Solve Symbolism Problems on the ACT Math Test

By Lisa Zimmer Hatch, Scott A. Hatch

Occasionally, you may encounter a few basic types of symbolism problems on the ACT Math Test. These problems aren’t difficult, but they may throw you off if you’re not expecting them. Really, they’re just odd substitution problems. To solve them, use this basic approach:

  • Substitute the number given for the variable in the explanation.

  • Talk through the explanation to see which constraint fits and then do the indicated operations.

Substitute for the variable in the explanation

You see a problem with a strange symbol. It may be a variable inside a circle, a triangle, a star, a tic‐tac‐toe sign, or something else. Realize that the symbol you see has no connection to the real world at all. Don’t panic, thinking that your math teachers forgot to teach you something. The ACT makes up the symbols for each problem.

The ACT includes a short explanation with the symbol. For example, it may give you this:

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Again, the # symbol doesn’t have any meaning in the outside world; it means only what just the one problem tells you it means.

Accompanying the ACT’s explanation is the question itself. For example, 3 # 2 # 1 =

To solve symbol questions like these, use substitution. Plug in a number for the variable in the equation. Which number do you plug in? The one that’s in the same position as that variable, like this:

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Because a is in the first position and 3 is in the first position, substitute 3 for a throughout the equation. Because b is in the second position and 2 is in the second position, substitute 2 for b throughout the equation. Because c is in the third position and 1 is in the third position, substitute 1 for c throughout the equation.

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This simpler of the two types of symbolism problems isn’t tough at all. Jump at the chance to solve this kind.

Talk through the explanation and do the operations

The other type of symbolism problem you’re likely to see on the ACT may seem more confusing until you’ve done a few. But then they become so easy that you wonder why you didn’t get them before. Here’s an example:

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To solve problems like this one, first talk through the explanation. You have something in a circle. If that something in the circle is odd, you multiply it by 3. If that something in the circle is even, you divide it by 2.

In the first half of the question, you have a 5 in the circle. Because 5 is odd, you multiply it by 3 to get 5 times 3 = 15. In the second half of the question, you have an 8 in a circle. Because 8 is even, you divide it by 2:

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Now all you have to do is add: 15 + 4 = 19.

You still may think of this type of symbolism problem as a plug‐in or substitution problem because you’re plugging the number into the equation for x and working through it. However, you first have to figure out which equation to plug the number into, and that requires talking things through.