84.

CASE II.

94. To divide a number into equal parts.

MENTAL EXERCISES.

1 What is one of the 4 equal parts of 24?

SOLUTION.-One of the 4 equal parts of 24 is 6, since 6 taken 4 times is

2. Find one of the 4 equal parts of 12; of 20; of 36; of 48; of 32; of 40; of 44.

3. Find one of the 5 equal parts of 15; of 25; of 30; one of the 6 equal parts of 42; of 48; of 60; of 72.

4. Find one of the 7 equal parts of 35; of 56; of 63; one of the 8 equal parts of 40; of 72; of 88; of 96.

5. What is one-third of anything? Ans. One third of anything is one of the 3 equal parts into which it can be divided.

6. What is one-half of anything? one-fourth? one-fifth? one-sixth? >ne-seventh? one-eighth? one-ninth? one-tenth?

7. What is one-fourth of 24 dollars?

SOLUTION.-One-fourth of $24 is $6, since $6 taken 4 times equals $24.

8. What is one-fourth of 20? of 36? of 48? of 56? of 72?

12. If 6 slates cost $1.20, how much is that apiece?

SOLUTION.-If 6 slates cost $1.20, one slate cost one-sixth of $1.20, which is 20 cents.

13. If 8 pairs of gloves cost $2.40, how much is that a pair? 14. If 6 cords of wood cost $48, what is the price per cord?

15. What do I pay a bushel for apples if 12 bushels cost me $4.80? 16. What does a lady pay a yard for lace if 12 yards cost her $7.201

WRITTEN EXERCISES.

1. Divide 235 into 5 equal parts.

SOLUTION 1ST.-If we divide 235 into 5 equal parts, each part is of 235: of 23 tens is 4 tens and 3 tens remaining; 3 tens and 5 units equal 35; of 35 is 7; hence of 235 is 47, or 47 is one of the 5 equal parts of 235.

OPERATION.

5)235
47

SOLUTION 2D.-One of the five equal parts of 5 is one, hence one of the 5 equal parts of 235 is as many times one as 5 is contained times in 235, which are 47. Therefore, etc.

2. Divide 212 into 4 equal parts.

3. Divide 222 into 6 equal parts.

Ans. 53.

Ans. 37.

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? Ans. 66 miles. 17. There are 1575 gallons in 25 hogsheads; how many gallons are there in 1 hogshead? 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; now 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 is $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 quotient by 6. Dividing by 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 sixes and 2 two, or 4, remaining; dividing by 4, we find hat 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.

2)791
3)395

1

4)131, 2 two8 = 4

32, 3 sixes=18

True remainder, 23.

Rule. Multiply each remainder by all the divisors pre ceding 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.

OPERATION.

6|00)82|54 13-454

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 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 figures at the left.

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

12, divide by long division.

2. Divide 876 by 50.

3. Divide 953 by 400.

Ans. 17; Rem. 26.

Ans. 2; Rem. 153.

4. Divide 1733 by 500. 5. Divide 2765 by 700. 6. Divide 7859 by 800. 7. Divide 9763 by 900. 8. Divide 14873 by 1900. 9. Divide 25075 by 2300. 10. Divide 187654 by 14700. 11. Divide 269856 by 237000. 12. Divide 5767220 by 4730000.

Ans. 3; Rem. 233. Ans. 3; Rem. 665. Ans. 9; Rem. 659. Ans. 10; Rem. 763. Ans. 7; Rem. 1573. Ans. 10; Rem. 2075.

Rem. 11254.

Rem. 32856.

Rem. 1037220.

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.

The vinculum, thus 8+6−4×3, is often used in place of the parenthesis.

1. What is the value of (12+9-7)X5?

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

Required the value

2. Of (25+23-18)×7.
3. Of (46+97-82) x 9.
4. Of (98-75+87) × 14.
5. Of (89+96-47)÷6.
6. Of (145-110+117)÷8.
7. Of (396-128+483) × 32.
8. Of (860+980-1120)÷45.

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)+59.

Ans. 210.

Ans. 549.

Ans. 1540.

Ans. 23. Ans. 19. Ans. 24032.

Ans. 16.

Ans. 158730.

Ans. 407028.

Ans. 9.

Ans. 8064.

NOTE. In a series of numbers connected with symbols, the sign X denotes the closest connection, the sign next, thus, 12+8÷2—5×2 =6; also, 16÷4×2=2, rather than 8.