Multiplying Fractions

Multiplying Fractions

Basic Concept

• To multiply fractions, you multiply the numerators together to get the new numerator, and you multiply the denominators together to get the new denominator.

  • For example: $\frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d}$

Example 1

• Problem: $\frac{1}{2} \times \frac{3}{4}$

• Step 1: Multiply the numerators: $1 \times 3 = 3$

• Step 2: Multiply the denominators: $2 \times 4 = 8$

• Solution: $\frac{1}{2} \times \frac{3}{4} = \frac{3}{8}$

Example 2

• Problem: $\frac{2}{5} \times \frac{7}{9}$

• Step 1: Multiply the numerators: $2 \times 7 = 14$

• Step 2: Multiply the denominators: $5 \times 9 = 45$

• Solution: $\frac{2}{5} \times \frac{7}{9} = \frac{14}{45}$

Example 3

• Problem: $\frac{4}{6} \times \frac{2}{3}$

• Step 1: Multiply the numerators: $4 \times 2 = 8$

• Step 2: Multiply the denominators: $6 \times 3 = 18$

• Initial Solution: $\frac{8}{18}$

• Step 3: Simplify the fraction (if possible).

  • Find a common factor for both the numerator and the denominator. In this case, both 8 and 18 are divisible by 2.

  • Divide both the numerator and the denominator by the common factor: $\frac{8 \div 2}{18 \div 2} = \frac{4}{9}$

• Simplified Solution: $\frac{4}{9}$

Simplifying Before Multiplying (Optional)

• Sometimes, it’s easier to simplify before multiplying.

• Example: $\frac{4}{6} \times \frac{2}{3}$ can be simplified by noticing that 4 and 2 share a common factor with 6 and 3 respectively.

• Simplify $\frac{4}{6}$ to $\frac{2}{3}$ by dividing both by 2.

• The problem becomes: $\frac{2}{3} \times \frac{2}{3}$

• Multiply the numerators: $2 \times 2 = 4$

• Multiply the denominators: $3 \times 3 = 9$

• Solution: $\frac{4}{9}$ (Same as simplifying after multiplying)

Multiplying More Than Two Fractions

• The same principle applies when multiplying more than two fractions.

• Example: $\frac{1}{2} \times \frac{2}{3} \times \frac{3}{4}$

• Multiply all numerators: $1 \times 2 \times 3 = 6$

• Multiply all denominators: $2 \times 3 \times 4 = 24$

• Initial Solution: $\frac{6}{24}$

• Simplify the fraction by dividing both numerator and denominator by their greatest common divisor, which is 6.

• Simplified Solution: $\frac{6 \div 6}{24 \div 6} = \frac{1}{4}$

Key Points

• Always multiply numerators together and denominators together.

• Simplify the fraction after multiplying, if possible, to get the answer in its simplest form.

• Simplifying before multiplying can make the calculation easier.