Maths Olympiad Prep

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Algebra Difficulty 5.0 AIME Find the answer

Three, (Full marks 12 points) At the foot of the mountain is a pond, the scene: a steady flow (i.e., the same amount of water flows into the pond from the river per unit time) continuously flows into the pond. The pond contains a certain depth of water. If one Type A water pump is used, it will take exactly 1 hour to pump out all the water in the pond; if two Type A water pumps are used, it will take exactly 20 minutes to pump out all the water in the pond. If three Type A water pumps are used simultaneously, how long will it take to pump out all the water in the pond exactly?

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

Three, let the amount of water flowing into the pond from the spring every minute be xx m3^{3}, each water pump extracts yy m3^{3} of water per minute, and the pond originally contains zz m3^{3} of water. It takes tt minutes for three water pumps to drain the pond. According to the problem, we have
{60y−60x=z,20y+20y−20x=z,3ty−tx=z. \left\{\begin{array}{l} 60 y-60 x=z, \\ 20 y+20 y-20 x=z, \\ 3 t y-t x=z . \end{array}\right.

Solving this, we get t=12t=12.

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Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.