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% higher than the weight of the third box, whose weight is 25% higher than the first box's weight. The fourth box at 350 kg is 30% lighter than the fifth box. Find the difference in the average weight of the four heaviest boxes and the four lightest boxes.

Options

A. 51.5 kg
B. 75 kg
C. 37.5 kg
D. 112.5 kg

Answer

To solve the problem step by step, we will first determine the weights of all five boxes based on the information provided.

### Step 1: Find the weight of the first box.
- The weight of the first box is given as **200 kg**.

### Step 2: Find the weight of the third box.
- The weight of the third box is **25% higher** than the weight of the first box.
- Calculation:
\[ \text{Weight of the third box} = 200 \text{ kg} + (25\% \text{ of } 200 \text{ kg}) = 200 + 50 = 250 \text{ kg} \]

### Step 3: Find the weight of the second box.
- The weight of the second box is **20% higher** than the weight of the third box.
- Calculation:
\[ \text{Weight of the second box} = 250 \text{ kg} + (20\% \text{ of } 250 \text{ kg}) = 250 + 50 = 300 \text{ kg} \]

### Step 4: Find the weight of the fifth box.
- The weight of the fourth box is given as **350 kg**, and it is **30% lighter** than the fifth box.
- Let the weight of the fifth box be \( x \).
- Thus, we can set up the equation:
\[ 350 \text{ kg} = x - (30\% \text{ of } x) = x - 0.3x = 0.7x \]
- Solving for \( x \):
\[ x = \frac{350}{0.7} = 500 \text{ kg} \]

### Step 5: Summarize the weights of all boxes.
- Box 1: 200 kg
- Box 2: 300 kg
- Box 3: 250 kg
- Box 4: 350 kg
- Box 5: 500 kg

### Step 6: Arrange the weights in ascending order.
- The weights in ascending order are: **200 kg, 250 kg, 300 kg, 350 kg, 500 kg**.

### Step 7: Find the average weight of the four heaviest boxes.
- The four heaviest boxes are: **250 kg, 300 kg, 350 kg, 500 kg**.
- Calculation of average:
\[ \text{Average of four heaviest} = \frac{250 + 300 + 350 + 500}{4} = \frac{1400}{4} = 350 \text{ kg} \]

### Step 8: Find the average weight of the four lightest boxes.
- The four lightest boxes are: **200 kg, 250 kg, 300 kg, 350 kg**.
- Calculation of average:
\[ \text{Average of four lightest} = \frac{200 + 250 + 300 + 350}{4} = \frac{1100}{4} = 275 \text{ kg} \]

### Step 9: Find the difference between the two averages.
- Calculation of difference:
\[ \text{Difference} = 350 \text{ kg} - 275 \text{ kg} = 75 \text{ kg} \]

### Final Answer:
The difference in the average weight of the four heaviest boxes and the four lightest boxes is **75 kg**.

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