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In aptitude simplification, solve the linear equation carefully: (2x/3) - (5*(4x/5 - 4/3))/2 = 1/3 What is the exact value of x? (Keep the answer in simplest fractional form.)

Difficulty: Medium

Correct Answer: 9/4

Explanation:


Introduction / Context:
This question tests algebraic simplification with fractions and brackets. The key skill is to simplify the bracketed term first, then solve the resulting linear equation without making sign mistakes.


Given Data / Assumptions:

  • Equation: (2x/3) - (5*(4x/5 - 4/3))/2 = 1/3
  • x is a real number.


Concept / Approach:
Simplify inside parentheses first, then simplify the multiplication and division, and finally collect x-terms on one side and constants on the other side.


Step-by-Step Solution:

Start with the inner bracket: (4x/5 - 4/3). Multiply by 5: 5*(4x/5 - 4/3) = 4x - 20/3. Divide by 2: (5*(4x/5 - 4/3))/2 = (4x - 20/3)/2 = 2x - 10/3. Substitute back: (2x/3) - (2x - 10/3) = 1/3. Distribute the minus: (2x/3) - 2x + 10/3 = 1/3. Write -2x as -6x/3: (2x/3) - (6x/3) = -4x/3. So -4x/3 + 10/3 = 1/3. Move constants: -4x/3 = 1/3 - 10/3 = -9/3 = -3. Multiply by 3: -4x = -9, so x = 9/4.


Verification / Alternative check:
Put x = 9/4 into the simplified form (2x/3) - (2x - 10/3). It becomes 3/2 - (9/2 - 10/3) = 3/2 - (7/6) = 1/3, which matches RHS.


Why Other Options Are Wrong:

-9/4 and -4/9 come from sign errors when distributing the minus. 4/9 is the reciprocal mistake after solving -4x = -9. 3/2 can appear if the bracket term is not simplified correctly.


Common Pitfalls:
Forgetting that the entire fraction term is subtracted, not just the first part; and mixing up 10/3 with 20/3 after division by 2.


Final Answer:
x = 9/4

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