Question

The product of the digits of a two-digit number is one-third that number. If we add 18 to the required number, we get a number consisting of the same digits written in the reverse order. Find the number.

Answer 1

Hint: Here we have 2 conditions on both digits of a two digit number. Every two digit number can be written as a sum of 10 times the tens digit and one time the units digit. So, now assume two variables to represent the units, say x and y. Using the data given in the question, form 2 equations.

Complete step-by-step solution:
First one would be $xy=\dfrac{z}{3}$ if the required number is z. Now we use a substitution method to solve these questions and get the value of 2 variables. And then solve this again as given in the question.
According to the question if we consider two digits of that number as x and y and the product of these will be $\dfrac{1}{3}$, if the required number is z, then we can write,
$xy=\dfrac{z}{3}$
And if we add 18 to the required number then we get a number consisting of the same digits written in the reverse order. So, let us assume that,
First digit = x
Second digit = y
Required number = z
So, we can write it as,
Product of digits = $\dfrac{1}{3}z$
And,
$\begin{align}
  & z=10x+y \\
 & \Rightarrow xy=\dfrac{1}{3}\left( 10x+y \right) \\
 & \Rightarrow 3xy=10x+y\ldots \ldots \ldots \left( i \right) \\
\end{align}$
Now, as given in the question, if we add 18, then,
$\Rightarrow z+18=$ number with same digits in reverse order.
$\begin{align}
  & \Rightarrow 10x+y+18=10y+x \\
 & \Rightarrow 9\left( y-x \right)=18 \\
 & \Rightarrow y-x=2 \\
 & \Rightarrow y=2+x \\
\end{align}$
If we put this in equation (i), then we get,
$\begin{align}
  & 3x\left( x+2 \right)=10x+x+2 \\
 & \Rightarrow 3{{x}^{2}}+6x=10x+x+2 \\
 & \Rightarrow 3{{x}^{2}}-5x-2=0 \\
\end{align}$
If we make the factored form here, then,
$\begin{align}
  & \Rightarrow \left( 3x+1 \right)\left( x-2 \right)=0 \\
 &\Rightarrow x=-\dfrac{1}{3},x=2 \\
 & y=4 \\
\end{align}$
Here we only consider x=2 as at x= $-\dfrac{1}{3}$ we do not get any integer number.
So, the required number is 24.

Note: While solving this question you should remember that if any two digit or three digit number is asked then we will write that in hundred’s multiple of tens multiply according to the digit of that question and then write all given conditions very much carefully otherwise the question will be wrong.