1. Classifying Numbers on ACT Math

Number classification is one of the first foundations students should review for ACT Math. Many ACT questions require you to understand what type of number you are working with before you calculate. A value may be real, imaginary, rational, irrational, integer, whole, or natural.

The key is to know which number sets include other number sets. For example, every natural number is also a whole number, every whole number is also an integer, and every integer is also rational. However, irrational numbers such as √3 and π cannot be written as fractions of integers.

2. Prime Numbers, Factorization, LCM, and HCF

A prime number is a positive integer greater than 1 that has exactly two positive factors: 1 and itself. The smallest prime number is 2, and 2 is the only even prime number. Numbers such as 9, 15, and 21 are not prime because they have factors other than 1 and themselves.

Prime Factorization

Prime factorization means rewriting a number as a product of prime numbers. This is especially useful for finding factors, least common multiples, greatest common factors, and simplifying expressions.

Since 210 can be divided by 2, then 3, then 5, and finally 7, its prime factorization is 2 × 3 × 5 × 7. ACT Math questions may not always ask directly for prime factorization, but the method can unlock harder factor and multiple problems.

Total Number of Factors

If a number can be written as N = x^ay^bz^c, where x, y, and z are prime numbers, then the total number of positive factors is found by adding 1 to each exponent and multiplying.

LCM and HCF/GCF

LCM means Least Common Multiple. It is the smallest number that is divisible by each number in a set. HCF, also called GCF, means Highest Common Factor or Greatest Common Factor. It is the largest number that divides each number in a set evenly.

3. Averages on ACT Math

Average questions are common on ACT Math because they look simple but often hide a sum-based setup. The average formula is one of the most important formulas to know.

If the average of 5 and 8 is needed, add the values and divide by 2. The average is (5 + 8) ÷ 2 = 6.5. For word problems, it is often more useful to rearrange the formula:

Average Word Problem Strategy

Suppose two people on an elevator have an average weight of 200 pounds. Their combined weight is not 200 pounds; it is 400 pounds because sum = average × count. If the elevator limit is 500 pounds, then a third person can weigh at most 100 pounds.

Average Change

Average change is different from total change and percent change. ACT questions often test whether students can distinguish between these ideas.

If a price increases from $1.50 to $2.50 over 4 years, the total increase is $1.00, but the average increase per year is $1.00 ÷ 4 = $0.25 per year.

Averages and Arithmetic Sequences

In an arithmetic sequence, the difference between consecutive terms is constant. A useful shortcut is that the average of an arithmetic sequence is the same as the median. For example, in the sequence 8, 12, 16, 20, 24, the average and median are both 16.

4. Ratios, Unit Rates, and Proportions

A ratio compares two quantities. Ratios can be written with a colon, as a fraction, or in words. For example, a ratio of 3 nurses to 4 medical assistants can be written as 3:4, 3/4, or “3 to 4.”

Reducing Ratios

Always reduce ratios when possible. If 8 hours of sleep are compared to 24 hours in a day, the ratio is 8:24, which reduces to 1:3.

Unit Rates

A unit rate compares a quantity to exactly one unit. For example, if a package costs $4.10 and contains 24 strips, divide 4.10 by 24 to find the cost per strip.

Proportions

A proportion is an equation showing that two ratios are equal. You can solve most proportions by cross multiplying.

Since 3 × 20 = 60 and 4 × 15 = 60, the ratios are equivalent.

5. Mixture Problems, Direct Variation, and Inverse Variation

Mixture problems often look harder than they are. The key is to convert the words into an equation. If a recipe has 9 parts flour and 2 parts water, then the total mixture has 11 parts.

Mixture Example

If you need 165 pounds of dough using a 9:2 flour-to-water ratio, write the equation 9x + 2x = 165. Then 11x = 165, so x = 15. That means flour = 9(15) = 135 pounds and water = 2(15) = 30 pounds.

Direct Variation

In direct variation, two variables increase or decrease together. If y varies directly with x, then:

Inverse Variation

In inverse variation, one variable increases while the other decreases. If y varies inversely with x, then:

6. Percentages and Percent Change

Percent means “out of 100.” This makes percentages another way to write fractions and decimals. For example, 25% = 25/100 = 0.25, and 99% = 99/100 = 0.99.

Percent Increase

To increase a quantity by a percent, multiply by 100% plus the increase.

For example, increasing 30 by 5% means multiplying by 105%, or 1.05. So 30 × 1.05 = 31.5.

Percent Decrease

To decrease a quantity by a percent, multiply by 100% minus the decrease.

For example, decreasing 70 by 30% means multiplying by 70%, or 0.70. So 70 × 0.70 = 49.

7. Square Roots and Radicals

A square root of a number is a value that produces the original number when multiplied by itself. For example, 4 is a square root of 16 because 4² = 16. Also, -4 is a square root of 16 because (-4)² = 16.

On ACT Math, the radical symbol √ usually refers to the principal, nonnegative square root. Therefore, √16 = 4, not ±4.

Zero is the only real number with exactly one square root. Since 0² = 0, we have √0 = 0.

Nth Roots and Rational Exponents

Roots can also be written using fractional exponents. This is important because ACT Math often moves between radical form and exponent form.

Laws of Radicals

Radical expressions can often be simplified using multiplication and division laws.

For example, √50 = √(25 × 2) = 5√2.

8. Exponent Rules for ACT Math

Exponents appear throughout ACT Math, especially in algebra, functions, radicals, and scientific notation. Students should memorize the basic exponent laws and practice applying them quickly.

Equations with Exponents

When an equation has exponents, a common strategy is to rewrite both sides using the same base. Once the bases are the same, you can set the exponents equal.

Since 4 = 2² and 128 = 2⁷, rewrite the equation as (2²)^x = 2⁷. Then 2x = 7, so x = 3.5.

How to Study These ACT Math Topics

The best way to study ACT Math is not to memorize pages of formulas without practice. Instead, learn one topic at a time, solve realistic ACT-style problems, review every mistake, and then practice under time limits.

Recommended Study Order

1. Review number types, integers, primes, factors, LCM, and HCF.
2. Practice averages, sums, and arithmetic sequences.
3. Master ratios, proportions, unit rates, and mixtures.
4. Review percentages and percent change.
5. Practice radicals, square roots, and rational exponents.
6. Memorize exponent rules and solve exponent equations.
7. Take timed quizzes and track the questions you miss.