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September 10th, 2018, 06:43 AM   #1
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Math Focus: Calculus
Question 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.
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September 10th, 2018, 12:11 PM   #2
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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.
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September 10th, 2018, 02:31 PM   #3
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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.
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