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PROCESS.

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5. What will 20 carriages cost at $346 each?

ANALYSIS.–Since 20 is 5 times 4, 20 $346 cost of 1 carriage.

carriages will cost 5 times as much as 4

4 carriages. 4 carriages wiil cost 4

times $346, or $1384, and 20 carriages $1384 cost of 4 carriages.

will cost 5 times as much as 4 carriages, 5

or 5 times $1384, which is $6920. $ 69 20 cost of 20carriages. Hence, 20 carriages will cost $6920.

RULE. Multiply the multiplicand by one factor of the multiplier, the product thus obtained by another factor, and so continue until all the factors have been used successively as multipliers. The las; product will be the product sought. Multiply in same manner, using the factors of the multiplier: 6. 425 by 32; by 36; by 48; by 72. 7. 1824 by 56; by 27; by 45; by 108. 8. What will be the cost of 35 cows at $64 each? 9. What will 21 cords of wood cost at $5.35 a cord ? 10. What will 72 yoke of oxen cost at $168 per yoke? 11. What will 36 boxes of lemons cost at $6.25

per

box? 12. What will 48 acres of land cost at $46

per

acre? 13. What will 24 paintings cost at $55 each? 14. What will 45 cases of boots cost at $36 a case ? 15. What will 56 barrels of salt cost at $2.35 a barrel?

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DIVISION BY FACTORS.

124. 1. What are the factors of 32? 25? 64? 96 ?

2. If a number is divided by 8, by what must the quotient be divided that the number

may be divided by 16? 3. If a number is divided by 8 and the quotient by 6, by what is the number divided ?

4. What factors may be used to divide a number by 36 ? 5. What factors may be used to divide a number by 48?

6. A miller put up 500 pounds of hominy in packages containing 4 pounds each, and packed them in boxes containing 10 packages each. How many packages and how many boxes did he have ? 7. Divide 888 by 24, using factors.

ANALYSIS.—24 is equal to 6 times 4. Hence to 4) 888 divide by 24 we may divide by 6 times 4. 888 :-4

222. But since we were to divide by 6 times 4, this 6) 222

quotient is 6 times too great, hence we must divide it by 37 6. 222 = 6=37 the true quotient.

PROCESS.

PROCESS.

8. Divide 5863 by 32, using factors.

ANALYSIS.—32 is equal to 4 X 4)5683

2 X 4. Dividing 5683 by 4 gives

a quotient of 1420 fours and 3 2)1420...3

units remaining. 4) 710

Dividing 1420 fours by 2 gives 177...2

a quotient of 710 eights. Dividing

710 eights by 4 gives a quotient 3+ (2 X 8)=19 true Rem.

of 177 thirty-tuos and 2 eights 177.19 Quotient.

remainder. The first partial remainder is 3 units, and the second, 2 eights, or 16 ; hence the entire remainder is 3 + 16, or 19, and the quotient is 17732.

RULE.Divide the dividend by one factor of the divisor, the quotient thus obtained by another factor, and so continue until all the factors have been used successively as divisors.

If there be remainders, multiply each remainder by all the preceding divisors except the one that produced it. The sum of these products will be the true remainder.

Divide, using factors: 9. 1704 by 24. 13. 1288 by 56. 10. 4725 by 15. 14. 3528 by 72. 11. 5710 by 28. 15. 3824 by 32. 12. 1428 by 42. 16. 2184 by 49.

17. 3275 by 56. 18. 3276 by 27. 19. 4104 by 45. 20. 7304 by 24.

21. A wholesale grocer put up 1120 pounds of tea in 35-pound packages, containing 5-pound canisters. How many packages and canisters were there?

22. A paper manufacturer put up his paper so that each quire contained 4 packages of 6 sheets each. How many packages and quires were made up from 912 sheets ?

CANCELLATION.

123. 1. How many times is 2 times 5 contained in 4 times 5.? 2 times 7 in 4 times 7? 2 times 9 in 4 times 9 ? 2 times 24 in 4 times 24? 2 times any number in 4 times that number?

2. How many čimes is 4 times 8 contained in 12 times 8? 4 times 25 in 12 times 25 ? 4 times 75 in 12 times 75 ? 4 times any number in 12 times the same number?

