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?

34. A trader failed in business, owing $11000, and having only $5000 to divide among his creditors. How 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? 36. A cubic foot of air weighs 1 ounce. pounds of air does a room contain, which is 16 feet long, 14 feet wide, and 10 feet high?

How many

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 has 1 500 000 inhabitants, New York 350 000, Philadelphia 220 000, New Orleans 115 000, Baltimore 110 000, and Boston 105 000; how many times greater is London, than each of the others?

When a scholar has reached this point, it will be well to consider how much more time he is likely to devote to study. If he have but a few months more to spend in school, the SUPPLEMENT will furnish for him the suitable exercises, with which to finish his course of study in arithmetic. If, however, he is likely to continue at school for several years, he may omit the Supplement, and enter immediately upon the exercises of PART THIRD.

In the preceding chapters, departments of business are not arranged under distinct heads. The arrangement is strictly arithmetical, and business examples are made incidental to the course. In the Supplement, departments of business are separately presented, in distinct articles. These articles, although brief, are rendered sufficient, by the learner's previous familiarity with the operations they require.

182

SUPPLEMENT.

ARTICLE I.

INDICATIVE CHARACTERS GNS.

+(Plus,) standing between numbers, indicates that they are to be added together; thus, 3+2 is 5.

(Minus,) indicates that the number after it is to be subtracted from the number before it; thus, 5-2 is 3. X (Into,) indicates that one number is to be multiplied into another; thus, 4×3 is 12.

(By,) indicates that the number on the left is to be divided by the number on the right; thus, 12÷3 is 4. (Equal to,) indicates that the number before it is equal to the number after it; for example, 4+2=6. 6-2 4. 5X3=15. 15÷÷3-5.

=

CANCELLATION OF FACTORS.

THE CANCELLATION OF FACTORS is the excluding of such factors from an operation as balance each other. Any two equal factors, one being a factor of a dividend, and the other a factor of the divisor, or, one a factor of a numerator, and the other of the denominator, may be cancelled, that is, crossed and omitted. For example, ofis reduced to a simple fraction, as follows

Here we cancel the two threes,

and multiply 1 by 1, and 4 by 2.

of

When one of two opposite factors will divide the other without a remainder, both may be cancelled, and the quotient retained in the place of the factor divided.

stance, let us find what is of of 1 of 1 of 20.

9 11 Ă
3

1. Reduce of
2. What is of
3. Reduce of

For in

of 1 of to a simple fraction. ofofofofof 100? of of to a simple fraction. When all of a term is cancelled off, the new term must be 1.

4. A merchant owning of of of a ship, sold of his share. What part of the ship did he sell?

5. 3 men owned equally a saw-mill; one sold of of of his share. What part of the mill did he sell

II.

DECIMAL FRACTIONS.

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 3; 316 is 1000 3164 is 3164

316

10000

100

1. Write upon the slate, the decimals expressing the following fractions. 46 708 1642 96041

3

100 1000 10000 100000 When a whole number and a decimal are written to gether, the decimal point is placed between them. Thus, 24.6 is 24; 5.71 is 5,7; 48.364 is 48,364.

71.

1000

2. Write the following mixed numbers, expressing the fractions decimally. 38. 51622. 8354. 247636.

10000

In whole numbers, 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 of a thousand. In decimals, this system is continued below the place of units. For example, 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 one next to the right of the 1-hundredth, is of a hundredth, 10 or 1-thousandth of a unit.

one unit.

one tenth. -one hundredth.

one thousandth.

1.1 1 1

Ciphers placed on the right hand of decimal figures, do not alter the value of the decimal; because, the figures

remain unchanged in their distance from the unit's place. For instance, .5, .50, and .500 are of equal value; being each equal to . But every cipher placed on the left of a decimal, renders it 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 [10‰0]; and so on.

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

6

8

TO READ DECIMAL FRACTIONS,-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, and read the following decimals.

.06

.065

.0007

.06264

.10809

.6500171

24.02

5.763084

160.052

712.3005

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 sub tracting the smaller decimal from the greater.

16. Find the difference between .45 and .31067. 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 76, 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.

8

72

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