Question

The age of a man is same as his wife's age with the digits reversed. The sum of their ages is $99$ years and the man is $9$ years older than his wife. How old is the man?

• A. $50$ years
• B. $49$ years
• C. $54$ years
• D. $44$ years

Answer 1

Answer: (C)

$54$ years

Let the man age be $10x+y$.

It is given that the age of man is same as his wife's age with the digits reversed, so wife's age would be $10y+x$

Now, sum of the ages given is $99$ which means that:

$10x+y+10y+x=99\Rightarrow 11x+11y=99\Rightarrow 11(x+y)=99\Rightarrow x+y=\frac{99}{11}\Rightarrow x+y=9.....\left(1\right)$

Also, the man is $9$ years older than his wife, therefore, man age = wife age$+9$ that is:

$10x+y=10y+x+9\Rightarrow 10x-x+y-10y=9\Rightarrow 9x-9y=9\Rightarrow 9(x-y)=9\Rightarrow x-y=\frac{9}{9}\Rightarrow x-y=1.....\left(2\right)$

Adding equations (1) and (2), we get

$(x+y)+(x-y)=9+1\Rightarrow 2x=10\Rightarrow x=\frac{10}{2}\Rightarrow x=5$

Substitute $x=5$ in equation (1) as follows:

$5+y=9\Rightarrow y=9-5=4$

Therefore, the age of man with $x=5$ and $y=4$ is:

$10x+y=(10\times 5)+4=50+4=54$

Hence, the man is $54$ years old.