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 information 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 information 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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This question provides more information 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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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 information into the rate formula:
- Time: x hours cycling
- Rate: 100 miles per hour
- Distance: 480 miles
So it takes the cyclist 4.8 hours to finish the race.
Work
In work questions, you will usually find the first quantity measured in time, the second quantity measured in work done, and the rate measured in work done per time. For example, if you knitted for 8 hours and produced two sweaters per hour, then:
Here is a sample work problem. It is one of the harder rate questions you might come across on the Math IC:
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First, let’s examine what that problem says: 4 men can dig a 40 foot well in 4 days. We are given a quantity of work of 40 feet and a time of 4 days. We need to create our own rate, using whichever units might be most convenient, to carry over to the 8-men problem. The group of 4 men dig 40 feet in 3 days. Dividing 40 feet by 4 days, you find that the group of 4 digs at a pace of 10 feet per day.
From the question, we know that 8 men dig a 60 foot well. The work done by the 8 men is 60 feet, and they work at a rate of 10 feet per day per 4 men. Can we use this information to answer the question? Yes. The rate of 10 feet per day per 4 men converts to 20 feet per day per 8 men, which is the size of the new crew. Now we use the rate formula:
- Time: x days of work
- Rate: 20 feet per day per eight men
- Total Quantity: 60 feet
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.