Q.Find $Q_1$, $Q_2$ and $Q_3$ for the continuous frequency distribution: classes $0\text{–}10, 10\text{–}20, 20\text{–}30, 30\text{–}40, 40\text{–}50$ with frequencies $5, 8, 15, 12, 10$.

Build the $cf$ column ($N = 50$), find each quartile class using $\frac{iN}{4}$, then interpolate.

The classes are already continuous. Build the less-than $cf$ column:

$N = 50$ (using $N$, not $N+1$, for grouped data).

$Q_1$: $\dfrac{N}{4} = 12.5$. First $cf \ge 12.5$ is $13$ → class $10\text{–}20$: $L = 10, cf = 5, f = 8, h = 10$.

$Q_2$: $\dfrac{2N}{4} = 25$. First $cf \ge 25$ is $28$ → class $20\text{–}30$: $L = 20, cf = 13, f = 15, h = 10$.

$Q_3$: $\dfrac{3N}{4} = 37.5$. First $cf \ge 37.5$ is $40$ → class $30\text{–}40$: $L = 30, cf = 28, f = 12, h = 10$. …

Each interpolation can be sanity-checked by its fractional part: for $Q_1$, the class $10\text{–}20$ must supply $12.5 - 5 = 7.5$ of its $8$ observations to reach the quarter mark, i.e. a fraction $\tfrac{7.5}{8} = 0.9375$ of the class width $10$ → $9.375$ …

Using $N + 1$ instead of $N$ for continuous data; taking $cf$ of the quartile class itself rather than the class before it; or forgetting to multiply …