# Write equivalent fractions with fraction 156/273 ,which have the property that for each , the sum of the numerator and denominator is a number S, so that 143 <S <209 .

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Equivalent fractions are fraction that have different numerators and denominators but have the same value.

One way of generating equivalent fraction is to multiply both numerator and denominator by a same whole number, other than 1, to get equivalent fractions. However this makes the fraction more difficult to understand.

The most easy to use equivalent fraction is one which has been reduced to a form so that the numerator and denominator have no common factors. This can be done by dividing both numerator and denominator by the largest common factor.

For the given fraction the largest common factor is 39. So we can reduce the given fraction as follows.

156/237 = (156/39)/(273/39) = 4/7

Sum of the numerator and denominator of these reduced equivalent fraction:

= S = 4 + 7 = 11

However this gives a value of S which does not meet the given condition for S.

If we reduce the fraction by dividing both numerator and denominator by 3 the fraction becomes:

52/91

For this value of S = 143

If we reduce the fraction by dividing both numerator and denominator by 13 the fraction becomes:

12/21

For this value of S = 33

To get a value of S between 143 and 209 we multiply the numerator and denominator of equivalent fraction 4/7 by 14.

Thus: 4/7 = (4*14)/(7*14) = 56/98

For this equivalent fraction value of S = 154.

Other equivalent fraction meeting the condition are:

60/105 ... (S = 165)

64/112 ... (S = 176)

68/119 ... (S = 187)

72/126 ... (S = 198)

The given fraction is 156/273 = 52/91 = p/q say, q>p.

Sum of numerator and denominator obeys 143< p+q < 209.

P/q = 52/91 ............(1) and

143< p+q <209........ (2) Or

156 and 273 has the HCF 39. Therefore

156/273 has the equivalent fraction in the lowest form (156/39)/273/39) = 4/7.

Now the equivalent forms of the fractions are

(4n)/(7n), where 143<4n+7n <209 Or

11n >143 Or 11n <209 Or

n> 143/11 =13 and n < 209/11 = 19 Or

4n/(7n) for n = 13,14,15,16,17,18,and 19.

By simplifying fraction (156/273), is resulting the irreducible fraction (4 / 7), and the sum of its numerator and denominator is 4 + 7 = 11.

The fraction (4 / 7), must be amplified with a number, so that S = 143. The number is 143:11 = 13.

The fraction (4 / 7), must be amplified with a number, so that S = 209, is 209:11 = 19.

But S must be between 143 and 209, so equivalent fractions will be obtained by amplifying the irreducible fraction, in turn, with numbers between 14 and 18.

**Equivalent fractions obtained are:**

**56/98, **

**60/105, **

**64/112, **

**68/119, **

**72/126.**