Ajay and Vijay each have some marbles. Ajay says to Vijay, "If you give me x marbles, then both of us will have an equal number of marbles." Vijay replies, "If you give me twice as many marbles, then I will have 30 more marbles than you will have." What is the value of x?

Introduction / Context:
This arithmetic reasoning question involves setting up and solving simple linear equations based on a word problem with marbles. Ajay and Vijay exchange marbles under two different hypothetical conditions, and both conditions must be satisfied for the same unknown x. By translating each statement carefully into an algebraic equation, you can solve for x and determine the number of marbles being discussed in each scenario.

Given Data / Assumptions:

• Ajay initially has A marbles.
• Vijay initially has V marbles.
• If Vijay gives x marbles to Ajay, they both end up with the same number of marbles.
• If Ajay gives twice as many marbles, that is 2x marbles, to Vijay, then Vijay ends up with 30 more marbles than Ajay.
• The variable x is a positive integer and one of the given options.

Concept / Approach:
Two conditions are described, so we set up two equations involving A, V, and x. From the first condition we express the equality of marbles after Vijay gives x to Ajay. From the second condition we express that Vijay has 30 more marbles than Ajay after Ajay gives 2x to Vijay. Since both statements describe the same starting amounts A and V, the expressions for V - A derived from both situations must be equal. This leads to a single equation in x, which we solve using basic algebra.

Step-by-Step Solution:
Step 1: From the first statement, if Vijay gives x marbles to Ajay, Ajay has A + x and Vijay has V - x. These are equal, so A + x = V - x. Step 2: Rearranging A + x = V - x gives V - A = 2x. Step 3: From the second statement, if Ajay gives 2x marbles to Vijay, Ajay has A - 2x and Vijay has V + 2x. Vijay then has 30 more marbles than Ajay, so V + 2x = (A - 2x) + 30. Step 4: Simplify V + 2x = A - 2x + 30 to get V - A = -4x + 30. Step 5: Both expressions for V - A must be equal: 2x = -4x + 30. So 2x + 4x = 30 gives 6x = 30, hence x = 30 / 6 = 5.

Verification / Alternative check:
If x = 5, then V - A = 2x = 10. So Vijay always has 10 more marbles than Ajay initially. Check the second scenario: V + 2x - (A - 2x) = (V - A) + 4x = 10 + 4*5 = 10 + 20 = 30. This matches the condition that Vijay has 30 more marbles after Ajay gives him 2x marbles, confirming that x = 5 is consistent.

Why Other Options Are Wrong:
If x were 4, then V - A would be 8 from the first condition and 30 - 4*4 = 14 from the second, which is inconsistent. Similar contradictions appear for x = 8 or x = 10. Each of these values fails to satisfy both equations simultaneously, so they cannot be correct.

Common Pitfalls:
A common mistake is to misinterpret the second statement and mix up who is giving marbles to whom. Another error is to equate A and V incorrectly instead of equating the expressions for V - A. Carefully tracking who gains and who loses marbles in each scenario avoids these algebraic mistakes.

Final Answer:
The correct value of x is 5.