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 its digits in the first number. Find the first number.

Answer 1

Hint: For the first number, let the unit place digit be x and tens place digit be y.
Then the two-digit number will be 10y + x. And the number formed by interchanging the unit place and tens place digits will be 10x + y. Add these and equate to 110 to get one equation. Now, using the other condition, form another equation: 5y = 14 + 4x. Use these two equations to find x and y and arrive at the final answer.

Complete step-by-step answer:

In this question, we are given that 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 its digits in the first number.
We need to find the first number.
For the first number, let the unit place digit be x and tens place digit be y.
Then the two-digit number will be 10y + x.
And the number formed by interchanging the unit place and tens place digits will be 10x + y.
We are given the question that the sum of these two numbers Is 110.
So, 10y + x + 10x + y = 110
11x + 11y = 110
Dividing the above equation by 11, we will get the following:
x + y = 10
x = 10 – y …(1)
Now according to the second condition, if 10 is subtracted from the first number, the new number is 10y + x - 10
Given that the new number is 4 more than 5 times the sum of its digits in the first number i.e.
the sum of its digits in the first number is x + y, now 5 times of it is, 5(x + y), and now 4 more that is 4 + 5(x + y)
therefore the new number = 4 + 5(x + y)
10y + x - 10 = 4 +5(x + y)
10y - 5y + x = 4 +10 +5x
5y = 14 + 4x …(2)
Substituting the value of x from equation (1) to equation (2), we will get the following:
5y = 14 + 4(10 - y)
5y = 14 + 40 - 4y
y = 6
Putting this in equation (1), we will get x = 4.
Then the first number is 10y + x = 10 $\times $ 6 + 4 = 64
Hence, the first number is 64.

Note: In this question, it is important to understand the language of the question perfectly and then form the equations accordingly. If there is a mistake in understanding the question or forming the equations, you will get the wrong answer though the method would be correct.