## FINDING THE REQUIRED FRACTION BY SOLVING WORD PROBLEMS

Finding the Required Fraction by Solving Word Problems :

We consider the required fraction as x/y.

Here x - numerator and y - denominator.

Example 1 :

The denominator of the fraction is 5 more than its numerator. The sum of the fraction and its reciprocal is 73/24. Find the fraction.

Solution :

Let x/y be the required fraction.

The denominator of the fraction is 5 more than its numerator.

Then,

y  =  x + 5

x / y  = x / (x + 5)

The sum of the fraction and its reciprocal is 73/24.

x / (x + 5) + (x + 5) / x  =  73 / 24

x2 + (x + 5)2 / x(x + 5)  =  73/24

(2x2 + 10x + 25)/(x+ 5x)  =  73/24

24(2x2 + 10x + 25)  =  73(x2 + 5x)

48x2 + 240x + 600  =  73x2 + 365x

73x² – 48x²  + 365x – 240x – 600  =  0

25x2 + 125x – 600  =  0

x2 + 5x – 24  =  0

(x + 8) (x – 3)  =  0

 x + 8  =  0x  =  -8 x - 3  =  0x  =  3

By applying the value of x in (1), we get

y  =  x + 5

y  =  3 + 5

y  =  8

Therefore the required fraction is 3/8.

Example 2 :

The denominator of the fraction is one more than twice the numerator. If the sum of the fraction and its reciprocal is 58/21, find the fraction.

Solution :

Let x/y be the required fraction.

The denominator is one more than twice the numerator.

Then,

y  =  2x + 1

x/y  =  x/(2 x + 1)

The sum of the fraction and its reciprocal is 58/21

x/(2x + 1) + (2x + 1)/x  =  58/21

x2 + (2x + 1)2 / x(2x + 1)  =  58/21

Simplifying the numerator :

x2 + (2x + 1) =  x2 + 4x2 + 4x + 1

x2 + (2x + 1)2  =  5x2 + 4x + 1

Simplifying the numerator :

x(2x + 1)  =  2x2 + x

(5x2 + 4x + 1) / (2x+ x)  =  58/21

21(5x2 + 4x + 1)  =  58(2x+ x)

105x2 + 84x + 21  =  116x2 + 58x

116x2 - 105x+ 58x - 84x - 21  =  0

11x2 - 26x - 21  =  0

(x – 3) (11 x + 7)  =  0

 x - 3  =  0x  =  3 11x + 7  =  0x  =  -7/11

When x  =  3,

y  =  2x + 1

y  =  2(3) + 1

y  =  7

Therefore the required fraction is 3/7

After having gone through the stuff given above, we hope that the students would have understood how to find the required fraction by solving word problems.

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