Question

The sum of a two-digit number and the number formed by interchanging the digits is 132. If 12 is added to the number, the new number becomes 5 times the sum of digits. Find the number.

Answer 1

Hello,

Let's assume a and b the 2 digits of the number.

$\overline{ab}+\overline{ba}=132\\ \Rightarrow\ (10*a+b)+(10*b+a)=132\\ \Rightarrow\ 11a+11b=132\\ \Rightarrow\ a+b=12\ (1)\\ \overline{ab}+12=5*(a+b)\\ 10a+b-5a-5b=-12\\ \\ So \\ \left \{ {{a+b=12} \atop {5a-4b=-12}} \right.\\\\ \left \{ {{a=4} \atop {b=8}} \right.$

Your number is48.

Proof: 48+84=132

and 48+12=5*12

Answer 2

Answer:

Let the number be xy = 10x + y

Number obtained by interchanging the digits = yx = 10y + x

From the given information, we have:

10x + y + 10y + x = 132 11x + 11y = 132 x + y = 12 ... (1)

Also, we have:

10x + y + 12 = 5(x + y) 10x + y + 12 - 5x - 5y = 0 5x - 4y + 12 = 0 ... (2)

Multiplying (1) by 4, we get 4x + 4y - 48 = 0 ... (3)

Adding (2) and (3), we get, 9x - 36 = 0 x = 4 So, y = 12 - 4 = 8

Thus, the number is 48.