EXERCISES

For ANALYTIC AND SYNTHETIC EXPLANATION.

1. What are the prime factors of 108 ?

Process.

Explanation. 2 108

Since the prime factors of a number are 2 54

exact divisors of the number, we find all the

prime numbers that exactly divide 108. 108 3 27

being an even number, is divisible by 2; 54 3 9

being even, is divisible by 2. Dividing by 3

3, and again by 3, the quotient is 3, a prime

number. Hence the prime factors of 108 are 108= 2 X 2 X 3 X

2, 2, 3, 3, 3. 3 X 3.

Note.In finding the prime factors of a number, use the prime numbers as divisors in order of their values, beginning with the lowest one that will divide the given number.

2. What are the prime factors :

1. Of 102, 105, 108, 221, 715, 845.
2. Of 84, 250, 735, 9800, 11,165.
3. Of 231, 78, 415, 852, 452, 1227.

REVIEW OF PRINCIPLES.

[See paye 72.] 1. Factor 54, and illustrate Principle 1. 2. Factor 36, and illustrate Principle 2. 3. Factor 108, and illustrate Principle 3. 4. Factor 144, and illustrate Principle 4. 5. Factor 231 and 154, and illustrate Principle 5. 6. Factor 360 and 320, and illustrate Principle 6. 7. Factor 1728, and illustrate Principle 7.

MULTIPLICATION BY FACTORS.

1. What will 24 carriages cost at $257 each ?

Process. 24 = 4 X 6; $257 X 6 = 1542; 1542 X 4= $6168. 2. In like manner find the cost of:

1. 35 cows at $53 each.
2. 22 violins at $10.35 each.
3. 72 cords of wood at $4.65 a cord.

a

4. 99 books at $2.18 apiece.
5. 123 hats at $5.65 a piece.
6. 51 acres of land at $125 an acre.
7. 49 barrels at $1.25 apiece.
8. 63 bags of salt at $1.875 a bag.
9. 21 shot-guns at $55.50 apiece.
10. 121 paper-weights at $.555 apiece.
11. 132 horses at $132 a head.
12. 144 rifles at $87.50 apiece.
13. 34 yards of silk at $2.56 a yard.
14. 81 bushels of wheat at $.95 per bushel.

DIVISION BY FACTORS.

When the divisor is a composite number, division may sometimes be readily performed by using factors of the divisor

1. Divide 3598 by 14.

Process.

Explanation. 23598

The factors of 14 are 2 and 7. Dividing 3598 into two 7 | 1799

equal parts, and each of those 2 equal parts into 7 equal

parts, we thus obtain 7 times 2 or 14 equal parts, each 257

equal to 257. 2. Divide, using factors : 1. 8445 ; 15.

7. 9345 ; 105. 2. 7776 ; 24.

8. 1152 ; 72. 3. 23,296 = 32.

9. 3648 ; 96. 4. 1152 ; 64.

10. 42,336 · 49. 5. 1855 ; 35.

11. 37,464 ; 42. 6. 16,340 = 38.

12. 153,160 = 56. The chief difficulty in dividing by factors is to find the true remainder. Notice the following explanation :

:

3. Divide 4753 by 140, using factors. Process.

Explanation. 4 4753

140 = 4 X 5 X 7. 5 1188 1 unit of 4753 remaining. 7 237 . . 3 units of 1188 remaining = 3 X 4 : 12 units of 4753. 33 . . 6 units of 237 remaining = 6 X 5 X 4 120 units of 4753

1 + 12 + 120 = 133, true remainder.

The true remainder must be a part of 4753. Why must partial remainder 3 be multiplied by 4? Why must partial remainder 6 be multiplied by 5 X 4 ? 4. Divide, using factors : 1. 7301 by 24.

5. 2184 by 49. 2. 4104 by 45.

6. 3824 by 32. 3. 3276 by 27.

7. 3548 by 72. 4. 3275 by 56.

8. 1299 by 56.

CANCELLATION.

Cancellation abridges the process of division by striking out a common factor from dividend and divisor.

Striking out a common factor is in effect dividing both dividend and divisor by the same number. [State the principle, page 68.] 1. Divide (96 X 9 X 8) by (12 X 16). Process.

Explanation. 4

Cancelling 12 and 96, we have 8

2. Divide 4 X 2 X 8 X 21 by 36 X 8 X 2. Process.

Explanation. 7

We first cancel 2 and 8 in both divi4 x 2 x 8 x 21 7 dend and divisor; next, 4 and 36, obtain36 x 8 x 2 3

ing 9 in the divisor; finally, we cancel 9 9

and 21, rejecting from each the factor 3,

and obtain 3 in the divisor and 7 in the 3

dividend. 3. Divide, using cancellation :

1. 18 x 24 x 35 by 6 X 8 X 7.
2. 30 X 10 X 9 X 4 by 8 X 5 X 6.
3. 6 X 7 X 9 X 11 by 2 X 3 X 7 X 3 X 21.
4. 10 X 6 X 84 X 42 by 12 X 5 X 24 X 7.
5. 144 x 75 X 15 X 32 X 23 by 432 X 25 X 8 X 30.
6. 400 X 18 X 30 X 42 by 270 X 20 X 30 X 14.
7. 28 x 56 x 400 by 112 x 280.

PROBLEMS.

1. A farmer exchanged 15 barrels of apples, each containing 3 bushels, at $.80 a bushel, for 8 pieces of cloth, each containing 30 yards. Find the price of the cloth per yard.

Process.

.10
13 X 3 X .80 3 x .10 .30

.15.
8 X 30

2

2 2

Explanation. The farmer had 15 times 3 bushels of apples worth 15 times 3 times $.80. He got therefor 8 times 30 yards of cloth, which cost him as much per yard as 8 X 30 is contained times in 15 X 3 X .80. By cancellation we obtain 15 cents as the cost per yard.

2. A miller bought 12 loads of wheat, each containing 130 bushels, at $1.25, and gave in exchange 8 loads of flour at $6.25 a barrel. How many barrels were there in a load?

a