Let X be the number of energy bars Leo did start with (the unknown value under the problem's question).

Leo eats 3.5 bars first.

The number of bars left is x-3.5.


Write the proportion as stated in the problem

     = .


From this proportion

    4x - 14 = 3x,

    4x - 3x = 14,

       x    = 14.


So, the answer to the problem's question is 14 energy bars.


You may check that this answer satisfies the problem's conditions,
Thus, all the problem's conditions are consistent and do not contradict each other.

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?
*******************************************
METHOD 1
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 , 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. 

******
CHECK
******
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  of remainder, plus  bar:
Number left after Jax ate  of NEW remainder, plus bar: 

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

METHOD 2
Let the number of bars Leo started with, be B
With  being eaten, remainder is: B - 3.5
With Sam eating  of remainder, plus  bar,remainder becomes:  =  = 
As Jax ate  of remainder, plus  bar, then NEW remainder becomes:  =  =  = 
Since 2 bars are now left, we get:                                                           
                                                           B - 6.5 = 8 ----Cross-multiplying 
   Original number of bars, or B = 8 + 6.5 = 14.5

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  bar, plus  of remainder:  = (2.5)*2 = 5
Number of bars before Sam ate  bar, plus  of remainder:  = (5.5)*2 = 11
Number of bars before Leo ate 3.5: 11 + 3.5 = 14.5 

******
CHECK
******
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  of remainder, plus  bar:
Number left after Jax ate  of NEW remainder, plus bar: 

VOILA!!!

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!!