Equation Writing Help for Age Problem

Question

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!

Answer 1

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.

Answer 2

$$M=3O =7Y$$

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 $14+36=50$.