LONG DIVISION.

91. Long Division is the method of dividing when the work is written out in full. It is generally used when the divisor exceeds 12.

1. Divide 5848 by 23.

OPERATION.

23)5848(254

46

124

115

98

92

6

SOLUTION.-23 is not contained in 5 thousands any thousands times, hence there are no thousands in the quotient. 5 thousands and 8 hundreds are 58 hundreds; 23 is contained in 58 hundreds 2 hundreds times: 2 hundreds times 23 are 46 hundreds, which subtracted from 58 hundreds leave 12 hundreds: 12 hundreds and 4 tens are 124 tens; 23 is contained in 124 tens 5 tens times: 5 tens times 23 are 115 tens, which subtracted from 124 tens leave 9 tens: 9 tens and 8 units are 98 units; 23 is contained in 98 units 4 times; 4 times 23 equals 92: subtracting there is a remainder of 6, which will not contain 23; hence the quotient is 2 hundreds, 5 tens, and 4 units, or 254, with a remainder of 6.

Rule.-I. Draw curved lines at both sides of the dividend, and place the divisor at the left.

II. Divide the number expressed by the fewest terms at the left that will contain the divisor, and place the quotient at the right.

III. Multiply the divisor by this quotient, write the product under the partial dividend, subtract, and to the remainder annex the next term of the dividend.

IV. Divide as before, and thus continue until all the terms of the dividend have been used.

V. If any partial dividend will not contain the divisor, place a cipher in the quotient, annex the next term of the dividend, and proceed as before.

VI. When there is a final remainder, annex it, with the divisor written beneath, to the integral part of the quotien

Proof.-Multiply the integral part of the quotient by the divisor, and add the remainder, if any, to the product; if the work is correct the result will be equal to the dividend.

NOTES.-I. The pupils will notice that there are five operations 1st. Trite the number; 2d. Divide; 3d. Multiply; 4th. Subtract; 5th. Bring down Pupils often have difficulty in finding the correct quotient figure: this difficulty can be greatly diminished by attention to the following suggestions:

1st. Notice how often the left-hand term of the divisor is contained in the term or terms of the partial dividend, as far from the right hand term as the left hand term in the divisor is from the right hand term.

2d. If, when we multiply, the product is greater than the partial dividend, the quotient term must be diminished.

3d. If, when we subtract, the remainder is greater than the divisor, the quotient term must be increased.

III. We commence at the left to divide, so that the remainder can be united to the number of units of the next lower order, giving a new partial Uvidend. The sign + is used to denote a remainder.

59. Divide 8074528 by 6328. 60. Divide 97547337 by 3891. 61. Divide 4223745376 by 23456. 62. Divide 170627676887 by 413071. 63. Divide 129652565329 by 360073. 64. Multiply 37602 by 608, and divide 304.

Ans. 1276. Rem. 3858. Ans. 180071.

Rem. 25846.

Ans. 360073.

the product by

Ans. 75204.

PROBLEMS IN DIVISION.

92. In Division there are two classes of practical problems:

1st. To find the number of equal parts of a number.

2d To divide a number into equal parts.

CASE I.

93. To find the number of equal parts of a number.

MENTAL EXERCISES.

1. At 8 cents a yard how many yards of ribbon can I buy for 72 cents?

SOLUTION.-If 1 yard of ribbon costs 8 cents, for 72 cents I can buy as many yards as 8 is contained times in 72, which are 9.

2. How many oranges can I buy for 48 cents, at the rate of 8 cents apiece?

3. A farmer expended $96 in buying sheep at $8 apiece; how many sheep did he buy?

4. How many melons at 7 cents apiece may be bought for 84 cents? 5. How many oranges at 5 cents apiece may be bought for 7 lcmong worth 10 cents apiece?

6. How many yards of lace at 8 cents a yard may be bought for 6 yards of muslin at 12 cents a yard?

7. How much wheat worth 6 dimes a bushel may be bought for 12 bushels of corn at 5 dimes a bushel?

8. If flour is worth $8 a barrel, how many barrels could be exchanged for $12 in money and 4 barrels of fish worth $12 a barrel? 9. A man gave 7 pencils, worth 5 cents each, for 4 packages of en· velopes, worth 10 cents each; what was the gain?

10. Divide 96 by 4.

SOLUTION.-4 is contained in 9 tens 2 tens times and 1 ten remaining; 1 ten and 6 units equal 16 units; 4 is contained in 16 units 4 units times; hence the quotient is 2 tens and 4 units, or 24.

11. Divide 68 by 2; 69 by 3; 56 by 4; 72 by 3; 75 by 5; 76 by 4; 126 by 6; 147 by 7; 176 by 8.

12. Divide 720 by 8; 350 by 7; 720 by 9, 156 by 12; 231 by 11; 1224 by 12; 2408 by 8; 2718 by 9; 1728 by 12.

13. If you earn $15 a week and pay $5 for board and $4 for other expenses, in what time could you save $72 ?

14. Two men start from places 180 miles apart, and travel towards each other, one going 4 and the other 5 miles an hour; in how many hours will they meet?

WRITTEN EXERCISES.

1. At 25 dollars each, how many cows can be bought for 575 dollars?

SOLUTION.-If 25 dollars will buy one cow, 575 dollars will buy as many cows as 25 dollars are contained times in 575 dollars, which are 23.

OPERATION.

25)575(23

2. In one hogshead there are 63 gallons; how many hogs heads in 15,435 gallons? Ans. 245 hhd. 3. How many horses can you get for 1824 dollars, at the rate of 152 dollars each? Ans. 12 horses. 4. If a boat sails 25 miles an hour, how long will it be in sailing 1800 miles? Ans. 72 hours. 5. How many years must a person labor to earn $13,140, at the rate of $730 a year? Ans. 18 years.

6. How many acres of land can you purchase for $11,696, at the rate of $86 an acre?

Ans. 136.

7. How many cows, at $37 each, can be bought for 74 horses, at $150 each? Ans. 300 cows. 8. How many oxen, at $54 each, can be bought for 108 mules, at $94 each? Ans. 188 oxen.

9. A labors 72 weeks, at $14 a week; how much wheat at 42 cents a bushel will pay him? Ans. 2400 bushels. 10. The circumference of the earth is 25,000 miles; how long would it take a vessel to sail around it, going at the rate of 125 miles per day? Ans. 200 days.

11. The distance from the earth to the sun is 93,000,000 miles; how long will it take light to reach us from the sun, moving 11,520,000 miles a minute? Ans. 8 min.+.

12. The moon is 240,000 miles from the earth; how long would it take a balloon to reach it, moving at the rate of 75 miles an hour? Ans. 3200 hours.

13. A builder received Western lands at $25 per acre and a balance in cash of $7300, in trade for a row of 15 houses at $2575 each; how many acres did he receive? Ans. 1253.

14. A person wishes to trade land worth 150 dollars an acre, for 125 acres, at 75 dollars an acre, and gain 675 dollars by the bargain; how many acres will be exchanged?

Ans. 58.