4
$\begingroup$

Hi this may be simple silly problem but it is bugging me as I am not able to devise a system of equations to solve it.

My husband's age," remarked a lady the other day, "is represented by the figures of my own age reversed. He is my senior, and the difference between our ages is one-eleventh of their sum.

The answer is 54 and 45, but not able to find way to get it.

  • 0
    Also, (594,495), (5454,4545), (5994,4995), etc...2011-05-13

1 Answers 1

5

Not sure if this totally solves what you want, but here's a way to get a stage where the numbers should be easier to guess and check. I assume the ages of the two people are two digit numbers. So we can represent the husband's age as $10x+y$, where $x$ and $y$ are the decimal digits, and the wife's age is then $10y+x$, with $10x+y\gt 10y+x$.

Since the difference of their ages is one eleventh of the sum, this translates to $ (10x+y)-(10y+x)=\frac{1}{11}(10x+y+10y+x) $ but this implies $ 9x-9y=\frac{1}{11}(11x+11y)=x+y. $

So you have $9(x-y)=x+y$. Since $0\leq x,y\leq 9$, it's not hard to experiment with these numbers to find that $x=5$ and $y=4$.

  • 3
    Actually, you don't need to do much trial and error, since you know that $x+y$ must be divisible by $9$, so either $x+y=9$ or $x+y=18.$ But $x+y=18$ only happens when $x=9, y=9$ which is obviously not a solution. If $x+y=9$, then $x-y=1$, and you get $45/54$2011-05-13