Substitution on the SAT/ACT

One of my favorite songs as a kid growing up was that great Stevie Wonder hit, “Substitution.” I can remember the lyrics like it was yesterday…

“When you need to solve problems

That you don’t understand,

Then don’t suffer,

Substitution is the way.”

Ok, maybe that’s not exactly the way the song went. And maybe Stevie was singing about superstitions instead. But using substitution on the ACT/SAT is still an excellent strategy that can help students overcome potentially difficult problems.

Not just one substitution

In general, substitution refers to replacing an unknown quantity with a known number. For most students, it’s far easier to work with real, tangible values than with x or y. And when it comes to the SAT and ACT, there are essentially three types of substitution students can use, depending on the situation.

When the question is the answer

The most obvious instance of a potential use of substitution is when the question provides a number to sub in. For example, when faced with a problem like this,

If f(x) = 6x – 3, what is the value of f(x) when x = 4?

the best strategy will be to plug 4 in for x and see what the result is.

f(4) = 6(4) – 3 = 24 – 3 = 21

And voila! The solution presents itself.

The second type of substitution involves using the answer choices provided on a multiple choice test–what some strategists call “backsolving.” This method is ideal when the question presents variables and the answers provided are actual numbers.

Given that x (x + 6) = 16, which of the following lists all solutions to this quadratic equation?

a) 8 and 2

b) -8 and -2

c) -8 and 2

d) 8 and -4

While some students would be able to solve this algebraically, those that struggle with this type of question could use the answer choices to their advantage. By plugging in 8 for x, for example, we see that 8 (8 + 6) does not equal 16. So choices a) and d) are out, leaving us with a 50-50 at worst. Seeing that -8 is featured in both choices b) and c), all we would have to do now is plug in either 2 or -2. Putting 2 in for x gives us 2 (2 + 6), which does equal 16, and we’re done! Easy peasy!

Substitutional creativity

The third and most strategic type of substitution is the student-generated variety. With many percentage questions or with questions featuring variables in both the question and the answer choices, coming up with a number that makes sense can be a great option. Consider the following:

A shoe store has a long standing sale on their best selling pair, selling the shoes for 20% off the original price. On Black Friday, they take an additional 10% off the sale price. What is the total percentage off from the original price?

a) 20%

b) 28%

c) 30%

d) 32%

e) 72%

This is a very common type of percentage question. And while it can be done in other ways, substitution can make it a snap.

Let’s assume for a moment that the original price of the shoes was \$100. Why \$100? Because it makes percentage calculation easier. That would make the long standing sale price equal to 20% off \$100, or \$80. Great! Now, the Black Friday sale took an additional 10% off the \$80, which would be another \$8, leaving the final sale price at \$72. How does 72 compare to 100? It’s 72/100 or 72%. And since the question asks for the total percentage off, that gives us a correct answer of 28% or choice b).

Notice the trap answers of 30% (20% + 10%) and 72% (if we didn’t pay attention to what the question is actually asking). But substitution made this potentially tricky question pretty simple.