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 September 10th, 2018, 06:43 AM #1 Member   Joined: Jun 2017 From: Lima, Peru Posts: 97 Thanks: 1 Math Focus: Calculus Is the method to solve this single variable equation used correctly? The problem is as follows: A king decides to split an award between his three best archers after organizing an accuracy contest in his court. The first one gets $\frac{2}{5}$ parts of the total minus $\frac{1}{5}$ of a pound, the second gets $\frac{2}{5}$ of the rest minus $\frac{1}{5}$ of a pound. If the third gets the $23$ remaining pounds. What was the total quantity of the cash prize that the king distributed?The alternatives given in my book are the following: - $53$ - $36$ - $38$ - $63$ - $72$ In my attempt to solve this problem, I stated the situation as follows: The total would be: $x$ pounds The first archer: $\frac{2}{5}x - \frac{1}{5}$ Second archer: $\frac{2}{5}\left(x-\left(\frac{2}{5}x - \frac{1}{5}\right ) \right)-\frac{1}{5}$ Third archer: $23$ So assuming that all summed up should become into the original prize, that would be: $\frac{2}{5}x - \frac{1}{5} + \frac{2}{5}\left(x-\left(\frac{2}{5}x - \frac{1}{5}\right ) \right)-\frac{1}{5} + 23 =x$ Doing up a little bit of algebra, I rearranged it in this form: $\left ( \frac{2}{5}x - \frac{1}{5} \right ) \left( 1 - \frac {2}{5} \right ) + \frac{2}{5}x-\frac{1}{5} = x - 23$ $\left ( \frac{2}{5}x - \frac{1}{5} \right ) \left( 1 - \frac {2}{5} + 1 \right )= x - 23$ $\left (\frac{2}{5}x - \frac{1}{5} \right ) \left( 2 - \frac {2}{5} \right ) = x - 23$ $\left (\frac{2}{5}x - \frac{1}{5} \right ) \left( \frac{8}{5} \right ) = x - 23$ Multiplying by $25$ both sides: $\left (\frac{2x-1}{5} \right ) \left( \frac{8}{5} \right ) = x - 23$ $\frac{8\left (2x-1 \right )}{25} \left ( 25 \right ) = 25x - 575$ $16x-8 = 25x - 575$ $9x = 567$ $x= 63$ Therefore the answer is $63$, which is what appears within the alternatives. But I was wondering whether this is the only way to solve this problem or could there be another way to solve it?. Last edited by skipjack; September 10th, 2018 at 01:18 PM. September 10th, 2018, 12:11 PM #2 Math Team   Joined: Oct 2011 From: Ottawa Ontario, Canada Posts: 14,597 Thanks: 1038 The first one gets 2/5 parts of the total minus 1/5 of a pound, the second gets 2/5 of the rest minus 1/5 of a pound. Unless above means: The first one gets 2/5 parts of (the total minus 1/5 of a pound), the second gets 2/5 of (the rest minus 1/5 of a pound). There are 4 possibilities if prize is less than 100: 13: 5,3,5 38: 15,9,14 63: 25,15,23 (yahoo!) 88: 35,21,32 Last edited by Denis; September 10th, 2018 at 12:43 PM. September 10th, 2018, 02:31 PM #3 Global Moderator   Joined: Dec 2006 Posts: 20,757 Thanks: 2138 Just work back twice, starting from 23. Each time, subtract 1/5 and then multiply by 5/3. (23 - 1/5)(5/3) = 38 and (38 - 1/5)(5/3) = 63. Or (equivalently) add 1, multiply by 5, divide by 3, and then subtract 2. Then repeat. (23 + 1) × 5/3 - 2 = 38 and (38 + 1) × 5/3 - 2 = 63. That's a bit longer to write, but quicker to do. For either way, dividing by 3 and then multiplying by 5 happens to be slightly easier than multiplying by 5 and then dividing by 3. Tags correctly, equation, method, single, solve, variable Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post w12 Calculus 2 September 7th, 2016 04:02 AM coenter Differential Equations 4 January 27th, 2015 10:24 PM mcl116 Calculus 2 November 10th, 2010 06:17 PM scherz0 Economics 0 November 15th, 2009 07:01 AM mikeduff9 Algebra 3 October 15th, 2007 10:57 AM

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