Rates are a special kind of ratio that deal with related quantities that have different units. The relationship between these two different quantities is defined by a rate:

That equation might seem a little complicated, but it will seem much simpler once we give some concrete examples. The most common rate questions on the SAT deal with speed, work, or price, so we’ll use those.

In rate questions dealing with speed, you will usually find the first quantity measured in time, the second measured in distance, and the rate in distance covered per second, minute, or hour, a unit known as speed. If you traveled for 7 hours at 30 miles per hour, then:

Notice how the units of “hour” canceled out, since the hour in the rate is at the bottom of the fraction, while the unit for the time is a normal number (meaning it could also be written as 7 ^hours/ ₁).

In rate questions dealing with work, you will usually find the first quantity measured in time, the second quantity measured in work done, and the rate in measured in work done per time. If you worked for 5 hours and dug 3 ditches an hour, then:

Notice how the “hour” units canceled out.

In rate questions dealing with price, you will usually find the first quantity measured in numbers of items, the second measured in price, and the rate in price per item. Let’s say you had 6 cucumbers, and you knew that cucumbers cost $.50 each.

Notice how the units of “cucumber” canceled out.

Simple SAT rate problems might test your ability to solve for any one of the three aspects of a rate equation: quantity 1, quantity 2, or the rate. The key to solving any rate problem is determining which values fit into which of the categories. Once you’ve figured out which information the question is giving you, all you have to do is plug in the numbers and work out the equation.

Take a look at the following example of a simple rate problem:

To answer this question let’s first define what we know:

First off, this problem contains a little trick. Did you notice that the units of time in the rate and input are not the same? The question states that the salesman sells .3 cars per hour, while asking you to figure out how many cars he will sell in 15 days. We included this trick in this sample problem because it is one that the SAT occasionally likes to play. So when you see rate questions dealing with time, be careful.

Before beginning to solve the problem, you must equalize the time units of hours and days. Since the salesman works 8 hours a day:

Now that the units are equalized, the problem can be answered using the rate equation:

so

The salesmen sold 36 cars. Notice that in this problem, the hours unit for quantity 1 cancels with the ^cars/_hours unit in the rate, yielding the proper unit of cars. If the units do not work out when you get your answer, you’ve probably made a mistake.

Complicated SAT rate problems can involve more than one rate. Such problems might ask you to compare in some way two different objects moving at a different rate or to determine the distance traveled by an object going at two different rates. There is no way for us to cover every single example of a complicated rate problem in this section. We will provide one good example here, but you should also pay attention to these rate problems when you come upon them in practice tests.

We know Griselda traveled at a rate of 3 km/hr on the way to the store and 2 km/hr back from the store. We also know it took her a total of 5 hours to make the trip. Finally, since the distance between the store and Griselda’s house remained constant for the trip there and back, we know that she must have spent a different amount of time traveling to the store and back. In fact, we know precisely the ratio between the time she spent walking to the store and the time she spent walking back. Since Griselda walked ³/ ₂ as fast on the way to the store as she did on the way back, we know (because she traveled the same distance on each trip) that she must have spent only ²/₃ as much time walking to the store as she spent walking back. Therefore, since we know she spent a total of 5 hours walking to and from the store, we can set up the equation:

where y stands for the number of hours Griselda spent walking home from the store. Now, to solve the equation for y

We now know Griselda spent 3 hours walking home from the store. If we plug that 3 into the equation for her walk home we get:

You can check this answer by looking back at the original question and making sure that the distance of 6 km works out. We know Griselda was walking at the speed of 3 km/hr to the store, and that the distance to the store was 6 km. Therefore, it took her 2 hours to walk to the store. We also know Griselda walked at the speed of 2 km/hr back from the store. Since the distance was 6 kilometers, it must have taken her 3 hours to walk back. In sum, then, it took her 5 hours to walk to and from the store, which fits the question perfectly.

The SAT will occasionally ask rate problems using variables rather than numbers. These questions can be difficult because the variables can make it hard for you to figure out what refers to quantity 1, quantity 2, and the rate, and also because they can cause some confusion about what the question is asking you to do.

Many test-prep books recommend that you try to answer this question by substituting in numbers for the variables and then working out all the answer choices to see which one works. This method will work, but it’s slow and laborious, and if you have a good understanding of rates, you would be better off just dealing with the variables directly. We’ll cover both methods here.

The key to answering this question is figuring out what the question wants and then figuring out how to give it what it wants. Looking at this question, we can see that the question is asking for the total value of the soap produced in x hours. Now, since we know that each bar of soap is worth 3 dollars, you know that the total value of soap produced in x hours is equal to:

Now all you have to do is figure out how many bars of soap are produced in x hours and multiply that expression by 3. To figure out the expression for soap produced in hours, you just need to identify what all those variables mean. Since the rate equation is:

we just have to figure out what variables go where. This is pretty easy. The question explicitly says that the rate is “b bars in h hours,” which means that the rate is ^b/ _h. Also, since you know you are looking for bars of soap produced, you can substitute the variable for hours worked, x, in for quantity 1.

You already know that so you just have to substitute to get:

(D) is the right answer. Now, all that looks like a lot of work, but please note that we explained every possible step to teach you how to answer this question. If you were really good at rates, you probably could have done many of those steps in your head and just written this:

and then substituted to get:

See? Very quick.

If you aren’t so comfortable with rates, your best bet is to substitute numbers in for the variables and then try to work out the question. We’ll make b = 4, h = 2, and x = 5. So the question that once read:

now reads as:

With the variables gone, the question immediately becomes much less difficult to comprehend. You can see that the rate is ⁴/₂ = 2 bars per hour. And you can see that to find out the total numbers of bars produced, you just have to multiply the rate by the hours given:

Then just multiply those 10 bars by the $3 charged per bar, and you get $30. To find the answer, you need to substitute the numbers into the answer choices and see which works out to $30:

(D) is still the right answer. This method is conceptually easier than the last, but you must do all the math to use this method, which makes it take quite some time.