The writers of the Math IC love word problems. These problems force you to show your range as a mathematician. They demand that you read and comprehend the problem, set up an equation or two, and manipulate the equations to find the solution. Luckily, the Math IC uses only a few types of word problems, and we have the nitty-gritty on all of them.
Rates
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 problem is correctly placing the given in the three categories. Then, you can substitute the values into the rate formula. We’ll look at the three most common types of rate: speed, work, and price.
Speed
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:
Note that the hour units canceled out, since the hour in the rate is at the bottom of the fraction. But you can be sure that the Math IC test won’t simply give you one of the quantities and the rate and ask you to plug it into the rate formula. Because rate questions are in the form of word problems, the that you’ll need to solve the problem will often be given in a less straightforward manner.
Here’s an example:
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Jim rollerblades 6 miles per hour. One morning, Jim starts rollerblading and doesn’t stop until he has gone 60 miles. How many hours did he rollerblade? | |
This question provides more than simply the speed and one of the quantities. We know unnecessary facts such as how Jim is traveling (by rollerblades) and when he started (in the morning). Ignore them and focus on the facts you need to solve the problem.
- Time a: x hours rollerblading
- Rate: 6 miles per hour
- Quantity b: 60 miles
So, we can write:
Jim was rollerblading for 10 hours. This problem requires a little analysis, but basically we plugged some numbers into the rate equation and got our answer. Here’s a slightly more difficult rate problem:
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At a cycling race, there are 50 cyclists in all, each representing a state. The cyclist from California can cumulatively cycle 528,000 feet per hour. If the race is 480 miles long, how long will it take him to finish the race? | |
Immediately, you should pick out the given rate of 528,000 feet per hour and notice that 480 miles are traveled. You should also notice that the question presents a units problem: the given rate is in feet cycled per hour, and the distance traveled is in miles.
Sometimes a question will give you inconsistent units, like in this example. Always read over the problem carefully and don’t forget to adjust the units—the answer choices are bound to include non-adjusted options, just to throw you off.
For this question, since we know there are 5,280 feet in a mile, we can find the rate for miles per hour:
We can now plug the into the rate formula:
