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The problem involves a faulty balance where the weights of the pans are in the ratio \(5:4\). When a cube is placed on one pan, it shows a weight of 15 kg, and when placed on the other pan, it shows a weight of 12 kg. The goal is to find the actual weight of the cube.

Let the actual weight of the cube be \( W \) kg.

### Step 1: Understanding the pan's imbalance
Since the balance is faulty and the ratio of the weights of the pans is \(5:4\), we need to account for this discrepancy when calculating the cube's true weight.

- If the cube is on the pan with the heavier weight (ratio 5), it will appear lighter, i.e., it shows 12 kg.
- If the cube is on the pan with the lighter weight (ratio 4), it will appear heavier, i.e., it shows 15 kg.

### Step 2: Setting up the equation
- Let the balance ratio be \( \frac{5}{4} \).
- The actual weight \( W \) is related to the shown weight on both sides of the balance.

From the given information:

1. When the cube shows a weight of 15 kg, it is being multiplied by the factor \( \frac{5}{4} \), i.e.,
   \[
   W \times \frac{5}{4} = 15
   \]

2. When the cube shows a weight of 12 kg, it is being multiplied by the factor \( \frac{4}{5} \), i.e.,
   \[
   W \times \frac{4}{5} = 12
   \]

### Step 3: Solving the system of equations

From the first equation:
\[
W = 15 \times \frac{4}{5} = 12 \, \text{kg}
\]

From the second equation:
\[
W = 12 \times \frac{5}{4} = 15 \, \text{kg}
\]

By averaging the two values:
\[
W = \frac{15 + 12}{2} = 13.5 \, \text{kg}
\]

Thus, the actual weight of the cube is \( \boxed{13.5} \, \text{kg} \).

### Answer: (C) 13.5 kg

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Would you like more details on any part of the solution or have any further questions? Here are some related questions to extend your understanding:

1. How would the answer change if the ratio of the pan weights were different, say \(6:5\)?
2. How do balance scales work, and how can inaccuracies arise from faulty pan ratios?
3. Can this same method be applied to other objects with different weights? How?
4. How can one calculate the percentage error in the readings given by the faulty balance?
5. How does the principle of a fulcrum play into the design of a traditional balance scale?

**Tip:** When dealing with faulty instruments, analyzing the ratio or discrepancy can often help recover the actual values from the erroneous ones.

A faulty balance has pans whose weights are in the ratio 5:4. A cube weighs 15 kg when kept on one pan but weighs 12 kg when kept on the other pan. Find the actual weight of the cube.

Learn how to find the actual weight of a cube using a faulty balance with pans in the ratio 5:4. The cube shows different weights of 15 kg and 12 kg on each pan, and this solution covers key concepts of algebra and ratios. Suitable for students in Grades 8-10.

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