The sum of digits of a two digit number is 13.
Question:

The sum of digits of a two digit number is 13. If the number is subtracted from the one obtained by interchanging the digits, the result is 45. What is the number?

Solution:

Let the digits at units and tens place of the given number be x and y respectively. Thus, the number is .

The sum of the digits of the number is 13. Thus, we have After interchanging the digits, the number becomes .

The difference between the number obtained by interchanging the digits and the original number is 45. Thus, we have

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

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

$\Rightarrow 9 x-9 y=45$

$\Rightarrow 9(x-y)=45$

$\Rightarrow x-y=\frac{45}{9}$

$\Rightarrow x-y=5$

So, we have two equations

$x+y=13$

$x-y=5$

Here x and y are unknowns. We have to solve the above equations for x and y.

Adding the two equations, we have

$(x+y)+(x-y)=13+5$

$\Rightarrow x+y+x-y=18$

$\Rightarrow 2 x=18$

$\Rightarrow x=\frac{18}{2}$

$\Rightarrow x=9$

Substituting the value of in the first equation, we have

$9+y=13$

$\Rightarrow y=13-9$

$\Rightarrow y=4$

Hence, the number is $10 \times 4+9=49$.