# Equation Writing Help for Age Problem

A mother's age is $3$ times older than her older son and $7$ times older than her younger son (She has only two sons). The older son will be $50$ years old when the younger son is same age as his mother's current age. What is the mother's current age?

Let's call them as $M$, $O$ and $Y$.

$$M = 3O \tag{1}$$ $$M=7Y \tag {2}$$

and

$$O - Y = 50 - M\tag{3}$$

This is where I'm stuck. Can you take a look?

With My Warnest Regards!

• What is $B$...? Commented Mar 10, 2018 at 20:53
• Oh, sorry for that! I meant Older son, which is $O$.
– user533031
Commented Mar 10, 2018 at 20:54
• Edited the question. Now it seems more clear.
– user533031
Commented Mar 10, 2018 at 20:55
• Check my edit on 3rd equation. Btw, what exactly is the problem, you got 3 equations and 3 unknowns, should be easy to solve imho. Commented Mar 10, 2018 at 20:58
• @mike239x Yes, but the most important thing is to specify how you found it.
– user533031
Commented Mar 10, 2018 at 20:58

$$M=3O =7Y$$

The younger will have the mother's age after $\color {green}{M-Y}$ years. after the same periode, the Older will have $O+\color {green}{M-Y}$. thus

$$50=O+(M-Y)$$ $$=O+6Y$$ $$150=3O+18Y=7Y+18Y$$ $$Y=6$$ $$M=42$$ $$O=14$$

In $36$ years, the younger will have $6+36=42$ the age of the mother, while the older will have $14+36=50$.

• Can you explain how you found this $50=O+(M-Y)$? I'd be grateful.
– user533031
Commented Mar 10, 2018 at 21:01
• @Busi Y becomes M after $(( M-Y))$ years.O will have $O+((M-Y))$ Commented Mar 10, 2018 at 21:03
• However, why we subtracted the ages and added $O$?
– user533031
Commented Mar 10, 2018 at 21:04
• For example , If you have 10 years. you will have 17 years in $17-10$. Commented Mar 10, 2018 at 21:05
• I still didn't get what you mean :/ Also why is it $150$? I wish you could give an explanation on your answer.
– user533031
Commented Mar 10, 2018 at 21:06

Let $x$ be some amount of time which has past. Surely this approach is more clear (at least in my opinion), although it is longer.

$50=O+x$

$x+Y=M$

So your $4$ equations are:

$$x+Y=M$$

$$50=O+x$$

$$M=3O$$ $$M=7Y$$

Substituting the 3rd and 4th equations:$$3O=7Y$$

$$O=\dfrac{7Y}3$$

Substituting for $O$ into the second equation:

$$50=\dfrac{7Y}3+x$$

And rewording the first equation by substituting $M$:$$x+Y=7Y \implies 0=x-6Y$$

Subtracting this from the equation you just got:$$50=\dfrac{7Y}3-(-6Y)\implies150=7Y+18Y\implies Y=6$$

You now know the age of the younger son is $6$, and you should be able to solve for the age of the mother and the older son.

• If you can explain how you wrote the equations, I'd be grateful. What I mean is I want to know how to write an equation.
– user533031
Commented Mar 10, 2018 at 21:01
• You had the 3rd and 4th equations.
– user535339
Commented Mar 10, 2018 at 21:02
• @busi Now you have to think of the first and second equations.
– user535339
Commented Mar 10, 2018 at 21:02
• The older son will be 50 after some amount of that time, let that time be $x$, $50=O+x$
– user535339
Commented Mar 10, 2018 at 21:03
• @Busi The younger son will be the mother's age ($M$) after that same amount of time, so $M=Y+x$. I hope I could provide a good explanation.
– user535339
Commented Mar 10, 2018 at 21:03