In three coloured boxes, Red, Green and Blue, a total of 108 balls are placed. There are twice as many balls in the green and red boxes combined as there are in the blue box, and there are twice as many balls in the blue box as there are in the red box. Based on these relationships, how many balls are there in the green box?

Introduction / Context:
This quantitative reasoning question involves three coloured boxes, each containing a certain number of balls. The question gives relationships between the numbers of balls in the red, green and blue boxes and the overall total. It tests the ability to translate verbal conditions into algebraic equations and then solve them systematically to find the exact count in one particular box, namely the green box.

Given Data / Assumptions:

• There are three boxes: Red, Green and Blue.
• The total number of balls in all three boxes together is 108.
• The combined number of balls in the green and red boxes is twice the number of balls in the blue box.
• The number of balls in the blue box is twice the number of balls in the red box.
• All ball counts are non negative integers.

Concept / Approach:
The most efficient approach is to introduce algebraic variables for the number of balls in each box and then convert the relationships into linear equations. This leads to a system of simultaneous equations that can be solved step by step. Once the value for one box is known, substitution allows us to compute the values for the other boxes and verify that their sum matches the total of 108 balls.

Step-by-Step Solution:
Step 1: Let R be the number of balls in the red box, G be the number of balls in the green box and B be the number of balls in the blue box.Step 2: The statement "twice as many balls in the green and red boxes combined as in the blue box" gives the equation R + G = 2B.Step 3: The statement "twice as many in the blue box as in the red box" gives the equation B = 2R.Step 4: The total number of balls is 108, so R + G + B = 108.Step 5: Substitute B = 2R into the equation R + G = 2B to get R + G = 2 * 2R = 4R, so G = 4R - R = 3R.Step 6: Now substitute B = 2R and G = 3R into the total equation: R + 3R + 2R = 108.Step 7: Simplify: 6R = 108, so R = 108 / 6 = 18.Step 8: From G = 3R, we get G = 3 * 18 = 54. So the green box contains 54 balls.Step 9: As a check, B = 2R = 36, and indeed 18 + 54 + 36 = 108.

Verification / Alternative check:
We can verify the relationships directly with the calculated values. The blue box has 36 balls, and the red plus green boxes together have 18 + 54 = 72 balls. This is exactly twice 36, so the first condition holds. The blue box has 36 balls, which is twice the 18 balls in the red box, so the second condition holds. The sum 18 + 54 + 36 is 108, so the total is also correct. This confirms that the solution is consistent.

Why Other Options Are Wrong:
If the green box had 76 balls, the total would exceed 108 when the other constraints are applied. If it had 64 balls or 48 balls, the relationships R + G = 2B and B = 2R cannot be satisfied simultaneously with an integer value for R that also makes the total equal 108. Only 54 balls in the green box satisfy all given conditions.

Common Pitfalls:
One common mistake is to misinterpret "twice as many in the green and red boxes combined as in the blue box" and instead write G = 2B or R = 2B. Another pitfall is to forget to include the total sum equation, which is essential for determining the exact values rather than just ratios. Care must also be taken not to mix up which box is twice which. Working systematically with variables and equations avoids these errors.

Final Answer:
The number of balls in the green box is 54.