In Grade 4 you made equivalent fractions by multiplying the top and bottom by the same number, and you listed multiples. Together they let you give two fractions pieces of the same size.
Key idea
A common denominator is a number both denominators divide into: a common multiple. Rename each fraction with it by multiplying the numerator and the denominator by the same number. The least common multiple keeps the numbers small.
Why it works
Thirds and fourths are different sizes, so 2 thirds and 3 fourths are hard to compare or add. Cut every third into 4 pieces and every fourth into 3 pieces, and both are cut into twelfths. Cutting each piece into smaller equal pieces changes how many pieces you have, not how much: $\frac{2}{3} = \frac{8}{12}$.
Worked examples
Worked example: Halves and thirds
Rename $\frac{1}{2}$ and $\frac{1}{3}$ with a common denominator.
- Multiples of 3: 3, 6, … 2 divides 6, so use 6.
- $\frac{1}{2} = \frac{1 \times 3}{2 \times 3} = \frac{3}{6}$.
- $\frac{1}{3} = \frac{1 \times 2}{3 \times 2} = \frac{2}{6}$.
Answer: $\frac{3}{6}$ and $\frac{2}{6}$
Worked example: Use the least common multiple
Rename $\frac{5}{6}$ and $\frac{3}{8}$ with the least common denominator.
- Multiples of 8: 8, 16, 24. The first one 6 divides is 24.
- $\frac{5}{6} = \frac{5 \times 4}{6 \times 4} = \frac{20}{24}$.
- $\frac{3}{8} = \frac{3 \times 3}{8 \times 3} = \frac{9}{24}$.
Answer: $\frac{20}{24}$ and $\frac{9}{24}$
Worked example: One denominator already works
Rename $\frac{3}{4}$ and $\frac{7}{12}$.
- 4 divides 12, so 12 works.
- $\frac{3}{4} = \frac{9}{12}$; $\frac{7}{12}$ stays as it is.
Answer: $\frac{9}{12}$ and $\frac{7}{12}$
Common mistake
Changing only the denominator, like writing $\frac{1}{2} = \frac{1}{6}$. That makes the pieces smaller without making more of them, so the amount shrinks. Whatever you multiply the bottom by, multiply the top by the same number.