14. A man divides $318 among 6 boys; how much will each one receive?

SOLUTION 1ST.-It will require $6 to give each boy $1; and in giving $318, each boy will receive as many dollars as $6 are contained times in $318, which are 53. Therefore, etc.

OPERATION.

6)318

53

SOLUTION 2D.-If 6 boys receive $318, one boy will receive one-sixth of $318, which, by division, we find is $53. Therefore, etc.

15. If 12 men earn $384 in a week, how much does one man earn?

Ans. $32. 16. A boat goes 1584 miles in 24 hours; how far will it go in 1 hour?

17. There are 1575 gallons in 25 gallons are there in 1 hogshead?

Ans. 66 miles. hogsheads; how many Ans. 63 gallons.

18. There are 8316 cubic inches in 36 gallons of wine; how many cubic inches are there in 1 gallon? Ans. 231. 19. There are 7614 cubic inches in 27 gallons of beer; how many cubic inches in one gallon? Ans. 282.

20. There are 221,760 feet in 42 miles; how many feet are there in one mile? Ans. 5280.

21. Sound moves 61,545 feet in 55 seconds; how far does it move in one second? Ans. 1119 feet.

22. If a turnpike 132 miles long cost $339,240, how much did it cost per mile? Ans. $2570.

23. The salary of the President of the United States ia $50,000 a year; what is it a day? Ans. $137 nearly. 24. A man having $20,000 buys 150 acres of land, at $75 an acre; how much land can he buy with what remains, at $125 an acre? Ans. 70 acres.

CONTRACTIONS IN DIVISION.

95. Contractions in Division are abbreviated forms of

dividing.

CASE I.

OPERATION.

96. When the divisor is a composite number. 1. Divide 2952 by 24, using the factors 4 and 6. SOLUTION 1ST.-To multiply by 24 we may multiply by 6, and then multiply that product by 4; hence, to divide by 24 we may divide by 4, and then divide that quctient by 6. Dividing by 4 we have 738, and dividing 738 by 6 we have 123; hence, etc.

4)2952

6)738

123

SOLUTION 2D.-Since 24 times a number equals 6 times 4 times the number, of the number equals of of the number; of 2952 is 738, and of 738 is 123; hence, etc.

Rule.-Divide the dividend by one factor of the divisor, the quotient by another factor, and thus continue for all the factors used; the last quotient will be the quotient required.

WRITTEN EXERCISES.

Divide the following, using the factors:

TO FIND THE TRUE REMAINDER.

97. The True Remainder in successive division, it is evident, is not the last remainder, nor the sum of all the remainders; it is necessary, therefore, to explain the method of finding the true remainder.

1. Divide 791 by 24, using the factors 2, 3, and 4.

SOLUTION. Dividing by 2 we find that 791 equals 395 twos and 1 remaining; dividing 395 twos by 3, we find 395 twos equals 131 sites and 2 twos, or 4, remaining; dividing by 4, we find that 131 sixes consists of 32 twenty-fours and 3 sixes, or 18, remaining. Hence the true remainder is 18+4+ 1, which is 23. Hence, to find the correct remainder we have the following

OPERATION.

1

2)791
3)395
4)131, 2 twos = 4

32, 3 sixes=18

True remainder, 23.

Rule. Multiply each remainder by all the divisors preceding the one which obtained it, and take the sum of the products and the remainder arising from the first division Divide the following and find the true remainder:

98. When there are ciphers at the right of the divisor.

1. Divide 8254 by 600.

SOLUTION.-6 hundreds are contained in 82 hundreds 13 times, and 400 remaining; 600 is not contained in 54, hence the entire remainder is 400+54, or 454. From this solution we may derive the following

OPERATION.

600)8254 13-454

Rule.-I. Cut off the ciphers at the right of the divisor, and as many terms at the right of the dividend.

II. Divide the remaining part of the dividend by the remaining part of the divisor.

III. Prefix the remainder to the part of the dividend cut off, and the result will be the true remainder.

NOTES.-1. When the divisor is a unit of any order with ciphers, the remainder will be the figures cut off at the right, and the quotient the dgures at the left.

2. When the part of the divisor at the left of the naughts is greater than

EXERCISE UPON THE PARENTHESIS.

99. The Parenthesis (), denotes that the quantities included are to be subjected to the same operation; thus, (8+6-4)×3 denotes that the value of 8+6-4, which is 10, is to be multiplied by 3.

1. What is the value of (12+9—7)×5?

SOLUTION.-12+9 equals 21, and 21 minus 7 equals 14, and 14 mul iplied by 5 equals 70. Therefore, etc.

Ans. 19. Ans. 24032.

Ans. 16.

Ans 158730.

Ans. 407028.

9. Of (320-98) x (860-145). 10. Of (689-327+986-397) × 428. 11. Of (729+487-244)÷(247-210+71). 12. Of (3014-2601)× (2477—1325)÷(295÷5).

Ans. 9.

Ans. 8064.

Ans. 912.

13. Of (2247+349–480)÷(3411—2882)+227 × 4.

WRITTEN EXERCISES.

ON THE FOUR FUNDAMENTAL RULES.

1. The minuend is 4160, and the subtrahend is 3425; what is the remainder? Ans. 735.

2. The minuend is 9164 and the remainder is 3426; what is the subtrahend? Ans. 5738.

3. The subtrahend is 3872 and the remainder 4648; what is the minuend?

Ans. 8520.

4. The multiplicand is 745 and the multiplier 456; what is the product?

Ans. 339720.

5. The multiplicand is 2463 and the product 854661; what is the multiplier?

Ans. 347.

6. The product is 881919 and the multiplier 981; what is the multiplicand?

Ans. 899.

7. The dividend is 518077 and the divisor 763; what is the quotient? Ans. 679. 8. The dividend is 801222 and the quotient 3257; what is the divisor? Ans. 246. 9. The divisor is 587 and the quotient 8723; what is the dividend? Ans. 5120401. 10. The dividend is 72987 and divisor 45; required the quotient and remainder. Ans 1621; 42.

11. The dividend is 7972, the quotient is 274, and remainder 26; what is the divisor?

Ans. 29.

12. The divisor is 26, the quotient 372, and remainder 23; what is the dividend?

Ans. 9695.

13. Thomas read 789 pages of history in a week, which lacks 324 of being as many as Walton read; how many did both read? Ans. 1902 pages.

14. A freight car ran 365 miles one week, and 3 times as far, lacking 246 miles, the next week; how far did it run the second week? Ans. 849 miles.

15. A sold 8318 bushels of wheat, then bought 2514 bushels, and then had 3146 bushels; how many bushels had be at first? Ans. 8950 bushels.

16. My barn cost $3156; my house cost 3 times as much as my barn, and my farm cost as much as both; what was the cost of all? Ans. $25,248.