A women says "my age is my husband age reversed. If you take the difference…

Clear algebraic setup:

Let the husband's age be 10x + y (two-digit number with digits x and y). The woman's age is the reverse, 10y + x.

The sum of their ages is 11(x + y), so one-eleventh of the sum equals x + y. The difference between the ages (depending on order) is 9|y - x|.

We must solve 9|y - x| = x + y. This gives two directed cases:

• Case 1 (woman's age minus husband's age = 1/11 of sum): 9(y - x) = x + y ⇒ 8y = 10x ⇒ 4y = 5x. The smallest digit solution is x = 4, y = 5, so husband's age = 10x + y = 45 and woman's age = 54.

• Case 2 (husband's age minus woman's age = 1/11 of sum): 9(x - y) = x + y ⇒ 8x = 10y ⇒ 4x = 5y. The smallest digit solution is y = 4, x = 5, so husband's age = 10x + y = 54 and woman's age = 45.

Conclusion: If the difference is taken as the woman's age minus the husband's age, the husband's age is 45. If the difference is taken as the husband's age minus the woman's age, the husband's age is 54. If the phrase meant absolute difference, both 45 and 54 satisfy the condition because the absolute difference is 9 and one-eleventh of the sum is 9.

Tip: To avoid ambiguity, write the equation explicitly as (woman's age) − (husband's age) = (1/11)(sum) before solving.