Source.

So here’s the thing with ratios and proportions on the SAT: they’re really easy. No, seriously, where are you going? Come back! They’re easy, I swear. All you have to do is keep very close track of your units, and you’ll be good to go. That means when you set up a proportion, actually write the units next to each number. Make sure you’ve got the same units corresponding to each other before you solve, and you’re home free. Pass Go, collect your \$200, and spend it all on Lik-M-Aid Fun Dip. So uh…let’s try one?

1. A certain farm has only cows and chickens as livestock. The ratio of cows to chickens is 2 to 7. If there are 63 livestock animals on the farm, how many cows are there?

(A) 13
(B) 14
(C) 16
(D) 18
(E) 49

The SAT writers would love for you to set up a simple proportion here and solve:

$\large&space;\inline&space;\dpi{300}&space;\fn_cm&space;\frac{2}{7}=\frac{x}{63}$

Hooray! = 18! That’s answer choice (D)! NOT SO FAST, SPANKY. You just solved for a terrifying hybrid beast, the COWNIMALKEN. Let’s look at that fraction more carefully, with the units included:

$\large&space;\inline&space;\dpi{300}&space;\fn_cm&space;\frac{2\:cows}{7\:chickens}=\frac{x\:cows}{63\:animals}$

So when you casually multiplied by 63 and solved, you solved for a unit that won’t do you any good: the [(cow)(animal)]/(chicken), or COWNIMALKEN. That’s terrifying. Nature never intended it to be so. Not only are you unwisely playing God, but you’re getting an easy question wrong. Before we can solve this bad boy, we need to make sure our units line up on both sides of the equal sign. So let’s change the denominator on the left to match the one on the right. Get rid of “7 chickens” and replace it with “9 animals.” Get it? Because cows count as animals, if there are 2 cows for every 7 chickens, that means there are 2 cows for every 9 animals.

$\large&space;\inline&space;\dpi{300}&space;\fn_cm&space;\frac{2\:cows}{9\:animals}=\frac{x\:cows}{63\:animals}$

Now, we can solve: x = 14 cows. That’s choice (B). See how the units cancel out nicely when you’ve properly set up a ratio question? That should make the hairs on your neck stand on end.

##### Look out for this tricky crap, too:

But but but! There’s one more thing you need to watch out for. Sometimes they’ll give you units that aren’t quite as easily converted. Like so:

1. The ratio of students to teachers at a certain school is 28 to 3. The ratio of teachers to cafeteria workers is 9 to 2. What is the ratio of cafeteria workers to students?

(A) 1 to 42
(B) 2 to 28
(C) 3 to 37
(D) 9 to 56
(E) 3 to 14

Here, we have a few options. It’s not too hard to find a number of teachers that will work with both ratios, so I’ll leave it to you to figure out how to solve it that way if you prefer. Instead let me point out that there’s a pretty elegant solution here that comes from simply multiplying the two ratios together, essentially solving for the expression we’re looking for. Peep the skillz:

$\large&space;\inline&space;\dpi{300}&space;\fn_cm&space;\frac{28\:&space;students}{3\:&space;teachers}\times&space;\frac{9\:teachers}{2\:cafeteria\:workers}$

What happens to the teachers? They cancel! So multiply, and simplify:

$\large&space;\inline&space;\dpi{300}&space;\fn_cm&space;=\frac{252\:students}{6\:cafeteria\:workers}=\frac{42\:students}{1\:cafeteria\:workers}$

Since the question asked for the ratio of cafeteria workers to students, just flip it and you’re done! 1 cafeteria worker to 42 students. That’s choice (A). Ah-mazing.

##### Mind your units, son.

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