Dividing of Proper and Mixed Fractions

Dividing fractions is easier than you think! Simply flip the divisor and turn division into multiplication. Below, you’ll find detailed explanations, examples, and a calculator to practice with.

Result

\[ \frac{5}{6}\]

Core Principle

Flip the second fraction (divisor) and turn division into multiplication—that’s the key to dividing fractions!

Real-Life Applications

From cooking recipes to dividing materials and budgeting, fraction division is surprisingly useful in daily life.

Basics of Fraction Division

Dividing fractions relies on a simple idea: we turn division into multiplication by using the reciprocal of the second fraction (the divisor).

Formula for dividing fractions:

\( \huge \frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} \)

Where:

Why flip the divisor?

Imagine you have 3/4 of a pizza and want to divide it into pieces that are each 1/4 of a pizza. How many pieces do you get?

\( \large \frac{3/4}{1/4} = 3 \) (you get 3 pieces)

That’s why dividing by a fraction is the same as multiplying by its reciprocal. It simplifies the process while giving the same result.

3 Simple Steps to Divide Fractions

1

Write the problem as \( \large \frac{a}{b} \div \frac{c}{d} \)

2

Flip the second fraction (divisor)—swap its numerator and denominator

3

Multiply the fractions and simplify the result to its lowest terms

Practical Examples

Examples of Dividing Fractions

Example 1: Simple Fraction Division

\[ \large \frac{2}{3} \div \frac{4}{5} \]
  1. Flip the second fraction: \( \large \frac{4}{5} \rightarrow \frac{5}{4} \)
  2. Turn division into multiplication: \( \large \frac{2}{3} \times \frac{5}{4} \)
  3. Multiply numerators and denominators: \( \large \frac{2 \times 5}{3 \times 4} = \frac{10}{12} \)
  4. Simplify the result: \( \large \frac{10}{12} = \frac{5}{6} \)

Example 2: Division with an Improper Fraction Result

\[ \large \frac{3}{7} \div \frac{2}{9} \]
  1. Flip the second fraction: \( \large \frac{2}{9} \rightarrow \frac{9}{2} \)
  2. Turn division into multiplication: \( \large \frac{3}{7} \times \frac{9}{2} \)
  3. Multiply numerators and denominators: \( \large \frac{3 \times 9}{7 \times 2} = \frac{27}{14} \)
  4. Convert to a mixed number: \( \large \frac{27}{14} = 1\frac{13}{14} \)

Example 3: Division by a Whole Number

\[ \large \frac{5}{8} \div 2 \]
  1. Write 2 as a fraction: \( 2 = \frac{2}{1} \)
  2. Flip it: \( \large \frac{2}{1} \rightarrow \frac{1}{2} \)
  3. Turn division into multiplication: \( \large \frac{5}{8} \times \frac{1}{2} \)
  4. Multiply: \( \large \frac{5 \times 1}{8 \times 2} = \frac{5}{16} \)

Common Mistakes to Avoid

Don’t forget to flip the divisor!

Wrong: \( \large \frac{2}{3} \div \frac{4}{5} = \frac{2 \times 4}{3 \times 5} = \frac{8}{15} \) ❌

Right: \( \large \frac{2}{3} \div \frac{4}{5} = \frac{2 \times 5}{3 \times 4} = \frac{10}{12} = \frac{5}{6} \) ✓

Multiplying Instead of Dividing

Always flip the second fraction before multiplying when dividing fractions.

Incorrect Simplification

Check if fractions can be simplified before multiplying to make calculations easier.

Not Simplifying the Result

Always check if the result can be reduced by finding a common factor.

Ignoring Signs

Pay attention to the signs of fractions—positive or negative—affecting the result.

Practice Exercises

Test Your Skills:

Exercise 1: Divide the fractions and simplify to lowest terms

  • \( \large \frac{3}{4} \div \frac{5}{6} \)
  • \( \large \frac{7}{9} \div \frac{2}{3} \)
  • \( \large \frac{1}{8} \div \frac{4}{5} \)

Exercise 2: Divide the fractions and convert the result to a mixed number

  • \( \large \frac{5}{6} \div \frac{3}{4} \)
  • \( \large \frac{8}{3} \div \frac{1}{2} \)
  • \( \large \frac{11}{7} \div \frac{4}{9} \)

Exercise 3: Real-Life Applications

  • You have \( \large \frac{3}{4} \) cup of flour. A recipe needs \( \large \frac{1}{8} \) cup per serving. How many servings can you make?
  • A board is \( 2\frac{1}{2} \) meters long. You need pieces that are \( \large \frac{3}{4} \) meter long. How many full pieces can you get?

Key Takeaways

Practice with our calculator to hone your fraction division skills!

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