To change Common Fractions to Decimal Fractions. 279. ILLUSTRATIVE EXAMPLE. Change š to a decimal fraction. Explanation. — The fraction is the same as of 8) 3.000 3, or of 3.000 (3000 thousandths), which is found by dividing 3.000 by 8 in the usual way (Art. 102). 0.375 WRITTEN WORK. 280. From the example above we derive the following Rule. To change a common fraction to a decimal fraction : Express the numerator as tenths, hundredths, thousandths, etc., by annexing as many zeros as may be required, and then divide it by the denominator. 281. Examples for the Slate. Change to decimals : (26.) H. (29.) 48 (32.) 14 (27.) 0. (30.) 5. (33.) 87. (28.) 16 (31.) (34.) 1714 (35.) 1.06. (36.) 0.047. (37.) 0.03. Change to decimals and add (Art. 45) the following: (38.) 4,, ), and 7o. (40.) 4, X, Y, and 3. (39.) }, }, 14, and to (41.) 1%, 15%, and 11: 42. A carpenter paid for a mantel-piece $ 275, for a grate $ 223, and for a hearth $426. How much did he pay in all ? (43.) 2 +34 + 874 +187 = what ? 44. A drover bought a cow and a calf for $38.85, and sold the cow for $324, and the calf for $10. How much did he gain? 45. A man owning 17.635 acres of land, sold 13 acres to one person, and 7 of an acre to another. How much had he left? 46. Change to seven decimal places, and add 1.8276 0.0097, and 0.10,76 WRITTEN WORK. E.cact sum, 15.21511 282. ILLUSTRATIVE EXAMPLE. What is the sum of 53 yards, 2Ă yards, and 734 yards ? Explanation. — In this example there are 5} = 5.125 fractions which cannot be completely ex23 = 2.6663 pressed as decimals; for, however far the division be carried, there will still be a 734= 7.42435 remainder. If we choose to stop dividing at thou sandths, the quotients are expressed accu5} = 5.125 rately by writing f of a thousandth and 23 = 2.667 of a thousandth, as in the margin. But 731 = 7.424 these results are no more convenient to add Approximate sum, 15.216 than the original numbers; hence nothing has been gained by changing the latter to the decimal form if our object was to find the exact sum. There are, however, many cases in which the error arising from the neglect of such small fractions as parts of a thousandth is of no importance. For such cases the second form of written work given in the margin is to be adopted. Here the decimal values are expressed to the nearest thousandth. This is done by increasing the last term of the decirnal by 1 whenever the neglected fraction is for more. Greater accuracy would be attained by carrying out the decimal to the nearest ten-thousandth, or to a still lower denomination. 283. Examples for the Slate. Note. Unless some other direction is given, the pupil will hereafter understand that decimal values are to be expressed to the nearest tenthousandth. 47. Find the decimal values of }, 72, 4, and add the results. 48. Change to ten thousandths, and add 93, 163, and 33: 49. Change it and 0.68 to ten thousandths, and find their difference. 50. Mr. Carpenter has worked for Mr. Bates 23 hours, 33 hours, and 5.5 hours. How many hours has he worked for him in all ? 51. How many rods are there in 253 rods, 0.4814 rods, 105) rods, and 8.62} rods? Other examples in addition and subtraction may be found on page 135. Circulating Decimals. 284. We have seen (Art. 282) that in expressing decimally (0.666...) the figure 6 is repeated again and again. So in expressing 14 decimally (0.4242...) the figures 4 and 2 are repeated again and again. Decimal fractions that are expressed by the same figures repeated again and again are called repeating or circulating decimals. NOTE. Circulating decimals arise from the reduction of common fractions whose denominators contain prime factors other than 2 and 5. 