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353 from which we infer that the value of 353 is between

1213'

and.

1213

Proceeding with the fraction of the denominator as before, we

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It follows that the value of the given fraction is between 4

and 2.

To continue the approximation, let us divide both terms of

7 19

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neglecting the fraction we

45

2+

7

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By continuing the process in a similar manner, we find other values which approximate more and more towards 353, until at last we obtain the given fraction. If the remaining fraction after each division be omitted, except when the numerator is unity, we have the following converging fractions:

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119. Ex. Find the fractions converging to 13.

347

The following process is convenient to determine the several fractions approximating to 135:

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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 7529 in a continued fraction, and find the converging fractions.

7529)11513(1
7529

3984)7529(1
3984

3545)3984(1

3545

439)3545(8

3512

33)439(13

429

10)33(3

30

3)10(3

9

1)3(3

3

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223 686 2281

and the approximating values are: 1, 1, 3, 13, 111, 105, 131,

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2. What are the converging fractions to ; also to

?

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 ✨=4§.

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, 4, 4, 11, 100, 4, 36°, 135, 20

24

171

3335 2009

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