ACT and SAT Math Strategy: Pick Your Own Numbers
There’s rarely just one route to the correct answer on ACT and SAT Math problems.
Solving algebraically will always work, but it may not be the fastest option. Besides, you don’t get extra points for solving the problem the “proper” way.
Graphing the problem on your calculator or Desmos can save time on many questions, especially those involving equations or functions. Testing the answer choices by plugging them into the original equation is another powerful strategy when the answer choices are numbers.
But what happens when none of these approaches seems like a good option? Some questions contain too many variables to graph easily. Others have variables in the answer choices, making it impossible to plug in the answers directly. You could, of course, solve the problem algebraically, but that may take longer than you’d like on a timed test.
On questions like these, try picking your own numbers.
Instead of solving for the variables, assign them simple values, substitute those values into the problem, and compare the results with the answer choices. In many cases, this lets you identify the correct answer much more quickly than solving the problem algebraically.
How the Pick Your Own Numbers Strategy Works
- Assign values to the variables. Choose values that are easy to work with, such as x = 1, y = 2, and z = 10. Leave any constants exactly as they appear in the problem.
- Substitute your values into the expression or equation in the question. Simplify until you have a numerical result.
- Substitute the same values into each answer choice. Simplify until you have a numerical result for each answer choice, and then choose the one that matches the value from the question. Be sure to test every answer choice, because it is possible to get the same value for more than one answer choice (especially if you pick 0 or 1).
- If necessary, choose new values and test again. If multiple answer choices still work, assign a different set of values to the variables and repeat the process. Usually, only the correct answer will continue to match.
Guidelines for Picking Numbers
Because the correct answer must work for every possible value of the variables, you can choose any values you like. It’s best to use numbers that are easy to work with, such as 0, 1, 2, 5, 10, or 100 (if you’re dealing with percentages). Avoid numbers that will make the equation or expression in the question undefined; for example, if the denominator is (x – 3), don’t set x equal to 3.
Choosing 0 and 1 is more likely to make multiple answer choices appear correct. When that happens, simply choose a different set of values and test again.
The best way to understand this strategy is to see it in action. Let’s work through a few ACT and SAT Math questions step by step:
Sample Math Problems
The following questions are from the 1,700+ SAT practice exercises and 2,250+ ACT practice exercises on the Test Innovators platform.
Sample Question 1
Which of the following is equivalent to the expression x (x + 5) – 2x (x – 3)?
- −x – 1
- −x 2 + 2
- −x 2 – x
- −x 2 + 11x
Test Innovators. SAT Practice Exercises: Equivalent Expressions - Easy #3
Let’s start with an easy question.
Let x = 1.
x (x + 5) – 2x (x – 3) = 1 (1 + 5) – 2(1)*(1 – 3) = 6 – 2 (-2) = 6 + 4 = 10
So, when x = 1, the original expression equals 10.
Now, plug x = 1 into all of the answer choices. Which answer choice equals 10?
A. –x – 1 = -1 – 1 = -2
B. –x² + 2 = –1² + 2 = -1 + 2 = 1
C. –x² – x = –1² – 1 = -1 – 1 = -2
D. –x² + 11x = –1² + 11(1) = -1 + 11 = 10
D is the correct answer. It is the only answer choice that also equals 10 when x = 1.
Now what happens if x = 0?
x (x + 5) – 2x (x – 3) = 0 (0 + 5) – 2(0)*(0 – 3) = 0.
A. –x – 1 = -0 – 1 = -1
B. –x² + 2 = –0² + 2 = 0 + 2 = 2
C. –x² – x = –0² – 0 = 0
D. –x² + 11x = –0² + 11(0) = 0 + 0 = 0
When x = 0, the original expression equals 0 and both C and D equal zero. We can narrow down the correct choice to either C or D but will need to plug in a new value to determine which is correct.
Sample Question 2
Which expression is equivalent to
(a 3b −5c 7)(a −1b 3c 2 + a 6b 5c −4), where a, b, and c are positive?
- a 8b 3c 5
- a −18b −75c −56
- a −3b −15c 14 + a 18b −25c −28
- a 2b −2c 9 + a 9c 3
Test Innovators. SAT Practice Exercises: Equivalent Expressions - Medium #2
Now, let’s try a more complicated question.
Let a = 2, b =1, and c = 1.
(a 3b −5c 7)(a −1b 3c 2 + a 6b 5c −4) = (2 31 −51 7)(2 −11 312 + 2 61 51 −4) = (2 3)(2 −1 + 2 6) = 516
A. a 8b 3c 5 = 2 81 31 5 = 2 8 = 256
B. a −18b −75c −56 = 2 −181 −751 −56 = = 2 −18 = 3.815 * 10 −6
C. a −3b −15c 14 + a 18b −25c −28 = 2 −31 −151 14 + 2 181 −251 −28 = 2 −3 + 2 18 = 262,144.125
D. a 2b −2c 9 + a 9c 3 = 2 21 −21 9 + 2 91 3 = 2 2 + 2 9 = 516
D is the correct answer.
Looking for more ACT and SAT practice?
Test Innovators has full-length practice tests and thousands of additional practice questions, giving you plenty of opportunities to practice this and other time-saving strategies.
Sara Laszlo
Sara Laszlo has nearly ten years of experience in private tutoring. An opera singer by training, Sara is especially interested in exploring better ways to practice and improve skills, whether musical or test-related. She holds a B.A. in Classical Civilization from Duke University and a Certificate of Merit in Voice from the New England Conservatory of Music.
