Question

A number consists of two digits whose sum is 8. If 18 is added to the number its digits get interchanged. The number is?
a) 53
b) 35
c) 92
d) 63

Answer 1

Hint: For solving this problem, let x and y be the unit and tens digit of a number respectively. We obtain the first equation by using the sum of digits. Another equation is formed by using the second statement. In this way, we have two variables and two equations, so we can easily obtain the answer.
Complete step-by-step answer:
Let the unit digit of a number be x and tens digit of a number be y. According to the problem statement, the sum of digits of a two-digit number is 8. This can be mathematically expressed as:
$\begin{array}{rl}& y+x=8\\ & y=8-x\dots \left(1\right)\end{array}$
Also, we are given that if 18 is added to the number, its digits get interchanged. The initial number is 10y + x and the interchanged number is 10x + y. By using this information, we form equation (2) as:
$10y+x+18=10x+y\dots \left(2\right)$
Replacing the value of y from equation (1) into equation (2), we get
$\begin{array}{rl}& \Rightarrow 10(8-x)+x+18=10x+8-x\\ & \Rightarrow 80-10x+x+18=9x+8\\ & \Rightarrow 98-9x=9x+8\\ & \Rightarrow 9x+9x=98-8\\ & \Rightarrow 18x=90\\ & \Rightarrow x={\displaystyle \frac{90}{18}}\\ & \therefore x=5\end{array}$
Therefore, the value of x is 5.
Substituting the value of x in equation (1), we get y as
y = 8 – 5 = 3
Therefore, the value of y is 3
Hence, the obtained number is $10\times 3+5=35$.
Therefore, option (b) is correct.
Note: The key steps involved in solving this problem is the manipulation of problem statements into equations. This problem can also be simplified by assuming digits as x and (8 - x) and then proceeding for solution. Substitution must be done carefully after obtaining the general form of numbers.