Question

A two-digit number is 4 times the sum of its digits the number obtained by interchanging the digits is less by 9 than twice the original number. Find the original number.

Answer 1

Hint: Here, first of all, take the original number as 10y + x. Then the number by reversing its digits would be 10x + y. Use the given information to find the value of x and y. Then finally substitute x and y in 10y + x to get the original number.

Complete step-by-step answer:
Here, we are given that a two-digit number is 4 times the sum of its digits and the number obtained by reversing its digits is less by 9 than twice the original number. Here, we have to find the original number. Let us consider the original two-digit number such that its units digit is x and the tens digit is y.
So, we get the original number = 10y + x…..(i)
We are given that this number is equal to 4 times the sum of its digits, so we get 10y + x = 4(x + y)
By simplifying the above equation, we get,
\[\Rightarrow 10y+x=4x+4y\]
\[\Rightarrow 10y-4y=4x-x\]
\[\Rightarrow 6y=3x\]
By dividing 3 on both the sides of the above equation, we get
\[\dfrac{6y}{3}=\dfrac{3x}{3}\]
\[\Rightarrow x=2y....\left( ii \right)\]
Now, by reversing the digits of the original number, we get the units digit of the new number as y and tens digits as x.
So, we get the new number as = 10x + y….(iii)
Now, we are given that this new number is less by 9 than twice the original number. So, we get,
New number = 2 (Original number) – 9
By substituting the values of the original number and the new number from equation (i) and (ii), respectively, we get,
\[10x+y=2\left( 10y+x \right)-9\]
By simplifying the above equation, we get,
\[10x+y=20y+2x-9\]
\[\Rightarrow 10x-2x=20y-y-9\]
\[\Rightarrow 8x=19y-9\]
By substituting the values of x = 2y from equation (ii), we get,
\[\begin{align}
  & \Rightarrow 8\left( 2y \right)=19y-9 \\
 & \Rightarrow 16y=19y-9 \\
 & \Rightarrow 16y-19y=-9 \\
 & \Rightarrow -3y=-9 \\
\end{align}\]
By dividing both the sides of the above equation by -3, we get,
\[y=\dfrac{-9}{-3}=3\]
By substituting the value of y in equation (ii), we get, x = 2(3) = 6.
So, we get our original number, 10y + x = 10(3) + 6 = 36.
Hence, we get the original number as 36.

Note: Here, students must note that any 2 digit number AB (where A is tens digit and B is units digit) is written as 10A + B. Similarly, any3 digit number ABC is written as 100A + 10B + C. All higher-order numbers are also written similarly according to the place value of the digits. Also, in this question, many students make this mistake of giving answers as 63 instead of 36 which is wrong. They must note that the original number is asked in the question which is 36 and not 63. So, this mistake must be taken care of.