32 times a two-digit number equals 23 times the number obtained by reversing…

For a two-digit number, write it as 10x + y, where x is the tens digit and y is the units digit; the number formed by reversing its digits is then 10y + x. Any relationship stated between a two-digit number and its reverse (a ratio, sum, or difference) becomes a pair of linear equations in x and y that can be solved together.

1. Let the two-digit number be 10x + y, so its reverse is 10y + x.

2. The digit-sum condition gives x + y = 15.

3. "32 times the number equals 23 times its reverse" gives 32(10x + y) = 23(10y + x).

4. Expanding both sides: 320x + 32y = 230y + 23x.

5. Rearranging: 320x - 23x = 230y - 32y, i.e. 297x = 198y.

6. Dividing both sides by 99: 3x = 2y, so y = 1.5x.

7. Substituting into x + y = 15: x + 1.5x = 15, so 2.5x = 15, giving x = 6.

8. Then y = 15 - 6 = 9.

9. So the number is 10x + y = 10(6) + 9 = 69.

Check: the reverse of 69 is 96. 32 x 69 = 2208 and 23 x 96 = 2208, so both sides match, confirming the ratio condition; and 6 + 9 = 15 confirms the digit-sum condition. So 69 satisfies both constraints.