Question

A woman says, “If you reverse my own age, the figures represent my husband’s age. He is, of course, senior to me and the difference between our ages is one-eleventh of their sum.” What is the woman’s age?

Answer 1

Hint:
For the husband’s age, take two variables x and y, where x can be the digit in the ten’s place and y be the digit in the ones’ place. Reversing the digits will be the age of the woman. Calculate their age by multiplying the ten’s digit with 10 and adding the one’s digit to it. Formulate a linear equation and solve it.

Complete step by step solution:
Let the husband age be \[xy\] where \[x\] is the digit in ten’s place and \[y\] is the digit in one’s place. Don’t confuse multiplication of \[x\] and \[y\].
Then, the age of husband is = \[(10x + y)\]
Therefore,
The woman says that if you reverse her age, the figure’s will represent the age of her husband, Keeping that in mind, the woman’s age will be \[yx\] where \[y\] is the digit in ten’s place and \[x\] is the digit in one’s place. Don’t confuse multiplication of \[x\] and\[y\] .
Hence, the age of the woman is = \[(10y + x)\]
She further says that the difference between their ages is one-eleventh of the sum of the ages. So, let us formulate the above statement into an equation.
According to the given condition
\[
  \{ (10x + y) - (10y + x)\} = \dfrac{1}{{11}} \times \{ (10x + y) + (10y + x)\} \\
   \Rightarrow (10x - x + y - 10y) = \dfrac{1}{{11}} \times (11x + 11y) \\
   \Rightarrow (9x - 9y) = (x + y) \\
   \Rightarrow 8x = 10y \\
   \Rightarrow 4x - 5y = 0 \\
\]
Therefore, for above equation to be zero (0)
\[x = 5\] and \[y = 4\]
Therefore, age of woman = \[10y + x = 10 \times 4 + 5 = 40 + 5 = 45\]

Hence, the woman is 45 years old.

Note:
For competitive exams or if you want to solve this question in less time, you can solve it intuitively too. Minimum difference of two figures reversed is 9, if both ages are not the same. 9 is 1/11 of 99, so 99 is the total of both their ages. That gives us 45 and 54 as they are reversed from each other and sum up to 99.