2. Reduce .6, 75, .089, to similar decimals. Reduce the following dissimilar decimals to similar decimals: 6. .0436, .04506, .82. 10. 5, .5, .005, 50. 206. To reduce a decimal to a common fraction. 1. . If 5 tenths be written as a common fraction what will be the numerator? What will be the denominator? 2. What is the numerator and what the denominator of the decimal 18 hundredths, when expressed as a common fraction? 3. Express the value of the decimal 50 hundredths, by a common fraction in its smallest terms. 4. Express by a common fraction in its smallest terms, the following decimals: 20 hundredths. 30 hundredths. 50 hundredths. 250 thousandths. 375 thousandths. 1. Reduce .75 to its equivalent common fraction. ANALYSIS.—.75 expressed as .75=26= fraction is too, which, being reduced to its smallest terms, equals 1. PROCESS. a common RULE.—Omit the decimal point, supply the denominator, and reduce the fraction to its lowest terms. Reduce the following decimals to equivalent common fractions in their smallest terms: 1.1 2. .054. 3. .03875. 4. .05625. 5. .4375. 6. 4.0125. 10. .354. 14. .5675. PROCESS. 5 3 350 100 100 100 thousa Godth Or che le 100 18. Reduce .154 to an equivalent common fraction. ANALYSIS.—The expres154 196 sion .154 is equal to 157, or .154 108 104. Reducing the denominator also to sevenths, the expression becomes 486, or 3 Change the following to equivalent common fractions, or to mixed numbers: 19. .121 25. .5624 28. .00031 20. .331 23. .045 26. .0033. 29. 2.756. 21. .163 24. :0371 27. .0783 30. 13.814. 53 22. .872 CASE III. 207. To reduce a common fraction to a decimal. 1. One half of an apple is equal to how many tenths of an apple ? 2. How many tenths are there in }? ? ? 5. How many hundredths are there in į, or 100 hundredths divided by 2? How many in 1? 6. How many hundredths are there in, or 400 hundredths divided by 5? How many in ? 7. How many thousandths are there in , or 5000 thousandths divided by 8? How many in 3? How many in f? TT RITTEN EXERCISES. PROCESS. 1. Reduce to an equivalent decimal. ANALYSIS.— is of 5, or 50 85.000 tenths; and í of 50 tenths is 6 tenths and 2 tenths remaining. .625 Or, 2 tenths are equal to 20 hunž=$488=1928=.625 dredths, and } of 20 hundredths is 2 hundredths and 4 hundredths remaining. 4 hundredths are equal to 40 thousandths, and f of 40 thousandths is 5 thousandths. Hence is equal to 6 tenths + 2 hundredths + 5 thousandths, or .625. Or we may multiply both terms of the fraction by 1000 and divide the resulting terms by 8, and obtain the decimal 6277 Tooo, or .625. RULE.—Annex ciphers to the numerator and divide by the denominator. Point off as many decimal places in the quotient as there are ciphers anneced. In many cases the division is not exact. In such instances the remainder may be expressed as a common fraction, or the sign + may be employed after the decimal to show that the result is not complete; thus: 1 = .1663, or .166 +. 14. . 20. 1. 3 20 17 21. 128 Reduce the following to equivalent decimals: 2.1 8. 1o. 9. 3. 15. s. 16. 17. : 18. 94 19. 13 Change the following to the decimal form: 26. 155 29. 3.41 32. .875 27. 243 30. .23 33. .431 28. .821 31. .621 34. 4.216 24. 199. 35. 37.5% 36. 20.03 37. .00018 8. 9. ADDITION. 11. 12. 13 . 208. 1. What is the sum of K and 1? and ? .3 and .7? 2. What is the sum of 10% and 100? 26 and 30%? .12 and .20? 3. What is the sum of 1 and 100? 1147 and Thi? .005 and .043? 4. Find the sum of so and 18. Of .5 and .06. .7 and .19. 5. Find the sum of .6, .31, .004. Of .5, .08 and .006. 209. PRINCIPLES.— The principles are the same as for addition of integers. WRITTEN EXERCISES. 16. PROCESS. 1. What is the sum of .36, 2.136 and 4.5004? ANALYSIS.—We write the numbers .36 = .3600 so that units of the same order shall 2.136 2.1360 stand in the same column, and add as 4.50 04=4.5004 in integers, separating the decimal part of the sum from the integral part by 6.99 64 6.99 64 the decimal point. The decimals are made similar by annexing ciphers until all the decimals have the same number of places. It is not usual to make the decimals similar, for if they are written so that decimals of the same order stand in the same column it is unnecessary to supply the ciphers. RULE.—The rule is the same as for addition of integers. What is the sum of 5. 6.843, 48.25 and 17.286? 8. What is the sum of $5.18, $3.09, $46. and $51.185? 12. A laborer earned $7.25 in one week, $7.122 in another, $9.18% in another, and $85 in another. How much did he earn during that time? 13. What is the sum of 18 thousandths, 15 millionths, 81 hundredths, 146 ten-thousandths, 834 hundred-thousandths ? 14. What is the sum of 8 dollars 5 cents, 13 dollars 19 cents, 18 dollars 3 cents 8 mills, 25 dollars 37 cents 5 mills, 125 dollars, and of a dollar? 15. Mr. A. paid the following bills for repairs upon his premises, viz: carpenter-work, $381.45; plastering, $215.385; plumbing, $323.94; and other expenses, $181.57. How much did he pay for repairs ? 16. A farmer purchased cloth for $137, boots for $86, crockery for $1011, and groceries for $15:49. How much did he pay for all his purchases ? SUBTRACTION. 210. 1. From to take . From .9 take .5. 2. Find the difference between To and 100; 1 and yo; .19 and .08. 3. Find the difference between 1ooo and 1000; 1oo and Todo; .007 and .005. 4. What is the difference between to and 187? .5 and .06? .7 and .09?. 5. What is the difference between .16 and .03? .15 ar .08? .45 and .3? 211. PRINCIPLES.—The principles are the same as for the subtraction of integers. |