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Math Help - Advanced Math Word Problem

  1. #1
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    Advanced Math Word Problem

    I've never had so much difficulty with a word problem, if someone could help I'd appreciate it. The word problem is below:







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    In the sketch above two Chinese junks are on the first leg of a race on a triangular course from buoy A to B to C, then back to A again.
    Three landlubbers on the winning junk tried to keep a record of the boat’s speed, but all three became violently seasick and their records suffered accordingly. Jian Pin observed that the junk sailed the first three-quarters of the race in three and a half hours. Hu Wei noted only that it did the final three-quarters in four and a half hours. Yao Li was so anxious to get back on land that the best he could do was observe that the middle leg of the race (from buoy B to C) took ten minutes longer than the first leg.
    Assuming that the buoys mark an equilateral triangle and that the junk had a constant speed on each leg, can you tell how long it took the junk to finish the race?
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    Re: Advanced Math Word Problem

    Quote Originally Posted by Kanika1989 View Post
    I've never had so much difficulty with a word problem, if someone could help I'd appreciate it. The word problem is below:







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    In the sketch above two Chinese junks are on the first leg of a race on a triangular course from buoy A to B to C, then back to A again.
    Three landlubbers on the winning junk tried to keep a record of the boat’s speed, but all three became violently seasick and their records suffered accordingly. Jian Pin observed that the junk sailed the first three-quarters of the race in three and a half hours. Hu Wei noted only that it did the final three-quarters in four and a half hours. Yao Li was so anxious to get back on land that the best he could do was observe that the middle leg of the race (from buoy B to C) took ten minutes longer than the first leg.
    Assuming that the buoys mark an equilateral triangle and that the junk had a constant speed on each leg, can you tell how long it took the junk to finish the race?
    1. Use the definition of speed: speed = \frac{distance}{time}~\implies~time = \frac{distance}{speed}

    2. Let u be the speed on the 1st leg;
    let v be the speed on the 2nd leg;
    let w be the speed on the 3rd leg.

    The length of one leg is a.

    3. You'll get a system of simulatneous equations in (u, v, w) with the parameter a:

    \left|\begin{array}{rcl}\frac au+\frac av +\frac{\frac14a}{w} &=& \frac72 \\ \frac{\frac14a}{u}+\frac av+\frac aw &=& \frac92 \\ \frac av &=& \frac au + \frac16\end{array}\right.

    4. Solve this system of equations for (u, v, w). I've got: u=\frac34 \cdot a \cdot h^{-1} , v=\frac23 \cdot a\cdot h^{-1} , w=\frac38 \cdot a\cdot h^{-1}.

    5. That means you'll get the times according to the definition at #1:

    t_1 = \frac43\ h
    t_2 = \frac32 \  h
    t_3 = \frac83\ h

    which sum up to a total time of 5 h 30 min.
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    Re: Advanced Math Word Problem

    I'm somewhat confused though. If it took 3.5 hours to sail the first 3/4, and 4.5 hours to sail the final 3/4, shouldn't the total time be greater than 8 hours? Or am I misunderstanding the question?
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    Re: Advanced Math Word Problem

    Quote Originally Posted by Kanika1989 View Post
    I'm somewhat confused though. If it took 3.5 hours to sail the first 3/4, and 4.5 hours to sail the final 3/4, shouldn't the total time be greater than 8 hours? Or am I misunderstanding the question?
    The first 3/4 and the last 3/4 overlap for 50% of the course, so the time to do the full course should be considerably less than the sum of the times for the first and last three quarters of the course.

    CB
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    Re: Advanced Math Word Problem

    Quote Originally Posted by Kanika1989 View Post
    I'm somewhat confused though. If it took 3.5 hours to sail the first 3/4, and 4.5 hours to sail the final 3/4, shouldn't the total time be greater than 8 hours? Or am I misunderstanding the question?
    1. In addition to Captain Black's reply I've marked the first and the last three quarters of the race in blue and red.

    2. Here is a short-cut of the calculations:

    Let t_1, t_2, t_3 be the times you need to sail the legs l_1, l_2, l_3 which all have the same length.

    Then you get a system of simultaneous equations in (t_1, t_2, t_3):

    \left|\begin{array}{rcl}t_1+t_2+\frac14 t_3&=&3.5\\\frac14 t_1+t_2+t_3&=&4.5\\t_2&=&t_1+\frac16\end{array}  \right.

    3. Solve this system and you'll get the periods of time directly.
    Attached Thumbnails Attached Thumbnails Advanced Math Word Problem-junkrace.png  
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    Re: Advanced Math Word Problem

    Oh I see! I had completely misinterpreted the question, I didn't notice the keyword "3/4 of the race", I was assuming that the first 3/4 was from A to B, and the final 3/4 was from C to A. However what you guys said makes complete sense, THANK YOU ALL SO MUCH!
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    Re: Advanced Math Word Problem

    You may find it easier if you let each leg = 4 (lengths don't matter here, so use an easy one!);
    then 3/4 lengths become 4 + 4 + 1 and 1 + 4 + 4; using u,v,w as speeds (as Earboth did),
    you can then work with these:
    Code:
     4/u + 4/v + 1/w = 7/2 [1]
     1/u + 4/v + 4/w = 9/2 [2]
    -4/u + 4/v       = 1/6 [3]
    Using elimination, you get rid of the w's first:
    say multiply [1] by -4, then add to [2]; will leave:
    15/u + 12/v = 19/2 [4]

    Similarly, eliminate v's using [3] and [4]:
    say multiply [3] by -3, then add to [4]; will leave:
    27/u = 9
    u = 3

    Substitute back, and you'll end up with v = 8/3 and w = 3/2.

    Since total time is 4/u + 4/v + 4/w, then:
    total time = 4/3 + 4/(8/3) + 4/(3/2) = 5.5 (as Earboth got).
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    Re: Advanced Math Word Problem

    Quote Originally Posted by Kanika1989 View Post
    I'm somewhat confused though. If it took 3.5 hours to sail the first 3/4, and 4.5 hours to sail the final 3/4, shouldn't the total time be greater than 8 hours? Or am I misunderstanding the question?
    I really agree with you i try and try again and again, still i got problem its really a real question or not.
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    Re: Advanced Math Word Problem

    Quote Originally Posted by dofseo View Post
    I really agree with you i try and try again and again, still i got problem its really a real question or not.
    Read the earlier responses in this thread and then ask specific questions about what points you do not understand.

    CB
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