Why must you find a common denominator before subtracting $6/4$ and $2/5$?

Conditions

• The fractions $\frac{6}{4}$ and $\frac{2}{5}$ have unlike denominators (4 and 5).
• The operation being performed is subtraction.

Reasoning, step by step

1. Observe that the denominators of $\frac{6}{4}$ and $\frac{2}{5}$ are 4 and 5, which are different.
2. Recognize that the subtraction rule for fractions requires a shared denominator so that numerators can be directly subtracted.
3. Find the least common multiple of 4 and 5, which is 20.
4. Rewrite $\frac{6}{4}$ as $\frac{30}{20}$ by multiplying numerator and denominator by 5.
5. Rewrite $\frac{2}{5}$ as $\frac{8}{20}$ by multiplying numerator and denominator by 4.
6. Now that both fractions have denominator 20, subtract the numerators: $\frac{30}{20} - \frac{8}{20} = \frac{22}{20}$.
7. Reduce $\frac{22}{20}$ to $\frac{11}{10}$.

Example

The instructor explicitly stops before subtracting because $\frac{6}{4}$ and $\frac{2}{5}$ do not share a denominator. He states, 'we have to do a common denominator here to subtract these two,' and then converts both to denominator 20 before performing the subtraction.

Common misconceptions

• Attempting to subtract the numerators and denominators separately (e.g., $\frac{6-2}{4-5}$), which is mathematically invalid.
• Believing that fractions with different denominators can be subtracted directly without conversion.

Watch the explanation

Explore next

• Why does multiplying by $5/5$ or $4/4$ keep a fraction equivalent?
• How does $22/20$ simplify to $11/10$ in this check?
• How do you verify that an ordered pair solves a linear equation with fractions?