Laura is three times as old as Luci.
In $5 years$ , the sum of their ages will be $30$ .
What are the current ages of Laura and Luci?

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Solution

Find the current ages of Laura and Luci:
Let $x years$ be the Luci current age.
Laura is three times as old as Luci, which $3 x years$ .
After $5 years$ , Laura and Luci ages will be $(3 x + 5) years$ and $(x + 5) years$ respectively.
In $5 years$ , the sum of their ages will be $30$ .
It follows that,
$\begin{array}{rcl} (3 x + 5) + (x + 5) & = & 30 \\ \Rightarrow & 3 x + x + 5 + 5 = 30 \\ \Rightarrow & 4 x + 10 = 30 \\ \Rightarrow & 4 x = 30 - 10 \\ \Rightarrow & 4 x = 20 \\ \Rightarrow & x = \frac{20}{4} \\ ∴ x & = & 5 years \end{array}$
Their current age of Luci is $x = 5 years$ .
Laura's current age, which is $3 x years$ .
Substitute $x = 5$ in $3 x$ :
$\begin{array}{rcl} 3 x & = & 3 (5) \\ = & 15 \end{array}$
Therefore, the current ages of Laura and Luci will be $15 years$ and $5 years$ respectively.

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