Question

The digit $5$ is written between the digits of a two-digit number to form a three-digit number. This number is $410$ more than the original two-digit number. If the sum of the digits of the three-digit number is $12$ then what is the difference between the digits of the two-digit number?

Answer 1

Hint: To solve this problem, first we will assume that $y$ is unit digit and $x$ is tens digit of two-digit number. That is, the two-digit number is of the form $10x + y$. Now we will use the given information to find required digits of the two-digit number.

Complete step-by-step solution:
To solve this problem, let us assume that $y$ is unit digit and $x$ is tens digit of two-digit number. Therefore, we can say that the two-digit number is of the form $10x + y$.
Now it is given that the three-digit number is formed by writing the digit $5$ between the digits of a two-digit number. That is, the digit $5$ is written at tens place between the digits of a two-digit number. Therefore, in this three-digit number unit digit is $y$, tens digit is $5$ and hundreds digit is $x$. Therefore, we can say that the three-digit number is of the form $100x + 50 + y$.
Also given that the three-digit number is $410$ more than the original two-digit number. Note that here the original two-digit number is $10x + y$. Therefore, we can write $100x + 50 + y = 10x + y + 410 \cdots \cdots \left( 1 \right)$.
Let us solve the equation $\left( 1 \right)$ by cancelling $y$ from both sides. Therefore, we get
$
  100x + 50 = 10x + 410 \\
   \Rightarrow 100x - 10x = 410 - 50 \\
   \Rightarrow 90x = 360 \\
   \Rightarrow x = \dfrac{{360}}{{90}} \\
   \Rightarrow x = 4 \\
 $
Now we have three digit number in which hundreds digit is $x = 4$, tens digit is $5$ and unit digit is $y$. Also given that the sum of the digits of this three-digit number is $12$. Therefore, we can write
$4 + 5 + y = 12 \cdots \cdots \left( 2 \right)$.
Let us solve the equation $\left( 2 \right)$ to find the value of $y$. Therefore, we get $9 + y = 12 \Rightarrow y = 3$.
Now we have a two-digit number in which tens digit is $x = 4$ and unit digit is $y = 3$. We need to find the difference between the digits of the two-digit number. That is, we have to find $x - y$. Therefore, we get $x - y = 4 - 3 = 1$.
Therefore, the difference between the digits of the two-digit number is $1$.

Note: Any two digit number can be written as $10x + y$ where $x$ is tens digit and $y$ is unit digit. For example, $99$ can be written as $10\left( 9 \right) + 9$. Any three digit number can be written as $100x + 10y + z$ where $x$ is hundreds digit, $y$ is tens digit and $z$ is unit digit. For example, $999$ can be written as $100\left( 9 \right) + 10\left( 9 \right) + 9$.