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33. What is the height of a steeple, whose shadow is 148 feet 4 inches, when a shadow 5 feet 4 inches long is projected from a post 6 feet 4 inches high?

How

34. A trader failed in business, owing $11000, and having only $5000 to divide among his creditors. much did he pay on a debt of $95.20 ?

35. A fox has 50 rods the start of a greyhound, but the hound runs 15 rods while the fox runs 9. How many rods must the hound run to catch the fox?

How many

36. A cubic foot of air weighs 14 ounce. pounds of air does a room contain, which is 16 feet long, 14 feet wide, and 10 feet high?

37. What number must that be, which, being increased by its half, and its third, becomes 88 ?

38. A and B hired a pasture for $30. A turned in 3 cows, and B turned in 12 sheep. Allowing 5 sheep to be equal to 1 cow, what must each pay?

39. Suppose London to contain 1 250 000 inhabitants, New York 203 000, Philadelphia 167 000, Baltimore 80 000, and Boston 61 000; how many times greater is London, than each of the other cities?

EXPLANATION OF CHARACTERS.

This character placed between numbers, shows that they are to be added together; thus, 6+4 is 10. This character between numbers, shows that the number on the right hand is to be subtracted from the number on the left hand; thus, 9-3 is 6.

× This character shows, that numbers are to be multiplied together; thus, 4 X 5 is 20.

This character shows, that the number on the left hand is to be divided by the number on the right hand; thus, 284 is 7.

This character shows, that the number on the right is equal to the number on the left; thus, 4+8=12 8—3—5. 7X3=21. 32÷4=8.

40. 1426026+706241 +360515 how much?
41. 514728650+643-426-how much?
42. 40673+62—40275 × 58=how much?
43. 8729516-1430 X 16+ 12 how much?

DECIMAL FRACTIONS.

SECTION 33.

A decimal fraction is a fraction whose denominator is 10, or 100, or 1000, &c. The denominator of a decimal fraction is never written: the numerator is written with a point prefixed to it, and the denominator is understood to be a 1, with as many ciphers annexed, as there are figures in the numerator. Thus, .3 is; .31 is 31 13.316 is 1000; .3164 is 3164

316

10000

1. Write upon the slate, the decimals expressing the following fractions. 10. 100 1000 10000 100000 708

46

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1642

96041

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When a whole number and a decimal are written together, the decimal point is placed between them. Thus, 24.6 is 24; 5.71 is 57; 48.364 is 48364

71

1000

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2. Write the following mixed numbers, expressing the fractions decimally. 38. 516-22. 8354. 24106366.

It is observable in our system of writing whole numbers, that any figure, wherever it may stand, expresses a quantity as great as it would express, if it were written one place further to the left. For instance, in the number 1111, the 1 hundred is of a thousand; the 1 ten is of a hundred, or of a thousand; the 1 unit is of a ten, or 10 of a thousand. In decimals, this system is continued below the place of units. For instance, in the number 1.111, the 1 next to the right of the unit is 1-tenth, that is, of a unit; the 1 next to the right of the 1-tenth is of a tenth, or 1-hundredth of a unit; the 1 next to the right of the 1-hundredth, is of a hundredth, or 1-thousandth of a unit.

one tenth one hundredth one thousandth

1.11 1

Ciphers placed on the right hand of decimal figures, do not alter the value of the decimal; because, the fig ures still remain unchanged in their distance from the unit's place. For instance, .5, .50, and .500 are of equal value, they are each equal to . But every cipher that is placed on the left of a decimal, renders its value ten times smaller, by removing the figures one

place further from the unit's place. Thus, if we prefix one cipher to .5, it becomes .05 [150]; if we prefix two ciphers, it becomes .005 [100]; if we prefix three ciphers, it becomes .0005 [1500]; and so on.

3. Write upon the slate, decimals expressing the following fractions. 100 1000 10000 100000 1000000

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8

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To read a decimal fraction,—Enumerate and read the figures, as if they were whole numbers, and conclude by pronouncing the name of the lowest denomination.

4. Copy upon the slate, the several numbers standing in the following columns, and then read them.

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ADDITION OF DECIMALS.

