Fraction Operations Review

If you have worked through our Arithmetic section, you already know the rules for fraction operations. This page is a fluency review — a condensed run-through of all four operations with an eye toward the patterns that matter most once you start algebra. The goal is speed and accuracy, not re-learning from scratch. If any operation still feels unfamiliar, follow the links to the full arithmetic lessons.

Adding and Subtracting Fractions

The rule is the same for both: you need a common denominator before you can combine the numerators.

$\frac{a}{c} + \frac{b}{c} = \frac{a + b}{c} \qquad \frac{a}{c} - \frac{b}{c} = \frac{a - b}{c}$

If the denominators differ, find the Least Common Denominator (LCD), rewrite each fraction, then combine.

For a full explanation of finding the LCD, see Adding Fractions and Subtracting Fractions in the Arithmetic section.

Example 1: Add $\frac{5}{6} + \frac{3}{8}$

Step 1 — Find the LCD. The LCM of 6 and 8 is 24.

Step 2 — Rewrite each fraction:

$\frac{5}{6} = \frac{5 \times 4}{6 \times 4} = \frac{20}{24} \qquad \frac{3}{8} = \frac{3 \times 3}{8 \times 3} = \frac{9}{24}$

Step 3 — Add:

$\frac{20}{24} + \frac{9}{24} = \frac{29}{24} = 1\frac{5}{24}$

Example 2: Subtract $\frac{7}{10} - \frac{1}{4}$

LCD of 10 and 4 is 20.

$\frac{7}{10} = \frac{14}{20} \qquad \frac{1}{4} = \frac{5}{20}$

$\frac{14}{20} - \frac{5}{20} = \frac{9}{20}$

Why This Matters for Algebra

In algebra you will add rational expressions like $\frac{2}{x} + \frac{3}{x+1}$, and the process is identical: find a common denominator, rewrite, combine. Building speed with numeric fractions now makes the algebraic version much easier later.

Multiplying Fractions

Multiplication is the simplest fraction operation. Multiply the numerators together and the denominators together.

$\frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d}$

Cross-cancel before multiplying to keep numbers small. Divide any numerator and any denominator by a common factor.

For the full lesson with more examples, see Multiplying Fractions.

Example 3: Multiply $\frac{9}{14} \times \frac{7}{12}$

Cross-cancel first: 9 and 12 share a factor of 3 (giving 3 and 4). Also 7 and 14 share a factor of 7 (giving 1 and 2).

$\frac{9}{14} \times \frac{7}{12} = \frac{3}{2} \times \frac{1}{4} = \frac{3}{8}$

Example 4: Multiply $\frac{5}{6} \times \frac{3}{10}$

Cross-cancel: 5 and 10 share 5 (giving 1 and 2). Also 3 and 6 share 3 (giving 1 and 2).

$\frac{5}{6} \times \frac{3}{10} = \frac{1}{2} \times \frac{1}{2} = \frac{1}{4}$

Why This Matters for Algebra

When you solve equations by multiplying both sides by a fraction, or simplify expressions like $\frac{2x}{3} \cdot \frac{9}{4x}$, the cross-canceling technique you practice here applies directly.

Dividing Fractions

Division is multiplication by the reciprocal. The method is often called Keep-Change-Flip:

1. Keep the first fraction
2. Change the division to multiplication
3. Flip the second fraction (take its reciprocal)

$\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}$

For the full lesson, see Dividing Fractions.

Example 5: Divide $\frac{4}{9} \div \frac{2}{3}$

Keep-Change-Flip:

$\frac{4}{9} \times \frac{3}{2}$

Cross-cancel: 4 and 2 share 2 (giving 2 and 1). Also 3 and 9 share 3 (giving 1 and 3).

$\frac{2}{3} \times \frac{1}{1} = \frac{2}{3}$

Example 6: Divide $\frac{5}{8} \div \frac{15}{16}$

$\frac{5}{8} \times \frac{16}{15}$

Cross-cancel: 5 and 15 share 5 (giving 1 and 3). Also 16 and 8 share 8 (giving 2 and 1).

$\frac{1}{1} \times \frac{2}{3} = \frac{2}{3}$

The Four Operations at a Glance

Real-World Application: Carpentry — Combining Measurements

A carpenter is building a shelf that needs two pieces of trim. The first piece is $\frac{7}{8}$ of an inch thick and the second is $\frac{3}{16}$ of an inch thick. What is the combined thickness?

Step 1 — Find the LCD. The LCM of 8 and 16 is 16.

Step 2 — Convert and add:

$\frac{7}{8} = \frac{14}{16}$

$\frac{14}{16} + \frac{3}{16} = \frac{17}{16} = 1\frac{1}{16} \text{ inches}$

The combined trim thickness is $1\frac{1}{16}$ inches. The carpenter needs to account for this when calculating the overall shelf depth.

Real-World Application: Nursing — Splitting a Dosage

A patient receives $\frac{3}{4}$ of a gram of medication per day, split equally across 3 doses. How much is each dose?

$\frac{3}{4} \div 3 = \frac{3}{4} \div \frac{3}{1} = \frac{3}{4} \times \frac{1}{3} = \frac{3}{12} = \frac{1}{4} \text{ gram per dose}$

Each dose is $\frac{1}{4}$ gram.

Common Mistakes to Avoid

1. Adding numerators AND denominators. Wrong: $\frac{1}{3} + \frac{1}{4} = \frac{2}{7}$. You must find a common denominator first. The correct answer is $\frac{7}{12}$.

2. Forgetting to flip when dividing. Division means multiply by the reciprocal of the second fraction, not the first.

3. Not simplifying. Always reduce your final answer. If you get $\frac{6}{8}$, simplify to $\frac{3}{4}$.

4. Skipping cross-cancellation. You can always simplify after multiplying, but cross-canceling first keeps the numbers manageable — especially important when you start working with algebraic fractions.

5. Using the LCD for multiplication or division. A common denominator is only needed for addition and subtraction. For multiplication and division, work directly with the original fractions.

Practice Problems

Test your fluency. Click to reveal each answer.

Key Takeaways

• Addition and subtraction require a common denominator; multiplication and division do not
• Cross-cancel before multiplying to keep numbers small — this same technique applies to algebraic fractions later
• Keep-Change-Flip turns every division problem into a multiplication problem
• Always simplify your final answer by dividing numerator and denominator by their GCF
• If any operation feels shaky, revisit the full lessons in Arithmetic before moving on

Return to Pre-Algebra for more topics in this section.