In three coloured boxes (Red, Green and Blue) there are 108 balls in total. The combined number of balls in the green and red boxes is twice the number of balls in the blue box, and the number of balls in the blue box is twice the number of balls in the red box. How many balls are there in the green box?

Introduction / Context:
This arithmetic reasoning problem deals with ratios and relationships between quantities in three different boxes: red, green, and blue. The question provides information about how the numbers of balls in the boxes relate to each other, and it asks specifically for the count in the green box. These types of questions are common in aptitude exams and test your ability to convert verbal relationships into algebraic equations.

Given Data / Assumptions:

• Total number of balls in all three boxes = 108.
• Let the number of balls in the red box be R, in the green box be G, and in the blue box be B.
• There are twice as many balls in the green and red boxes combined as there are in the blue box: R + G = 2B.
• There are twice as many balls in the blue box as there are in the red box: B = 2R.
• We must find G (the number of balls in the green box).

Concept / Approach:
The key idea is to transform the relationships into equations and then solve them systematically. By substituting B = 2R into the other equations, we can express everything in terms of a single variable R. After finding R, we back-calculate B and G, and finally verify that their sum is 108. If the computed G does not match any of the numerical options given, then the correct choice is “None of these”.

Step-by-Step Solution:
Step 1: From the condition “twice as many in the blue box as in the red box”, we have B = 2R.Step 2: From “green and red combined are twice the blue”, we have R + G = 2B.Step 3: Substitute B = 2R into R + G = 2B, giving R + G = 2 * (2R) = 4R.Step 4: Simplify to get G = 4R - R = 3R.Step 5: Use the total balls condition: R + G + B = 108.Step 6: Substitute G = 3R and B = 2R into this equation: R + 3R + 2R = 108.Step 7: This simplifies to 6R = 108, so R = 18.Step 8: Then B = 2R = 36 and G = 3R = 54.

Verification / Alternative check:
Check the total: R + G + B = 18 + 54 + 36 = 108, which matches the given total. Check the conditions: blue is twice red (36 is twice 18), and green plus red equals 54 + 18 = 72, which is indeed twice blue (2 * 36 = 72). Thus, all relationships are satisfied, and the number of balls in the green box is 54. Since 54 is not listed among the numerical options given (18, 36, 45), we must select “None of these” as the correct answer.

Why Other Options Are Wrong:
Option A (18): If G were 18, you would not be able to satisfy all the relationships while keeping the total at 108.
Option B (36): If G were 36, the equations would not hold consistently when you try to satisfy the given doubling relationships.
Option C (45): Similarly, G = 45 does not lead to a consistent solution with the specified ratios and the total of 108 balls.

Common Pitfalls:
Some students incorrectly assume that one of the simple numeric options must be correct and do not consider “None of these” seriously. Others may make algebraic mistakes when substituting B = 2R into the equations. It is important to follow each substitution carefully and verify the final numbers against all given conditions, not just one of them.

Final Answer:
The number of balls in the green box is 54, which is not among the numeric options. Therefore, the correct option is None of these.