Fractions. .28 + EXAMPLE III. Explanation. It will be observed that, however far this division may be carried, there is always a re. mainder. The fact that there is a remainder 72.000 is indicated by writing the plus sign after the last figure of the decimal. The first quotient .285 + may be read 28 hundredths, plus. Observe, too, that the error in the first 7)2.0000 answer is less than 1 hundredth, since the .2857 + true quotient is more than 28 hundredths and less than 29 hundredths. We may therefore say that the first result is true to hundredths; the second, true to thousandths, etc. 1. Reduce to decimals, true to thousandths. 10.5 11. 12. Determine which of the following fractions can be reduced to terminating decimals † and which cannot. 1. 3 10. 13. * 2. 4 5. 2 / 옿 8. 11. is 14. 3 6. 9. À 9 3 10 5 12 5 3. * 12. 15. 27 Observe that if a fraction is in its lowest terms and the denoininator contains any other prime factor besides 2's and 5's, the fraction cannot be reduced to a “terminating” decimal. Can you tell why? * Observe that the difference between a aud b must be less than 12 thousandths. Why? + See footnote, page 113. Fractions. 203. TO REDUCE A COMPLEX DECIMAL* TO A COMMON FRACTION. 1. .384 7. .61} 8. .16% 9. .544 5. .3511 6. .40341 NOTE.-If the division be carried sufficiently far in any non termiDating decimal there will be found a certain figure or set of figures that is constantly repeated: thus, we may have, .3666666, or .27272727 or .5236236236. The part repeated is called a repetend, and may be written thus: .38, 27, .5236. It is a curious fact that the real denominator of any repetend is as many 9's as there are figures in the repetend; .36 = .38, 27 = }, .5236 .533.. * Decimals that are complete without the annexation of a common fraction are said to be terminating decimals. .24 is a terminating decimal. .66.666 + is a nonterminating decinial. A decimal with a common fraction annexed, as .3333, is sometimes called a complex decimal. Observe that this work is not different from that given under example II., page 111., except that the number of zeros to be annexed to the numerator is always two. Why? I. Reduce to hundredths. 16. 17. 13. 10 18. 2 19. 15. 20. 3 5 14 II. Reduce to hundredths. 7. 14 5. 14:26 8. 16 9. 118 3 5 176 Fractions. 36 300 III. Reduce to hundredths. NOTE.-Such fractions as the following may be easily reduced to hundredths by dividing the numerator and the denominator of each by that number which will change the denominator to 100.* 1. 37 5. 106 9. 10.30 11. 86 12. 138 3. 2046 7. 120 IV. Reduce to hundredths. NOTE.- Multiply the numerator and the denominator of each fraction by that number which will change the denominator to 100*. 7 8 13 7. 20 12} (a) Find the sum of the nine decimals. * Every common fraction can be changed to hundredths by annexing two zeros to the numerator and dividing by the denominator; but this method of reduction is not always the most simple. + Take of the numerator and of the denominator. Multiply the numerator and the denominator by 24. Fractions. 205. DENOMINATE FRACTIONS. 1. One half inch is what part of a foot ? 2. Two and I inches are what part of a foot ? 3. Five and inches are what part of a foot? 4. One half foot is what part of a rod ? 5. Three and one half feet are what part of a rod ? 6. Ten and one half feet are what part of a rod ? 7. Sixty-four rods are what part of a mile ? 8. Ninety-six rods are what part of a mile? 9. One hundred eighty rods are what part of a mile? 10. One and one half quarts are what part of a peck ? 11. Two and one half quarts are what part of a gallon ? 12. Twenty-four quarts are what part of a bushel ? 13. Fourteen ounces are what part of a pound? 14. Seven and one half ounces are what part of a pound? 15. One and one fourth ounces are what part of a pound? 16. Six hundred pounds are what part of a ton ? 17. Four hundred fifty pounds are what part of a ton ? 18. Six hundred twenty-five pounds are what part of a ton? 19. Seventy-five square rods are what part of an acre? 20. Forty-five square rods are what part of an acre? 21. One hundred square rods are what part of an acre? 22. Thirty-two cubic feet are what part of a cord? 23. Fifty-six cubic feet are what part of a cord ? 24. One hundred cubic feet are what part of a cord ? 25. Seven and one half minutes are what part of an hour? 26. Forty minutes are what part of an 8-hour day? 27. Ninety minutes are what part of an 8-hour day? |