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WRITTEN EXERCISES.

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1. From 34.634 take 5.6857. PROCESS.

ANALYSIS.— We write the numbers so that units 34.6340 of the same order shall stand in the same column,

5.68 57 and subtract as in integers, separating the decimal 28.9483

part of the remainder from the integral part by a

decimal point. Or,

In the first process the decimals are made simi

lar by annexing a cipher to the minuend. 34.6 34

In the second process, which is the one com5.6857

monly employed, the cipher is not written, but we 28.9483

suppose it to be annexed. RULE.-The rule is the same as for subtraction of integers.

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(2.) From 48.356 Subtract 23.453

(3.) 39.82 13.856

(4.)
$43.25
$18.375

(5.)
$118.375
$ 43.50

9.

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6. What is the difference between .7134 and .50645 ?
7. What is the difference between 8.34 and 6.3168 ?.
8. What is the difference between 100 and .03846?
9. From 84 millionths take 84 ten-millionths.
10. From 80 thousand take 80 thousandths.
11. From 29 dollars 3 cents take 17 dollars 9 cents.
12. From 27 dollars 8 cents take 9 dollars 37 cents 5 mills.

13. If I spend $45.892 for merchandise, how much will I have left after paying for it with a fifty-dollar bill?

14. A gentleman's income was $12384.16, and his expenses the same year were $9864.18. How much of his income was left?

15. The receipts of a reaper manufactory for the year 1876 were $1374837.64, and the expenditures $1298369.58. What was the surplus ?

MULTIPLICATION.

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212. 1. What is the product of 10 X 2? * X3? .4 X 2?

2. How many decimal figures are there in the product of tenthis by units?

3. What is the product of 16 X 4? To X 42.04 X 2?

4. How many decimal figures are there in the product of hundredths by units?

5. What is the product of 4 X iç? 16X? .AX.2?

6. How many decimal places are required to express the product of tenths multiplied by tenths ?

7. What is the product of * X 18? Bo X 167? .4X.02?

8. How many decimal places are required to express the product of tenths multiplied by hundredths ? Tenths by thousandths? Hundredths by thousandths?

9. If the multiplicand has two decimal places, and the multiplier three, how many will there be in the product?

10. How does the number of places required to express the product of two decimals compare with the number of decimal places in the factors ?

213. PRINCIPLE.—The product of two decimals contains as many decimal places as there are decimal places in both factors.

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WRITTEN EXERCISES.

PROCESS.

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1. Multiply .312 by .24.

ANALYSIS.—.312 X.24=1366 X 36 =171887 .312 .07488. Hence .312 X 24 = .07488. Or, .24

We may multiply as if the numbers were integers;

and since the multiplicand has three decimal places, 1248

and the multiplier two places, the product must have 624 five places. (Prin.) Or, thousandths multiplied by .07488

hundredths give hundred-thousandths, the denomi. nation of the product.

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RULE.--Multiply as if the numbers were integers, and from the right of the product point off as many figures for decimals as there are decimal places in both factors.

If the product does not contain as many figures as there are decimals in both factors the deficiency must be supplied by prefixing ciphers.

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Multiply:

2. .65 by..31.
3. .45 by 4.5.
4. .436 by .46.
5. 348 by .44.
6. 3.48 by .64.
7. 31.8 by .74.
8. 8.75 by 8.5.
9. .579 by .035.
10. 486 by 3.75.
11. 2.48 by 2.37.
12. 3.94 by 3.84.
13. 5384 by .0064.

Multiply:
14.

$2.75 by 84
15. $31.16 by 5%.
16. 34.165 by 3
17. 3.845 by 7.3.
18. $15.18 by .6664.
19. 500.15 by 5.36.
20. 37.856 by 30.04.
21. 70.05 by .0405.
22. 63.18 by 2.308.
23. 64.032 by .0634.
24. 51.27 by 5.321.
25. of .55 by of 6.5.

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PROCESS.

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26. Multiply 4.639 by 100.

ANALYSIS.-Since each removal of a figure one 4.639 place to the left increases its value tenfold, the 100

removal of the decimal point one place to the right

multiplies by 10, and the removal of the point two 463.900

places to the right multiplies by 100. Hence, RULE.-To multiply by 1 with any number of ciphers annexed, remove the decimal point as many places to the right as there are ciphers annexed.

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27. Multiply 384.64 by 100. By 10. By 1000.
28. Multiply 1.8465 by 100. By 1000. By 10000.
29. What will be the cost of 34.5 yards of cloth at $3.15

per yarı?

30. When land is worth $137.18 per acre, how much must be paid for a farm of 38 acres ?

31. Since 16.5 feet make a rod, how many feet are there in 23.7 rods?

32. What will be the cost of 9 houses at $3847.93 each?

33. When wheat is worth $1.621 per bushel, what will 37.3 bushels cost?

34. What is the value of 57 barrels of flour at $8.371 a barrel ?

35. Mr. Orr sold 8 horses at an average price of $213.27 each. How much did he receive for them ?

36. A lady made the following purchases, viz: 37 yards of bleached sheeting at $.13} per yard, 8 yards of velvet ribbon at $.374 per yard, 27 yards of silk at $2.35 per yard. What was the entire cost of her purchases?

DIVISION.

214. 1. What is the product of .6 by 8?

2. 4.8 is the product of two factors, one of which is 8: what is the other factor?

3. What is the product of .6 by .8?

4. .48 is the product of two factors, one of which is .6: what is the other factor?

5. What is the product of .06 by .8?

6. .048 is the product of two factors, one of which is .06. what is the other factor?

7. What is the product of .06 by .08?

8. .0048 is the product of two factors, one of which is .06: what is the other factor ?

9. How many decimal places are there in the quotient when tenths are divided by units ? Hundredths by tenths? Thousandths by hundredths? Ten-thousandths by hundredths ?

10. How many decimal places are there in the product of any two factors ?

11. If the product and one of two factors are given, how may the number of decimals in the other factor be found ?

12. How may the factor be found ?

13. Since the factor sought will be the quotient, how many decimal places will there be in the quotient ?

215. PRINCIPLE.-The quotient will contain as many decimal places as the number of decimal places in the dividend exceeds those in the divisor.

16.

WRITTEN EXERCISES.

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1. Divide 8.88 by 2.4. PROCESS.

ANALYSIS.-8.88 2.4=188 = 24=188 2.4) 8.88(3.7 X 19 = 17=3.7. Or, 72

We divide as if the numbers were inte

gers; and since the dividend has two deci168

mal places, and the divisor one, the quotient 168

will have one. (Prin.)

RULE.—Divide as if the numbers were integers, and from the right of the quotient point off as many figures for decimals as the number of decimal places in the dividend exceeds the number of those in the divisor.

1. If the quotient does not contain a sufficient number of decimal places the deficiency must be supplied by prefixing ciphers.

2. Before commencing the division, the number of decimal places in the dividend should be made at least equal to the number of decimal places in the divisor.

3. When there is a remainder after using all the figures of the dividend, annex decimal ciphers and continue the division.

4. For the ordinary purposes of business it is not necessary to carry the division further than to obtain four or five decimal figures in the quotient.

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