# System of Equations Algebraic Equations Word Problem

The problem is

"In a math class that contains both 7th and 8th graders, each student must do a class presentation on a famous mathematician. Each student may do the presentation alone or with a class partner. A 7th grader's partner must be an 8th grader and an 8th grader's partner must be a 7th grader. If two-thirds of the 7th graders and three-fiths of the 8th graders work with partners, what fraction of the class works alone? Solve using a system of equations."

I've come up with the equations $$\frac{2}{3}x = \frac{3}{5}y$$ and $$z = \frac{1}{3}x + \frac{2}{5}y$$ but I'm not entirely sure whether these are correct/where to go form here.

You are halfway there!

The fraction of students working alone is $$q= \frac{z}{z+\frac{2}{3} x+ \frac{3}{5}y}$$

This is $$q=\frac{\frac{1}{3} x + \frac{2}{5} y }{x+y}$$

Your first equation implies $$x=\frac{9}{10} y$$, so

$$q = \frac{\left(\frac{9}{10}\right)\left(\frac{1}{3}\right) + \frac{2}{5} }{\frac{19}{10}}= \frac{7}{19}.$$

• I believe you're missing an $x$ after the $\frac{2}{3}$ in the first equation. Mar 29, 2019 at 4:24
• Nice catch! Thank you!
– mjw
Mar 29, 2019 at 14:39