Proportional Resource Sharing
Three players start a Minecraft world and agree to share all resources equally. Player 1 collects 30 apples; Player 2 collects 18 apples; Player 3 collects no apples but instead provides 48 gold nuggets. The 48 gold nuggets are to be split between Player 1 and Player 2 in proportion to how many apples each contributed to Player 3.
How many more gold nuggets does Player 1 receive than Player 2?
Hints
- First determine how many apples each player ends up with after equal sharing -- this tells you what each player contributed to Player 3.
- Total apples = 48; split equally means each player keeps 16. Player 1 gives 14 apples to Player 3; Player 2 gives 2.
- Split 48 gold nuggets in the ratio $14 : 2$ (or $7 : 1$). Player 1 gets $\frac{7}{8} \times 48 = 42$; Player 2 gets $\frac{1}{8} \times 48 = 6$.
Worked Solution
How to Think About It: The three players share apples equally, so figure out how many apples each player ends up with, and then compute how many apples each player gave to Player 3. That gives the contribution ratio, which determines how the 48 gold nuggets are split.
Quick Estimate: Total apples: $30 + 18 = 48$. Split equally among 3 players: 16 each. Player 1 has 30, keeps 16, gives 14 to Player 3. Player 2 has 18, keeps 16, gives 2 to Player 3. Ratio of contributions: 14 to 2, or 7 to 1. Player 1 gets $\frac{7}{8} \times 48 = 42$ nuggets; Player 2 gets $\frac{1}{8} \times 48 = 6$ nuggets. Difference: $42 - 6 = 36$.
Formal Solution:
Step 1: Find the equal share of apples. $$\text{Total apples} = 30 + 18 = 48 \implies \text{each player gets } \frac{48}{3} = 16 \text{ apples}$$
Step 2: Find each player's contribution to Player 3. $$\text{Player 1 gives: } 30 - 16 = 14 \text{ apples}$$ $$\text{Player 2 gives: } 18 - 16 = 2 \text{ apples}$$ $$\text{Total given to Player 3: } 14 + 2 = 16 \text{ apples}$$
Step 3: Split 48 gold nuggets proportionally. $$\text{Player 1 receives: } \frac{14}{16} \times 48 = \frac{7}{8} \times 48 = 42 \text{ nuggets}$$ $$\text{Player 2 receives: } \frac{2}{16} \times 48 = \frac{1}{8} \times 48 = 6 \text{ nuggets}$$
Step 4: Compute the difference. $$42 - 6 = 36$$
Answer: Player 1 receives $\boxed{36}$ more gold nuggets than Player 2.
Intuition
The key step that trips people up is identifying what "proportion of contribution" means. The gold nuggets compensate Players 1 and 2 for the apples they gave up for Player 3's benefit -- not for their total apple count. Player 1 gave 14 apples; Player 2 gave only 2. That 7-to-1 ratio drives the lopsided gold split.
This pattern -- proportional compensation for unequal contributions to a shared good -- appears in cooperative game theory (the Shapley value formalizes it) and in real-world settings like cost allocation among business units, IP royalties split by inventor contribution, and fee structures in fund manager arrangements. The instinct to split gold in proportion to total apples (30:18 ratio) rather than contributed apples (14:2 ratio) is the classic error here.