Question

A two digit number is obtained by multiplying the sum of the digit by 8. Also it is obtained by multiplying the difference of the digits by 14 and adding 2. Find the number.

Answer 1

Hint: Firstly we have to assume the tens and ones term in the problem as x and y, so the number will be 10x+y. Then after reading the question we have to create the first equation by multiplying the sum of the digits by 8. Then after reading the second part we have to multiply the difference of the digits by 14 and add 2 with them. By doing this we can create a second equation. After getting two equations we have to just solve them so that we can get the number.

Complete step by step solution:
Let us assume that the tens digit is x and ones digit is y.
Then we get that the original number will be \[10x+y\].
According to the question the two digit number is obtained by multiplying the sum of the digit by 8.
So, we can write in mathematical form as below,
\[\begin{align}
  & 10x+y=\left( x+y \right)\times 8 \\
 & \Rightarrow 10x+y=8x+8y \\
 & \Rightarrow 2x-7y=0..............................\left( i \right) \\
\end{align}\]
We marked this equation as “i”.
In the question it is also said that the number is obtained by multiplying the difference of the digits by 14 and adding 2 in it.
Now, again we can express mathematically and we have the below equation,
\[\begin{align}
  & 10x+y=\left( x-y \right)\times 14+2 \\
 & \Rightarrow 10x+y=14x-14y+2 \\
 & \Rightarrow 4x-15y+2=0...............................\left( ii \right) \\
\end{align}\]
We marked this equation as “ii”.
By multiply equation “i” with 2 and subtract “ii” from it, we get,
\[\begin{align}
  & \left( 4x-14y \right)-\left( 4x-15y+2 \right)=0 \\
 & \Rightarrow 4x-14y-4x+15y-2=0 \\
 & \Rightarrow y-2=0 \\
 & \Rightarrow y=2 \\
\end{align}\]
Putting the value of \[y=2\] on equation “i” we get,
\[\begin{align}
  & 2x-7y=0 \\
 & \Rightarrow 2x-7\times 2=0 \\
 & \Rightarrow 2x=14 \\
 & \Rightarrow x=7 \\
\end{align}\]

In the beginning of this problem we assume the original number will be \[10x+y\]. Now after getting the value of \[x=7\] and \[y=2\]. We simply put these values to get the required number.
The number will be,
\[\begin{align}
  & 10x+y \\
 & \Rightarrow 10\times 7+2 \\
 & \Rightarrow 72 \\
\end{align}\]

Note: For calculating the equation we can take another route. In this problem we did this by multiplying equation “i” with 2 and subtracting it from “ii” to get the required value. We can simply evaluate equation “i” a little bit to get the value of 2x.
\[\begin{align}
  & 2x-7y=0 \\
 & \Rightarrow 2x=7y...................\left( iii \right) \\
\end{align}\]
And put the value of x in equation on “ii” and get,
\[\begin{align}
  & 4x-15y+2=0 \\
 & \Rightarrow 2\times 2x-15y+2=0 \\
 & \Rightarrow 2\times 7y-15y+2=0 \\
 & \Rightarrow -y+2=0 \\
 & \Rightarrow y=2 \\
\end{align}\]
And put the value of y in “iii” we get,
\[\begin{align}
  & 2x=7y \\
 & \Rightarrow 2x=7\times 2 \\
 & \Rightarrow x=7 \\
\end{align}\]
By this method we can solve this problem also.