The modal class for the weight of 150 pupils is 40-50 kg. The frequencies of the other classes are 30, 12, 16, 21. Find the least possible value of P.

The modal class is the class interval with the highest frequency.

The problem states that the modal class is $40-50 \text{ kg}$. From the table, the frequency for this class is $P$.

For $40-50 \text{ kg}$ to be the modal class, its frequency $P$ must be greater than the frequencies of all other classes.
The frequencies of the other classes are:

• $20-35 \text{ kg}$: 30 pupils
• $55-60 \text{ kg}$: 12 pupils
• $65-70 \text{ kg}$: 16 pupils
• $75-85 \text{ kg}$: 21 pupils

So, $P$ must satisfy the following conditions:

To satisfy all these conditions, $P$ must be greater than the largest of the other frequencies. The largest frequency among $30, 12, 16,$ and 21 is 30.
Therefore, $P > 30$.
Since $P$ represents the number of pupils, it must be an integer. The least integer value greater than 30 is 31.

The least possible value of $P$ is: