A number consists of two digits, the sum of the digits is 10. If 72 is subtracted from the number, the digits are interchanged. Find the number.

Let's use algebra to solve this problem. Let the two-digit number be represented as 10a + b, where 'a' is the tens digit and 'b' is the units digit.

We are given two conditions:

The sum of the digits is 10, so we can write the equation: a + b = 10

When 72 is subtracted from the number, the digits are interchanged. This means that 10a + b - 72 becomes 10b + a. We can write this as another equation: 10a + b - 72 = 10b + a

Now, let's solve this system of equations:

From equation 1, we can express 'b' in terms of 'a': b = 10 - a

Substitute this expression for 'b' into equation 2: 10a + (10 - a) - 72 = 10(10 - a) + a

Now, simplify and solve for 'a': 10a + 10 - a - 72 = 100 - 10a + a

Combine like terms: 9a + 10 - 72 = 100 - 9a

Subtract 10 from both sides: 9a - 62 = 100 - 9a

Add 9a to both sides: 18a - 62 = 100

Add 62 to both sides: 18a = 162

Divide by 18: a = 9

Now that we have found 'a' to be 9, we can find 'b' using the first equation: a + b = 10 9 + b = 10

Subtract 9 from both sides: b = 10 - 9 b = 1

So, the tens digit 'a' is 9, and the units digit 'b' is 1. Therefore, the two-digit number is 91.