There are five boxes in a cargo hold. The weight of the first box is $200$ kg and the weight of the second box is $20\%$ more than the weight of the third box, whose weight is $25\%$ more than the first box's weight. The fourth box at $350$ kg is $30\%$ lighter than the fifth box. The difference in the average weight of the four heaviest boxes and the four lightest boxes is

### Concept & Strategy This problem requires a sequential calculation of weights based on percentage relationships. Once all weights are found, the shortcut to find the difference between the averages of the highest four and lowest four of a five-item set relies on recognizing overlapping values. ### Step-by-Step Solution * **Given:** Box 1 ($B_1$) = $200$ kg $B_2$ = $1.20 \times B_3$ $B_3$ = $1.25 \times B_1$ $B_4$ = $350$ kg $B_4$ = $(1 - 0.30) \times B_5 = 0.70 \times B_5$ * **Calculation:** Step 1: Calculate $B_3$. $B_3 = 1.25 \times 200 = 250$ kg Step 2: Calculate $B_2$. $B_2 = 1.20 \times 250 = 300$ kg Step 3: Calculate $B_5$. $350 = 0.70 \times B_5$ $B_5 = \frac{350}{0.70} = 500$ kg Step 4: List all weights in ascending order. $200, 250, 300, 350, 500$ Step 5: Calculate averages. Four heaviest ($250, 300, 350, 500$): Sum = $1400$. Average = $\frac{1400}{4} = 350$ kg. Four lightest ($200, 250, 300, 350$): Sum = $1100$. Average = $\frac{1100}{4} = 275$ kg. Step 6: Find the difference. $\text{Difference} = 350 - 275 = 75$ kg. ### Exam Strategy & Shortcut **The Overlap Cancellation Trick:** When you have $5$ ordered items ($x_1, x_2, x_3, x_4, x_5$), the difference between the average of the top $4$ and the bottom $4$ is simply the difference of the extreme values divided by $4$. The middle three items ($x_2, x_3, x_4$) overlap and perfectly cancel out when you subtract the two sums! $\text{Difference} = \frac{x_5 - x_1}{4}$ $\text{Difference} = \frac{500 - 200}{4} = \frac{300}{4} = 75$ kg. This bypasses the need to sum the heavy and light groups entirely! ### Common Pitfall Misinterpreting "$30\%$ lighter than the fifth box." Students often calculate $30\%$ of the fourth box ($350$) and add it back, doing $350 \times 1.3 = 455$ kg for the fifth box. This is mathematically incorrect. The percentage is based on the *fifth* box's weight, so it must be modeled as $0.70 \times B_5 = 350$. ### Final Answer **Therefore, the correct answer is 75 kg.**