Fractions & Rational Expressions
A complete, test-focused breakdown of every Fractions & Rational Expressions concept on the ACT — from basic fraction arithmetic to simplifying algebraic rational expressions and solving rational equations — with worked examples and strategy notes.
In This Lesson
- 01 Fraction Fundamentals
- 02 Adding & Subtracting Fractions
- 03 Multiplying & Dividing Fractions
- 04 Mixed Numbers & Improper Fractions
- 05 Complex Fractions
- 06 Rational Expressions — Simplifying
- 07 Operations with Rational Expressions
- 08 Rational Equations
- 09 Word Problems with Fractions
- 10 Test Day Strategy
- → Practice Quizzes
Fraction Fundamentals
A fraction represents a part of a whole. Every fraction has a numerator (top) and a denominator (bottom). The denominator can never be zero — this is the single most important rule for rational expressions on the ACT.
Fraction: a/b where b ≠ 0
Proper fraction: |numerator| < |denominator| e.g. 3/7
Improper fraction: |numerator| ≥ |denominator| e.g. 9/4
Mixed number: integer + proper fraction e.g. 2¼
Equivalent Fractions
Two fractions are equivalent if they represent the same value. You create equivalent fractions by multiplying or dividing both numerator and denominator by the same nonzero number.
// Example: 2/3 = 4/6 = 8/12 = 10/15
Simplifying (Reducing) Fractions
Divide numerator and denominator by their Greatest Common Factor (GCF). ACT answers are always in simplest form — unsimplified forms won't appear as answer choices.
Comparing Fractions
The fastest method: cross-multiply. Compare a/b and c/d by comparing a×d with b×c. The fraction whose numerator contributes the larger cross-product is bigger.
// 3/7 vs 5/11: 3×11=33 vs 7×5=35 → 33<35 → 3/7 < 5/11
Sign trap: The fraction −3/4, 3/(−4), and −(3/4) are all equal to −¾. A negative sign anywhere in a fraction makes the whole fraction negative.
Adding & Subtracting Fractions
You can only add or subtract fractions that share the same denominator. If denominators differ, find the Least Common Denominator (LCD) first, convert both fractions, then operate on the numerators only.
Same Denominator
a/c − b/c = (a − b)/c
// Keep the denominator — never add denominators together!
Different Denominators — Finding the LCD
The LCD is the smallest number divisible by both denominators. Two reliable methods:
First one divisible by 4 is 12. Done.
LCD = 2² × 3 = 12
Quick Cross-Method for Two Fractions
For a/b ± c/d when you need speed, use the butterfly method — no need to find the LCD separately:
a/b − c/d = (ad − bc) / bd (then simplify)
ACT speed tip: The butterfly method always works but may give you a fraction that needs more simplification. For three or more fractions, finding the LCD explicitly is faster and cleaner.
Multiplying & Dividing Fractions
Multiplying Fractions
Multiply straight across — no common denominator needed. Numerator × numerator, denominator × denominator, then simplify.
Cross-Canceling Before Multiplying
Before multiplying, cancel any common factor between any numerator and any denominator (diagonally or straight across). This keeps numbers small and avoids large arithmetic.
Dividing Fractions — KCF
Division by a fraction is the same as multiplication by its reciprocal. Remember KCF: Keep, Change, Flip.
// Keep the first fraction · Change ÷ to × · Flip the second fraction
Common mistake: Students flip the wrong fraction. You always flip the divisor — the fraction you are dividing by (the second one). Never flip the first fraction.
Mixed Numbers & Improper Fractions
Converting Between Forms
3⅖ = (3×5 + 2)/5 = 17/5
23/6 = 3 remainder 5 = 3 5/6
Arithmetic with Mixed Numbers
The safest ACT strategy: always convert mixed numbers to improper fractions first, then apply standard fraction rules. Converting back at the end is optional — ACT answer choices appear in both forms.
Don't distribute mixed numbers: 2⅓ × 5 ≠ 2×5 + ⅓×5 is technically correct but error-prone. Always convert to improper first: 7/3 × 5 = 35/3. It's faster and safer.
Complex Fractions
A complex fraction has a fraction in its numerator, denominator, or both. These appear frequently on the ACT — they look intimidating but follow a predictable process.
Method A — Rewrite as division: (a/b)/(c/d) = (a/b) ÷ (c/d) → apply KCF.
Method B — Multiply by the LCD: Identify the LCD of ALL mini-fractions, multiply every term top and bottom by it to clear all fractions at once.
Method B is faster when the complex fraction has sums or differences inside. Method A is faster when it's a single fraction over a single fraction.
Rational Expressions — Simplifying
A rational expression is a fraction where the numerator and/or denominator are polynomials. The same rules that apply to numeric fractions apply here — but instead of finding the GCF of two numbers, you factor polynomials.
Step 1: Factor the numerator completely.
Step 2: Factor the denominator completely.
Step 3: Cancel any common factors (not terms — factors only).
