Zé Roberto has five children, two are twins and the other three are triplets. It is known that today Zé's age is equal to the sum of the ages of his five children. In 15 years, if we add up the ages of the five children, we will have double the age that Zé will have at the same time and the sum of the ages of the twins will be equal to the sum of the ages of the triplets.

(a) What is Zé's current age?

(b) What is the current age of the triplets?

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Solution

(a) Let $z$ be the current age of Zé, $g$ the current age of the twins, and $t$ the current age of the triplets. The information from the problem can be translated into three equations:

$$
\begin{aligned}
z & =2 g+3 t \\
2(z+15) & =2(g+15)+3(t+15) \\
2(g+15) & =3(t+15)
\end{aligned}
$$

To determine $z$, it is enough to substitute the value of $2 g+3 t$ from the first equation into the second:

$$
\begin{aligned}
z & =2 g+3 t \\
2 z+30 & =2 g+30+3 t+45 \\
2 z & =z+45 \\
z & =45
\end{aligned}
$$

(b) From the third equation, we have $2 g=3 t+15$. Substituting this equation and the value of $z$ from the previous item into the first equation, we get:

$$
\begin{aligned}
45 & =3 t+15+3 t \\
45 & =6 t+15 \\
30 & =6 t \\
t & =5
\end{aligned}
$$