12. Reduce the complex decimal .2 to a common fraction. SOLUTION.-.24 is 21 tenths, which, by writing 21 the denominator, becomes which equals or OPERATION. = 10 10 = =Ans. 10' .2}= which reduced to its lowest terms equals 1. Therefore, etc. Reduce the following to common fractions or mixed num 206. To reduce a common fraction to a decimal. 1. Reduce to a decimal. SOLUTION. equals of 5. 5 equals 50 tenths; f of 50 tenths is 6 tenths and 2 tenths remaining; 2 tenths equal 20 hundredths; of 20 hundredths equals 2 hundredths and 4 hundredths remaining; 4 hundredths equal 40 thousandths; of 40 thousandths is 5 thousandths. Therefore, equals .625. OPERATION. }=} of 5 =8)5.000 .625 Rule.-I. Annex ciphers to the numerator, and divide by the denominator. II. Point off as many decimal places in the quotient as there are ciphers used. NOTES.-1. In many cases the division will not terminate; the common fraction cannot then be exactly expressed by a decimal. Such decimals are called interminate or infinite decimals. 2. The symbol + annexed to a decimal indicates that it contains other decimal terms. The symbol annexed to a decimal indicates that the This is often done when the nex last decimal term is increased by 1. term is greater than 5. ADDITION OF DECIMALS. 207. Addition of Decimals is the process of finding the sum of two or more decimals. 1. What is the sum of 11.96, 25.075, 84.306, 90.728? OPERATION. 11.96 25.075 84.306 90.728 SOLUTION.-We write the numbers so that terms of the same order shall stand in the same column, and begin at the right to add. 8 thousandths, plus 6 thousandths, plus 5 thousandths, are 19 thousandths, which equals 1 hundredth and 9 thousandths; we write the 9 thousandths, and add the 1 hundredth to the next column: 2 hundredths, plus 7 hundredths, plus 6 hundredths, equal 15 hundredths, and the Í hundredth added is 16 hundredths, which equals 1 tenth and 6 nundredths; we write the 6 hundredths, etc. 212.069 Rule.-I. Write the numbers so that terms of the same order stand in the same column. II. Add as in whole numbers, and place the decimal point between the units and tenths of the sum. 2. Find the sum of 79.76, 85.08, 36.125, 140.309. Ans. 341.274. 3. Find the sum of 87.09, 5837, 95.42, 237.675. Ans. 478.555 4. Add 18.79, 147.072, 856.709, 185.8761, 397.05784. Ans. 1605.50494 5. Add 59.874, 435.095, 672.328, 976.309, 8467.500843. Ans. 10611.106843. 6. What is the sum of $254, $371, $28.37, $50.06, 815 371, $573, $157, and $23.87? Ans. $253.884. 7. Add 9 and 7 tenths, 41 and 8 hundredths, 75 and 54 hundredths, 128 and 187 thousandths. Ans. 254.507. 8. Add 187 and 5 thousandths, 49 and 9 hundred-thousandths, 1876 and 245 millionths, 187 ten-thousandths, and 999 ten-millionths. Ans. 2112.0241349. 9. Add 798 and 9 ten-thousandths, 17 millionths, 18 thousandths and 98 ten-millionths, 67 hundred-thousandths and 95 ten-millionths. Ans. 798.0196063. 10. Find the sum of 3 tenths, 6 hundredths, 4 thousandths, 3 ten-thousandths, and 63 hundred-thousandths. Ans. .3696245. SUBTRACTION OF DECIMALS. 208. Subtraction of Decimals is the process of finding the difference between two decimals. OPERATION. 972.163 856.235 115.928 1. From 972.163 take 856.235. SOLUTION.-We write the numbers so that terms of the same order stand in the same column, and begin at the right to subtract. We cannot subtract 5 thousandths from 3 thousandths, hence we add ten thousandths to 3 thousandths, which equals 13 thousandths; 5 thousandths from 13 thousandths leaves 8 thousandths, which we write in the order of thousandths: since we have added 10 thousandths or 1 hundredth to the minuend, we must add 1 hundredth to the subtrahend; 1 hundredth and 3 hundredths are 4 hundredths; 4 hundredths from 6 hundredths leaves 2 hundredths, etc. Rule.-I. Write the subtrahend under the minuend, so that terms of the same order stand in the same column. II. Subtract as in whole numbers, and place the decimal point between the units and tenths of the remainder. 