6. Add the following numbers into one sum. 63.75 and 524.0764 and .23 and 261.803.

63.75 524.0764

.23 261.803

849.8594

In arranging decimals for addition, we place tenths under tenths, hundredths under hundredths, &c.

We

then begin with the lowest denomination, and proceed to add the columns as in whole numbers.

7. What is the sum of 2.164, 870.31, 756, 9.18, 157.0008, 26.104, and .3728?

8. What is the sum of 2706, 58.2, .2065, 6.441, 75, 14.2, and 990.752 ?

In Federal Money, the dollar is the unit; that is, dollars are whole numbers; dimes are tenths, cents are hundredths, and mills are thousandths. See page 124.

9. Add together $24.6, $9.07, $5.009, and 5 cents

10. Write the following sums of money in the form of decimals, and add them together. $46 and 9 cents, 14 cents, $7 and 8 mills, 6 dimes, 8 dimes and 7 mills.

SUBTRACTION OF DECIMALS. 11. Subtract 52.6087 from 406.91.

406.91
52.6087
354.3013

After placing tenths under tenths, &c., we subtract as in whole numbers. The blank places over the 7 and 8, are viewed as ciphers.

12. Subtract 943.076 from 8270.54. 13. Subtract 1084.72 from 5603.0626. 14. Subtract 146.1706 from 16094.

15. Find the difference between .8 and .08, by subtracting the smaller decimal from the greater.

16. Find the difference between .45 and .31067, subtracting the smaller decimal from the greater.

by

17. What is the difference between 1 and .046 ? 18. Write 4 dollars and 8 mills in decimal form, and subtract therefrom, 6 dimes and 5 mills.

19. Subtract 7 cents and 3 mills from 10 dollars.

MULTIPLICATION OF DECIMALS.

Multiplying by any fraction, is taking a certain part of the multiplicand for the product; consequently, multiplying one fraction by another, must produce a fraction smaller than either of the factors. For example, multiplied by is 10%, or, decimally, .9 multiplied by .8 is 72. Hence you may observe, that the number of decimal figures in any product, must be equal to the number of decimal figures in both the factors.

72

20. Multiply 531 by .52. 65.7 by .43. 7.06 by .24. .439 by .38 .149 by .26.

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RULE FOR MULTIPLICATION OF DECIMALS. Multiply as in whole numbers; and in the product, point off as many figures for decimals, as there are decimal places in both factors. If the number of figures in the product be less than the number of decimal places in both factors, prefix ciphers to supply the deficiency

21. Multiply 1608 by .4,-that is, find .4 of 1608. 22. Multiply .45 of a dollar by 8.

23. How much is 36 times .495 of a dollar?

24. What cost 18 yards of cloth, at $4.072 per yd.? 25. What cost 28.7 yards of cloth, at $9 per yd.? 26. What cost 9.3 acres of land, at $8.41 per acre? 27. If 1 yard of silk cord cost 7' mills, [.007], what is the price of .9 of a yard ?

28. What is 6 per cent. or .06 of 340.4 ?
29. Multiply 42.863 by 70.28.
30. Multiply 2046 by .932.
31. Multiply .7253 by .0423.
32. Multiply 6.5431 by .402.

33. What is the product of .04 multiplied by .07? 34. What is the product of .005 by .009?

35. Multiply 7 and 5-hundredths by 6-thousandths.

DIVISION OF DECIMALS.

RULE FOR DIVISION OF DECIMALS. Divide as in whole numbers; and in the quotient, point off as many figures for decimals, as the decimal places in the dividend exceed those in the divisor; that is, make the decimal places in the divisor and quotient counted together, equal to the decimal places in the dividend.

If there be not figures enough in the quotient to point off, prefix ciphers to supply the deficiency.

When there are more decimal places in the divisor, than in the dividend, render the places equal, by annexing ciphers to the dividend, before dividing.

After dividing all the figures in the dividend, if there be a remainder, ciphers may be annexed to it, and the division continued. The ciphers thus annexed, must be counted with the decimal places of the dividend.

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