Question

The numerator of a fraction is 1 less than its denominator. If the numerator is increased by 1 and denominator is increased by 5, the new fraction becomes 4/5. What is the original fraction?

Answer 1

$\sf \Large Solution!!$

Let the denominator be x

Given, numerator is 1 less than denominator

The numerator will be x-1

The fraction is $\frac{x-1}{x}$

The numerator is increased by 1 => (x-1)+1

The denominator is increased by 5 => x+5

The new fraction => $\frac{(x-1)+1}{x+5}=\frac{4}{5}$

=> $\frac{x}{x+5}=\frac{4}{5}$

=> 5x=4(x+5) => 5x=4x+20

=> 5x-4x=20 => x=20

The new fraction => $\frac{(x-1)+1}{x+5}$

=> $\frac{(20-1)+1}{20+5}$

=> $\frac{20}{25}$

=> $\frac{4}{5}$

The original fraction => $\frac{x-1}{x}$

=> $\frac{20-1}{20}$

=> $\frac{19}{20}$

Answer 2

Given:

• The numerator of a fraction is 1 less than its denominator.
• If the numerator is increased by 1 and denominator is increased by 5, the new fraction becomes 4/5.

To find:

• Original Fraction?

Solution:

☯ Let Denominator of fraction be x.

Then, Numerator of fraction will be (x - 1).

⠀⠀⠀⠀

$\underline{\bigstar\:\boldsymbol{According\:to\:the\:question\::}}$

$:\implies\sf \dfrac{(x - 1) + 1}{x + 5} = \dfrac{4}{5}\\ \\ \\:\implies\sf \dfrac{x}{x + 5} = \dfrac{4}{5}$

$:\implies\sf 5(x) = 4(x + 5)\qquad\qquad\bigg\lgroup\bf Cross\: Multiplication \bigg\rgroup$

$:\implies\sf 5x = 4x + 20\\ \\ \\ :\implies\sf 5x - 4x = 20\\ \\ \\:\implies{\underline{\boxed{\frak{\purple{x = 20}}}}}\;\bigstar$

Therefore,

• Denominator of fraction, x = 20
• Numerator of fraction, (x - 1) = 20 - 1 = 19

⠀⠀⠀⠀

$\therefore\:{\underline{\sf{Hence,\:The\: original\:fraction\:is\: \bf{ \dfrac{19}{20}}.}}}$

⠀⠀⠀⠀━━━━━━━━━━━━━━━━━━━━━

$\qquad\qquad\boxed{\bf{\mid{\overline{\underline{\pink{\bigstar\: Verification :}}}}}\mid}$

Given that,

⠀⠀⠀⠀

• If the numerator is increased by 1 and denominator is increased by 5, the new fraction becomes 4/5.

⠀⠀⠀⠀

$:\implies\sf \dfrac{19 + 1}{20 + 5} = \dfrac{4}{5}\\ \\ \\:\implies\sf \cancel{ \dfrac{20}{25}} = \dfrac{4}{5}\\ \\ \\:\implies\sf \dfrac{4}{5} = \dfrac{4}{5}$

$\qquad\qquad\qquad\dag\:{\underline{\underline{\sf{\purple{Hence\: Verified!}}}}}$