In the first three football games of the seasons, Carlos gained 3 times as many yards as Alden. Travis gained 10 years more than Carlos. Altogether the three players gained a total of 430 yards. How many did Carlos gain?

Aug 26, 2017

Explanation:

Let $a$ be the number of yards gained by Alden, $c$ for Carlos and $t$ for Travis.

$c = 3 a$ , $t = c + 10$ and $a + c + t = 430$

$a + 3 a + 3 a + 10 = 430$

$\therefore 7 a = 420$

$a = 60$

$c = 3 a = 180$

Aug 26, 2017

See a solution process below:

Explanation:

First, let's name:

The number of yards Carlos gained: $c$

The number of yards Alden gained: $a$

The number of yards Travis gained: $t$

From the problem we know:

$c = 3 a$

and

$t = c + 10$

and

$c + a + t = 430$

We can solve the first equation in terms of $a$ as follows:

$c = 3 a$

$\frac{c}{\textcolor{red}{3}} = \frac{3 a}{\textcolor{red}{3}}$

$\frac{c}{3} = \frac{\textcolor{red}{\cancel{\textcolor{b l a c k}{3}}} a}{\cancel{\textcolor{red}{3}}}$

$\frac{c}{3} = a$

$a = \frac{c}{3}$

We can now substitute for $a$ and $t$ in the 3rd equation and solve for $c$:

$c + a + t = 430$ becomes:

$c + \frac{c}{3} + c + 10 = 430$

$\left(\frac{3}{3} \times c\right) + \frac{c}{3} + \left(\frac{3}{3} \times c\right) + 10 = 430$

$\frac{3 c}{3} + \frac{c}{3} + \frac{3 c}{3} + 10 = 430$

$\frac{3 c}{3} + \frac{1 c}{3} + \frac{3 c}{3} + 10 = 430$

$\frac{\left(3 + 1 + 3\right) c}{3} + 10 = 430$

$\frac{7 c}{3} + 10 = 430$

$\frac{7 c}{3} + 10 - \textcolor{red}{10} = 430 - \textcolor{red}{10}$

$\frac{7 c}{3} + 0 = 420$

$\frac{7 c}{3} = 420$

$\frac{\textcolor{red}{3}}{\textcolor{b l u e}{7}} \times \frac{7 c}{3} = \frac{\textcolor{red}{3}}{\textcolor{b l u e}{7}} \times 420$

$\frac{\cancel{\textcolor{red}{3}}}{\cancel{\textcolor{b l u e}{7}}} \times \frac{\textcolor{b l u e}{\cancel{\textcolor{b l a c k}{7}}} c}{\textcolor{red}{\cancel{\textcolor{b l a c k}{3}}}} = \frac{\textcolor{red}{3}}{\cancel{\textcolor{b l u e}{7}}} \times \textcolor{b l u e}{\cancel{\textcolor{b l a c k}{420}}} 60$

$c = 180$

Carlos gained 180 yards