How to Multiply a Negative Fraction by a Positive Fraction
Multiplying a negative fraction by a positive fraction follows the same basic rule as multiplying any two fractions—multiply the numerators, multiply the denominators—with one extra step: apply the sign. Because a negative times a positive always equals a negative, the product will be negative. So the direct answer is: multiply straight across as usual, then put a minus sign in front of the result.
The Core Rule: Multiply Across, Then Apply the Sign
When you multiply fractions, you don’t need a common denominator (that’s only for adding and subtracting). Instead, you multiply the top numbers together and the bottom numbers together. This basic rule is covered by Khan Academy.
The sign is determined separately. A negative value times a positive value is negative, so the answer is always negative when exactly one of the two fractions is negative.
| First factor | Second factor | Product sign |
|---|---|---|
| Positive | Positive | Positive |
| Negative | Positive | Negative |
| Positive | Negative | Negative |
| Negative | Negative | Positive |
Since this article deals with one negative and one positive fraction, you land in the “Negative” rows every time. For example:
[ -\frac{2}{3} \times \frac{4}{5} = -\frac{8}{15} ]
Multiply (2 \times 4 = 8) for the numerator and (3 \times 5 = 15) for the denominator, then attach the negative sign.
Working Through a Full Example
Let’s multiply (-\frac{3}{4} \times \frac{2}{9}).
Step 1: Multiply the numerators. (3 \times 2 = 6). (Ignore the sign for now.)
Step 2: Multiply the denominators. (4 \times 9 = 36).
Step 3: Combine and apply the sign. You get (-\frac{6}{36}).
Step 4: Simplify. Both 6 and 36 divide by 6, so (-\frac{6}{36} = -\frac{1}{6}).
The final answer is (-\frac{1}{6}). Simplifying at the end keeps the numbers manageable, and you can also simplify before multiplying using cross-cancellation. Math is Fun explains simplifying fractions as dividing the numerator and denominator by their greatest common factor—here, that factor was 6.
Tips to Avoid Common Mistakes
Most errors with negative fractions come from sign slips or forgetting to simplify, not from the multiplication itself.
- Don’t lose the negative sign. It’s easy to multiply correctly and then forget the minus. Circle it or write it to the side until the final step.
- Only one sign matters here. Because your second fraction is positive, there’s no double-negative to worry about.
- Simplify fully. An answer like (-\frac{6}{36}) is correct but not finished. Reduce it to lowest terms.
- Check with estimation. (-\frac{3}{4}) is close to (-1), and (\frac{2}{9}) is a small positive number, so the product should be a small negative number. (-\frac{1}{6}) fits.
You can also place the negative sign in the numerator instead of in front of the fraction: (-\frac{1}{6} = \frac{-1}{6}). Both mean the same thing. Keeping the sign in front is the most common convention.
What About Mixed Numbers or Whole Numbers?
The rule doesn’t change if the numbers aren’t written as simple fractions—you just convert first.
If a fraction is paired with a whole number, write the whole number over 1. For instance, (-\frac{2}{5} \times 3) becomes (-\frac{2}{5} \times \frac{3}{1} = -\frac{6}{5}).
If either value is a mixed number, turn it into an improper fraction before multiplying. For example, to compute (-\frac{1}{2} \times 2\frac{1}{3}):
- Convert (2\frac{1}{3}) to (\frac{7}{3}).
- Multiply: (-\frac{1}{2} \times \frac{7}{3} = -\frac{7}{6}).
- Convert back if needed: (-\frac{7}{6} = -1\frac{1}{6}).
This conversion step is the same whether the numbers are positive or negative. The Khan Academy lessons on multiplying fractions cover these conversions in detail if you want more practice.
Key Takeaways
- To multiply a negative fraction by a positive fraction, multiply the numerators together and the denominators together, then make the result negative.
- A negative times a positive is always negative, so the product of one negative and one positive fraction is negative.
- The sign rule is simple: opposite signs give a negative product; matching signs give a positive product.
- Always simplify the final answer to lowest terms, and convert mixed numbers to improper fractions before multiplying.
- Double-check your sign at the end—forgetting the minus is the most common error.
