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Operations with Fractions

Adding, subtracting, multiplying, and dividing fractions

Written and reviewed by the Study Mondo Education TeamLast updated
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Operations with Fractions

Adding and Subtracting Fractions

Same denominator: Add or subtract numerators, keep denominator ac+bc=a+bc\frac{a}{c} + \frac{b}{c} = \frac{a + b}{c}

Example: 27+37=57\frac{2}{7} + \frac{3}{7} = \frac{5}{7}

Different denominators: Find common denominator first!

Example: 13+14\frac{1}{3} + \frac{1}{4}

LCD = 12: 13=412,14=312\frac{1}{3} = \frac{4}{12}, \quad \frac{1}{4} = \frac{3}{12} 412+312=712\frac{4}{12} + \frac{3}{12} = \frac{7}{12}

Multiplying Fractions

Multiply numerators, multiply denominators ab×cd=a×cb×d\frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d}

Example: 23×45=815\frac{2}{3} \times \frac{4}{5} = \frac{8}{15}

Tip: Simplify before multiplying when possible!

Dividing Fractions

Multiply by the reciprocal ab÷cd=ab×dc\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}

Example: 23÷45=23×54=1012=56\frac{2}{3} \div \frac{4}{5} = \frac{2}{3} \times \frac{5}{4} = \frac{10}{12} = \frac{5}{6}

Memory aid: "Keep, Change, Flip"

Simplifying Fractions

Divide numerator and denominator by their GCF: 1218=12÷618÷6=23\frac{12}{18} = \frac{12 \div 6}{18 \div 6} = \frac{2}{3}

📚 Practice Problems

1Problem 1easy

❓ Question:

Calculate: 38+18\frac{3}{8} + \frac{1}{8}

💡 Show Solution

Same denominator: add numerators, keep denominator.

38+18=3+18=48\frac{3}{8} + \frac{1}{8} = \frac{3 + 1}{8} = \frac{4}{8}

Simplify: 48=12\frac{4}{8} = \frac{1}{2}

Answer: 12\frac{1}{2}

2Problem 2easy

❓ Question:

Calculate: 38+18\frac{3}{8} + \frac{1}{8}

💡 Show Solution

Same denominator: add numerators, keep denominator.

38+18=3+18=48\frac{3}{8} + \frac{1}{8} = \frac{3 + 1}{8} = \frac{4}{8}

Simplify: 48=12\frac{4}{8} = \frac{1}{2}

Answer: 12\frac{1}{2}

3Problem 3medium

❓ Question:

Calculate: 25×34\frac{2}{5} \times \frac{3}{4}

💡 Show Solution

Multiply numerators and denominators:

25×34=2×35×4=620\frac{2}{5} \times \frac{3}{4} = \frac{2 \times 3}{5 \times 4} = \frac{6}{20}

Simplify (divide by 2): 620=310\frac{6}{20} = \frac{3}{10}

Answer: 310\frac{3}{10}

4Problem 4medium

❓ Question:

Calculate: 25×34\frac{2}{5} \times \frac{3}{4}

💡 Show Solution

Multiply numerators and denominators:

25×34=2×35×4=620\frac{2}{5} \times \frac{3}{4} = \frac{2 \times 3}{5 \times 4} = \frac{6}{20}

Simplify (divide by 2): 620=310\frac{6}{20} = \frac{3}{10}

Answer: 310\frac{3}{10}

5Problem 5hard

❓ Question:

Calculate: 34÷23\frac{3}{4} \div \frac{2}{3}

💡 Show Solution

Keep, Change, Flip: Multiply by the reciprocal

34÷23=34×32\frac{3}{4} \div \frac{2}{3} = \frac{3}{4} \times \frac{3}{2}

Multiply: 3×34×2=98\frac{3 \times 3}{4 \times 2} = \frac{9}{8}

Convert to mixed number: 98=118\frac{9}{8} = 1\frac{1}{8}

Answer: 98\frac{9}{8} or 1181\frac{1}{8}

6Problem 6hard

❓ Question:

Calculate: 34÷23\frac{3}{4} \div \frac{2}{3}

💡 Show Solution

Keep, Change, Flip: Multiply by the reciprocal

34÷23=34×32\frac{3}{4} \div \frac{2}{3} = \frac{3}{4} \times \frac{3}{2}

Multiply: 3×34×2=98\frac{3 \times 3}{4 \times 2} = \frac{9}{8}

Convert to mixed number: 98=118\frac{9}{8} = 1\frac{1}{8}

Answer: 98\frac{9}{8} or 1181\frac{1}{8}

7Problem 7medium

❓ Question:

Calculate: 2/3 × 3/4

💡 Show Solution

Step 1: Multiply numerators. 2 × 3 = 6

Step 2: Multiply denominators. 3 × 4 = 12

Step 3: Write the result. 6/12

Step 4: Simplify. GCF of 6 and 12 is 6 6/12 = 1/2

Answer: 1/2

8Problem 8hard

❓ Question:

A recipe calls for 2 1/4 cups of flour. You want to make 2/3 of the recipe. How much flour do you need?

💡 Show Solution

Step 1: Convert mixed number to improper fraction. 2 1/4 = 9/4

Step 2: Multiply by 2/3. 9/4 × 2/3

Step 3: Multiply numerators and denominators. (9 × 2)/(4 × 3) = 18/12

Step 4: Simplify. GCF of 18 and 12 is 6 18/12 = 3/2

Step 5: Convert to mixed number. 3/2 = 1 1/2

Answer: 1 1/2 cups of flour

Explain using:

⚠️ Common Mistakes: Operations with Fractions

Avoid these 3 frequent errors

🌍 Real-World Applications: Operations with Fractions

See how this math is used in the real world

📝 Worked Example: Solving a Quadratic by Factoring

Problem:

Solve x2−5x+6=0x^2 - 5x + 6 = 0.

2Factor the quadratic
3Set each factor equal to zero
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Decimals and Percents
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❓ Frequently Asked Questions

What is Operations with Fractions?▾
Adding, subtracting, multiplying, and dividing fractions
How can I study Operations with Fractions effectively?▾
Start by reading the study notes and working through the examples on this page. Then use the flashcards to test your recall. Practice with the 8 problems provided, checking solutions as you go. Regular review and active practice are key to retention.
Is this Operations with Fractions study guide free?▾
Yes — all study notes, flashcards, and practice problems for Operations with Fractions on Study Mondo are free to access. No account is needed.
What course covers Operations with Fractions?▾
Operations with Fractions is part of the Pre-Algebra course on Study Mondo, specifically in the Fractions and Decimals section. You can explore the full course for more related topics and practice resources.
Are there practice problems for Operations with Fractions?▾
Yes, this page includes 8 practice problems with detailed solutions. Each problem includes a step-by-step explanation to help you understand the approach.