ACT Math: Fractions & Rational Expressions
Fractions & Rational Expressions is a foundational ACT Math topic, and it covers both numeric fractions and their algebraic cousins, rational expressions built from variables. This free, complete lesson covers simplifying fractions and mixed numbers, adding, subtracting, multiplying, and dividing fractions, simplifying rational expressions, and operations with rational expressions, including complex fractions. Each idea is followed by a fully worked example, and then a set of original practice problems with instant feedback, in the same four-answer-choice, calculator-allowed format as the current ACT. Everything here is free, and you can keep practicing afterward with the full ACT Math Question Bank linked below.
1. Simplifying Fractions and Mixed Numbers
To simplify a fraction, divide the numerator and denominator by their greatest common factor (GCF). An improper fraction (numerator larger than denominator) can be rewritten as a mixed number, a whole number plus a proper fraction, and vice versa.
Worked Example: Simplify 1824.
The GCF of 18 and 24 is 6.
18 ÷ 624 ÷ 6 = 34
1. Simplify: 2436
Explanation: The GCF of 24 and 36 is 12: 24 ÷ 1236 ÷ 12 = 23.
2. Convert 72 to a mixed number.
Explanation: 7 ÷ 2 = 3 with a remainder of 1, so 72 = 312.
3. Convert 435 to an improper fraction.
Explanation: Multiply the whole number by the denominator and add the numerator: 4 × 5 + 3 = 23, so the result is 235.
4. Simplify: 4560
Explanation: The GCF of 45 and 60 is 15: 45 ÷ 1560 ÷ 15 = 34.
5. Order the following fractions from least to greatest: 35, 710, 23
Explanation: Rewrite with a common denominator of 30: 35 = 1830, 710 = 2130, and 23 = 2030. Ordering the numerators: 18 < 20 < 21, so 35 < 23 < 710.
2. Operations with Fractions
To add or subtract fractions, first rewrite them with a common denominator. To multiply fractions, multiply straight across, numerator by numerator and denominator by denominator. To divide fractions, multiply by the reciprocal of the second fraction.
Worked Example: Compute 23 + 56.
The least common denominator is 6: 23 = 46.
46 + 56 = 96 = 32
1. Compute: 14 + 38
Explanation: 14 = 28, and 28 + 38 = 58.
2. Compute: 56 − 13
Explanation: 13 = 26, and 56 − 26 = 36 = 12.
3. Compute: 23 × 34
Explanation: Multiply straight across: 2 × 33 × 4 = 612 = 12.
4. Compute: 35 ÷ 27
Explanation: Multiply by the reciprocal of the second fraction: 35 × 72 = 2110.
5. Compute: 112 + 213
Explanation: As improper fractions: 32 + 73 = 96 + 146 = 236 = 356.
3. Simplifying Rational Expressions
A rational expression is a fraction with a polynomial in the numerator, denominator, or both. Simplify one by factoring the numerator and denominator completely, then canceling any factors they share.
Worked Example: Simplify x² − 9x + 3.
(x − 3)(x + 3)x + 3 = x − 3
1. Simplify: 6x²9x
Explanation: The GCF of 6x² and 9x is 3x: 6x² ÷ 3x9x ÷ 3x = 2x3.
2. Simplify: x² − 16x − 4
Explanation: (x − 4)(x + 4)x − 4 = x + 4.
3. Simplify: x² + 5x + 6x + 2
Explanation: Factor the numerator: (x + 2)(x + 3)x + 2 = x + 3.
4. Simplify: 4x + 8x + 2
Explanation: Factor the numerator: 4(x + 2)x + 2 = 4.
5. For what value of x is the expression x + 5x − 3 undefined?
Explanation: A rational expression is undefined wherever its denominator equals 0: x − 3 = 0, so x = 3.
4. Operations with Rational Expressions
Rational expressions follow the exact same rules as numeric fractions: multiply straight across (then simplify), divide by multiplying by the reciprocal, and add or subtract only after finding a common denominator. A complex fraction, a fraction within a fraction, is simplified the same way as dividing two fractions.
Worked Example: Simplify 3x × x²6.
3x²6x = x2
1. Simplify: x4 ÷ x²8
Explanation: Multiply by the reciprocal: x4 × 8x² = 8x4x² = 2x.
2. Add: 3x + 2x
Explanation: These fractions already share a denominator, so just add the numerators: 3 + 2x = 5x.
3. Add: 1x + 1y
Explanation: The common denominator is xy: yxy + xxy = x + yxy.
4. Simplify the complex fraction: 1/23/4
Explanation: A complex fraction is division: 12 ÷ 34 = 12 × 43 = 46 = 23.
Common Mistakes to Avoid
- Adding or subtracting fractions without a common denominator. Unlike multiplication, addition and subtraction of fractions require the same denominator first. Never add numerators and denominators separately.
- Forgetting to flip the second fraction when dividing. Dividing by a fraction means multiplying by its reciprocal, flipping only the second fraction, not the first.
- Canceling terms instead of factors in a rational expression. You can only cancel a factor that multiplies the entire numerator and entire denominator, never a term that's only added or subtracted within a larger expression.
- Forgetting to factor before canceling. An expression like 4x + 8x + 2 can't be simplified by canceling the x + 2 pieces directly, it must be factored first, into 4(x + 2)x + 2, before anything cancels.
- Treating a complex fraction like a sum instead of a division. A fraction stacked over another fraction represents division between the two, not addition, so multiply by the reciprocal of the bottom fraction.
Frequently Asked Questions
Do I need to find the least common denominator, or will any common denominator work?
Any common denominator technically works, but the least common denominator keeps the numbers smaller and the arithmetic simpler, which matters on a timed test like the ACT.
Are rational expressions harder than numeric fractions?
The rules are identical, add/subtract with a common denominator, multiply straight across, divide by the reciprocal. The extra step with rational expressions is factoring first, since the "common factors" you're canceling are algebraic expressions rather than plain numbers.
Why does a rational expression have restricted values?
Division by zero is undefined, so any x-value that makes the denominator of a rational expression equal zero must be excluded from its domain, even after the expression has been simplified.
Where can I practice more problems like these?
The Fractions & Rational Expressions quizzes in the ACT Math Question Bank include additional original problems on this topic, along with quizzes covering every other topic tested on the ACT Math section.