4. Annexing a cipher to the right of a decimal, or removing a cipher from the right of a decimal, does not alter the value, as it simply multiplies or divides both numerator and denominator by the same number. SECTION II. REDUCTION OF DECIMALS. CASE I. Reduction to Higher or Lower Terms. ORAL EXERCISE. 1. How many tenths in 5 units? In 12 units? 2. How many hundredths in 6 tenths? In 8 tenths? In 7 tenths? 3. How many WRITTEN PROBLEMS. 1. Reduce .12 to thousandths. Ans. .120. PROCESS.-.12=.120. 2. Reduce .18 to thousandths. 3. Reduce .05 to hundred-thousandths. 5. Reduce 6.5 to tenths; to hundredths. 7. Reduce 18.75 to ten-thousandths. 8. Reduce .06600 to hundred-thousandths. .9. Reduce .0300 to hundredths. 10. Reduce .3000 to tenths. CASE II. Reduction of Decimals to Common Fractions. ORAL EXERCISE. 1. In .6 how many fifths? 2. How many fourths in 25? In .50? In .75? 100 3. How many 20ths in 15? In .25? In .35? In .70? 100 NOTE.-Omit the decimal point, supply the denominator and reduce to the lowest terms. Reduce the following fractions to common fractions in their ORAL EXERCISE. 1. How many tenths in,,? 2. How many hundredths in 4, 1, 4? 3. How many thousandths in 1, 1, 1, 26, 40? WRITTEN PROBLEMS. 1. Change to a decimal. PROCESS. 40)3.000(.075 280 200 200 Ans. .075. Explanation.=} of 3; 3=18, or 188, or 1888; 40 From the foregoing we have the RULE. Annex ciphers to the numerator, and divide by the denomi nator, pointing off as many decimal places as there are ciphers annexed to the numerator. NOTE 1.-When the denominator of the common fraction contains other prime factors than 2 or 5, the division will not terminate. 2. When a sufficient number of decimal places is obtained, the remainder may be expressed by a common fraction. 3. When the quotient repeats the same figure or set of figures, it is called a circulating or repeating decimal. SECTION III. ADDITION OF DECIMALS. WRITTEN PROBLEMS. 1. Add 25.75, 35.04, 1.648, 442.613. PROCESS. 25.75 35.04 1.648 442.613 505.051 Explanation. The decimals are so written that units of the same order stand in the same perpendicular column, and the addition is then performed the same as the addition of simple numbers. RULE. Write the numbers so that the decimal points shall stand in a perpendicular column, and add as in simple numbers. NOTE.-Place the separatrix under those in the column. 2. Add 25.002, 206.31, 505.05, 1.015, 8.33. Ans. 745.707. Ans. 600.5371. Ans. 787.428. Ans. 87.474. Ans. 931.5315. 3. Add 89.07, 76.79, 31.375, 403.3021. Ans. 285.2717. Ans. 119.6078. Ans. 124.9659. Ans. 179.0064. 11, Four fields contain as follows: 16.375 acres, 12.6125 acres, 14.003 acres, 16.5 acres: how much do the four fields contain? Ans. 59.4905 acres. 12. Five pieces of silver weigh as follows: .33 pounds, 1.275 pounds, .127 pounds, .5 pounds and 1.324 pounds: how much do the five pieces weigh? Ans. 3.556 pounds. SECTION IV. SUBTRACTION OF DECIMALS. WRITTEN PROBLEMS. 1. From 78.75 take 23.125. PROCESS. 78.750 23.125 55.625 Ans. 55.625. Explanation. Since annexing a cipher to the right of a decimal does not alter its value, 78.75 may be written 78.750. Writing the numbers so that the decimal points are immediately one over the other, the subtraction is performed the same as in simple numbers. 2. From 274.684 take 217.423. 3. From 186.004 take 87.621. 4. From 8648.3684 take 1764.8342. 5. From 621 take 3.17. 6. From 8.4062 take .6434. 7. From 13.3175 take 9.61478. 8. From 4256.85 take .00564. Ans. 57.261. Ans. 98.383. Ans. 6883.5342. Ans. 617.83. Ans. 7.7628. Ans. 3.70272. Ans. 4256.84436. 6.13 7.24 24 52 122 6 4291 44.3812 Ans. 44.3812. Explanation 1.-6.13 or 1 multiplied by 7.24, or 38, equals 443812, or 44.3812. Explanation 2.-6.13 multiplied by 724 =4438.12, and by 724 hundredths, one one-hundredth as much, or 44.3812. From the foregoing we have the rule for multiplication of decimals: RULE. Multiply decimals as simple numbers, and point off from the right as many decimal places as there are in the multiplier and the multiplicand. NOTE.-If the number of figures in the product is less than the number in the two factors, prefix as many ciphers as may be necessary to make the number of decimal places in the product equal to the number in both factors. What is the product 2. Of 25.75 by 5.6? Ans. 144.200, or 144.2. Ans. 3115.20258. Ans. 3.5912. |