Question

 The sum of the digits of the two digit number is 9. If nine times this number is equal to twice the number obtained by reversing the order of digits of the number. Find the numbers.
A. 18 and 76
B. 18 and 1
C. 98 and 81
D. 18 and 81

Answer 1

Hint: Here, consider the two digit number as p with tens place digit x and unit’s place digit
y. Generally, the two digit number can be represented as $p=10x+y$ .Given that the sum of the digits is 9. Now, by reversing the number we get the relation, $9(10x+y)=2(10y+x)$. Solve the above equation to get the values of x and y.

Complete step-by-step answer:
 Let the two digit number be p.
Here, it is given that the sum of the two digit numbers is 9.
Let the digits of the number p be x and y.
Then we have:
$x+y=9$ …… (1)
Here, let us assume that x be at the ten’s place and y be at the unit’s place. Then, the number p can be written as:
$p=10x+y$.
The above is the usual representation of a two digit number.
Here, it is also given that nine times this number p is equal to twice the number obtained by reversing the order of digits of the number.
By reversing the number we get:
 $q=10y+x$
It means that by reversing the number, the ten's digit has become y and the unit's digit has become x. By the given data we can write:
$\begin{align}
  & 9p=2q \\
 & 9(10x+y)=2(10y+x) \\
\end{align}$
Next, by multiplying we get:
$\begin{align}
  & 9\times 10x+9\times y=2\times 10y+2\times x \\
 & 90x+9y=20y+2x \\
\end{align}$
Now, by taking 2x to the left side it becomes -2x and 9y to the right side it becomes -9y. Hence, we will get:
$\begin{align}
  & 90x-2x=20y-9y \\
 & 88x=11y \\
\end{align}$
Now, by cross multiplication, we get:
$\dfrac{88}{11}x=y$
Next, by cancellation we get:
$8x=y$
By substituting $8x=y$ in equation (1), we obtain:
$\begin{align}
  & x+8x=9 \\
 & 9x=9 \\
\end{align}$
Now, by cross multiplication we get:
$x=\dfrac{9}{9}$
Next, by cancellation we get:
$x=1$
Now, substitute $x=1$ in $8x=y$ to obtain y. Thus we get:
$\begin{align}
  & 8\times 1=y \\
 & 8=y \\
\end{align}$
Hence, we got the values of $x=1$ and $y=8$.
Therefore, the number p will be:
$\begin{align}
  & p=10x+y \\
 & p=10\times 1+8 \\
 & p=10+8 \\
 & p=18 \\
\end{align}$
Hence, the required number is 18 and the number obtained by reversing the order of digits is 81.
Therefore, the correct answer for this question is option (d).
Note: Here, when you reverse the number the ten’s place and the unit’s place changes. That is, from $10x+y$ it becomes $10y+x$. So, don’t forget to change the places while reversing the number.