Slow and methodical does it... (Big Bang Theory)
Word problems on the SAT can be very intimidating. A lot of students cruise along the rest of the math section and then freeze as soon as they see a word problem. Why am I being quizzed on reading comprehension??? But don’t panic! The trick is to figure out how to translate these word problems into the familiar math problems you’ve been seeing all along.
In this post, I’ll teach you a strategy for approaching any word problem. The key word to remember here is CUBE, which stands for:
- Circle the numbers
- Underline the important words
- Box the question
- Eliminate the unnecessary information.
Let’s take a problem and work through this method step by step.
EXAMPLE PROBLEM #1
Jim rollerskates 6 miles per hour. One morning, Jim starts rollerskating and doesn’t stop until he has gone 60 miles. How many hours did he spend rollerskating?
CUBE the Problem
C) Circle the numbers
There are two numbers in this problem. Remember to circle units too: 6 miles per hour and 60 miles.
U) Underline the important words
What’s the relevant information here?
- Jim rollerskates 6 miles per hour
- Doesn’t stop until he has gone 60 miles
B) Box the question
The question should almost always, if not always, be the last sentence: “How many hours did he spend rollerskating?”
E) Eliminate the unnecessary information
Everything else you can cross out. So we’re left with:
Basically, if Jim rollerskates at 6 miles an hour, how many hours does it take him to go 60 miles?
Systematically Solving the Problem
So this is a pretty simple problem, but for the sake of solving problems systematically as practice for harder problems, let’s work out the equation that this translates to. When you see the word “per” in a unit, remember that:
Rate = Distance/Time
If you ever forget the formula for rate, just look at the units in the rate given to you: 6 miles per hour. Miles are units of distance, “per” can be translated into a division sign, and hours are units of time. Now plug the numbers into the equation:
6 miles/1 hour = 60 miles/x hours
Note that while this could have been written as 6 = 60/x, I wrote out the units and even the unnecessary 1 on the rate side of the equation for clarity. This is good practice for two reasons:
- Sometimes the units don’t match up in the problem and this can lead you to a wrong answer if you forget to conver
- The later step of cross-multiplication becomes a lot clearer and you won’t accidentally multiply the wrong numbers
So the next step is to check that the units match. They do: Great! Now remember also that we can reduce fractions horizontally or vertically if there is a common factor. We can also cross out units that match horizontally or vertically. Looking at this problem, we can cross out all the units and reduce 6 and 60 by their greatest common factor (GCF)--6, leaving us with:
1/1 = 10/x
Cross-multiplying, we get that x = 10. Check the problem by multiplying the time (10 hours) by the rate (6 miles per hour), and indeed you get 60 miles.
When we have a simple problem with the same units and two numbers, it’s also good to use common sense to get the answer quickly to save time. For rate problems, there’s going to be either multiplication or division involved. If we multiply 6 and 60, we get a large number--360 hours--which just seems like a ridiculously long time for Jim to rollerskate!
We could also divide 6 by 60 to get 1/10 of an hour, but that just seems ridiculously short, not to mention that SAT word problems will also tend towards whole numbers (but be careful, not always!). The last choice is to divide 60 by 6 to get 10. Jim must be a pretty athletic fellow to rollerblade for that long, but out of the three possibilities, it makes the most sense!
EXAMPLE PROBLEM #2
Okay, let’s practice with a harder problem.
At a cycling race, the cyclist from California can cycle 528,000 feet per hour. If the race is 480 miles long, how long will it take her to finish the race? (1 mile = 5280 feet)
CUBE the Problem
C) Circle 528,000 feet per hour, 480 miles, and 1 mile = 5280 feet.
U) Underline “the cyclist can cycle 528,000 feet per hour” and “the race is 480 miles long.”
B) Box the question: “how long will it take her to finish the race?”
E) Eliminate everything else.
At a cycling race, the cyclist from California can cycle (528,000 feet per hour). If the race is (480 miles long), [how long will it take her to finish the race]? (1 mile = 5280 feet)
Now restate the question in terms of the information retained:
If the cyclist can cycle 528,000 feet per hour, how long does it take her to cycle 480 miles?
Systematically Solving the Problem
Again, we’ll set it up with the rate equation:
528,000 feet/1 hour = 480 miles/x hours
Notice that the units for distance are difference, so let’s convert feet into miles using the information in the parentheses. To set up the equation right, we want the same units to be across from one another so that they can all be crossed out.
528,000 feet/x miles = 5280 feet/1 mile
We can then divide by the GCF (5280 feet) to get:
528/x = 1
Thus, x is 100 and therefore the cyclist is cycling at 100 miles per hour. Rewriting the original equation as:
100 miles/1 hour = 480 miles/x hours
the units may be crossed out and the numbers horizontal from each other can be divided by the GCF (20) to get:
5/1 = 24/x
Cross multiply to get 5x = 24 and then divide to get that x = 24/5, which yields the answer as 4.8 hours.
As you get used to these problems, you’ll be able to become more efficient and skip simple steps, but in the beginning, it’s easy to be distracted by extra information, or to cross multiply incorrectly. Solve word problems by first getting the information you need (via the CUBE method), and then systematically setting them up-- and you’ll greatly increase the number of careless mistakes you might make!
Need more help with the SAT? Check out other blog posts on our site, written by our private SAT tutors in New York and Boston: 4 Vital Rules for Beating the SAT Essay, ACT Math vs. SAT Math: What's the Difference?, Why You Should Wear a Bathrobe on Test Day. Feel free to contact us if you'd like to work with a private SAT tutor!
*Example problems are from Sparknotes