You can already compare fractions with the same numerator or the same denominator. For other pairs, a quick trick is to compare each one with a benchmark: $\dfrac{1}{2}$ or 1.
Key idea
A fraction is more than $\dfrac{1}{2}$ when its numerator is more than half its denominator, so when double the numerator is more than the denominator. It is more than 1 when the numerator is more than the denominator. If one fraction is below a benchmark and the other is above, you know which is bigger.
Why it works
Half of 8 eighths is 4 eighths. $\dfrac{5}{8}$ has 5 eighths, more than half, while $\dfrac{2}{5}$ has 2 fifths, less than half of 5 fifths. Since one is above the middle and one is below, $\dfrac{5}{8} > \dfrac{2}{5}$, and you never needed a common denominator.
Worked example: Compare with one half
Is $\dfrac{3}{7}$ more or less than $\dfrac{1}{2}$?
- Half of 7 is 3 and a half.
- 3 is less than 3 and a half (double check: $2 \times 3 = 6 < 7$).
Answer: Less than $\dfrac{1}{2}$
Worked example: Use the benchmark to compare
Compare $\dfrac{5}{8}$ and $\dfrac{2}{5}$.
- $\dfrac{5}{8}$: $2 \times 5 = 10 > 8$, so it is more than $\dfrac{1}{2}$.
- $\dfrac{2}{5}$: $2 \times 2 = 4 < 5$, so it is less than $\dfrac{1}{2}$.
Answer: $\dfrac{5}{8} > \dfrac{2}{5}$
Benchmarks only settle it when the two fractions land on different sides. If both are more than $\dfrac{1}{2}$, use another method, like a common denominator.