Three pumpkins are weighed two at a time in all possible ways. The weights of the pairs of pumpkins are $12 \mathrm{~kg}, 13 \mathrm{~kg}$ and $15 \mathrm{~kg}$. How much does the lightest pumpkin weigh?
(A) $4 \mathrm{~kg}$
(B) $5 \mathrm{~kg}$
(C) $6 \mathrm{~kg}$
(D) $7 \mathrm{~kg}$
(E) $8 \mathrm{~kg}$
Solution 1

In kilograms, let the weights of the 3 pumpkins in increasing order be $A, B$ and $C$.

The lightest combined weight, $12 \mathrm{~kg}$, must come from weighing the two lightest pumpkins.

That is, $A+B=12$.

The heaviest combined weight, $15 \mathrm{~kg}$, must come from weighing the two heaviest pumpkins.

That is, $B+C=15$.

Then the third given weight, $13 \mathrm{~kg}$, is the combined weight of the lightest and heaviest pumpkins.

That is, $A+C=13$.

Systematically, we may try each of the 5 possible answers.

If the lightest pumpkin weighs $4 \mathrm{~kg}$ (answer (A)), then $A+B=12$ gives $4+B=12$, or $B=8$. If $B=8$, then $8+C=15$ or $C=7$.

Since $C$ represents the weight of the heaviest pumpkin, $C$ cannot be less than $B$ and therefore the lightest pumpkin cannot weigh $4 \mathrm{~kg}$.

If the lightest pumpkin weighs $5 \mathrm{~kg}$ (answer (B)), then $A+B=12$ gives $5+B=12$, or $B=7$.

If $B=7$, then $7+C=15$ or $C=8$.

If $A=5$ and $C=8$, then the third and final equation $A+C=13$ is also true.

Therefore, the weight of the smallest pumpkin must be $5 \mathrm{~kg}$.

## Solution 2

In kilograms, let the weights of the 3 pumpkins in increasing order be $A, B$ and $C$.

The lightest combined weight, $12 \mathrm{~kg}$, must come from weighing the two lightest pumpkins.

That is, $A+B=12$.

The heaviest combined weight, $15 \mathrm{~kg}$, must come from weighing the two heaviest pumpkins.

That is, $B+C=15$.

Then the third given weight, $13 \mathrm{~kg}$, is the combined weight of the lightest and heaviest pumpkins.

That is, $A+C=13$.

Since $A+B=12$ and $A+C=13$, then $C$ is one more than $B$.

Since $B+C=15$ and $C$ is one more than $B$, then $B=7$ and $C=8$.

Since $A+B=12$, then $A+7=12$ or $A=5$.

When $A=5, B=7$ and $C=8$, the weights of the pairs of pumpkins are $12 \mathrm{~kg}, 13 \mathrm{~kg}$ and $15 \mathrm{~kg}$ as was given.

Therefore, the weight of the lightest pumpkin is $5 \mathrm{~kg}$.

ANSWER: (B)