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.

Hint – In order to solve this problem assume the variables for first number in ones and tens place then make equations and solve according to the condition provided in the question. Doing this will give you the right answer.

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
According to the first condition given in the question that is, the sum of two numbers is 110.
So, we can do,
$\Rightarrow$10y + x + 10x + y = 110
$\Rightarrow$11x + 11y = 110
Divide the above equation by 11 we get
$\Rightarrow$x + y = 10
$\Rightarrow$x = 10 – y …………….....(i)

Now according to the second equation, if 10 is subtracted from the first number the new number will be 10y + x - 10

Given that the new number is 4 more than 5 time the sum of its digits in the first number that is the sum of its digits in the first number is x + y.

Now 5 times of it is 5(x + y)
And 4 more than it is, 4 + 5(x + y)
Therefore new number = 4 + 5(x + y)

$\Rightarrow$10y + x - 10 = 4 +5(x + y)
$\Rightarrow$10y - 5y + x = 4 +10 +5x
$\Rightarrow$5y = 14 + 4x ………………….....(ii)

Substitute the value of x from (i) to (ii)
We get,
$\Rightarrow$5y = 14 + 4(10 - y)
$\Rightarrow$5y = 14 + 40 - 4y
So, y = 6
And from (i)
$\Rightarrow$x = 4
Then the first number can be written as
10y + x = 10(6) + 4 = 64 (On putting the value of x and y)
First number is 64.

Note – Whenever you face such type of problems you have to first assume the variables and then make the equations according to the question. Here in this problem the two digit number has been asked, therefore a two digit number can be written as 10x + y if x is at tens place and y is at one place. Here we have made two equations according to the condition provided as the number of variables is also two. Proceeding like this will give you the right answer.