Question

When one is added to each of two given numbers, their ratio becomes $3:4$ and when $5$ is subtracted from each, the ratio becomes $7:10$. The numbers are

• A. $8,11$
• B. $11,15$
• C. $26,35$
• D. $27,36$

Answer 1

The correct option is C $26,35$

Let the two numbers be $x$ and $y$.

According to the given condition,

$\frac{x+1}{y+1}=\frac{3}{4}$
$\Rightarrow 4(x+1)=3(y+1)$

$\Rightarrow 4x+4=3y+3$

$\Rightarrow 4x-3y=3-4$
$\Rightarrow 4x-3y=-1$ ........$\left(i\right)$

Also,

$\frac{x-5}{y-5}=\frac{7}{10}$
$\Rightarrow 10(x-5)=7(y-5)$

$\Rightarrow 10x-50=7y-35$

$\Rightarrow 10x-7y=-35+50$
$\Rightarrow 10x-7y=15$ .......$\left(ii\right)$

Multiply equation $\left(i\right)$ by $7$, we get

$[4x-3y=-1]\times 7$

$\Rightarrow 28x-21y=-7$.............$\left(4\right)$

Mutliply equation $\left(ii\right)$ by $3$, we get

$[10x-7y=15]\times 3$
$\Rightarrow 30x-21y=45$...............$\left(5\right)$

Subtract equation $\left(4\right)$ from $\left(5\right)$

$30x-21y=45$

$\underset{–}{(-)28x-21y=-7}$

$\Rightarrow 2x=45-(-7)=45+7$
$\Rightarrow 2x=52$
$\Rightarrow x=\frac{52}{2}$

$\Rightarrow x=26$

Putting the value of $x$ in $\left(i\right)$

$4\left(26\right)-3y=-1$
$\Rightarrow 104-3y=-1$

$\Rightarrow -3y=-1-104$
$\Rightarrow -3y=-105$

$\Rightarrow y=\frac{-105}{-3}=35$

$\therefore$ Numbers are $26,35$.