Question

The sum of a two digit number and the number formed by interchanging its digits is $110$. If $10$ is subtracted from the first number, the new number is $4$ more than $5$ times the sum of the digits in the first number. Find the first number.

Answer 1

Let the digits at units and tens place in the given number be $x$ and $y$ respectively. Then,
Number $=10y+x$ (i)

Number formed by interchanging the digits $=10x+y$

According to the given conditions, we have
$(10y+x)+(10x+y)=110$
and, $(10y+x)-10=5(x+y)+4$

$\Rightarrow 11x+11y=110$
and, $4x-5y+14=0$

$\Rightarrow x+y-10=0$
and, $4x-5y+14=0$

By using cross-multiplication, we have

$\frac{x}{14-50}=\frac{y}{-40-14}=\frac{1}{-5-4}$

$\Rightarrow \frac{x}{-36}=\frac{y}{-54}=\frac{1}{-9}$

$\Rightarrow x=\frac{-36}{-9}$ and $y=\frac{-54}{-9}$

$\Rightarrow x=4$ and $y=6$.

Putting the values of $x$ and $y$ in equation (i), we get

Number $=10\times 6+4=64$.