The sum of the present ages of a mother and her son is $42$. What was the mother's age $15$ years ago before her son was born?

Question

The sum of the present ages of a mother and her son is $42$. When the mother was same age as her son now, her son wouldn't be born until $15$ years later. What was the mother's age when her son was born?

This question had seemed a bit complex. However, we can say that

$$M + S = 42 \tag {1}$$

where $M = \text{Mother}$, $S = \text{Son}$

Mother's age $15$ years ago before her son was born

$$ M-t = -S-15 \tag{2}$$

$$t = \text{passed time}$$

This is where I'm stuck. I'll be waiting for your professional helps.

Answer 1

We have $M+S=42$, assume that the mother is $t$ years older than her son.

Currently, the mother is $M$ years old and the son is $S$ years old.

$\Rightarrow$ $t$ years ago, the mother was $M-t=S$ years old and her son was $S-t$ years old. Note that the son wouldn't be given birth until $15$ years later, so at this point we consider that her son was $-15$ years old or $S-t=-15$.

We have this set of equations:

$${\begin{cases}M+S=42\\M-S-t=0\\S-t=-15\end{cases}}$$

After solving this, you can check the answer below.

The mother is currently $33$ years old, the son is currently $9$ years old.

Answer 2

You don't have enough information to solve the problem. Your $M+S=42$ is correct, but you need another equation to sort them out. Please check the problem.

Added after the edit: When the mother was as old as the son is now is $M-S$ years ago. In the second sentence we are told $S-(M-S)=2S-M=-15$, where the $-15$ represents $15$ years until the son is born. We now have two equations in two unknowns. $$M+S=42\\2S-M=-15$$ adding $$3S=27\\S=9\\M=33$$ The age of the mother at the son's birth is the difference in ages, which is $24$.

Answer 3

Let define the following uknowns

• $M$ mother's present age
• $S$ son's present age

then we have

• $M + S = 42\implies S=42-M$

from the condition "when the mother was same age as her son, there was 15 years her son to be born" corresponds to

• mother's age when son born: $S+15=M-S \implies M=2S+15$

then

• $M=84-2M+15 \implies 3M=99 \implies M=33$