We observe that the approximation is alternately greater and less than the real value, and that the denominators of the continued fractions are the quotients, which are found by the method of ascertaining the greatest common measure of two numbers. This last observation leads us to a short process of finding the denominator of the continued fractions, the numerators of which are unity. 11513 120. Ex. Express in a continued fraction, and find the converging fractions. 7529)11513(1 3984)7529(1 3545)3984(1 439)3545(8 3512 33)439(13 10)33(3 30 3)10(3 9 3 223 686 2281 and the approximating values are: 1, 1, 3, 13, 111, 106, 1311, 7529 11513 121. EXERCISES. 7; also of 1103' 1. Find the approximating values of 335 3. On an average of 100 years, it is found that the lunar month consists of 27.321661 days. From this datum, it would follow that the moon describes 1000000 revolutions round the earth in 27321661 days. Express, approximately, the fractions which give its daily revolution. 4. It is found that the circumference of a circle, whose diameter is 1, is 3.1415926535, nearly. Find the converging fractions expressing nearly the ratio of the diameter to the circumference. 5. The planet Mercury revolves in 87.969255 days, and Venus in 224.700817. Express these times of revolution approximately, by small numbers. 122. Suppose it was required to reduce into a whole number, or a mixed quantity; that is, into a whole number and a fraction. Since 9-ninths make 1 whole, 18-ninths are equal to 2 wholes, 27-ninths to 3 wholes; therefore, as many times as 9 is contained in 44, so many wholes will there be, now 9 is contained in 44, 4 times+the remainder §, then ✨=48. Therefore, to reduce an improper fraction to a whole or mixed number, divide the numerator by the denominator; the quotient expresses the whole number, and if there be a remainder, place it over the denominator for the fractional part. 123. Exercises: Reduce the following fractions to a whole or mixed number:-1, 3, 4, 11, 10°, 47, 31°, 2335, 2009 24 171 124. Let us consider the converse of the preceding case; for instance, suppose it were required to reduce the mixed quantity 52 to an improper fraction. Since unity is supposed to be divided into 7 equal parts, every unit=7-sevenths; therefore, 5 units=5 x 7-sevenths, or 35-sevenths, and 2-sevenths make altogether 37-sevenths. Therefore, 5=37. Hence, to reduce a mixed quantity to an improper fraction, multiply the integer by the denominator of the fraction, and to the product add the numerator of the fraction; the sum is the new numerator, and the denominator of the fraction will be its denominator. 125. Exercises: Express the following mixed quantities as improper fractions:-33, 74, 163, 147, 268, 3511, 48††, 1243}. 126. RECAPITULATION.-EXERCISES. 1. Express that an object is divided into 25 equal parts, and 17 of them taken away. 2. Exhibit the quantity represented by an object broken into 134 equal parts, and 21 of them taken away. 35 3. Write down the following fractions in words:-1, 4, 6, 17, 144. What is the meaning of, 4, 3, 1), 11? 13 5. Write down in figures the following expressions-five-sixths, one-thirteenth, seven-tenths, thirteen-fifteenths, three-fifths, four-ninths, three-nineteenths, fifteen-twenty-eighths. 6. Which is the greater or .? Why? Why? Why? 7. Which is the greater of, 5 6 10 ? 13, 13, 13, 13, 13 8. Express in figures that a workman has done but threequarters of his work. 9. Two workmen are in a factory, one works five-sevenths of the time, and the other five-sixths. Express in figures the time each worked, and also mention which worked the longer time. 10. Two men work in the same manufactory, the first performs one-fifth less than his daily work, and the second one-fifth more. How much does each perform? 31 11. What name is given to quantities of this form: 15, 11 21, 76, 100, 134 ? and express them in mixed quantities. 12. Which quantity is twice as great as ?? three times 19 47 13. The treble of is required; the quadruple of the double of? 14. A mason builds, in one day, of a wall. How much will he do in 7 days? 15. A man goes, in one day, of his journey. How much will he accomplish in 8 days? 17. How much is one-quarter of one quarter? 19. In 5 days, of a work is performed; how much is done in one day? In nine days, &; how much in one day? In seven days,; how much in one day? 20. A and B divide a sum of money equally between them; A divides his share equally between his three children, and B divides his equally between his four children. What part of the whole money does each of the children of A and B receive? 21. A person received of an inheritance, which was trebled in trade; his son quadrupled that fortune; but the grandson, failing in business, lost one-seventh of his inheritance. What part of the whole inheritance was left? 22. Required, the greatest common measure of 28 and 98; of 345 and 2415; of 22893 and 79245. 23. Reduce to their lowest terms, if possible, 720 1664 18 317 8739 504 12969 24. A shopkeeper was asked for the 31 of a yard of cloth, but not understanding the question, what must be done? 24. A boy who was offered the part of an orange, asked that the quantity be put in a simpler form. How is this to be done? ADDITION 127. If to of a yard The sum of § and 3 is 7. OF FRACTIONS. be added, the sum is of a yard. Find the sum of ++ +}=}; } + } ==1} 13 Therefore, when fractions have the same denominator, add all the numerators together, and place their sum over the denominator, the result will be the sum of the fractions, which must be reduced to a whole or mixed quantity, when possible. 128. How much are +? These fractions, not having the same denominator, cannot be put together without transforming them to fractions of the same denominator, for instance : Hence, +3+3=5. We observe that sixths by multiplying the terms by 3: formed by multiplying the terms by 2: is transformed into 3 x 1 3 × 2 6 129. The sum of 3, 4, and § is required. Here the fractions can all be transformed into twelfths : Therefore, ++} = A +++8=}}=2135=24. |