Question

The sum of a two digit number and the number obtained by interchanging its digits is 99. Find the number.

Answer 1

Hint: As we need to find out a two digit number then for that we will consider the two digit number in the form of 10x + y which is after changing the variables will give us the interchanged digit as of the form 10y + x. By this we will solve the question and find the required two digit number.

Complete step-by-step solution -
We are going to suppose that the two digit number is 10x + y. And after interchanging the digits x and y we will get the number as 10y + x. As we are informed by the question that the sum of a two digit number and the number obtained by interchanging its digits is 99 therefore, we get that
10x + y + 10y + x = 99
$\Rightarrow $ 11x + 11y = 99
Now we will divide both the equations by 11. Therefore, we will get x + y = 9. Now, we can clearly see that the equation cannot be simplified further and that we do not have any other equation so that we can eliminate any one of the two variables and get to the answer. Therefore, we will have a few numbers as choices which can be obtained by substituting the numbers from 1 to 9 into the equation x + y = 9. So, if we put x = 1 then, we will get 1 + y = 9. On further solving we will get that y = 9 – 1 or y = 8. Thus, the number can be 10x + y = 10(1) + 8 or 10 + 8 which means 18. Continuing in a similar manner we will get a two digit numbers as 18, 27, 36, 45, 54, 63, 72 and 81 which after interchanging the digits will imply 81, 72, 63, 54, 45, 36, 27 and 18.
Hence, the required number can be one out of 18, 27, 36, 45, 54, 63, 72 and 81.

Note: We cannot take a two digit number in the form of xy. Because if we do this then we will get the interchanged digit as yx and after adding them together we will get an equation like xy + yx = 99. This will result in 2xy = 99 or xy = $\dfrac{99}{2}$ which will give us a decimal number. As any two digit number can be split in terms of 10 for example, 18 as 10 + 8 or 35 as 3(10) + 5 that is why we have considered the number as of the form of 10x + y. After forming the right equation one can solve the question easily and get to the right answer.