469. PROB. III.—To multiply by 9, 99, 999, etc.

Move the decimal point in the multiplicand as many places to the right as there are nines in the multiplier, annexing ciphers when necessary, and subtract the given multiplicand from the result.

Observe that by moving the decimal point as directed, we multiply by a number 1 greater than the given multiplier; hence the multiplicand is subtracted from the result. To multiply by 8, 98, 998, and so on, we move the decimal point in the same manner, and subtract from the result twice the multiplicand.

Perform the following multiplication :

470. PROB. IV.—To divide by 10, 100, 1000, etc.

Move the decimal point in the dividend as many places to the left as there are ciphers in the divisor, prefixing ciphers when necessary.

Perform the division in the following:

471. PROB. V. To divide where there are ciphers at the right of the divisor.

Move the decimal point in the dividend as many places to the left as there are ciphers at the right of the divisor, prefixing ciphers when necessary (140), and divide the result by the significant figures in the divisor (142).

472. PROB. VI. To multiply one fraction by another.

Cancel all factors common to a numerator and a denominator before multiplying (185—II).

Perform the following multiplications by canceling common

473. PROB. VII.-To divide one fraction by another.

Cancel all factors common to both numerators or common to both denominators before dividing (291). Or,

Invert the divisor and cancel as directed in Prob. VI.

Perform the division in the following, canceling as directed:

474. PROB. VIII.—To divide one number by another.

Cancel the factors that are common to the dividend and divisor before dividing (185-II).

Perform the following divisions, canceling as directed:

ALIQUOT PARTS.

475. An Aliquot Part of a number is any number, integral or mixed, which will exactly divide it.

Thus, 2, 21, 34, are aliquot parts of 10.

476. The aliquot parts of any number are found by dividing by 2, 3, 4, 5, and so on, up to 1 less than the given number. Thus, 100-250; 100÷333; 100 ÷ 4 = 25. Each of the quotients 50, 33, and 25, is an aliquot part of 100.

477. The character @ is followed by the price of a unit or one article. Thus, 7 cords of wood @ $4.50 means 7 cords of wood at $4.50 a cord.

478. Memorize the following aliquot parts of 100, 1000, and $1.

479. PROB. IX. To multiply by using aliquot parts.

1. Multiply 459 by 331.

3) 45900

15300

EXPLANATION.-We multiply by 100 by annexing two ciphers to the multiplicand, or by moving the decimal point two places to the right. But 100 being equal to 3 times the multiplier 33, the product 45900 is 3 times as large as the required product; hence we divide by 3.

Solve the following examples orally, by aliquot parts.

11. What cost 48 lb. butter @ 25 ct.? @ 50 ct.? @ 331 ct.? SOLUTION.-At $1 a pound, 48 would cost $48. Hence at 331 cts. a pound, which is } of $1, 48 pounds would cost } of $48, which is $16.

12. What cost 96 lb. sugar @ 12 ct.? @ 14 ct.? @ 16 ct.? 13. What is the cost of 24 bushels wheat @ $1.331?

SOLUTION.-At $1 a bushel, 24 bushels cost $24; at 331 ct., which is of $1 a bushel, 24 bushels cost $8. Hence at $1.33 a bushel, 24 bushels cost the sum of $24 and $8, which is $32.

14. What cost 42 yards cloth @ $1.16? @ $2.14?

15. What cost 72 cords of wood @ $4.12? @ $3.25?

Find the cost of the following, using aliquot parts for the cents in the price.

16. 834 bu. wheat @ $1.331; @ $1.50; @ $1.25; at $1.16. 17. 100 tons coal @ $4.25; at $5.50; @ $6.121; @ $5.33. 18. 280 yd. cloth @ $2.14; @ $1.121; @ $3.25; @ $2.50. 19. 150 bbl. apples @ $4.20; @ $4.50; @ $4.331.

20. 2940 bu. oats @ 33 ct.; @ 50 ct.; @ 25 ct.
21. 896 lb. sugar @ 124; @ 144; @ 16.
22. What is the cost of 2960 yd.

cloth at 37 ct. a yard?

EXPLANATION.-At $1 a yard, 2960 yd. will cost $2960. But 25 ct. is of $1, hence of $2960 which is $740, is the cost at 25 ct. a yd.

2. Again, 123 ct. is the of

25 ct., hence $740, the cost at 25 cts., divided by 2, gives the cost at 12 ct., which is $370. But 25 ct. + 12 ct. = 37, hence $740 + $370 or $1110 is the cost at 37

ct.

23. 495 bu. barley @ 75 ct.; @ 621 ct.; @ 871 ct. 24. 680 lb. coffee @ 37 ct.; @ 75 ct. ; @ 60 ct. 25. 4384 yd. cloth @ 12 ct.; @ 15 ct.; @ 30 ct.; @ 35 ct. Observe, that 10 ct. = of 100 ct., and 5 ct. = of 10 ct.

26. 870 lb. tea @ 60 ct.; @ 62 ct.; @ 80 ct.; @ 87 ct.

480. PROB. X.—To divide by using aliquot parts.

1. Divide 7258 by 331.

72.58

3

217.74

EXPLANATION.-1. We divide by 100 by moving the decimal point two places to the left.

2. Since 100 is 3 times 331, the given divisor, the quotient 72.58 is only of the required quotient; hence we multiply the 72.58 by 3, giving 217.74, the required quotient.

Perform by aliquot parts the division in the following:

2. 8730÷31.

3. 9764÷5.

4. 8.375-16.

5. 379.6÷331.

6. 98.5450.

7. 394.8125.

8. 460.85250.

9. 90.63825. 10. 73096-3331.

Solve the following examples orally, using aliquot parts. 11. At 33 ct., how many yards of cloth can be bought for $4?

SOLUTION.-Since $1, or 100 ct., is 3 times 33 ct., we can buy 3 yards for $1. Hence for $4 dollars we can buy 4 times 3 yd., which is 12 yd.

Observe, that in this solution we divide by 100 and multiply by 3, the number of times 33, the given price, is contained in 100. Thus, $4=400 ct., 400+100-4, and 4×3=12. In the solution, the reduction of the $4 to cents is omitted, as we recognize at sight that 100 ct., or $1, is contained 4 times in $4.

12. How many yards of cloth can be bought for $8 @ 124 ct. ? @ 14 ct.? @ 33 ct.? @ 16 ct. ? @ 25 ct.? @ 10 ct.? @50 ct.? @ 8 ct.? @ 5 ct.? @ 4 ct. ?

13. How many pounds of butter @ 334 ct. can be bought for $7? For $10? For $40?

14. How much sugar can be bought at 12 ct. per pound for $3? For $8? For $12? For $30? For $120 ?