Question

If the numerator and denominator of a fraction are each increased by 4 the fraction become 2 and when numerator and denominator of the same fraction are each decreased by 6 the fraction become 12. What is the sum of the numerator and denominator ​

Answer 1

Solution

Given :-

• If the numerator and denominator of a fraction are each increased by 4 the fraction become 2
• numerator and denominator of the same fraction are each decreased by 6 the fraction become 12.

Find :-

• sum of the numerator and denominator

Explanation

Let,

• Numerator = x
• Denominator = y

Case(1).

(If the numerator and denominator of a fraction are each increased by 4 the fraction become 2)

==> (x+4)/(y+4) = 2

==> (x+4) = 2*(y+4)

==> x - 2y = 8 - 4

==> x - 2y = 4 _________(1)

Case(2).

(numerator and denominator of the same fraction are each decreased by 6 the fraction become 12.)

==> (x-6)/(y-6) = 12

==> (x-6) = 12*(y-6)

==> x - 12y = -72 + 6

==> x - 12y = -66 __________(2)

Subtract equ(1) & equ(2)

==> -2y + 12y = 66 + 4

==> 10y = 70

==> y = 70/10

==> y = 7

Keep value of y in equ(1)

==> x - 2*7 = 4

==> x = 4 + 14

==> x = 18

Hence

• Numerator be (x) = 18
• Denominator be (y) = 7

Now , calculate sum of Numerator & Denominator

• (x+y) = (18 + 7) = 25

__________________

Answer 2

Solution :

Let the numerator place digit be r & denominator place digit be m respectively;

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

$\underbrace{\bf{1^{st}\:case\::}}}$

$\longrightarrow\sf{\dfrac{r+4}{m+4} =2}\\\\\longrightarrow\sf{1(r+4) = 2(m+4)}\\\\\longrightarrow\sf{r+4 = 2m+8}\\\\\longrightarrow\sf{r-2m = 8-4}\\\\\longrightarrow\sf{r-2m = 4}\\\\\longrightarrow\sf{r=4+2m........................(1)}$

$\underbrace{\bf{2^{nd}\:case\::}}}$

$\longrightarrow\sf{\dfrac{r-6}{m-6} =12}\\\\\longrightarrow\sf{1(r-6) = 12(m-6) }\\\\\longrightarrow\sf{r-6 = 12m - 72 }\\\\\longrightarrow\sf{r-12m = -72 + 6}\\\\\longrightarrow\sf{r-12m = -66}\\\\\longrightarrow\sf{4+2m-12m = -66\:\:[from(1)]}\\\\\longrightarrow\sf{4 - 10m = -66}\\\\\longrightarrow\sf{-10m = -66 -4}\\\\\longrightarrow\sf{-10m = -70}\\\\\longrightarrow\sf{m=\cancel{-70/-10}}\\\\\longrightarrow\bf{m=7}$

Putting the value of m in equation (1),we get;

$\longrightarrow\sf{r=4+2(7)}\\\\\longrightarrow\sf{r=4+14}\\\\\longrightarrow\bf{r=18}$

Now;

$\boxed{\sf{Numerator = 18}}}\\\boxed{\sf{Denominator= 7}}}$

⇒ Numerator + Denominator

⇒ 18 + 7

⇒ 25

Thus;

The sum of numerator & denominator will be 25 .