Multiplying Fractions

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

Problem: $\frac{1}{2} \times \frac{3}{4}$ $$\frac{1}{2} \times \frac{3}{4}$$
Step 1: Multiply the numerators: $1 \times 3 = 3$ $$1 \times 3 = 3$$
Step 2: Multiply the denominators: $2 \times 4 = 8$ $$2 \times 4 = 8$$
Solution: $\frac{1}{2} \times \frac{3}{4} = \frac{3}{8}$ $$\frac{1}{2} \times \frac{3}{4} = \frac{3}{8}$$

Problem: $\frac{2}{5} \times \frac{7}{9}$ $$\frac{2}{5} \times \frac{7}{9}$$
Step 1: Multiply the numerators: $2 \times 7 = 14$ $$2 \times 7 = 14$$
Step 2: Multiply the denominators: $5 \times 9 = 45$ $$5 \times 9 = 45$$
Solution: $\frac{2}{5} \times \frac{7}{9} = \frac{14}{45}$ $$\frac{2}{5} \times \frac{7}{9} = \frac{14}{45}$$

Problem: $\frac{4}{6} \times \frac{2}{3}$ $$\frac{4}{6} \times \frac{2}{3}$$
Step 1: Multiply the numerators: $4 \times 2 = 8$ $$4 \times 2 = 8$$
Step 2: Multiply the denominators: $6 \times 3 = 18$ $$6 \times 3 = 18$$
Initial Solution: $\frac{8}{18}$ $$\frac{8}{18}$$
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: $\frac{8 \div 2}{18 \div 2} = \frac{4}{9}$ $$\frac{8 \div 2}{18 \div 2} = \frac{4}{9}$$
Simplified Solution: $\frac{4}{9}$ $$\frac{4}{9}$$

Sometimes, it’s easier to simplify before multiplying.
Example: $\frac{4}{6} \times \frac{2}{3}$ $$\frac{4}{6} \times \frac{2}{3}$$ can be simplified by noticing that 4 and 2 share a common factor with 6 and 3 respectively.
Simplify $\frac{4}{6}$ $$\frac{4}{6}$$ to $\frac{2}{3}$ $$\frac{2}{3}$$ by dividing both by 2.
The problem becomes: $\frac{2}{3} \times \frac{2}{3}$ $$\frac{2}{3} \times \frac{2}{3}$$
Multiply the numerators: $2 \times 2 = 4$ $$2 \times 2 = 4$$
Multiply the denominators: $3 \times 3 = 9$ $$3 \times 3 = 9$$
Solution: $\frac{4}{9}$ $$\frac{4}{9}$$ (Same as simplifying after multiplying)

The same principle applies when multiplying more than two fractions.
Example: $\frac{1}{2} \times \frac{2}{3} \times \frac{3}{4}$ $$\frac{1}{2} \times \frac{2}{3} \times \frac{3}{4}$$
Multiply all numerators: $1 \times 2 \times 3 = 6$ $$1 \times 2 \times 3 = 6$$
Multiply all denominators: $2 \times 3 \times 4 = 24$ $$2 \times 3 \times 4 = 24$$
Initial Solution: $\frac{6}{24}$ $$\frac{6}{24}$$
Simplify the fraction by dividing both numerator and denominator by their greatest common divisor, which is 6.
Simplified Solution: $\frac{6 \div 6}{24 \div 6} = \frac{1}{4}$ $$\frac{6 \div 6}{24 \div 6} = \frac{1}{4}$$

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.

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Multiplying Fractions

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

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: $\frac{4}{9}$

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)

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

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.