Step 1: The Pearl index expresses how often a contraceptive method fails, scaled to 100 women using it for one year (100 woman-years).
Step 2: It is computed as $\dfrac{\text{pregnancies} \times 1200}{\text{woman-months of use}}$, the 1200 coming from 100 women followed for 12 months.
Step 3: With 10 pregnancies over 100 women followed for 24 months, the total exposure is $100 \times 24 = 2400$ woman-months, so \[\frac{10 \times 1200}{2400} = 5.\]
Step 4: The failure rate therefore works out to 5 pregnancies per 100 woman-years.
\[\boxed{5}\]