The 5 Trickiest Math Problems on the CPS HSAT (and How to Solve Them)
On the CPS HSAT Math section, some problems aren't just designed to test what you know; they're designed to be tricky. They look confusing, have tempting "trap" answers, and can cause students to waste precious time.
The good news is that these tricky problems follow predictable patterns. Once you learn to spot these traps, you can turn them into easy points. This guide will break down 5 of the most common tricky problems and give you a clear strategy for each, building on the skills from our main Conquering the Math Section guide.
Trick #1: The Shaded Area Geometry Problem
The Trap: You see a bizarre shape—like a square with a strange curve cut out of it—and panic because there's no formula for that specific shape.
The Strategy: The secret is always Subtraction. You find the area of the larger, simpler shape, and then subtract the area of the smaller, unshaded shape that has been "cut out."
Example: A circle with a radius of 5 cm has a square inscribed inside it. What is the area of the region inside the circle but outside the square?
How to Solve: Find the area of the whole circle (pir2). Then, find the area of the square inside. Finally, subtract the square's area from the circle's area to find the leftover "shaded" region.
Trick #2: The Ratio-to-Total Problem
The Trap: A problem gives you a ratio and a total, and you accidentally use the wrong denominator in your fraction.
The Strategy: Always add the parts of the ratio to find the "total" part for your fraction.
Example: A classroom has a ratio of boys to girls of 2:3. If there are 30 students in total, how many are boys?
How to Solve: The trap is to say boys are 2/3 of the class. This is wrong. Add the ratio parts: 2 + 3 = 5 "total parts." Therefore, boys are 2/5 of the total, and girls are 3/5.
(2/5) of 30 = 12 boys.
Trick #3: The Percent of a Percent Problem
The Trap: A problem involves multiple percentage changes, and you incorrectly add or subtract the percentages directly.
The Strategy: You must calculate each percentage change one step at a time.
Example: A jacket costs $200. It is on sale for 20% off. You then pay a 10% sales tax on the discounted price. What is the final cost?
How to Solve: The trap is to do 20% - 10% = 10% and take 10% off $200. This is wrong.
Step 1: Calculate the discount. 20% of $200 is $40. The sale price is $200 - $40 = $160.
Step 2: Calculate the tax on the new price. 10% of $160 is $16.
Step 3: Add the tax. $160 + $16 = $176.
Trick #4: The Mean vs. Median Problem
The Trap: Not understanding how an outlier (a very high or very low number) affects the mean (average) and the median (middle number) differently.
The Strategy: Remember this rule: Outliers have a big effect on the mean but very little effect on the median.
Example: The set is {10, 12, 15, 18, 20}. The mean is 15 and the median is 15. What happens if we add the number 100 to the set?
How to Solve: The new set is {10, 12, 15, 18, 20, 100}. The median is now just the average of 15 and 18, which is 16.5 (a small change). The mean, however, will jump dramatically to about 29.2 (a huge change).
Trick #5: The "Solve for the Expression" Algebra Problem
The Trap: You are given an equation and asked to solve for an expression (like 3x + 3), but you waste time solving for x first.
The Strategy: Look for a shortcut! Often, you can manipulate the equation to get right to the expression.
Example: If 4x + 8 = 20, what is the value of x + 2?
How to Solve: The trap is to solve for x: 4x = 12, so x = 3. Then plug it in: 3 + 2 = 5. This works, but it's slow.
The Shortcut: Look at the original equation: 4x + 8 = 20. Notice that every term is divisible by 4. Divide the entire equation by 4:
(4x/4) + (8/4) = (20/4)
x + 2 = 5. You've solved directly for the expression in one step.
How to Become a "Trap-Spotter"
Reading about these tricks is one thing, but spotting them under pressure is a skill built through practice. When you work through our CPS HSAT Practice Tests, actively hunt for these problem types. Each time you find one, circle it. By repeatedly exposing yourself to these traps in a realistic testing environment, you build a "trap detector" that will protect your score on test day and help you master the advanced concepts that lead to success.