Multiple choice

A boat goes 30 km upstream and 44 km downstream in 10 hours. In 13 hours, it can go 40 km upstream and 55 km downstream. If x represents the speed of the boat in still water in km/hr and y represents the speed of the stream in km/hr, then _________.

  1. $x + y = 11, x-y= 5$
  2. $x + y = 5, x - y = 11$
  3. $x + y = 6, x - y = 10$
  4. $x + y = 10, x - y = 6$
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A Correct answer
Explanation

Let u = 1/(x-y) and v = 1/(x+y). 30u + 44v = 10 and 40u + 55v = 13. Solving the system: 120u + 176v = 40 and 120u + 165v = 39. Subtracting gives 11v = 1, so v = 1/11. Thus x+y = 11. Substituting v into the first eq: 30u + 4 = 10, 30u = 6, u = 1/5. Thus x-y = 5.

AI explanation

Let x be the speed of the boat in still water and y be the speed of the stream, so the downstream speed is x + y and the upstream speed is x - y. From the given information, we set up two equations: 30 divided by the quantity x minus y plus 44 divided by the quantity x plus y equals 10, and 40 divided by the quantity x minus y plus 55 divided by the quantity x plus y equals 13. Solving this system of equations by letting x - y = a and x + y = b yields a = 5 and b = 11, meaning the correct relationships are x + y = 11 and x - y = 5.

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