The block will travel the farthest because applying conservation of energy, the only force acting on the block is peg and Ke but in the ball you have peg, ke and ke rotational acting on the ball thus since all the energy from the block the peg is converted into KE, this will cause an increase in the speed of the block which will cause an increase in the blocks height.
Good Job! A couple of tips for wording here.
 You start with conservation of energy, then immediately talk about the only force being PEg. PEg isn’t a force, Gravity is.
So a better way to word this (not just for this situation but for any conservation law situation is to refer to external vs internal forces.
For example:
For both cases, there are no external forces acting on the object  earth  ramp system. Because of that, the total amount of energy remains constant …

You differnetiate between KE translation and KE rotational for the sphere, but in the next sentance you refer to just KE. It can get a bit confusing for the reader (I know you mean translational KE), but in both cases PEg gets converted to KE, just not 100% translational for the sphere.
A better way to approach this is to avoid using the generic KE term in this case and instead use KErot and KEtrans (or KEr and KEt) just to be clear. 
You could also work on the last sentence a bit more and establish why an increase in velocity results in a higher height (KEtrans > PE, or v^2 = vo^2 + 2ax for examples).
As far as scoring goes, I’m quibbling over maybe 12 points on a 7 point question for this, you’ve got the main concept down well. We just need to polish the arguement up a bit.
Hey Peter! I would really appreciate it if you gave me feedback on my answer. Thanks!
The block will travel higher than the ball. Based on the Law of Conservation of Energy, the total initial energy of each system will equal the total final energy of each system (each systen has the same total energy because they have identical mass and start with the same PEg). Each case starts out with only PEg. The block leaves the ramp with some PEg, and some KElin. The ball leaves the ramp with the same amount of PEg as the block, some KElin, and, because it rolls without slipping, it has some KErot. Because the KElin of the block must be equal to the KElin+KErot of the ball because of the Law of Conservation of Energy, the final velocity of the ball will be less than that of the block. After leaving the ramp (at the same angle), the objects both become projectiles. Because the block has a greater starting velocity (as a projectile), it will travel higher than the ball, ignoring air resistance.
Nice job on this! You did a good job distinguishing between the two types of KE! The only thing I can think to improve this (and I’m being really picky here) is to mention that both objects leave the ramp at the same angle, which lets us justify the equivalent projectile motion statement. It’s obvious in the image that the ramps are identical, but doesn’t hurt to mention it in your reasoning.
As an alternative last few sentences, you could stick to energy in your reasoning. For example:
The block leaves the ramp with some KElin which can get converted back into PEg as the block rises. The ball leaves the ramp with some KElin and KErot, but only the KElin can be converted back into PEg. Therefore the block will end up with more PEg and a greater height than the ball.
a) The block
Both the ball and the block start with the same GPE relative to the lowest point of the ramp. As the ball rolls down the ramp, GPE is converted into translational KE and rotational KE as it heads toward the lowest point on the ramp, and then some of that rotational KE and translational kinetic energy is converted back into GPE by the time it reaches the end of the ramp. The block, on the other hand, has all of its GPE converted to translational KE as it heads toward the lowest point on the ramp, and then some of that KE is converted back into GPE by the time the block reaches the end of the ramp. Right when both objects leave the ramp, they have the same total amount of energy and the same GPE relative to the lowest point of the ramp. However, the ball has rotational KE in addition to translational kinetic energy, meaning that it must have a smaller linear speed. The block, on the other hand, has all of its mechanical energy that isn’t GPE in the form of translational kinetic energy, meaning it is moving faster at the end of the ramp. Once both blocks leave the ramp, the same force of gravity acts on both of them, so their accelerations are the same. Since voy is greater for the block since it has a greater initial velocity, it will travel a greater height, delta y, than the ball:
In the y direction:
v^2 = vo^2+2ay
2ay = vo^2
y = vo^2/2a, understood negative of “a” cancels with negative in front of vo:
y = vo^2/2a
vo is in the numerator, so a greater vo (ydirection) means a greater height, delta y, for the block.
Nice response John!
You did a great job with distinguishing between KEr and KEt for the ball & block and for explaining the projectile motion part. My only nitpick would be that you should mention that the ramp angles are the same, so that their voy would be the same fraction of their total v.
You could also approach this by looking at friction doing work on the ball but not on the block. The work done by friction on the ball removes translational KE so at the end of the ramp the ball has less KEt and a smaller v when it leaves the ramp.
Overall though, very nice job that’s at least a 6/7 on this question!