Question

A number consists of two digits whose sum is five. When the digits are reversed, the number becomes greater by nine. Find the number.

Answer 1

Hint: First of all we will assume the unit and tenth digits of the number to be a variable. Then we will make equations between them according to the given condition in the question and thus solve to find the digits of the number and we get the number.

Complete step-by-step answer:
Let us suppose the units digit to be y and the tenth digit to be x of a two digit number. Then the number is as follows:
Number = \[10\times x+y\times 1\]
Here, x is at the tenth place and y is at the units place. We have been given that the sum of the digits is five, i.e.
\[\Rightarrow x+y=5.....(1)\]
We also have been given that if the digits are reversed, the number becomes greater by nine.
If we reversed the digits, we will have x at units place digit and y at tenth place.
So the number \[=10\times y+x\times 1\]
Here, y is at the tenth place and x is at the units place.
\[\Rightarrow 10\times y+x\times 1=10\times x+y\times 1+9\]
Here, it is given that the number obtained after we reversed the digits is greater by nine.
\[\Rightarrow 10y+x=10x+y+9\]
On taking y to the left hand side of the equation, we get as follows:
\[\begin{align}
  & \Rightarrow 10y-y+x=10x+9 \\
 & \Rightarrow 9y+x=10x+9 \\
\end{align}\]
On taking x to the right hand side, we get as follows:
\[\begin{align}
  & \Rightarrow 9y=10x-x+9 \\
 & \Rightarrow 9y=9x+9 \\
\end{align}\]
On taking 9x to the left hand side, we get as follows:
\[\Rightarrow 9y-9x=9\]
Now taking 9 as common, we get as follows:
\[\Rightarrow 9(y-x)=9\]
On dividing the equation by 9, we get as follows:
\[\begin{align}
  & \Rightarrow \dfrac{9}{9}(y-x)=\dfrac{9}{9} \\
 & \Rightarrow y-x=1.....(2) \\
\end{align}\]
Now adding equation (1) and (2) we get as follows:
\[\begin{align}
  & \Rightarrow x+y+y-x=5+1 \\
 & \Rightarrow 2y=6 \\
 & \Rightarrow y=3 \\
\end{align}\]
On substituting y = 3 in equation (2) we get as follows:
\[\begin{align}
  & \Rightarrow 3-x=1 \\
 & \Rightarrow 3-1=x \\
 & \Rightarrow 2=x \\
\end{align}\]
Hence x = 2 and x = 3.
So the number,
\[\Rightarrow 10x+y=10\times 2+3=23\]
Therefore the required number is 23.

Note: Be careful while formulating the equation using the given conditions, while solving the equations and take care of signs also. Do not write the number as xy or yx there will be a lot of confusion and also it is not the correct way to represent a number.