285. The repeating figures of a circulating decimal are called a repetend. A repetend is marked by placing dots over the first and last of the figures that repeat. Thus, 14 = 0.297297 ... = 0.297; 1 공동 = 0.4242 ... = 0.42; 31 0.42; 31 = 3.166 ... = 3.16. 286. Change the following fractions to decimals till the figures repeat, and mark the repetends : (52.) } (55.) 5. (58.) 18 (61.) 24 (53.) 6 (56.) 11. (59.) 22. (62.) 135 (54.) 4. (57.) 4. (60.) 11: (63.) 337. WRITTEN WORK. To change a Circulating Decimal to a Common Fraction. 287. ILLUSTRATIVE EXAMPLE I. Change 0.63 to a common fraction. To change a circulating decimal to a common fraction: Take the repetend for the figures of the numerator, and for the figures of the denominator as many 9's as there are figures in the repetend. Change the fraction thus expressed to its smallest terms. 0.63 = 3 = 11 For an explanation of this rule, see Appendix, page 305. Change to common fractions in their smallest terms: (64.) 0.3 (65.) 0.6 (66.) 0.42 (67.) 0.39 (70.) 0.016 (71.) 0.62i (72.) 0.108 (73.) 0.1881 (74.) 0.428571 (75.) 0.571428 WRITTEN WORK. To change a Mixed Circulate to a Common Fraction. 288. ILLUSTRATIVE EXAMPLE II. Change 0.263 to a common fraction. To change a mixed circulate to a com mon fraction: Take for the numerator the 263 tft=difference between the mixed circulate and 2 the part which does not repeat, both regarded 261 as integers, and take for the figures of the denominator as many 9's as there are figures in the repetend, with as many zeros annexed as there are figures in that part of the circulate which does not repeat. (See Appendix, p. 305.) Change to common fractions in their smallest terms: (76.) 1.86 (77.) 2.73 (78.) 0.033 (79.) 0.027 (80.) 0.016 (81.) 0.042 (82.) 2.07671 MULTIPLICATION. WRITTEN WORK. In Articles 82 and 86 the multiplication of decimals by integers has been taught. These the pupil may now review. 289. ILLUSTRATIVE EXAMPLE I. Multiply 175 by 0.01. Multiply 175 by 0.5. Explanation. — (1.) To multiply 175 by 0.01 is to take 1 hundredth of it, which we (1.) 175 < 0.01 =1.75 express by placing the decimal point so that the figures 175 may express hundredths ; (2.) 175 thus, 1.75. 0.05 (2.) To multiply 175 by 0.05 is to take 5 hundredths of it. One hundredth of 175 is 8.75 1.75, and 5 hundredths is 5 times 1.75, which equals 8.75. Ans. 8.75. 290. ILLUSTRATIVE EXAMPLE II. Multiply 0.4 by 0.9. WRITTEN WORK. 0.4 0.9 Explanation. - To multiply 0.4 by 0.9 is to take 9 tenths of 4 tenths. One tenth of 0.4 is 4 hundredths, and 9 tenths of 4 tenths is 9 times 0.04, which equals 0.36. Ans. 0.36. 0.36 291. From the written work above may be derived the following Rule. To multiply by decimals : Multiply as in integers, and print off as many places for decimals in the product as there are decimal places in the multiplicand and the multiplier counted together. NOTE. If there are not figures enough in the product, prefix zeros. 292. Examples for the Slate. Multiply (84.) 0.048 by 9. (93.) 40.5 by 0.016 (85.) 0.027 by 34. (94.) 1842 by 0.07 (86.) 0.075 by 20. (95.) 0.0758 by 20. (87.) 84 by 0.056 (96.) 6.6 by 33} (88.) 600 by 0.07 (97.) 10.75 by 83 (89.) 8.4 by 0.56 (98.) 187 by 0.054 (90.) 4.65 by 2.2 (99.) 561 by 2.73 (91.) 0.8 by 0.0206 (100.) 1.7 by 2727 (92.) 7.06 by 0.053 (101.) 66.63 by 5.7 102. What is the sum of 75 x 100 and 0.001 x 1000 ? 103. What is the sum of 7.5 x 1000 and 0.0001 ~ 0.001 ? 104. How many are 56.8 x 0.01 + 5.29 1000 + 0.7 x 0.001 ? 105. How many are 48.125 x 8.331 +8169.5 x 0.09 ? 106. What is the cost of paving 146.74 squares at $ 16.84 per square For other examples in multiplication of decimals, see page 135. |