Word problems come in all shapes and sizes. But each and every year, the SAT includes certain particular varieties. We’ve got the skinny on ’em.

A rate is a ratio of related qualities that have different units. For example, speed is a rate that relates the two quantities of distance and time. Here is the general rate formula:

No matter the specifics, the key to a rate word problem is in correctly placing the given information in the three categories: A, r, and B. Then, you can substitute the values into the rate formula. We look at the three most common types of rate: speed, work, and price.

In the case of speed, time is quantity A and distance is quantity B. For example, if you traveled for 4 hours at 25 miles per hour, then

Usually, the new SAT won’t simply give you one of the quantities and the rate and ask you to plug it into the rate formula. Since rate questions are always in the form of word problems, the information that you’ll need to solve the problem often given in the befuddling complicated manner you’ve grown to know and hate.

Here’s an example:

This question provides more information than simply the speed and one of the quantities. You get unnecessary facts such as how Jim is traveling (by rollerskates) and when he started (in the morning). Ignore them and focus on the facts you need to solve the problem.

Here’s a more difficult rate problem:

You should immediately pick out the given rate of 528,000 feet per hour and notice that the total distance traveled is 480 miles. You should also notice that the question presents a units problem: The given rate is in feet per hour, while the total distance traveled is given in miles.

Sometimes a question gives you inconsistent units, as in this example. Always read over the problem carefully and don’t forget to adjust the units—the SAT make sure that the answer you would come to if you had forgotten to correct for units appears among the answer choices.

For the cycling question, since the question tells you that there are 5,280 feet in a mile, you can find the rate for miles per hour:

Now you can plug the information into the rate formula:

Work sucks. You’re there from 9 to 5 and, at best, you get two weeks off per year, and you’ve got a boss constantly checking up on you. Work word problems on the SAT are a breeze in comparison. On work word problems, you’ll usually find the first quantity measured in time (t), the second quantity measured in work done (w), and the rate measured in work done per time (r). For example, if you knitted for 8 hours and produced 2 sweaters per hour, then

Here’s a sample work problem. It’s one of the harder rate word problems you might come across on the SAT:

First, examine what that problem says: 4 workers can dig a 40-foot well in 4 days. You know how much total work was done and how many people did it, you just don’t know the rate at which the workers worked. You need that rate, since the 8 workers digging the 60-foot wells are working at the same rate. Since , you can get the rate by dividing 40 by 4, which equals 10. The workers together dig at a pace of 10 feet per day.

Now for that group of 8 workers digging a 60-foot well. The total work (w) done by the 8 workers is 60 feet, and they work at a rate (r) of 10 feet per day per 4 workers. Can you use this information to answer the question? Oh yeah. The rate of 10 feet per day per 4 workers converts to 20 feet per day per 8 workers, which is the size of the new crew. Now you can use the rate formula:

This last problem required a little bit of creativity—but nothing you can’t handle. Just remember the classic rate formula and use it wisely.

In rate questions dealing with price, you’ll usually find the first quantity measured in numbers of items, the second measured in price, and the rate in price per item. If you have 8 basketballs, and you know that each basketball costs $25,

Exponential growth and decay problems are like percent change problems on steroids: You must perform a percent change over and over again. You can use exponents on these repeated percent change questions. Here’s an example:

You could do two things to solve this problem. You could multiply each successive generation by 5% fifty times to get the final answer, or you could use this formula:

The formula is probably the better bet. So, to solve this problem,

Exponential decay only slightly modifies the formula:

Exponential decay is often used to model population decreases as well as the decay of physical mass.

We’ll work through a few example problems to get a feel for both exponential growth and decay problems.

You’ve got an original population of 100, a growth rate of .35 every hour, and 6 hours. To solve the problem, you just need to plug the appropriate values into the formula for exponential growth.

Since the beach ball loses air, you know this is an exponential decay problem. The decay rate is .06, the original amount is 4000 cubic centimeters of air, and the time is 10 days. Plugging the information into the formula,

This problem is a bit tricky because the interest rate is per month, while the time period is given in years. You need to make the units match up. In the two-year time period given by the question, there will be 2 12 = 24 months

Here’s another compounding problem:

Ben’s account will have $2000 1.05¹⁵ ≈ $4157.85 in it after 15 years. Justin’s account will have $2500 1.04¹⁵ ≈ $4502.36 in it. Justin’s account will still have more money in it than Ben’s after 15 years. Notice, however, that Ben’s account is gaining on Justin’s account.

And with that, you’ve covered everything you need to know to rock SAT algebra. Geometry’s next.