Factoring Toolkit
| Pattern | Form | Example |
|---|---|---|
| GCF | ax + ay = a(x+y) | 6x² + 9x = 3x(2x+3) |
| Difference of Squares | a²−b² = (a+b)(a−b) | x²−25 = (x+5)(x−5) |
| Perfect Square Trinomial | a²±2ab+b² = (a±b)² | x²+6x+9 = (x+3)² |
| Trinomial (leading coeff = 1) | x²+bx+c = (x+p)(x+q) where p+q=b, pq=c | x²+5x+6 = (x+2)(x+3) |
| Trinomial (leading coeff ≠ 1) | ax²+bx+c — factor by grouping or AC method | 2x²+5x+3 = (2x+3)(x+1) |
| Sum/Diff of Cubes | a³±b³ = (a±b)(a²∓ab+b²) | x³−8 = (x−2)(x²+2x+4) |
Critical trap — canceling terms vs. factors:
WRONG: (x+3)/(x+5) → 3/5 by "canceling x" — this is illegal!
RIGHT: (x(x+3))/(x(x+5)) → (x+3)/(x+5) — x is a factor of both, so it cancels.
You can only cancel factors that multiply the entire numerator or denominator.
Opposite Factors
Watch for factors that are negatives of each other: (a − b) and (b − a). Factor out −1 from one of them: (b − a) = −(a − b). Then cancel the (a − b) factor and keep the −1.
// This shows up frequently — don't miss it on the ACT!
Operations with Rational Expressions
The same four operations (add, subtract, multiply, divide) apply to rational expressions exactly as they do to numeric fractions. The extra skill needed is factoring to find the LCD.
Multiplying Rational Expressions
Dividing Rational Expressions
Same as numeric fractions: KCF — keep the first, change to multiplication, flip the second. Then factor and cancel.
Adding & Subtracting Rational Expressions
You must have a common denominator. For rational expressions, find the Least Common Denominator (LCD) by factoring each denominator and taking the highest power of each unique factor.
Subtraction sign trap: When subtracting a fraction, the negative distributes to every term in the numerator being subtracted. In step 03 above, −2(x−1) = −2x+2, not −2x−2. This is the #1 error on rational expression subtraction.
Rational Equations
A rational equation has one or more rational expressions and an equals sign. The strategy is to eliminate all denominators by multiplying both sides by the LCD, then solve the resulting polynomial equation.
1. Factor all denominators.
2. Identify the LCD.
3. Multiply every term on both sides by the LCD.
4. Solve the resulting equation.
5. Check for extraneous solutions — values that make any original denominator zero.
Always check for extraneous solutions. The ACT sometimes includes extraneous roots as trap answer choices. Any value that zeros out an original denominator must be rejected, even if the algebra says it works.
Proportion Equations (Cross-Multiply)
When the equation is a single fraction equal to another single fraction, cross-multiply directly.
// Only valid when each side is a single fraction — not when there are added terms!
Word Problems with Fractions
Rate & Work Problems
The most common ACT fraction word problem type. Two workers (or pipes, or machines) complete a job at different rates. The key formula:
If A completes the job in a hours: A's rate = 1/a (job per hour)
Combined rate = 1/a + 1/b
Time together = 1 / (1/a + 1/b) = ab/(a+b)
Part-of-a-Whole Problems
These ask what fraction of a group satisfies a condition. Set up carefully — the denominator is always the total, not a subset.
Splitting & Sharing Problems
A quantity is divided according to a ratio or fraction. Find the total parts first, then multiply each share by the total.
Test Day Strategy
Top ACT Traps to Avoid
- Canceling terms instead of factors: (x+3)/(x+7) ≠ 3/7.
- Forgetting to distribute the negative when subtracting a rational expression.
- Flipping the wrong fraction when dividing (always flip the divisor).
- Adding denominators: a/c + b/c ≠ (a+b)/(2c).
- Keeping extraneous solutions that zero out a denominator.
- Forgetting to convert mixed numbers before multiplying or dividing.
Speed Strategies
- Cross-cancel before multiplying to keep numbers small.
- For two-fraction addition, use the butterfly method: (ad±bc)/bd.
- For complex fractions with sums inside, multiply by the LCD immediately.
- Plug in numbers to verify your simplified rational expression matches the original.
- On proportion problems (a/b = c/d), cross-multiply directly — no LCD needed.
- For work problems, always use: combined rate = 1/a + 1/b.
Quick Reference Formulas
| Situation | Formula / Rule |
|---|---|
| Simplify a fraction | Divide numerator and denominator by GCF |
| Add/subtract fractions | (ad ± bc)/bd — then simplify |
| Multiply fractions | ac/bd — cancel before multiplying |
| Divide fractions | KCF: keep first, change to ×, flip second |
| Complex fraction (single/single) | Rewrite as division, apply KCF |
| Complex fraction (sums inside) | Multiply every term by LCD |
| Simplify rational expression | Factor completely, cancel common factors |
| Rational equation | Multiply by LCD, solve, check for extraneous roots |
| Combined work rate | Time = ab/(a+b) |
| Proportion | a/b = c/d → ad = bc |
Put It to the Test
You've covered the full topic. Now lock in the skills with ACT-style timed questions. Start with Quiz 1 and work through to Quiz 3 — difficulty increases with each set.