Pre-Algebra
Numbers and operations
Multiplying and dividing rational numbers
20 practice questions
0 video lessons
Theory + worked examples
Multiplying and Dividing Rational Numbers
Texas Pre-Algebra (TEKS) • Standard 7.3(A) • Numbers & Operations
Multiplying and Dividing Rational Numbers is a topic in Numbers & Operations in the Texas Essential Knowledge and Skills. It is aligned to Standard 7.3(A), which requires students to multiply and divide rational numbers, including fractions and decimals.
Multiply rational numbers across, and divide by multiplying by the reciprocal, applying the sign rules.
Theory
To multiply rational numbers, multiply numerators and denominators; to divide, multiply by the reciprocal.
Apply the integer sign rules: same signs positive, different signs negative.
Multiplying and dividing fractions.
Divide by the reciprocal.
The rules:
\[\dfrac{a}{b}\cdot\dfrac{c}{d}=\dfrac{ac}{bd},\qquad \dfrac{a}{b}\div\dfrac{c}{d}=\dfrac{a}{b}\cdot\dfrac{d}{c}\]
Simplify before or after multiplying.
How to multiply or divide rationals
- For division, flip the second fraction (reciprocal).
- Multiply numerators and denominators.
- Decide the sign with the integer rules.
- Simplify.
Example 1 — Multiply
Find \(\dfrac{2}{3}\cdot\dfrac{3}{5}\).
Solution
Multiply across, then simplify.
| \(\dfrac{2\cdot3}{3\cdot5}\) | \(=\) | \(\dfrac{6}{15}=\dfrac{2}{5}\) |
Example 2 — Divide
Find \(\dfrac{4}{5}\div\dfrac{2}{3}\).
Solution
Multiply by the reciprocal.
| \(\dfrac{4}{5}\cdot\dfrac{3}{2}\) | \(=\) | \(\dfrac{12}{10}=\dfrac{6}{5}\) |
Example 3 — Signs
Find \(-\dfrac{3}{4}\cdot\dfrac{2}{9}\).
Solution
Different signs: negative.
| \(-\dfrac{6}{36}\) | \(=\) | \(-\dfrac{1}{6}\) |
Example 4 — Decimals
Find \((-0.5)(-6)\).
Solution
Same signs: positive.
| \((-0.5)(-6)\) | \(=\) | \(3\) |
Common pitfalls
Divide by the reciprocal, don't divide straight across.
Multiply straight across — no common denominator needed.
Apply the sign rules to the result.
Frequently asked questions
How do I multiply fractions?
Multiply numerators together and denominators together.
How do I divide fractions?
Multiply by the reciprocal of the second fraction.
Do I need a common denominator?
No, only for adding and subtracting.
What is \(\dfrac{2}{3}\cdot\dfrac{3}{5}\)?
\(\dfrac{2}{5}\).
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Dividing fractions by fractions
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