ACTStudy Topic
ACT Math Foundations Study Guide
Review geometry, probability, functions, sequences, statistics, and quantitative models in original practice problems.
Topic Overview
Map the situation to a relationship
For a geometry problem, label the dimensions and identify what is asked: length, area, volume, angle, or distance. A right triangle uses the Pythagorean relationship only when the right angle is established. In trigonometry, opposite and adjacent depend on the chosen angle, while the hypotenuse always lies opposite the right angle. Convert degrees to radians by multiplying by pi/180.
Count and summarize carefully
For equally likely outcomes, probability is favorable outcomes divided by all outcomes. List a small sample space before assuming two events are equivalent. In a union, shared elements count once. For a median, sort first; for an even number of values, average the two middle values. Adding a constant to every observation adds that constant to the mean.
Track order and rates
An arithmetic sequence adds a constant difference, while a geometric sequence multiplies by a constant ratio. An inverse function reverses an input-output rule; it is not the same as a reciprocal. In a trip with changing speed, compute each segment’s time, add the times, and divide total distance by total time. For example, equal distances at two speeds do not generally produce an average speed equal to the arithmetic mean of those speeds.
- Using diameter as radius in an area formula.
- Averaging speeds without accounting for travel time.
- Counting shared set elements twice.
- Confusing an inverse function with one divided by the function.
Checkpoint Quiz
Test your understanding of ACT Math Foundations
These questions are for study practice only and are not official exam questions.
1. A fair coin is tossed twice. What is the probability of exactly one head?
2. For an acute angle in a right triangle, the opposite side is 6 and the hypotenuse is 10. What is the sine of the angle?
3. Two similar triangles have corresponding side lengths 4 and 10. If another side of the smaller triangle is 6, what is its corresponding side in the larger triangle?
4. What is the value of 5!/(3!2!)?
5. A data set has mean 14. Adding 3 to every value produces what new mean?
6. A rectangle's length is doubled and its width is tripled. Its area is multiplied by what factor?
7. A circle has radius 5 cm. What is its area?
8. A bag contains 4 red, 3 blue, and 5 green beads. One bead is selected at random. What is the probability it is not green?
9. What is the distance between points (-1, 2) and (5, 10)?
10. What is the midpoint of the segment with endpoints (2, -4) and (8, 6)?
Frequently asked questions
How do I choose a trigonometric ratio?
Identify the given and requested sides relative to the angle, then use sine, cosine, or tangent accordingly.
What is the safest way to find average speed?
Divide total distance by total elapsed travel time.
How do I recognize a geometric sequence?
Check whether each nonzero term is multiplied by the same ratio to obtain the next term.
What does a union include?
Every distinct element that belongs to either set, with overlaps counted only once.