3. How many times is 6 x 48 contained in 24 X 48? 6 X 144 in 24 X 144 ?

4. In determining the quotient, what numbers may be omitted from both dividend and divisor?

126. Cancellation is the process of shortening computations by rejecting equal factors from the dividend and divisor.

127. PRINCIPLE. — Rejecting equal factors from both dividend and divisor does not alter the value of the quotient. 1. Divide 66 times 36 by 24 times 11.

ANALYSIS. We write 66 X 36 11X6 X 4 X3 X3

PROCESS.

the numbers as in divis

9 ion, the dividend above, 24 X 11 6 X 4 X 11

the divisor below a line. instead of multiplying 66 by 36 we resolve 66 into its factors 11 and 6, and 36 into its factors 4, 3 and 3, and in the divisor resolve 24 into the factors 6 and 4.

PROCESS.

=

Cancelling equal factors from both dividend and divisor, which is the same as dividing both by the same number, and does not alter the value of the quotient, we have remaining in the dividend the factors 3 and 3, or 9, which is the quotient. 2. Divide 72 X 66 X 49 by 63 X 40 X 21.

ANALYSIS.—We write the nuni8 22 7

bers as before. Since 9 is a factor 72 X 66 X 49 22

of both 72 and 63 it may be rejected

4. from both, leaving 8 instead of 72 in 63 X 40 X 21 5

the dividend, and 7 instead of 63 in 7 5 3

the divisor. We next cancel 8 from 8 and 40, leaving 5 instead of 40 in the divisor. We next cancel 7 from 7 and 49, leaving 7 instead of 49 in the dividend, and 7 again from 7 and 21, leaving 3 instead of 21. Rejecting the factor 3 from both 66 and 3, there is left for a dividend 22, and for a divisor 5, which gives a quotient of 4.

RULE.- Reject from the dividend and divisor all factors common to both, and then divide the product of the remaining factors of the dividend by the product of the remaining factors of the divisor.

When all the factors of both dividend and divisor are cancelled, the quotient is 1, for the dividend will then exactly contain the divisor

once.

EXAMPLES.
Divide, using cancellation:

3. 7 X 5 X 3 X 11 by 5 x 11 x 3.
4. 12 X 14 by 6 X 7 X 2.
5. 6 X 3 X 5 X 2 by 3 X 5 X 2 X 2.
6. 4 X2 X 8 X 24 by 36 X 8 X 2.
7. 24 X 32 by 8 X 6 X 4.
8. 45 X 60 X 7 by 49 X 12 X 9.
9. 2 X3 X5 X 8 X7 by 6 X 5 X 2 X 7.
10. 5 X 8 X 12 X 6 by 20 X 16 x 2.
11. 12 X 60 X 36 X 35 by 7 X 30 X 18 X 24.
12. 30 x 49 x 64 x 25 by 35 X 15 X 24.

13. Divide the product of 26 times 18 times 35, by 78 times 30.

14: Find the quotient of 99 times 360 times 365, divided by 11 times 72.

15. Find the quotient of 175 X 28 X 72 times 363, divided by 12 X 11 x 9.

16. Four farms containing 80 acres each, worth $65 per acre, were exchanged for 5 farms containing 95 acres each. What was the value per acre of the farms received in exchange?

17. A farmer buys 3 pieces of muslin each containing 44 yards at 11 cents a yard, and pays for it in wheat at $2 per bushel. How many bushels are required ?

18. A merchant bought 13 tubs of butter, each containing 39 pounds, at 32 cents a pound, paying for it in 4 patterns of silk of 13 yards each. How much was the silk a yard?

COMMON DIVISORS.

128. 1. What numbers will exactly divide 12? 15? 20?

2. What numbers will exactly divide both 12 and 15?. 15 and 20? 24 and 48? 63 and 72?

3. What numbers will exactly divide both 12 and 24? What is the largest number that will exactly divide them?

4. What is the largest number that will exactly divide both 15 and 30? 16 and 32? 16 and 24? 24 and 32?

5. Name all the divisors common to 15 and 30.

6. Name all the prime divisors or factors common to 15 and 30?

7. How is the greatest divisor common to 15 and 30 found from the prime factors of those numbers?

8. What is the greatest divisor common to 24 and 30?

9. How is the greatest divisor common to 24 and 30 obtained from the prime factors of those numbers?

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