# Fun with Ratios — Quantitative Reasoning Example Problem

Hope everyone had a fun Halloween! Let’s kick off November with a super fun ratios problem.

The ratio of men to women in a room is 5:6. If there are 121 people in the room, how many of them are men?

This is a pretty basic ratios problem, and there are several ways to solve it. The method I’m going to use is based on algebra, and it’s probably the easiest to remember.

So, you have this room. In this room, there are M men and W women. You know two things about the men and women in this room.

1.  There are a total of 121 people
2. The ratio of men to women is 5:6

Let’s translate these two facts into algebraic formulas.

1. M + W = 121
2. M/W = 5/6

We have two variables, and we have two equations. If you remember from your algebra classes, we have sufficient information to solve this system of equations. The easiest way to solve this system of equations is by substitution. Take one of the equations, solve for one of the two variables, and then plug it into the other.

It doesn’t really matter which variable you solve for or which equation you start with, because, eventually, you can get to your final answer either way. I’m going to start with equation 2, and I’m going to solve for M because it’s easy.

To solve for M, we multiply both sides by W.

M = 5W/6

Now that I have M, I’m going to plug it back into the first equation and solve for W.

5W/6 + W = 121     (plug in M = 5W/6)
5W/6 + 6W/6 = 121     (find a common denominator)
11W/6 = 121     (combine like terms)
11W = 726     (multiply both sides by 6)
W = 66     (divide both sides by 11)

There are 66 women in the room. But hold on! The question is asking for men! So the final step is to subtract 66 from 121.

121 – 66 = 55.

So the final answer is 55. See? Wasn’t that fun?

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