Question

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The numerator of a fraction is 7 less than the denominator. If 3 is added to the numerator and 9 is subtracted from the denominator, the new fraction is equal to 3/2. Find the original fraction.​

Answer 1

The numerator of a fraction is 7 less than the denominator.

Let numerator be "N" and denominator be "D".

According to question,

$\Rightarrow\:N\:=\:D\:-\:7$ _____ (eq 1)

If 3 is added to the numerator and 9 is subtracted from the denominator, the new fraction is equal to 3/2.

Now,

• Numerator = N + 3
• Denomination = D - 9

According to question,

$\Rightarrow\:\dfrac{N\:+\:3}{D\:-\:9}\:=\:\dfrac{3}{2}$

Cross multiply them

$\Rightarrow\:2(N\:+\:3)\:=\:3(D\:-\:9)$

$\Rightarrow\:2N\:+\:6\:=\:3D\:-\:27$

$\Rightarrow\:2(D\:-\:7)\:+\:6\:=\:3D\:-\:27$

[From (eq 1)]

$\Rightarrow\:2D\:-\:14\:+\:6\:=\:3D\:-\:27$

$\Rightarrow\:2D\:-\:8\:=\:3D\:-\:27$

$\Rightarrow\:2D\:-\:3D\:=\:-\:27\:+\:8$

$\Rightarrow\:-\:D\:=\:-\:19$

$\Rightarrow\:\boxed{D\:=\:19}$

Put value of D in (eq 1)

$\Rightarrow\:N\:=\:19\:-\:7$

$\Rightarrow\:\boxed{N\:=\:12}$

Now,

$\rightarrow\:Fraction\:=\:\dfrac{Numerator}{Denomination}\:=\:\dfrac{N}{D}$

From above calculations we have N = 12 and D = 19

So,

$\rightarrow\:Fraction\:=\:\dfrac{12}{19}$

∴ Original fraction is 12/19.

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☆ Verification :

From above calculations we have N is 12 and D is 17.

Put value of N and D in (eq 1)

=> 12 = 19 - 7

=> 12 = 12