Question

The difference between the two digits of a number less than 100 is 2. If 3 upon2 times the sum of the digits be diminished from it ,the digits will be reversed. Find the number.

Answer 1

Let the two digit number be such that the tens digit should be greater than the units digit.

Since the difference between the digits of the number is 2, let our two digit number be $\sf{10x+(x-2).}$

Here the digits of the number are $\sf{x}$ and $\sf{x-2,}$ whose sum is equal to $\sf{2x-2.}$

Given that, $\sf{\left(\dfrac{3}{2}\right)^{th}}$ of the sum of the digits, when subtracted from the number, makes the number interchange its digits. So,

$\longrightarrow\sf{10x+(x-2)-\dfrac{3(2x-2)}{2}=10(x-2)+x}$

$\longrightarrow\sf{11x-2-(3x-3)=11x-20}$

$\longrightarrow\sf{11x-2-3x+3=11x-20}$

$\longrightarrow\sf{8x+1=11x-20}$

$\longrightarrow\sf{3x=21}$

$\longrightarrow\sf{x=7}$

This is the tens digit of our number. So the units digit should be,

$\longrightarrow\sf{x-2=5}$

Hence our two digit number is,

$\longrightarrow\sf{\underline{\underline{10x+(x-2)=75}}}$

Verification:-

• 75 is a number less than 100.

• Digits of 75, i.e., 7 and 5, differ each other by 2.

• Sum of digits is 7 + 5 = 12, whose $\sf{\left(\dfrac{3}{2}\right)^{th}}$ is 18. When 18 is subtracted from 75, we get 57, i.e., the digits are interchanged.

Hence 75 is the answer.