Multiplying Fractions
Multiplying fractions is generally more straightforward than adding or subtracting because you do not need a common denominator.
The Steps for Multiplying Fractions
- Convert (if needed): Change any mixed numbers into improper fractions.
- Multiply Numerators: Multiply the top numbers (numerators) straight across.
- Multiply Denominators: Multiply the bottom numbers (denominators) straight across.
- Simplify: Reduce the resulting fraction to its lowest terms.
Pro Tip: Simplify BEFORE You Multiply!
Look for any common factors between any numerator and any denominator in the fractions being multiplied.
Divide both the numerator and the denominator by their common factor. This is sometimes called "cross-cancelling" or "cross-reducing".
Doing this makes the numbers smaller and the final multiplication and simplification much easier!
Example: In $\frac{\cancel{3}^1}{8} \times \frac{4}{\cancel{9}^3}$, 3 and 9 share factor 3. In $\frac{1}{\cancel{8}^2} \times \frac{\cancel{4}^1}{3}$, 4 and 8 share factor 4. Result: $\frac{1}{2} \times \frac{1}{3}$
Reminder: Sign Rules for Multiplication
- Positive × Positive = Positive (+)
- Negative × Negative = Positive (+)
- Positive × Negative = Negative (-)
- Negative × Positive = Negative (-) Essentially, if the signs are the same, the result is positive. If the signs are different, the result is negative.
Examples with Step-by-Step Solutions
Example 1: Calculate $\frac{2}{3} \times \frac{5}{7}$
Multiply straight across:
$$ \begin{align*} & \frac{2}{3} \times \frac{5}{7} \\ = & \frac{\boxed{2 \times 5}}{\boxed{3 \times 7}} \quad & \text{(Multiply numerators; Multiply denominators)} \\ = & \frac{10}{21} \end{align*} $$Result: $\frac{10}{21}$ (already simplified)
Example 2: Calculate $\left(\frac{3}{8}\right) \times \left(-\frac{4}{9}\right)$
Method: Simplify Before Multiplying (Recommended)
$$ \begin{align*} & \left(\frac{3}{8}\right) \times \left(-\frac{4}{9}\right) \quad & \text{(Look for common factors)} \\ & \quad \text{3 and 9 share factor 3: } \frac{\cancel{3}^1}{8} \text{ and } -\frac{4}{\cancel{9}^3} \\ & \quad \text{4 and 8 share factor 4: } \frac{1}{\cancel{8}^2} \text{ and } -\frac{\cancel{4}^1}{3} \\ = & \left(\frac{1}{2}\right) \times \left(-\frac{1}{3}\right) \quad & \text{(Multiply the simplified fractions)} \\ = & \frac{\boxed{1 \times (-1)}}{\boxed{2 \times 3}} \\ = & -\frac{1}{6} \end{align*} $$Method: Multiply First, Then Simplify
$$ \begin{align*} & \left(\frac{3}{8}\right) \times \left(-\frac{4}{9}\right) \\ = & \frac{\boxed{3 \times (-4)}}{\boxed{8 \times 9}} \\ = & -\frac{12}{72} \quad & \text{(Now simplify, GCF = 12)} \\ = & -\frac{12 \div 12}{72 \div 12} \\ = & -\frac{1}{6} \end{align*} $$Result: $-\frac{1}{6}$ (Both methods yield the same answer)
Example 3: Calculate $(-1\frac{1}{4}) \times \frac{6}{5}$
Convert mixed number, then simplify before multiplying:
$$ \begin{align*} & \boxed{(-1\frac{1}{4})} \times \frac{6}{5} \quad & \text{(Convert mixed to improper: } -\frac{5}{4}) \\ = & \left(-\frac{5}{4}\right) \times \left(\frac{6}{5}\right) \quad & \text{(Look for common factors)} \\ & \quad \text{5 and 5 share factor 5: } -\frac{\cancel{5}^1}{4} \text{ and } \frac{6}{\cancel{5}^1} \\ & \quad \text{6 and 4 share factor 2: } -\frac{1}{\cancel{4}^2} \text{ and } \frac{\cancel{6}^3}{1} \\ = & \left(-\frac{1}{2}\right) \times \left(\frac{3}{1}\right) \quad & \text{(Multiply simplified fractions)} \\ = & \frac{\boxed{-1 \times 3}}{\boxed{2 \times 1}} \\ = & -\frac{3}{2} \end{align*} $$Result: $-\frac{3}{2}$ or $-1\frac{1}{2}$