Given that 0 < a < b < c < d, which of the following the largest ?
2025
Given that 0 < a < b < c < d, which of the following the largest ?
- A.
(c+d) / (a+b)
- B.
(a+d) / (b+c)
- C.
(b+c) / (a+d)
- D.
(b+d) / (a+c)
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Correct answer: A
Answer: (c+d)/(a+b) is the largest.
Reasoning (proof). For 0 < a < b < c < d compare (c+d)/(a+b) with each other ratio:
Compare with (b+d)/(a+c): (c+d)(a+c) - (b+d)(a+b) = (c-b)(a+b+c+d) > 0, so (c+d)/(a+b) > (b+d)/(a+c).
Compare with (b+c)/(a+d): (c+d)(a+d) - (b+c)(a+b) = (d-b)(a+b+c+d) > 0, so (c+d)/(a+b) > (b+c)/(a+d).
Compare with (a+d)/(b+c): (c+d)(b+c) - (a+d)(a+b) = (c-a)(a+b+c+d) > 0, so (c+d)/(a+b) > (a+d)/(b+c).
Since (c+d)/(a+b) is greater than each of the other three expressions, it is the largest.