The following table of common fractions with their decimal equivalents should be carefully committed to memory: = .50. 1 = .331. = .25. = .75. = .20. } = .40. 38 3. Reduce OPERATION. 3)1.0000 = .60. .33331 = = .831. = .14%. 4 = .284. = .429. = .574. = .714. to a decimal. .51351319 Or, .513513513513+ = = .80. $ = .855. 16.12. = .621. } = .871. 6)1.000 - SOLUTION. - Proceeding as before, we find that each successive figure in the quotient is 3 with a remainder of, which, reduced to the next lower denomination, gives another quotient of 3 with a remainder of. Hence we cannot reduce to an exact decimal, and the answer must be an infinite number of 3's with a remainder. 4. Reduce to a decimal. OPERATION. 37)19.000000 SOLUTION. Proceeding as before and performing the division to millionths, we obtain for a quotient .51351319. This is not an exact decimal, but since the remainder is the same as the fraction originally reduced, we know that the quotient figures of the lower denominations will be 513 repeated. Hence we may add as many of these figures as we desire, adding the sign +, after the last. 5. Reduce to millionths. OPERATION. 26.05. 25.04. SOLUTION. Proceeding as before, we have for a quotient .166 with a remainder, and we readily see that this remainder reduced to any number of lower denominations will repeat the quotient figure 6. Hence we add as many 6's as are necessary to reduce the number to the required denomination, and place the sign+, after the result. .166666+ PRAC. AR. -9 151. Decimals like those in Examples 3, 4, 5, in which certain figures or sets of figures constantly repeat, are called repeating decimals, or circulating decimals. The figure, or set of figures, repeated is called the repetend. A repetend need be written but once, and when it consists of a single figure a point is placed over it. When it consists of more than one figure, points are placed over the first and last figures. Thus, .666 is written .6; .297297 is written .297. When the decimal commences with a repetend, as .3 in Example 3 and .513 in Example 4, it is called a pure circulating decimal; but when it is preceded by one or more decimal figures which do not repeat, as by .1 in Example 5, it is called a mixed circulating decimal. Every pure circulating decimal is equal to a fraction whose numerator is the repetend and whose denominator consists of as many 9's as there are places in the repetend. Thus .3+, or }; .3333+ = 3333; .5555+ = 55, etc. 9999 to a decimal. 6. Reduce 64 9. Reduce to a decimal. 80 10. Reduce to a decimal. 800 11. Reduce to a pure circulating decimal. 12. Reduce to a mixed circulating decimal. Ans. .259. Ans. .83. 13. Reduce .7 to a common fraction. 18. Reduce to a decimal; ; . 19. Reduce 20. Reduce to a decimal. 17/ 21. Change to a pure circulating decimal. Ans. .7. 22. Change to a pure circulating decimal. Ans. .324. 23. Change 47 to a mixed circulating decimal. 50 Ans. .313. 24. Reduce to a decimal. 25. Reduce to a decimal. 26. Reduce to a decimal. 27. Reduce .324 to a common fraction. 28. Reduce .0117 to a common fraction. 97 Ans. f. Ans. 13. Ans. 4. Ans. .1. 29. Reduce 226 to a decimal. 1130 Ans. .01; .05. Tell by inspection the fractional equivalents of: 30. .66; .574; .09; .163. 31. .40; .42; .33; .80. 32. .111; .081; .555; .04. 33. .87; .50; .441; .061. 34. .83; .71; .621; .284. 35. .777; .855; .05; .371. 36. .144; .22; .75; .121. 37. .60; .20; .888; .25. Give instantly the decimal equivalents of: 38. 臺;告;♀;音;舌;春;舌; b 3 39. ; ; ; 8; 1; 7; 8; 1 40. ; ; ; ; t ; & ; † ; Te ADDITION. EXAMPLES. 152. 1. What is the sum of 3.703, 621.57, .672, and 20.0074 ? SOLUTION. We write the numbers so that figures of like orders of units stand in the same column; that is, units under units, tenths under tenths; hundredths under hundredths, etc. This brings the decimal points directly under one another. Commencing at the right hand we add each column separately, carry as in whole numbers, and in the result place a decimal point between units and tenths. OPERATION. 3.703 621.57 .672 20.0074 645.9524 RULE. -I. Write the numbers so that the decimal points stand directly under each other. II. Add as in whole numbers, and place the decimal point, in the result, directly under the points in the numbers added. 2. Add .199 2.7569 .25 .654 Sum, 3.8599 3. Add 4.015 6.75 27.38203 375.01 2.5 Amount, 415.65703 4. Add 1152.01, 14.11018, 152348.21, 9.000083. Ans. 153523.330263. 5. Add 37.03, .521, .9, 1000, 4000.0004. Ans. 5038.4514. 6. What is the sum of twenty-six, and twenty-six hundredths; seven tenths; six, and eighty-three thousandths; four, and four thousandths? Ans. 37.047. 7. What is the sum of thirty-six, and fifteen thousandths; three hundred, and six hundred five ten thousandths; five, and three millionths; sixty, and eighty-seven ten-millionths? Ans. 401.0755117. 8. What is the sum of fifty-four, and thirty-four hundredths; one, and nine ten-thousandths; three, and two hundred seven millionths; twenty-three thousandths; eight, and nine tenths; four, and one hundred thirty-five thousandths? Ans. 71.399107. 9. How many yards are there in three pieces of cloth, the first piece containing 18.375 yards, the second piece 41.625 yards, and the third piece 35.5 yards? 10. A's farm contains 61.843 acres, B's contains 143.75 acres, C's 218.4375 acres, and D's 21.9 acres. How many acres are there in the four farms? 10 11. My farm consists of 7 fields, containing 12 acres, 183 acres, 9 acres, 24 acres, 418 acres, 8 acres, and 151 acres respectively. How many acres are there in my farm? Ans. 93.6375. Reduce the common fractions to decimals before adding. 12. A grocer has 2 barrels of A sugar, 5 barrels of B sugar, 3ğ barrels of C sugar, 3.0642 barrels of crushed sugar, and 8.925 barrels of pulverized sugar. How many barrels of sugar has he? Ans. 23.8642. 13. A tailor made 3 suits of clothes; for the first suit he used 2 yards of broadcloth, 3 yards of cassimere, and yard of satin; for the second suit 2.25 yards of broadcloth, 2.875 yards of cassimere, and 1 yard of satin; and for the third suit 5 yards of broadcloth, and 1 yards of satin. How many yards of each kind of goods did he use? How many yards of all? Ans. to last, 18.375 yards. 14. A dressmaker bought in the course of a week 3g yards cloth, 5 yards, 97 yards, 4 yards, and 7 yards. How much did she buy altogether during the week? Ans. 30.87 yards. |