Pre-Algebra
Numbers and operations
Dividing fractions by fractions
20 practice questions
0 video lessons
Theory + worked examples
Dividing Fractions by Fractions
Texas Pre-Algebra (TEKS) • Standard 6.3(E) • Numbers & Operations
Dividing Fractions by Fractions is a topic in Numbers & Operations in the Texas Essential Knowledge and Skills. It is aligned to Standard 6.3(E), which requires students to divide fractions by fractions by multiplying by the reciprocal.
Dividing by a fraction asks how many fit into the first number and is done by multiplying by the reciprocal (keep–change–flip).
Theory
Dividing by a fraction asks “how many of the divisor fit into the dividend?” and is done by multiplying by the reciprocal.
Keep–change–flip: keep the first, change \(\div\) to \(\times\), flip the second.
Dividing by a fraction.
Keep–change–flip.
The rule:
\[\dfrac{a}{b}\div\dfrac{c}{d}=\dfrac{a}{b}\cdot\dfrac{d}{c}\]
The reciprocal flips numerator and denominator.
How to divide fractions
- Keep the first fraction.
- Change division to multiplication.
- Flip the second fraction.
- Multiply and simplify.
Example 1 — How many halves
Find \(3\div\dfrac{1}{2}\).
Solution
How many halves in \(3\)? Multiply by \(2\).
| \(3\cdot 2\) | \(=\) | \(6\) |
Example 2 — Fraction by fraction
Find \(\dfrac{2}{3}\div\dfrac{1}{6}\).
Solution
Keep–change–flip.
| \(\dfrac{2}{3}\cdot\dfrac{6}{1}\) | \(=\) | \(\dfrac{12}{3}=4\) |
Example 3 — Simplify
Find \(\dfrac{3}{4}\div\dfrac{9}{8}\).
Solution
Multiply by the reciprocal, then simplify.
| \(\dfrac{3}{4}\cdot\dfrac{8}{9}\) | \(=\) | \(\dfrac{24}{36}=\dfrac{2}{3}\) |
Example 4 — Word problem
How many \(\dfrac{1}{4}\)-cup scoops are in \(\dfrac{3}{2}\) cups?
Solution
Divide.
| \(\dfrac{3}{2}\div\dfrac{1}{4}\) | \(=\) | \(\dfrac{3}{2}\cdot 4=6\) |
Common pitfalls
Flip the second fraction, not the first.
Change the sign of the operation to \(\times\) too.
Dividing by a fraction \(<1\) makes the result larger.
Frequently asked questions
How do I divide by a fraction?
Multiply by its reciprocal.
What is keep–change–flip?
Keep the first, change to times, flip the second.
What is \(\dfrac{2}{3}\div\dfrac{1}{6}\)?
\(4\).
Why does the answer get bigger?
Dividing by a number less than one increases the result.
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