Hence, the RULE.-Express the decimal as a common fraction, and reduce it to its lowest terms. Reduce the following to common fractions in their lowest terms: = 216. To reduce common fractions to decimals. 1. Reduce to a decimal. = OPERATION. 8)3000 EXPLANATION. -330 tenths; 30 tenths ÷ 8 3 tenths and 6 tenths over. 6 tenths 60 hundredths; 60 hundredths8=7 hundredths and 4 hundredths over. 4 hundredths = 40 thousandths, and this ÷ 8 5 thousandths. Hence, the answer is 3 tenths, 7 hundredths, 5 thousandths .375. Hence, the = RULE.-Annex ciphers to the numerator, divide by the denominator, and from the right of the quotient point off as many decimal figures as there are ciphers annexed. ADDITION OF DECIMALS. 217. Since decimals increase and decrease regularly by the scale of ten, they are evidently added like whole numbers. Hence, the RULE. Write the numbers so that points shall stand under points, tenths under tenths, hundredths under hundredths, etc., and add as in whole numbers. 2. 53.246, 44.82, 706.4, 49.82, and .5. 4. 9.74, 16.07, 924., 75.24, and .879. Ans. 542.0889. 6. 23 and 7 hund'ths, 5 and 9 tenths, 271 and 46 thous'ths, and 133 and 575 ten-thous'ths. Ans. 433.0735. 7. 27 hund'ths, 83 thous'ths, 984 thous'ths, 7 and 8 hund'ths, and 74 and 125 ten-thous'ths. Ans. 82.4295. 8. 43 and 9 tenths, 13 and 13 thous'ths, 61 hund'ths, 5 and 17 thous'ths, and 425 and 78 hund'ths. Ans. 488.320. NOTE.-In the following, first reduce the fractions to decimals by Art. 216, and then add. Find the sum of: 9.,, and . 10. 31, 98, 71, and 161. Ans. 1.875. Ans. 36.55. Ans. 22.96875. Ans. 540.025. SUBTRACTION OF DECIMALS. 218. Decimals are subtracted like whole numbers for the same reason that they are added as such. Hence, the RULE. Write the subtrahend under the minuend, so that points shall stand under points, tenths under tenths, hundredths under hundredths, etc., and subtract as in whole numbers. 23.35 17.65 We write ciphers above the 3 and 5, as there are no tenths and no hund'ths in the minuend. 9. 23 and 5 tenths take 12 and 176 millionths. Ans. 11.499824. 10. 184 and 7 thous'ths take 137 and 68 thous'ths. 11. 1 thousand take 1 thousandths. 12. 7 hunds. take 7 hund'ths. Ans. 46.939. Ans. 999.999. Ans. 699.93. NOTE.-In the following, first reduce the fractions to decimals, and then subtract. What is the value of: 13. 88-71? 14. 7-8? 32 Ans. 1.25. | 16. -? Ans. .25625. Ans. 0.275. 17. 1%-470? Ans. .17. 15. 1-? Ans. 0.65.18. 243-13? Ans. 11.325. RULE.-Multiply as in whole numbers, and in the product point off as many decimal figures from the right as there are decimal places in both factors, prefixing ciphers when necessary to supply the deficiency. 5. 125.08 by .25. Ans. 31.27. 6. .0061 by .05. Ans. 000305. 10. .1806 by 5.4. 11. .0586 by .75. 12. The distance around a circle is 3.1416 times the distance through it; how far is it around a circular garden if it is 75 yards through it? Ans. 235.62 yards. DIVISION OF DECIMALS. 220. Since Division is the reverse of Multiplication, by reversing the rule of the latter we get the RULE.-Divide as in whole numbers, and in the quotient point off as many decimal figures from the right as the dicimal places of the dividend exceed those of the divisor, prefixing ciphers, if necessary, to supply the deficiency NOTES.-I. When there are more decimal places in the divisor than in the dividend, make them equal by annexing ciphers to the dividend before dividing. II. If there is a remainder, ciphers may be annexed to it as decimals, and the division continued at pleasure. III. When there is a remainder at the close of the operation, the sign should be annexed to the quotient to show that it is not complete. Divide: EXERCISES. What is 1. A Decimal Fraction? 2. The decimal point? 3. A mixed number? How do decimals decrease from left to right? On what does the value of a decimal figure depend? The first place on the right of the decimal is that of what? The second place on the right? The third? The fourth? The fifth ? Is the denominator of a decimal written? What is it? How are decimals read? Repeat the three principles, p. 168. How are: 1. Decimals reduced to fractions in their lowest terms? 2. Common fractions reduced to decimals? How are decimals: 1. Added? 2. Subtracted? 3. Multiplied? 4. Divided? |