The questions on the SAT math section increase in difficulty towards the end of each section. As a result, the last two or three questions will seem exceedingly difficult and the fact that you tend to run out of time towards the end of a section does not help matters. However, most of these types of problems are hard not because they test very difficult math concepts or strange formulas. Instead, they are difficult because they require a great deal of puzzle-solving and high-order reasoning. These kinds of tough math problems are difficult for one or more of the following reasons:
- A traditional high-school math formula will not help you on these questions
- These questions require outside-of-the-box thinking and problem solving
- You must find a hidden pattern or convenient method to simplify the problems
In order to successfully navigate these kinds of problems, you need to remain calm and be comfortable with some exploratory problem-solving approaches. You can’t memorize your way to success on these problems and you’ll need to think creatively to reduce the problem to simpler terms to cut through some of the complexity.
Consider the following free-response question:
An integer greater than 0 is considered “tricon-factorable” if it can be expressed as the product of three consecutive integers. How many positive integers less than 10,000 are tricon-factorable?
First off, there’s no such thing as a “tricon-factorable” integer. Sometimes, the test-makers will throw in these random, made-up terms and define them on the spot so that they purposely put you in a position of zero information going into a problem. The test does this to see if you can reason your way out of a strange problem and showcase your creative problems-solving skills.
Here’s the approach you’ll want to take with this problem:
- Make sure you understand any weird definitions. In this case, “tricon-factorable” is our strange term. Since it means that an integer can be expressed as the product of 3 consecutive integers, 6 is tricon-factorable since it is the product of 1 * 2 * 3 = 6.
- See if you can find a pattern or simplified version of the problem. Here, the number 10,000 is pretty large and intimidating. But every SAT problem could be solved in less than 2-3 minutes at most, so there is always a trick or pattern to take advantage of. What if we tried to do it for 100 instead to try and find this pattern?
- Solve the larger problem. After you’ve found a pattern or approach to solving a simplified version of the problem, you can typically extend it to the larger problem.
Using this approach, let’s solve this problem. Let’s see if we can find the number of tri-con factorable numbers under 100. A good start might be to start listing out all the products of consecutive integers:
1 x 2 x 3 = 6
2 x 3 x 4 = 24
3 x 4 x 5 = 60
4 x 5 x 6 = 120 STOP!
It looks like there are just 3 tri-con factorable integers less than 100 since after 4, the product exceeds 100. We can also see that the 3 term in the “3 x 4 x 5” expression also corresponds to the number of tri-con factorable integers we’ve found. Looks like as long as we can find the set of consecutive integers for the largest tri-con factorable number that is less than our max, we will also find the total number of tri-con factorable integers! Let’s do this for 10,000. Using our calculator, we can find that:
20 x 21 x 22 = 9240,
but 21 x 22 x 23 = 10626
As a result, our largest term under 10,000 is 20 x 21 x 22 = 9240. Since the smallest factor, 20, corresponds to the number of total tri-con factorable integers so far, we have found our answer: there are 20 tri-con factorable integers under 10,000.
When you come across a problem like this on the test, remember to stay calm, try to understand all the definitions they give you, and finally look for patterns and a way to solve a simplified version of the problem. Don’t just panic or try to remember some formula from high school math classes; neither will be of much help. Learn to rely on your reasoning skills and make sure to practice on more of these kinds of problems!
Jason Sun is the Director of College Prep for Veritas Prep. When he’s not in the office, he can be found competing in swing dance competitions or defending his title as a table tennis champion.