System of Equations Problem

Question

When you reverse the digits in a certain two-digit number you decrease its value by 72. Find the number if the sum of its the sum of its digits is 8 

Answer 1

This is very interesting math word problem, which could be solved by applying simple quantitative reasoning rather than going deep into equation solving. The logic is as following.

 

In order to satisfy the first condition ("decreased by 72 when digits swap"), the first digit could be only 9 or 8 to get the positive number or null after subtracting 72. Number 9 is not acceptable because it will fail the second condition "sum of digits equal 8" (it exceeds 8 regardless of second digit).

 

Thus the only choice is 8 as the first digits leading to only one acceptable option of "0" as the second digit.

 

Number 80 satisfies both conditions; it is the answer.

Answer 2

10u + t = 10t + u -72 => 9u - 9t = -72 => t - u = 8
and
t + u = 8

2t = 16 => t = 8 and u = 0

Answer 3

The number must be 10n + m

 

Given:

 

(E1)  10m + n = 10n + m - 72

(E2)  n + m = 8

 

Solve the 2 equations above for n and m.

 

(E1)  -9n + 9m = - 72

-n + m = -8

 

(E2) n + m = 8

 

Add (E1) and (E2) and get

 

2m = 0

m = 0

n = 8 - 0 = 0

 

The number is

 

10(8) + 1(0) = 80

 

Test the answer.

 

80 → 08 = 8, and 8 is 72 less than 80 [good]

8 + 0 = 8 [good]