REDUCTION.

199. To reduce dissimilar decimals to similar decimals.

1. Reduce .3, .75, 1.36, and 4.0432 to similar fractions.

EXPLANATION-The lowest order of decimals in the given numbers is ten-thousandths. Hence, all of the decimals must be reduced to ten-thousandths. This may be done by annexing ciphers (Prin. 1, Art. 190).

RULE Give all the decimals the same number of places by annex

200. To reduce a common fraction to a decimal.

Reduce to a decimal.

8(3.000

of 30

EXPLANATION— equals of 3. 3 equals 30 tenths, and .375 tenths is 3 tenths and 6 tenths remaining. 6 tenths equals 60 hun

dreds and of 60 hundredths is 7 hundredths and 4 hundredths remaining. 4 hundredths equals 40 thousandths, and of 40 thousandths is 5 thousandths. Hence, equals .375.

RULE-Annex ciphers to the numerator and divide by the denomi

nator.

Point off as many decimal places in the quotient as there are ciphers used.

1. In many cases the division will not terminate. Such decimals are called interminate, infinite, or circulating decimals. The remainder may be expressed as a common fraction or the sign + may be used to show that the division is not exact. Thus, } = .33}, or .333+.

2. Common fractions in their lowest terms cannot be reduced to pure decimals when their denominators contain any prime factors besides 2 or 5.

EXPLANATION.-.35 expressed as a common fraction %, or when reduced to its lowest terms.

is 35

RULE. Write the denominator under the decimal, omitting the decimal point, and reduce the fraction to its lowest terms.

EXPLANATION.-.37 is hundredths, which,

written with its denominator becomes

or when reduced to its lowest terms.

100'

or 200

= 375 which reduced to its

Or, .37 .375 lowest terms equals 3.

10009

ADDITION.

202. 1. What is the sum of 14.8, 120.35, .025, and 4.05?

14.8

120.35

.025

4.05

139.225

EXPLANATION-We write the numbers so that units of the same order stand in the same column. Then beginning at the right, we add as in whole numbers, and place the decimal point in the result directly under those of the numbers.

RULE-Write the numbers so that units of the same order stand in the same column.

Add as in whole numbers, and place the decimal point between the units and tenths of the sum.

When there are complex decimals, extend, if they do not terminate, to at least five decimal places before beginning to add.

17. Find the sum of thirty-seven thousandths; two hundred sixty-five, and fifty-four hundred-thousandths; seventy-five, and seventy-five thousandths.

18. Find the sum of seven hundred fifty, and one hundred one ten-thousandths; three thousand sixty-seven, and twenty-two millionths; eight, and five thousandths.

19. In five piles of wood there are respectively 4.418 cords, 6.214 cords, 7 cords, 94 cords, and 10% cords. How many cords in all?

20. A farmer sold at one time 2.35 tons of hay, at another time 11.8 tons, at another time 4.05 tons, and at another time 6 tons. How many tons of hay did he sell?

766079

SUBTRACTION.

203. 1. From 57.24 subtract 39.625.

57.24 39.625 17.615

EXPLANATION.-We write the subtrahend under the minuend so that units of the same order stand in the same column. We then subtract as in whole numbers, and place the decimal point in the result directly under the points of the numbers.

RULE.- Write the subtrahend under the minuend so that units of the same order stand in the same column.

Subtract as in whole numbers, and place the decimal point between the units and tens of the remainder.

If the minuend has not so many decimal places as the subtrahend, annex ciphers or subtract as if decimal ciphers were annexed.

16. From thirty-seven, and six thousandths subtract thirteen, and fifteen millionths.

17. From five, and five thousandths subtract five hundred seven thousandths.

18. Mr. Brown's income last year was $3465. $2347.76. How much money did he save?

He spent

19. From a cistern containing 500 barrels of water 189.075 barrels were pumped out. How many barrels of water remained in the cistern?

20. The receipts of a railroad company for a certain year were $1340769.23, and the expenditures for the same year were $1123002.89. What were the profits?

21. A dealer bought flour to the amount of $3647.37, and sold it for $5000. How much did he gain?

MULTIPLICATION.

204. Multiply 4.27 by .32.

4.2 7 .3 2

8 54 1281

1.3 6 6 4

RULE.

EXPLANATION.-4.27 multiplied by 32 equals 136.54; and multiplied by 32 hundredths the product is 1 hundredth as great. Expressing this by removing the decimal point two places to the left (Prin. 4, Art. 190), the result becomes 1.3664. Hence, 4.27 .32 = 1.3664.

-Multiply as in whole numbers, and from the right of the product point off as many figures as there are decimal places in both factors.

If there are not as many figures in the product as there are decimal places in both factors, prefix ciphers to make as many.

Find the products:

2. .23 X .15

3. .35 X 2.4

4. 2.304 × 12}

5. .325 X .004

10. Multiply 3.4256 by 200.

3.4256

6. 31.42 X 1.05

7. 21.35 X .035

8. .0416 X .0015

9. 121.48 X .00025

EXPLANATION.--Since moving the decimal point two places 200 to the right multiplies the decimal by 100 (Prin. 3, Art. 190), we need only multiply 3.4256 by 2, and place the decimal point two places to the right.

685.12

17. Multiply thirty-one, and forty-two thousandths by seven, and six hundredths.

18. Multiply one thousand two hundred fifty by twenty, and two ten-thousandths.

19. A farmer sold 41.36 bushels of wheat at $1.12 per bushel. How much did he receive for the wheat?

20. Mr. C. sold 10.48 cords of wood at $3.37 per cord. What was the amount of the sale?

21. A grocer sold 6 pounds of rice at 7 cents a pound, 241 pounds of lard at 54 cents a pound, 8 pounds of butter at 20 cents a pound, and 25 pounds of sugar at 64 cents a pound. How much did he receive for all?