Advanced Math Tricks

SAT/ACT math sections often contain questions requiring advanced techniques, and students are expected to creatively problem-solve their way out of these unfamiliar boxes (this is especially true of the difficult second module of the SAT).

Although these questions are challenging and unpredictable by nature, two solution methods have emerged that can be studied.

Compound Expressions

These are simple expressions with more than one part. Such questions might ask students to find 4x + y, for example, rather than simply x or y.

When encountering compound expressions questions, students are usually tempted to first find the values of the unknowns, one at a time, and then substitute them into the compound expression itself. But on the SAT/ACT, this is never the best way to solve the problem. It’s faster and easier to deduce a clever way to create the compound expression directly from what’s given.

For example, suppose you’re given that 15n – 5m = 70, and asked to find the value of 3n – m. Noticing the similarity between the left side of the given equation and the compound expression, one sees it’s easy to derive the expression itself directly simply by dividing both sides of the given equation by 5, yielding 3n–m = 14.

Matching Forms

The trick here is to match the form of expression on the right side of a given equation with that on the left, and then equate corresponding terms. The forms typically required on the SAT/ACT are standard quadratic and linear forms.

For instance, suppose we’re given i^2 = -1, a and b are real numbers, and a + b + 5i = 9 + ai, and asked to find b. Both sides can be written in the same linear form: (a + b) +5i = 9 + ai. Since the expressions on either side of the equation showing matching forms are equal, corresponding terms must likewise be equal. Thus a + b = 9 and 5i = ai, which implies a = 5 and b = 4.

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