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:
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 = 9/11Verification / 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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