Question

A certain number between $10$ and $100$ is $8$ times the sum of its digits, and if $45$ be subtracted from it the digits will be reversed. Find the number.

Answer 1

We have,

Suppose that the number$=x$ and $y$

Then, the value of number is $10x+y$

According to given question,

$10x+y=8(x+y)$

$10x+y=8x+8y$

$10x-8x=8y-y$

$2x=7y$

$x=\frac{7y}{2}......\left(1\right)$

Again, according to given question,

$10x+y-45=10y+x$

$10x+y-x-10y-45=0$

$9x-9y-45=0$

$x-y-5=0......\left(2\right)$

Put the value of x by equation (1) in equation (2) and we get,

$\frac{7y}{2}-y-5=0$

$\frac{7y-2y-10=0}{2}$

$5y=10$

$y=2$

Put the value of y in equation (1) and we get,

$x-y-5=0$

$x-2-5=0$

$x=7$

The number is

$10x+y$

$=10\times 7+2$

$=72$

Hence, this is the answer.