Given that 0 < a < b < c < d, which of the following is the largest?

To compare two ratios of positive numbers, p/q and r/s (with q, s > 0), cross-multiply: p/q is greater than r/s exactly when ps is greater than rq, because multiplying both sides by the positive product qs preserves the direction of the inequality.

Placing larger numbers in the numerator and smaller numbers in the denominator increases a ratio, since it simultaneously raises the top and lowers the bottom of the fraction.

1. The four expressions to compare are (c+d)/(a+b), (a+d)/(b+c), (b+c)/(a+d), and (b+d)/(a+c).

2. Since every denominator here (a+b, b+c, a+d, a+c) is positive, any two of these expressions can be compared by cross-multiplying: p/q is greater than r/s exactly when ps is greater than rq.

3. Compare (c+d)/(a+b) with (a+d)/(b+c): (c+d)(b+c) − (a+d)(a+b) = (c−a)(a+b+c+d), which is positive because c > a and a+b+c+d > 0. So (c+d)/(a+b) > (a+d)/(b+c).

4. Compare (c+d)/(a+b) with (b+c)/(a+d): (c+d)(a+d) − (b+c)(a+b) = (d−b)(a+b+c+d), which is positive because d > b. So (c+d)/(a+b) > (b+c)/(a+d).

5. Compare (c+d)/(a+b) with (b+d)/(a+c): (c+d)(a+c) − (b+d)(a+b) = (c−b)(a+b+c+d), which is positive because c > b. So (c+d)/(a+b) > (b+d)/(a+c).

6. Since (c+d)/(a+b) is greater than each of the other three expressions individually, it is the greatest of all four.

As an independent check, substitute a concrete set of values satisfying 0 < a < b < c < d, say a = 1, b = 2, c = 3, d = 4: (c+d)/(a+b) = 7/3 ≈ 2.33, (a+d)/(b+c) = 5/5 = 1, (b+c)/(a+d) = 5/5 = 1, and (b+d)/(a+c) = 6/4 = 1.5. This numeric example is consistent with the general proof above — (c+d)/(a+b) is the largest.

Therefore, (c+d)/(a+b) is the largest of the four expressions for every 0 < a < b < c < d.