From the above illustrations we derive the following RULE. To divide decimal fractions: Divide as in whole numbers. If the divisor is a whole number, point off as many decimal places in the quotient as there are decimal places in the dividend. If the divisor is not a whole number, make it a whole number before dividing, by removing the decimal point to the right. Remove the decimal point in the dividend as many places to the right; divide, and point off as many decimal places in the quotient as there are in the altered dividend. NOTE I. When there is a remainder after all the figures in the dividend are exhausted, zeros may be annexed, and the division continued. In pointing off, the annexed zeros must be considered as places in the div idend. NOTE II. - In the examples in this book, when there is a remainder, the quotient may be continued to the fifth decimal place, if no other direction is given. 1. 14.917=? 2. .0726 = EXAMPLES. Ans. 2.13. 16. 6807771.66=? Ans. 950. ? Ans. 5.5. 4. 3.24.81=? 5. .00468.013 =? 6. 5446.7768 = ? 7. 180.375 1.625 = ? 18. .17064.2368=? 20. 10588.1.4606=? 8. 579.075=? Ans. 7720. 11. 1.29.32 =? 12. .7057.5=? Ans. .094. 28. 49.2654756.0759 =? 13. 329.9? Ans..10033.29. 2464.176 ÷ 57.2 =? 14. 20.013 =? 30. 164.6156 ÷ 1334=? Ans. 1538.46153+. 31. .07991997 83497 =? 15. 4066.2 ÷ .648=? 32. 20339.82009 1.07001=? 33. Divide 93.75 by 3265096.575, and give three significant figures in the quotient. 34. Find the product of the quotients of the following to 6 places: .650849583.69; 405000. 35. What is the quotient of 1.497 (260.401 — 13.02)? 36* Required the product of the quotients of the following: 1021 ten millionths 107 ten thousandths; 2012 millionths → 1.006. X 37. Divide 600 by .006, multiply the quotient by .05, and by that product divide .005. 38 (1.002) X (.250)=? * -39 (80.48182589.325) × (9617.5168.47896)=? 40 Required the product of the sum and difference of the following: 856494839.7; .00946589.4. (To 6 places.) 41. Divide the difference of the above quotients by their sum. For Dictation Exercises, see Key. 238. REDUCTION OF COMMON FRACTIONS TO DECIMAL FRACTIONS. ILL. EX. Reduce to a decimal fraction. OPERATION. 8) 7.000 .875, Ans. of 7; of 7 no whole ones, with a re mainder of 7, which reduced 70 tenths (7.0); of 70 tenths: .8 with a remainder of .6; .6= 60 hundredths; of.60.07 with a remainder of .04; .04 40 thousandths; of .040.005 ;.. .875. Hence the RULE. To reduce a common fraction to a decimal fraction: Annex zeros to the numerator, and divide it by the denominator. Point off as many decimal places as there are zeros annexed. 855. 14.3. 15.. 16. Reduce to decimals, and add, 1, 18825, 3400. 17. Reduce to seven places, and add, 1.82, .009, and 10. For Dictation Exercises, see Key. RULE. To reduce a decimal fraction to a common fraction: Represent the decimal fraction in the form of a common fraction having for its denominator 1 with as many zeros annexed as there are decimal places in the decimal fraction, and reduce the common fraction to its lowest terms. 240. To add or subtract decimal fractions terminated by common fractions: Reduce all the decimals to the same denomination; then add or subtract as by Art. 143 and 144; thus, .34+ .83what? 31.831.333.8381.16g, Ans. EXAMPLES. 1. Add .087, 9.04, .71, 2758, and .04. Ans. 285.2549 2. Add 19.37, 10.0,, and .0416. 3. Subtract .05555 from .3333. 4. Subtract 1.2076244 from 1§. 5.3+.63+.833+.285714+.5714284+.637=? 241 CIRCULATING DECIMALS. If the denominator of a common fraction (when the fraction is in ita lowest terms) contains any prime factor besides 2 and 5, the fraction i> not capable of being entirely reduced to a decimal form. In reducing such fractions, if the division be continued, the same figures will recur again and again in the decimal fraction. These fractions are called Repeating or Circulating Decimals. The figures which repeat are called a Repetend. A Repetend is distinguished by two dots written over the first and last of the figures that repeat; thus,=.297297+=.297. 243 REDUCTION OF CIRCULATING DECIMALS TO COM It can be proved that the Repetend of a Circulating Decimal equals a fraction whose numerator is the repetend, and whose denominator is as many 9's as there are places in the repetend. Hence the RULE. To reduce a Circulating Decimal to a common fraction: Express the repetend as a common fraction having as many 9's for the denominator as there are figures in the repetend, and reduce. If any part of the decimal fraction does not repeat, annex the reduced repetend to it, and change the complex fraction thus obtained to a simple fraction. NOTE. - Circulating decimals may be added, subtracted, multiplied, and divided, by first reducing them to common fractions. Other processes might here be given, but the reasoning is too abstruse for an elementary treatise. ILL. EX., I. Reduce .09 to a common fraction. 244. TO REDUCE COMPOUND NUMBERS TO DECIMAL FRACTIONS OF Higher DENOMINATIONS. OPERATION. 43.00 qr. d. ILL. EX., I. Reduce 2 d. 3 yr. to the decimal of a shilling. Since 4 qr. equal 1 d., there will be as many as qr., or d., which equals .75 d.; this, with the 2 d. given, equals 2.75 d.; since 12 d. equals 1 shilling, there will be as many shillings as d., &c. 12 2.75000 d. .22916 S., Ans. ILL. EX., II. What is the value of 3 rds. 4 yds. 2 ft. in the decimal of a rod? From the above, we deduce the following RULE. To reduce compound numbers to decimal fractions of higher denominations: Divide the number of the lowest denom ination by what it takes of that denomination to make one of the next higher; place the quotient as a decimal fraction at the right of that higher; so continue till all the terms are reduced to the denomination required. EXAMPLES. 1. Reduce 7 d. 3 qr. to the decimal of a £. Ans. £.03229+ 2. Reduce 3 da. 22 h. 4 m. 48 sec. to the decimal of a week. Ans. .56 wk. 3. Reduce 5 cwt. 3 qr. 10 lb. to the decimal of a ton. |