Quick Fraction Comparisons
- Trick / Shortcut
- Quick Fraction Comparisons
- Topic
- Fractions
- How It Works
To compare two fractions $\frac{a}{b}$ and $\frac{c}{d}$ without finding a common denominator, use cross-multiplication:
- Multiply the numerator of the first fraction by the denominator of the second ($a \times d$).
- Multiply the numerator of the second fraction by the denominator of the first ($b \times c$).
- Compare the two products:
- If $a \times d > b \times c \rightarrow \frac{a}{b} > \frac{c}{d}$
- If $a \times d < b \times c \rightarrow \frac{a}{b} < \frac{c}{d}$
- If $a \times d = b \times c \rightarrow \frac{a}{b} = \frac{c}{d}$
General Tip: For proper fractions, the fraction whose numerator and denominator have a smaller difference (i.e., closer to 1) is often larger, especially when comparing fractions with similar numerical ranges. This is a quick mental check, not a formal method.
- Example
Compare $\frac{5}{7}$ and $\frac{7}{10}$:
- Cross-multiply $5 \times 10 = 50$
- Cross-multiply $7 \times 7 = 49$
- Compare: $50 > 49$. Therefore, $\textbf{\frac{5}{7} > \frac{7}{10}}$
Compare $\frac{3}{8}$ and $\frac{2}{5}$:
- Cross-multiply $3 \times 5 = 15$
- Cross-multiply $8 \times 2 = 16$
- Compare: $15 < 16$. Therefore, $\textbf{\frac{3}{8} < \frac{2}{5}}$