yard there are 9 square feet. How many square yards in 2115 square feet?

4. In a certain box there were 1728 eggs. How many dozen did it contain?

5. At $6 a barrel, how many barrels of flour can be purchased for $1200?

6. The salaries of 7 men amounted to $5838. What was the average salary?

7. If a vessel sails 9 miles an hour, how long at this rate will it take her to sail 1089 miles?

8. I bought 6 lots for $5484. What was the average price of each lot?

9. At 8 cents a yard, how many yards of muslin can be bought for $3.04?

10. A father divided $19480 equally among his 8 children. How much did each child receive?

11. A man bought 7 cows for $171.50. price paid for each cow?

What was the average

DRILL EXERCISES.

101. 1. How many 10's are there in 40? In 50? In 60? In 70? In 100? In 150? In 200? In 300?

2. What must be cut off from 40 to make the number 4?

50 to make it 5? From 100 to make it 10?

it 30?

From

From 300 to make

3. How many ciphers in the divisor 10? How many figures cut off from the right of the dividend will give the quotient when divided by 10?

4. How many 100's in 400? In 500? In 1500? In 2500?

5. What must be cut off from 400 to make the number 4? From 500 to make it 5? From 1500 to make it 15?

it 25%

From 2500 to make

6. How many ciphers in the divisor 100? And how many figures are cut off from the right of the dividend in dividing by 100? 7. How many 1000's in 2000? In 6000? In 10000? In 18000 ? 8. How many ciphers in 1000? And how many figures are cut off from the right of the dividend to secure the quotient in dividing by 1000?

9. How many figures did you cut off in dividing numbers by 10? In dividing by 100? In dividing by 1000?

10. How then may a number by divided by 10, 100, 1000, etc. 102.-A number may be divided by 10, 100, 1000, etc., by cutting off from the right of the dividend as many figures as there are ciphers at the right of the divisor.

19. Divide 54743 by 300. 300)54743 182-143,

or 182148

300

EXPLANATION.-We first cut off the ciphers at the right of the divisor and as many figures at the right of the dividend. The remaining part of the dividend is then divided by 3, the remaining part of the divisor, giving a quotient of 182 and a remainder of 1 hundred. 1 hundred plus 43, the partial remainder cut off, gives the entire remainder 143. Hence, the quotient is 182148.

43

41. Divide 4819 by 23.

Divisor. Dividend. Quotient

23) 4819 (2091

46

EXPLANATION.-23 is not contained in 4 thousands any thousands times; hence, there are no thousands in the quotient. 4 thousands and 8 hundreds are 48 hundreds. 23 is contained in 48 hundreds 2 hundreds times. We write the 2 hundreds in the quotient at the right of the dividend and multiply the divisor by them. 2 hundreds times 23 are 46 hundreds, which subtracted from 48 hundreds leave 2 hundreds.

219

207

12

2 hundreds and 1 ten are 21 tens. 23 is not contained in 21 tens any ten times; therefore we write 0 in the tens's place in the quotient.

21 tens and 9 units are 219 units. 23 is contained in 219 units 9 units times. We write the 9 units in the quotient and multiply the divisor by them. 9 units times 23 units are 207 units, which subtracted from 219 units leave a remainder of 12 units. To indicate the division of the remainder by 23, we write the 12 over the 23 with a line between them.

Hence, the quotient is 209132.

=

In practice we say: 23 in 48, 2 times; write 2 in the quotient. 2 × 23 = 46, which subtracted from 48 leaves 2. Bring down 1. 23 in 21, 0 times; write 0 in the quotient. Bring down 9. 23 in 219, 9 times; write 9 in the quotient. 9×23=207, which subtracted from 219 leaves 12. Answer, 209}} Proof. (20923) + 12 = 4819. Hence, the work is correct (Prin. 4).

When in dividing the work is written out in full, as above, the process is called Long Division.

104. Long Division is the method of dividing when the work is written out in full.

Long division is generally employed when the divisor exceeds 12.

RULE.—Draw a curve line at both sides of the dividend, and write the divisor at the left.

Divide the number expressed by the fewest figures at the left of the dividend that will contain the divisor, and write the quotient at the right of the dividend.

Multiply the divisor by the quotient; subtract the product from the part of the dividend used, and to the remainder bring down the next figure of the dividend.

If any partial dividend will not contain the divisor, place a cipher in the quotient, bring down the next figure of the dividend, and proceed as before.

When there is a final remainder, write it after the quotient, or with the divisor beneath it as part of the quotient.

PROOF.-The proof is the same as in Short Division.

1. The student should carefully observe the four steps in the process of long division: 1st, Divide; 2d, Multiply; 3d, Subtract; 4th, Bring down.

2. To find the quotient figure, see how often the first figure at the left in the divisor is contained in the first figure, or first two figures, at the left of the partial dividend, making allowance for the addition of the tens from the product of the second figure of the divisor.

3. If after multiplying the divisor by the quotient figure, we find the product greater than the partial dividend from which it is to be subtracted, the quotient figure is too large, and must be diminished.

4. Each remainder must be less than the divisor. When the remainder is as great as the divisor, the quotient figure is too small, and must be in

creased.

5. When there is no remainder, the divisor is called an exact divisor of the dividend.

105. 1. At $35 each, how many cows can be bought for $840?

35) 840 (24

70

140

140

SOLUTION.-If $35 will buy one cow, $840 will buy as many cows as $35 are contained times in $840, which are 24.

2. If a steamboat sails 18 miles an hour, how long will it take to go 576 miles?

3. There are 32 quarts in a bushel.

there in 3936 quarts?

How many bushels are

4. How many days are there in 4128 hours?

பப

5. I wish to place 1440 pens in boxes of 36 pens each. How many boxes will be required?

6. How many farms of 50 acres each can be made from a tract of land containing 1200 acres?

7. A certain cistern contains 4725 gallons of water. How many hogsheads of 63 gallons each would be required to hold it?

8. A field of 80 acres produced 2160 bushels of wheat. What was the average yield per acre?

9. There are 56 pounds in a bushel of corn. How many bushels in 19320 pounds of corn?

10. At the rate of 136 miles a day, how many days would it require to travel 5712 miles?

11. There are 128 cubic feet in one cord. How many cords are there in 10112 cubic feet of wood?

12. A gallon contains 231 cubic inches. How many gallons in vessel that contains 7392 cubic inches?

13. A cotton grower raised 154224 pounds of cotton. How many bales averaging 324 pounds each could be made from it?

14. The stock of a railroad company is $3750000. If it is divided into shares of $500 each, how many shares are there?

14. If the combined wealth of 2135 men is $6419945, what is the average wealth of each man?

16. The product of two numbers is 515522, and the smaller is 322. What is the larger number.

17. The product of two numbers is 433825428, and the larger is 71873. What is the smaller number.

18. The state of Pennsylvania has an area of 45215 square miles. Rhode Island has an area of only 1250 square miles. How many states of the size of Rhode Island could be formed out of Pennsylvania, and how many square miles would be left over?

19. In 1900, the inhabitants of Pennsylvania numbered 6302115. About how many persons, on an average, were living on a square mile?

THE SIGNS OF AGGREGATION.

106. The Signs of Aggregation are the Parenthesis, (); the Vinculum, ; the Brackets, []; and the Braces, {}. They show that the numbers included by them are to be subjected to the same operation.

Thus, each of the expressions (3 + 4) × 5, 3 + + × 5, [3 +4] × 5, and {3+4} X5, shows that the sum of 3 and 4, or 7, is to be multiplied by 5; but 3+4X5eans that 3 is to be added to 4 times 5.