ACT Math formulas: what to know and practice

The ACT does not provide a general Math formula sheet. Learn the basic relationships you need, along with what their symbols mean and when they apply. ACT's design framework for the enhanced test also explains that questions requiring a complex or unfamiliar formula may include it in the question.

Our free ACT Math formulas and practice guide gives you a selected reference, worked examples and questions to try with the guide closed. Use this article alongside it to practice a different decision: how the setup changes when the question asks for a different quantity.

Which ACT Math formulas should you learn?

ACT says its Math section assumes knowledge of basic formulas, while recall of complex formulas is not required. Start by checking the relationships below against your own practice. If you cannot use one without looking it up, include it in your next study session.

Source: Sharp's selected teaching relationships, not a frequency ranking or a complete syllabus. Checked November 5, 2026. Worked examples and additional formulas appear in the ACT Math practice guide.

The companion guide also covers distance and midpoint, sequences, exponents, volume and other selected relationships. A formula list still leaves out part of your preparation. ACT's Math content description includes interpreting functions, working with matrices and analyzing data. Our ACT format and timing guide puts Math in the context of the full test.

How the question changes the formula you use

These are original teaching examples, not official ACT questions. In each pair, the quantities are closely related, but the requested answer changes. Write the setup before doing the arithmetic.

A discount going forward and backward

A backpack costs $80. Its price falls by 25%, then by another 10% of the reduced price. Find the final price.

After the first discount, you pay 75% of the original. After the second, you pay 90% of that amount:

$80 × 0.75 × 0.90 = $54.

Now reverse the question. A backpack costs $54 after those same two discounts. Find its original price.

Let p be the original price. The relationship is still p × 0.75 × 0.90 = 54, but now you need to solve for p:

p = 54 ÷ (0.75 × 0.90) = $80.

Adding 25% and then 10% to $54 gives $74.25, which does not undo the discounts. Those increases use the reduced amounts as their bases. To reverse a multiplication, divide by the multiplier. Check your answer by applying the original discounts to $80 again.

An arc's length and a sector's area

A circle has radius 6 cm. A central angle measures 120°. Find the length of its intercepted arc, then the area of its sector.

The angle selects 120/360 = 1/3 of the circle. That fraction stays the same for both questions.

For the arc length, take one third of the circumference:

(1/3) × 2π(6) = 4π cm.

For the sector area, take one third of the circle's area:

(1/3) × π(6²) = 12π cm².

The unit helps you choose. An arc is a distance around the circle, measured in centimeters here. A sector covers part of its interior, measured in square centimeters. Writing “cm” or “cm²” beside the requested quantity before calculating can expose a wrong choice of formula.

Two green counters, or a first green already given

A bag holds 4 green counters and 3 yellow counters. Two counters are drawn at random, one at a time, without replacement, with each remaining counter equally likely to be selected. Find the probability that both are green.

The first draw has 4 green counters among 7. If it is green, the second draw has 3 green counters among the 6 left:

P(both green) = (4/7) × (3/6) = 2/7.

Now change the question: given that the first counter drawn was green, what is the probability that the second is green?

• A. 2/7
• B. 3/7
• C. 1/2
• D. 4/7

C is correct. The first green counter has already been drawn. Of the 6 counters left, 3 are green, so the answer is 3/6 = 1/2.

A answers the earlier question about both draws. B removes a green counter from the numerator but leaves it in the total. D uses the bag's original composition. The words “given that” change the information you start with; multiplying by the chance of the first draw would answer a different question.

Does a calculator replace formula study?

A calculator can evaluate 54 ÷ (0.75 × 0.90), but you still need to decide why division answers the question. Make that decision on paper first, then use your calculator for the arithmetic if it helps.

ACT's calculator policy permits approved calculators on Math and says every question can be solved without one. Online ACT Math includes a built-in Desmos graphing calculator. Practice with the calculator and testing interface you plan to use.

The downloadable guide is for studying before the test. ACT prohibits notes in the testing room. Its calculator policy also requires removing all documents and all programs with computer algebra system functionality from devices that can hold them. Keep your preparation separate from anything you plan to bring into the room.

How to use the formula guide in your next session

Open the free ACT Math practice guide, review a relationship you need, then close it and attempt the practice questions. Keep your setup visible when you check the explanations. A wrong answer with the correct formula needs a different repair from an answer you could not begin.

If you forgot the relationship, write what each symbol means and try another problem that uses it. If you remembered it but chose the wrong quantities, compare the two versions of an example above. Identify the words that changed the setup. If the setup was correct, locate the arithmetic or algebra step where your work first diverged from the solution.

For example, if you found the discounted price correctly but could not recover the original, assign yourself another reverse-percent question. Recopying the formula would not show whether you can decide which amount is the original.

Build that assignment into your ACT study plan, then return to unfamiliar questions from our ACT practice test resource guide. Check whether you can select the relationship when the problem does not name it for you. Your next session should address the error you can now describe.

For support with your preparation, explore Sharp for ACT students.