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MULTIPLICATION OF DECIMALS.

209. Multiplication of Decimals is the process of finding the product when one or both factors are decimals.

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9. How many places in the product of units multiplied by tenths! Tenths by tenths? Units by hundredths? Tenths by hundredths? Hundredths by hundredths?

10. How many decimal places in the product if there are two in the multiplicand and one in the multiplier? If there are two in each factor? If three in multiplicand and two in multiplier?

WRITTEN EXERCISES.

1. Multiply 7.23 by .46.

SOLUTION.-7.23 multiplied by 46 equals 332.58, and multiplied by 46 hundredths the product is 1 hundredth as great, which, by removing the decimal point two places to the left, becomes 3.3258. Hence, 7.23 multiplied by .46 equals 3.3258.

OPERATION.

7.23

.46

4338

2892

3.3258

SOLUTION 2D.-7.23.46-788×45=$8888=1000033258-3.3258. From either of these solutions we derive the following

Rule.-Multiply as in whole numbers, and from the right of the product point off as many decimal places as there are in both factors, prefixing ciphers when necessary.

2. Multiply 27.13 by .67. 3. Multiply 43.08 by 2.36. 4. Multiply 79.52 by .019. 5. Multiply 8.534 by 20.074. 6. Multiply 123.107 by 1.52. 7. Multiply 512.073 by 35.08. 8. Multiply 54.0079 by 7.072. 9. Multiply 1.08096 by 3.5702. 10. Multiply .03507 by .005873. 11. Multiply 2.0709 by .000246. 12. Value of 9.08×10?

Ans. 18.1771. Ans. 101.6688.

Ans. 1.51088. Ans. 171.311516.

Ans. 187.12264. Ans. 17963.52084. Ans. 381.9438688.

Ans. 3.859243392. Ans. .00020596611. Ans. .0005094414.

Ans. 8.3125.

13. Value of .5 of 7.05?

14. Value of .07 of × 300×.09 ?

15. Value of $25.67×.05}+$15×6.031?

Ans. .0175.

Ans. 1.65.

Ans. $91.8575.

16. 200+.02 − × .81+ (.81 − .12 × 3 of 71)? Ans. 164.7432.

17. Multiply 1 hundredth by 1 thousandth, and add 1 tenth to the product. Ans. .10001. 18. What is the product of one-tenth by one-tenth? one hundred by one-hundredth? one million by one-millionth ?

DIVISION OF DECIMALS.

210. Division of Decimals is the process of finding the quotient when one or both terms are decimals.

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7. .84? 8.4? 8.04? 8. 7.2.6? .72÷.6? .72÷.06? 9. 8.4.7? .84.7? .84÷.07? 10. 96.8? 9.6÷.8? .96÷.8? What then is the

11. What is the product of tenths by tenths? quotient of hundredths by tenths? What is the product of tenths by hundredths? What is the quotient of thousandths divided by tenths! 12. If there are three decimal places in the multiplicand and two in the multiplier, how many in the product? If there are five in the dividend and three in the divisor, how many in the quotient?

WRITTEN EXERCISES.

1. Divide 12.3084 by 2.34.

SOLUTION.-Dividing as in whole numbers, we obtain the quotient 526; now, since the dividend is the product of the divisor and quotient, the number of decimal places in the dividend should equal the number in the divisor, plus the number in the quotient; hence the number of decimal places in the quotient equals the number of decimal places in the dividend, minus the number of places in the

OPERATION.

2.34)12.3084(5.26 1170

608

468

1404

1404

divisor; hence there should be four minus two, or two decimal places in the quotient, therefore the quotient is 5.26.

100X234

SOLUTION 2d.-12.30842.34-128084 ÷ 284—128084×100=1230; =100x123004=100 of 526=5.26. From either of these solutions we derive the following

Rule.-I. Annex ciphers to the dividend, if necessary to make the number of decimals equal to the number of decimal places in the divisor.

II. Divide as in whole numbers, annexing ciphers to the dividend when needed to continue the division.

III. Point off as many decimals in the quotient as the number of decimal places in the dividend exceeds the number in the divisor.

NOTES.-1. When there are ciphers at the right of the divisor, cut them off, divide by the significant part, and then point off as many decimal places as before, pius the number of ciphers cut off.

2. Make complex decimals pure or divide them like common mixed numbers, or multiply both by the L. C. M. of the denominators, and then divide.

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16. .004 2.5? Ans. .0016. 21. 25.00625. Ans. 4000,

22. (2.04÷17+47÷200×5000)—}?

Ans. 1174.245.

23. (789-.789)÷(.75—.075×.75 of 8)? Ans. 194.62. 24. 45.181+3.6×.3÷(3.5—3.48)? Ans. 8.543611† 25. ($347.84 $10.87)x.0025+.01 × 50? Ans. 32. 26. ($1080 x 3.27)+$10.90-($790÷$3.95)? Ans. 124.

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6. Multiply 1.25 of .8 of 9o by .8 of 1 of 8.75.

.2

Ans. 3075

7. Divide 118 by .04; also 2.4001g by 1.56

1.25
61

256

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241 28

9. Divide £2401 by § of 24 of 4 of 14. Ans. £285}.

10. What number multiplied by 7 will give 6 for a product? Ans.

11. Divide seven millionths by twelve and a half ten-mil. lionths.

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Ans. 5.6.

12. Add 1, .375, and .5, and multiply the sum by 6.21.

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14. Multiply the sum of.5 and 3 by the difference be

tween and

81 10

17 26

621 1000

14

Ans. 1.

15. Multiply by 25 millionths, and divide the pro

dact by 125 hundred-thousandths.

Ans. .00125.

16. The product of two numbers is, and one of them is of of 2; what is the other?

Ans. 1

17. Add 2.5 cwt., 1.25 cwt., 5 cwt., 4.375 cwt., 1.875 cwt., 2.5 cwt., 1.225 cwt., and 1.275 cwt. Ans. 20 cwt.

18. What number multiplied by 3 of 1 of 5 is equal to of 27? Ans. 86.

of

19. Divide the sum of six thousandths and six milliontha by their difference, to 6 decimal places. Ans. 1.002002 +. 20. From the sum of,, and, take the remainder ob tained by subtracting from of 5 of 4.

21. What number must be divided by make a quotient equal to the value of 63 of

Ans. 188.

of 24 of

1층

10

5

18

of 6 ?

Ans. 24.

times $88 more; he had $186§

22. John had $225 and earned of 3 having lost part of his money, he found that remaining; what amount did he lose?

Ans. $978.

23. A ton of iron ore from the mines of Cornwall yields .65 of a ton of pure iron; how much iron will 578.8 tons of ore yield? Ans. 376.22 tons.

24. An oil refiner has on hand 16,745 gallons of coal oil; how many casks, each containing 42.5 gallons, can be filled with it? Ans. 394 casks.

25. How many dress patterns of 12.25 yards each, can be cut from a piece of French percale, containing 50 yards? Ans. 4+ patterns.

26. An engine pumped 41.25 barrels of 31.5 gallons each from a tank containing 1500 gallons; how many gallons remained? Ans. 200.625 gal.

27. I bought 3 loads of wood, the first containing 1.02 cords, the second 1.05 cords, the third .945 cords; what did it cost at $3.60 a cord? Ans. $10.854.

28. A dealer bought 1086 bushels of wheat; how many bins, each holding 20.25 bushels, will it fill, and how many bushel-bags can be filled from what remains?

Ans. 53 bins; 12.75 bags. 29. Two speculators bought 4320 acres of Western land,

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