Results 1 to 2 of 2

Math Help - Distance puzzle

  1. #1
    Newbie
    Joined
    Oct 2007
    Posts
    2

    Distance puzzle

    "Calum was floating down a river on a raft, when, half a mile downstream, his brother duncan set off in a canoe. Duncan paddled downstream as quickly as he could, then turned round and paddles back again, still at his best pace. He arrived back at his starting point just as Calum floated by. Assuming Duncans best pace in still water is 10 times that of the river current, how far did he paddle ?"


    Thats all the info ive got, it could be taken at face value - (and equal 5 miles) but im afraid its meant to be harder than that...

    Thanks.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Super Member

    Joined
    May 2006
    From
    Lexington, MA (USA)
    Posts
    11,864
    Thanks
    744
    Hello, rossco21!

    We will use: . \text{Time} \:=\:\frac{\text{Distance}}{\text{Speed}}


    Calum was floating down a river on a raft, when a half-mile downstream,
    his brother Duncan set off in a canoe. .Duncan paddled downstream as
    quickly as he could, then turned round and paddled back again, still at
    his best pace. He arrived back at his starting point just as Calum floated by.
    Assuming Duncan's best pace in still water is 10 times that of the river current,
    how far did he paddle?
    Code:
          C      
          * - - - - *
    
                    D        x        P
          + - - - - * → → → → → → → → *
                    * ← ← ← ← ← ← ← ← *
                             x

    Let r be the speed of the river's current.

    Calum starts at C and drifts a half-mile to D.
    . . At r mph, this takes him: . \frac{\frac{1}{2}}{r} \:=\:\frac{1}{2r} hours.


    Duncan's speed is 10r mph.
    Paddling with the current, his speed is: 10r + r \,=\,11r mph.
    Paddling against the current, his speed is: 10r - r \,=\,9r mph.

    Duncan paddles x miles downstream from D to P at 11r mph.
    . . This takes him: . \frac{x}{11r} hours.
    Duncan paddles x miles upstream back to D at 9r mph.
    . . This takes him: . \frac{x}{9r} hours.
    Duncan's total time is: . \frac{x}{11r} + \frac{x}{9r} \:=\:\frac{20x}{99r} hours.


    Since Calum and Duncan meet at point D, their times are equal.

    Our equation is: . \frac{20x}{99r} \;=\;\frac{1}{2r}
    Multiply by 198r\!:\;\;40x \:=\:99\quad\Rightarrow\quad x \:=\:\frac{99}{40} miles.

    Therefore, Duncan paddled a total of: . 2x \:=\:2\left(\frac{99}{40}\right) \:=\:4.95 miles.

    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 1
    Last Post: August 9th 2010, 04:03 PM
  2. Need Help With This Puzzle.... Thanks!
    Posted in the Algebra Forum
    Replies: 2
    Last Post: August 8th 2009, 10:35 AM
  3. Distance vs. Total distance
    Posted in the Calculus Forum
    Replies: 5
    Last Post: January 5th 2009, 04:22 PM
  4. Distance Formula with given Distance but not X!
    Posted in the Pre-Calculus Forum
    Replies: 9
    Last Post: November 5th 2008, 10:48 PM
  5. A puzzle
    Posted in the Math Challenge Problems Forum
    Replies: 7
    Last Post: October 30th 2007, 12:59 AM

Search Tags


/mathhelpforum @mathhelpforum