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Let the initial volume of liquid A and B be 5x and 7x, respectively. After 15 liters of liquid A is replaced by liquid B, the volume of liquid A becomes (5x - 15) and the volume of liquid B becomes (7x + 15). We are told the new ratio of A to B is 1: 2, so: (5x - 15) / (7x + 15) = 1/2. Cross-multiply: 2(5x - 15) = (7x + 15), 10x - 30 = 7x + 15, 10x - 7x = 30 + 15, 3x = 45, x = 15 Thus, the initial volume of liquid A = 5x = 5 × 15 = 75 liters.
The smallest fraction, which should be added to the sum of2(1/2), 3(1/3), 4(1/4) and 5(1/5) to make the result a whole number, is
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