Question 1210591
<pre>
Three brothers—Leo, Sam, and Jax—are sharing a supply of energybars during a hike. Leo starts the trip with a backpack
full of bars. He eats 3 1/2 bars. After eating them, the ratio ofthe bars he has left to the bars he started with is
exactly 3:4. Sam then takes the remaining bars. He eats half ofthem plus half a bar more. Jax takes what is left. He eats
half of that amount plus half a bar more. After Jax is finished,there are exactly 2 bars left in the backpack. How many
energy bars did Leo start with?
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<font color = red><u>METHOD 1</u></font>
Let the multiplicative factor be x
Since the ratio of the bars he has left to the bars he started with is exactly 3:4, then the number of bars
he has left is 3x, and the number of bars he started with is 4x
With the number of eaten bars being {{{3&1/2}}}, or 3.5, we get: 4x - 3x = 3.5, and x = 3.5
This means that the number Leo starts with is 4(3.5) = 14. 

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CHECK
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Original number of bars: 14
Number eaten by Leo: 3.5
Number remaining after Leo ate 3.5: 14 - 3.5 = 10.5
Number left after Sam ate {{{1/2}}} of remainder, plus {{{1/2}}} bar:{{{(1/2)10.5 - .5 = 5.25 - .5 = 4.75}}}
Number left after Jax ate {{{1/2}}} of NEW remainder, plus{{{1/2}}} bar: {{{(1/2)4.75 - .5 = 2.375 - .5 = 1.875}}}

As seen above, a 2-bar end-result didn't ensue. So, itw's then decided to apply a different method.

<font color = red><u>METHOD 2</u></font>
Let the number of bars Leo started with, be B
With {{{matrix(1,3, 3&1/2, or, 3.5)}}} being eaten, remainder is: B - 3.5
With Sam eating {{{1/2}}} of remainder, plus {{{1/2}}} bar,remainder becomes: {{{(1/2)(B - 3.5) - .5}}} = {{{(B - 3.5 - 1)/2}}} = {{{(B -4.5)/2}}}
As Jax ate {{{1/2}}} of remainder, plus {{{1/2}}} bar, then NEW remainder becomes: {{{(1/2)((B - 4.5)/2) - .5}}} = {{{(B - 4.5)/4 - .5}}} = {{{(B - 4.5 - 2)/4}}} = {{{(B - 6.5)/4}}}
Since 2 bars are now left, we get: {{{(B - 6.5)/4 = 2}}}                                                          
                                                           B - 6.5 = 8 ----Cross-multiplying 
   <font color =red>Original number of bars</font>, or B = 8 + 6.5 = <font color =red>14.5</font>

OR

Some like to go from the end to the beginning. In other words, from the 2-bar end result to the original number of bars.
This is illustrated below.

End-Number of bars: 2
Number of bars before Jax ate {{{1/2}}} bar, plus {{{1/2}}} of remainder: {{{(2 + .5)/(1/2)}}} = (2.5)*2 = 5
Number of bars before Sam ate {{{1/2}}} bar, plus {{{1/2}}} of remainder: {{{(5 + .5)/(1/2)}}} = (5.5)*2 = 11
Number of bars before Leo ate 3.5: 11 + 3.5 = 14.5 

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CHECK
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Original number of bars: 14.5
Number eaten by Leo: 3.5
Number remaining after Leo ate 3.5: 14.5 - 3.5 = 11
Number left after Sam ate {{{1/2}}} of remainder, plus {{{1/2}}} bar:{{{(1/2)11 - .5 = 5.5 - .5 = 5}}}
Number left after Jax ate {{{1/2}}} of NEW remainder, plus{{{1/2}}} bar: {{{(1/2)5 - .5 = 2.5 - .5 = 2}}}

<font size = 4>VOILA!!!</font>

There's an obvious CONTRADICTION, as one method produces an orignal amount of 14, while the other finds
the count to be 14.5. As a result of this. there doesn't seem to be a SOLUTION to this problem, at all!!</font></font></font></b></pre>