Question

6. The sum of numerator and denominator of a fraction is 3 less than twice the denominator. If each of the numerator and denominator is decreased by 1 the fraction becomes ½. Find the fraction. Answer In x and y instead of a and b.

Answer 1

GIVEN:

• The sum of numerator and denominator of a fraction is 3 less than twice the denominator.
• If each of the numerator and denominator is decreased by 1 the fraction becomes ½.

TO FIND:

• What is the fraction ?

SOLUTION:

Let the numerator be 'x'

and

Denominator be 'y'

✎ The fraction become x/y

CASE: 1)

The sum of numerator and denominator of a fraction is 3 less than twice the denominator.

According to question:-

$\rm{\hookrightarrow x + y = 2y - 3 }$

$\rm{\hookrightarrow x +3 = 2y -y }$

$\bf{\hookrightarrow x + 3 = y...1) }$

CASE: 2)

If each of the numerator and denominator is decreased by 1 the fraction becomes ½.

According to question:-

$\rm{\hookrightarrow \dfrac{x-1}{y-1} = \dfrac{1}{2}}$

$\rm{\hookrightarrow 2x- 2 = y-1 }$

$\rm{\hookrightarrow 2x -y = -1+2 }$

$\bf{\hookrightarrow 2x - y = 1...2) }$

Put the value of 'y' from equation 1 in equation 2)

$\rm{\hookrightarrow 2x -x -3 = 1 }$

$\rm{\hookrightarrow x = 1+3}$

$\bf{\hookrightarrow x = 4 }$

Put the value of 'x' in equation 1)

$\rm{\hookrightarrow 2 \times 4 - y = 1 }$

$\rm{\hookrightarrow 8 -y = 1 }$

$\rm{\hookrightarrow 8-1 = y }$

$\bf{\hookrightarrow 7 = y }$

• x = Numerator = 4
• y = Denominator = 7

$\large{\boxed{\bf{\star \: FRACTION = \dfrac{4}{7} \: \star}}}$

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