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Math Help - Emptying a tank

  1. #1
    Member Jonboy's Avatar
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    Emptying a tank

    Hello everyone! Boy these problems are so interesting, but I'm struggling to get the equation setup correct. Either way I find these problems so tasty.

    A water tank can be emptied by using one pump for 5 hours. A second, smaller pump can empty the tank in 8 hours. If the larger pump is started at 1:00 p.m. at what time should the smaller pump be started so that the tank will be emptied at 5:00 p.m. ?

    So 5 hours passes.

    In one hour the bigger pump empties 1/5 of the tank

    In one hour the smaller pump empties 1/8 of the tank

    Let x be the amount of hours.

    So I thought: \frac{1}{5}x\,+\,\frac{1}{8}(5\,-\,x)\,=\,\frac{1}{5}

    But that make no sense. Can someone show me how to get the equation?
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  2. #2
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    The larger pump empties \frac{4}{5} of the tank in those 4 hours so... The smaller tank must empty the remaining \frac{1}{5} of the tank.

    So how long does it take to remove that much of the tank with the smaller pump? It is \frac{8}{5} hours because. At a rate of \frac{1}{8} per hour, the equation is:

    \frac{1}{8} x = \frac{1}{5}

    So that's 1.6 hours and we find we must start the smaller pump at \boxed{\text{3:24 pm}}
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  3. #3
    Member Jonboy's Avatar
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    Question

    How do you know the tanks empties 4/5 in 4 hours? I'm totally lost.
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  4. #4
    Eater of Worlds
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    Jonboy:

    Here's another way to look at it.

    Pump A can empty 1/5 of the tank in 1 hour, because it can empty the whole thing in 5 hours. See?.

    Therefore, it starts at 1:00 pm and empties \frac{1}{5}x of the tank by x hours past 1:00 pm.

    After the other pump kicks in x hours after 1:00 pm, the both of them pump

    \frac{1}{5}+\frac{1}{8}=\frac{13}{40} of the tank per hour.

    The remaining time is 4-x hours.

    So, we have \frac{1}{5}x+\frac{13}{40}(4-x)=1

    The 1 on the right side is the entire tank.

    Solve for x and we get:

    x=\frac{12}{5} hours past 1:00 pm.

    That's 144 minutes past 1:00 pm or 3:24 pm

    Just as Aradesh said. Except, he/she subtracted 96 minutes from 5:00 pm. Same thing.

    Did that explain it a little better?.
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  5. #5
    Grand Panjandrum
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    Quote Originally Posted by Jonboy View Post
    Hello everyone! Boy these problems are so interesting, but I'm struggling to get the equation setup correct. Either way I find these problems so tasty.

    A water tank can be emptied by using one pump for 5 hours. A second, smaller pump can empty the tank in 8 hours. If the larger pump is started at 1:00 p.m. at what time should the smaller pump be started so that the tank will be emptied at 5:00 p.m. ?

    So 5 hours passes.

    In one hour the bigger pump empties 1/5 of the tank

    In one hour the smaller pump empties 1/8 of the tank

    Let x be the amount of hours.

    So I thought: \frac{1}{5}x\,+\,\frac{1}{8}(5\,-\,x)\,=\,\frac{1}{5}

    But that make no sense. Can someone show me how to get the equation?

    The larger pump empties 1/5 of a tank per hour, the smaller 1/8 of a tank per hour.

    The larger pump runs for 4 hours so empties 4/5 of the tank, leaving 1/5
    to be emptied by the smaller pump. So the smaller pump will take (1/5)(8)
    hours to do its share, or 1.6 hours or 1 hour and 36 minutes, so it will have
    to start pumping at 3:24.

    RonL
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  6. #6
    Member Jonboy's Avatar
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    Quote Originally Posted by galactus View Post

    Did that explain it a little better?.
    Yes thanks
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