Equation Writing Help (Age Problem)

Question

A mother's age is $8$ times older than her son. When her son's age is half ($\frac{1}{2}$) of mother's current age, the sum of their ages is $60$. What was the mother's age when her son was born?

I only could write that $M = 8S$, $t$ = passed time.

$$S + t = \frac {M}{2}$$

$$S + t + M= 60$$

$$\frac{M}{2} +M = 60$$

$$2M = 120$$

$$M = 60$$

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EDIT: We can easily find mother out.

$$M = 8S, S + t = \frac {M}{2}, M = 40$$

Now we need to know what her son's age is.

$$M = 8S, 40 = 8S, S = 5$$

Why is it wrong? The right answer seems $18$, which I didn't get why.

Answer 1

Edit since OP just edited the question

So $M = 8S$. We have $$S+t = \frac{M}{2} \iff t = \frac M2-S$$ and thus the son's age will be half the mother's current age in $\frac M2 -S$ years.

The sum is then $60$ so $$\frac M2 + M + (\frac M2 -S) = 60$$

Indeed, the son's age is then $\frac M2$ and the mother's age $M + (\frac M2 - S)$.

Thus $$2M-S = 60$$

Now using $M=8S$, you should be able to conclude.

Answer 2

The problem statement is very poorly written, and the edit seems to make it worse. The first and second sentences must refer to two different points in time. There's no way to be 8 times and 2 times someone's age on the same day. The first sentence refers to today and the second to some other time (which must be in the future). So, just taking the second sentence, all by itself, we find that the son's age plus twice his age is 60. He's 20 and his mother is 40. She was 20 when he was born.

The first sentence is just a source of confusion. The problem doesn't ask his age, today, fortunately. Assuming he wasn't born on his mother's birthday, she will turn 24 while he's still 3.