The sum of 3 times Darlene's age and 7 times Sharon's is 173. Darlene is 2 years less than twice as old as Sharon is. How do you find each of their ages?

Sep 6, 2015

With the question as given Darlene is $13 \frac{10}{13}$ and Sharon is $25 \frac{7}{13}$.

If the $173$ is a typo for $163$, then Darlene is $13$ and Sharon is $24$.

Explanation:

Let $d$ be Darlene's age and $s$ be Sharons's age.

We are given:

$3 d + 7 s = 173$

$d = 2 s - 2$

Substitute this second equation into the first to get an equation in $s$:

$173 = 3 d + 7 s = 3 \left(2 s - 2\right) + 7 s = 6 s - 6 + 7 s = 13 s - 6$

Add $6$ to both ends to get: $13 s = 173 + 6 = 179$

Divide both ends by $13$ to get: $s = \frac{179}{13} = 13 \frac{10}{13}$

Hence $d = 2 s - 2 = 2 \cdot \frac{179}{13} - 2 = \frac{358}{13} - \frac{26}{13} = \frac{332}{13} = 25 \frac{7}{13}$

I suspect the $173$ in the question should be $163$, which would yield: $s = 13$ and $d = 24$.