A mother's age is $19$ years more than the sum of the ages of her two sons. $5$ years ago, the mother's age was $4$ times than the sum of the ages of her two sons. What is the age of the older child?
Let's say the sum of the ages of her sons is $x$, the mother's age will be $x+19$. $5$ years ago, $x-5$ = $x+19-5$. However, I believe that I've gone too wrong.
What kind of methods can I use to solve this question?
I'll be waiting for your professional helps.
My Kindest Regards!
On
hint
Today $$y=x+19$$ five years ago
$$y-5=4 (x-5n) $$ $$14=3x-20n$$ $n $ is the number of sons.
The sum $x $ must satisfy $x>5n $
thus $n=2$ and $x=18$ years . the mother has $37$ years .
On
$$\begin{align} M &= \text{mother’s current age in years} \\ S &= \text{older son’s current age in years} \\ s &= \text{younger son’s current age in years} \end{align}$$
The first statement translates to
$$M = 19+S+s$$
The second statement translates to
$$M-5 = 4\bigl( (S-5)+(s-5) \bigr)$$
We also know logically that any solutions, if they exist, should be restricted to
$$\begin{align} M &> S >0 \\ S &> s >0\\ M &> s >0 \end{align}$$
Can you take things from there? How does the number of variables compare to the number of equations?
There isn't enough information to solve for the age of the older child, even if there are two children. I can only solve for the following:
$\text{Ages of sons combined} = x$
$\text{Mother's age}=y$
$$x+19=y$$ $$4(x-10)=y-5$$ $$4x-40=x+19-5$$
$$3x=54$$
$$x=\dfrac {54}{3}=18$$
You know the sums of the ages of the two sons is $18$. But, you need to find the age of one of the children to find the age of the other child.
Note that the younger child must be older than $5$ years but younger than $9$ years. For example, if the younger child is $7$ years old, then the older child is $11$ years old.