Problem 15

Explanation

We will find the value of the replaced number (quantity $P$) first.

Analyzing Quantity P

Let the three numbers be $a, b,$ and $c$. The initial average is given as $24$:

Suppose the number $c$ is replaced by the number $14$. The new average decreases by $3$ from the initial average, so the new average becomes $24 - 3 = 21$.

We substitute the value $a + b = 49$ into the initial equation to find the value of $c$:

Thus, the replaced number is $23$. So, $P = 23$.

Comparing P and Q

Given $Q = 42$.

It is clear that $23 < 42$.

Thus, $P < Q$.