2. From 707.325 take 623.452. 3. From 826.438 take 734.936. 4. From 78.3057 take 29.084. 5. From 1230.207 take 384.1231 6. From 2.07 take 1.432765. 7. From .3 take 3 hundred-millionths. Ans. 83.873. Ans. 91.502. Ans. 49.2217.. Ans. 846.0839. Ans. .637235. Ans. .29999997. 8. From 1 and .001 take .01 and .000001. Ans. .990999. 9. From 2 take 2 thousandths and 2 billionths. Ans. 2.4974999975 MULTIPLICATION OF DECIMALS. 209. Multiplication of Decimals is the process of finding the product when one or both factors are decimals. MENTAL EXERCISES. What is the value of 4. times? .4x.8? 5. X.6? .6x.7? 6. 6Xrj? BX08? 9. How many places in the product of units multiplied by tenths! Tenths by tenths? Units by hundredths? Tenths by hundredths! Hundredths by hundredths? 10. How many decimal places in the product if there are two in the multiplicand and one in the multiplier? If there are two in each factor? If three in multiplicand and two in multiplier? WRITTEN EXERCISES. 1. Multiply 7.23 by .46. SOLUTION.-7.23 multiplied by 46 equals 332.58, and multiplied by 46 hundredths the product is 1 hundredth as great, which, by removing the decimal point two places to the left, becomes 3.3258. Hence, 7.23 multiplied by .46 equals 3.3258. OPERATION. 7.23 .46 4338 2892 3.3258 SOLUTION 2D.-7.23.46-788×48888=10000×33258—3.3258. From either of these solutions we derive the following Rule.-Multiply as in whole numbers, and from the right of the product point off as many decimal places as there are in both factors, prefixing ciphers when necessary. 2. Multiply 27.13 by .67. 3. Multiply 43.08 by 2.36. 4. Multiply 79.52 by .019. 5. Multiply 8.534 by 20.074. 6 Multiply 123.107 by 1.52. 7. Multiply 512.073 by 35.08. 8. Multiply 54.0079 by 7.072. 9. Multiply 1.08096 by 3.5702. 10. Multiply .03507 by .005873. 11. Multiply 2.0709 by .000246. 12. Value of 9 ×.08 × 10? Ans. 18.1771. Ans. 101.6688. Ans. 1.51088. Ans. 171.311516. Ans. 187.12264. Ans. 17963.52084. Ans. 381.9438688. Ans. 3.859243392. Ans. .00020596611. Ans. .0005094414. Ans. 8.3125. 13. Value of .5 of×.05? 14. Value of .07 of x 300 x .094 ? 15. Value of $25.67×.05}+$15×6.034?
Ans. .0175. Ans. 1.65. Ans. $91.8575. 16. 200+.02 −1 × .81+ (.84 − .127 × § of 74)? Ans. 164.7432. 17. Multiply 1 hundredth by 1 thousandth, and add 1 tent) to the product. Ans. .10001. 18. What is the product of one-tenth by one-tenth? one hundred by one-hundredth? one million by one-millionth? DIVISION OF DECIMALS. 210. Division of Decimals is the process of finding the quotient when one or both terms are decimals. MENTAL EXERCISES. What is the value of 1. A÷2? .4÷2?. 2. 6? .24÷6? 3. .502? .5÷2? 4. .150÷2? .15÷2? 5. 5÷4? .5÷4? .05÷4? 6. 6÷? 6÷÷.3? 6÷.03? 7. .8:41 8÷.41 8÷.04? 8. 7.2.6 .72÷.6? .72÷.06? 9. 8.4.7? .84.7? .84÷.071 10. 96.8 9.5÷.8? .96÷.8? What then is the 11. What is the product of tenths by tenths? quotient of hundredths by tenths? What is the product of tenths by hundredths? What is the quotient of thousandths divided by tenths! 12. If there are three decimal places in the multiplicand and two in the multiplier, how many in the product? If there are five in the dividend and three in the divisor, how many in the quotient? WRITTEN EXERCISES. 1. Divide 12.3084 by 2.34. SOLUTION.-Dividing as in whole numbers, we obtain the quotient 526; now, since the dividend is the product of the divisor and quotient, the number of decimal places in the dividend should equal the number in the divisor, plus the number in the quotient; hence the number of decimal places in the quotient equals the number of decimal places in the dividend, minus the number of places in the OPERATION. 2.34)12.3084(5.26 1170 608 468 1404 1404 divisor; hence there should be four minus two, or two decimal places in the quotient, therefore the quotient is 5.26. |