What is the reciprocal of $\frac{4}{5}$?

The reciprocal is $\frac{5}{4}$. Swap the numerator and denominator.

What does "flip'n'times" mean?

When dividing fractions, flip the second fraction (take its reciprocal) and change the division to multiplication.

What is the reciprocal of a fraction?

The reciprocal is the fraction with numerator and denominator swapped. For $\frac{a}{b}$, the reciprocal is $\frac{b}{a}$.

Why can you cancel $\frac{3}{25} \times \frac{10}{11}$ before multiplying?

The 25 and 10 share a common factor of 5. Cancelling gives $\frac{3}{5} \times \frac{2}{11} = \frac{6}{55}$. The result is the same but the numbers are smaller.

Why does $\frac{1}{3} \div \frac{4}{5} = \frac{1}{3} \times \frac{5}{4}$?

Dividing by a fraction is the same as multiplying by its reciprocal. So $\div \frac{4}{5}$ becomes $\times \frac{5}{4}$.

Do you need a common denominator to multiply fractions?

No. Unlike addition, multiplication only requires multiplying numerators and denominators. No common denominator is needed.

Compare multiplying $\frac{2}{3} \times \frac{3}{4}$ with adding $\frac{2}{3} + \frac{3}{4}$.

Multiplication: $\frac{2}{3} \times \frac{3}{4} = \frac{6}{12} = \frac{1}{2}$ (no common denominator). Addition: need LCD 12, so $\frac{8}{12} + \frac{9}{12} = \frac{17}{12}$.

What is the difference between $\frac{1}{2} \div \frac{1}{4}$ and $\frac{1}{2} \times \frac{1}{4}$?

Division: $\frac{1}{2} \div \frac{1}{4} = \frac{1}{2} \times \frac{4}{1} = 2$. Multiplication: $\frac{1}{2} \times \frac{1}{4} = \frac{1}{8}$. Division gives a larger result.

How does cancelling in multiplication differ from simplifying after addition?

In multiplication, you can cancel factors across the fractions (numerator of one with denominator of another). In addition, you simplify the final result by dividing numerator and denominator by their HCF.

What error gives $\frac{2}{3} \div \frac{4}{5} = \frac{2}{3} \times \frac{4}{5}$?

Forgetting to flip the second fraction. The correct step is to multiply by the reciprocal: $\frac{2}{3} \times \frac{5}{4} = \frac{10}{12} = \frac{5}{6}$.

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Multiplying & Dividing Fractions

Summary

To multiply fractions, multiply numerators together and denominators together, cancelling common factors first if possible. To divide fractions, multiply by the reciprocal of the divisor (flip the second fraction and multiply). Mixed numbers must be converted to improper fractions before operating.

1. Definition & Core Concepts

2. Underlying Principles

Multiplying fractions is simpler than adding: no common denominator is needed.

Division by a fraction is equivalent to multiplication by its reciprocal.

Cancelling common factors preserves the result while simplifying the calculation.

3. Methods & Techniques

Visual showing multiplication of fractions: 2/3 × 3/4 = 6/12 = ½, and the rule for division.

4. Key Distinctions

5. Exam Strategy & Tips

Always cancel common factors before multiplying to simplify; it reduces the chance of arithmetic errors.

Remember "flip'n'times" for division: flip the second fraction and multiply.

Convert mixed numbers to improper fractions before multiplying or dividing.

6. Common Pitfalls

7. Connections

Multiplying & Dividing Fractions

Summary

1. Definition & Core Concepts

Multiplying fractions: Multiply the numerators and multiply the denominators. Cancel common factors before multiplying to simplify.

Dividing fractions: Multiply by the reciprocal of the divisor. The reciprocal of $\frac{a}{b}$ is $\frac{b}{a}$ .

The rule "flip and multiply" (or "flip'n'times") applies to division: $\frac{1}{3} \div \frac{4}{5} = \frac{1}{3} \times \frac{5}{4}$ .

Common factors can appear in either fraction; cancel them before multiplying to avoid large numbers.

2. Underlying Principles

Multiplying fractions is simpler than adding: no common denominator is needed.

Division by a fraction is equivalent to multiplication by its reciprocal.

Cancelling common factors preserves the result while simplifying the calculation.

3. Methods & Techniques

Multiply: Cancel any common factors between numerators and denominators (across both fractions), then multiply numerators and multiply denominators.

Divide: Flip the second fraction (swap numerator and denominator), change $\div$ to $\times$ , then multiply as above.

Mixed numbers: Convert to improper fractions first, operate, then convert back to mixed if required.

Example: $\frac{4}{15} \times \frac{25}{11}$ . Cancel 5: $\frac{4}{3} \times \frac{5}{11} = \frac{20}{33}$ .

Visual showing multiplication of fractions: 2/3 × 3/4 = 6/12 = ½, and the rule for division.

4. Key Distinctions

Operation	Rule	Example
Multiply	$\frac{a}{b} \times \frac{c}{d} = \frac{ac}{bd}$	$\frac{2}{3} \times \frac{3}{4} = \frac{6}{12} = \frac{1}{2}$
Divide	$\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}$	$\frac{1}{3} \div \frac{4}{5} = \frac{1}{3} \times \frac{5}{4} = \frac{5}{12}$

Unlike addition, multiplication and division do not require a common denominator.

5. Exam Strategy & Tips

Always cancel common factors before multiplying to simplify; it reduces the chance of arithmetic errors.

Remember "flip'n'times" for division: flip the second fraction and multiply.

Convert mixed numbers to improper fractions before multiplying or dividing.

6. Common Pitfalls

Finding a common denominator for multiplication: This is unnecessary. Multiply numerators and denominators directly.

Forgetting to flip when dividing: $\frac{1}{3} \div \frac{4}{5}$ is NOT $\frac{1}{3} \times \frac{4}{5}$ . You must flip to $\frac{5}{4}$ .

Multiplying mixed numbers directly: Convert to improper fractions first.

7. Connections

Cancelling uses the same idea as simplifying fractions.

The reciprocal is related to division of whole numbers: dividing by 2 is the same as multiplying by $\frac{1}{2}$ .

Fractions of amounts use multiplication: $\frac{2}{3}$ of 60 = $\frac{2}{3} \times 60$ .