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Find the original fraction: A fraction becomes 7/8 when 5 is added to both numerator and denominator. If 3 is added to both numerator and denominator, it becomes 6/7. What is the original fraction?

Difficulty: Medium

Correct Answer: 9/11

Explanation:


Introduction / Context:
This problem converts two transformation statements about a fraction into linear equations in its numerator and denominator. Solving the resulting system yields the original fraction exactly, with a quick verification step to ensure both conditions hold.


Given Data / Assumptions:

  • Let the fraction be x/y with y ≠ 0.
  • (x + 5)/(y + 5) = 7/8.
  • (x + 3)/(y + 3) = 6/7.


Concept / Approach:
Cross-multiply both relations to produce two linear equations in x and y. Solve them simultaneously (elimination works well). Reduce the final fraction to simplest terms if necessary.


Step-by-Step Solution:

8(x + 5) = 7(y + 5) ⇒ 8x + 40 = 7y + 35 ⇒ 8x − 7y = −5 … (1) 7(x + 3) = 6(y + 3) ⇒ 7x + 21 = 6y + 18 ⇒ 7x − 6y = −3 … (2) Multiply (2) by 8: 56x − 48y = −24 Multiply (1) by 7: 56x − 49y = −35 Subtract: (−49y) − (−48y) = −35 − (−24) ⇒ −y = −11 ⇒ y = 11 Then from (2): 7x − 6*11 = −3 ⇒ 7x − 66 = −3 ⇒ 7x = 63 ⇒ x = 9 Original fraction = 9/11


Verification / Alternative check:
(9 + 5)/(11 + 5) = 14/16 = 7/8 and (9 + 3)/(11 + 3) = 12/14 = 6/7. Both conditions are satisfied exactly.


Why Other Options Are Wrong:
8/11 and 10/11 fail at least one of the two transformation checks; “Cannot be determined” is incorrect because the system has a unique solution.


Common Pitfalls:
Arithmetic slips when cross-multiplying or subtracting equations can lead to near misses; always verify both given transformations on your final fraction.


Final Answer:
9/11

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