Lots And Lots Of Pawns

I think the answer should be 64 because pawns of the same colour can't attack each other.

Relevant wiki: Pigeonhole Principle

Consider breaking the $8\times8$ chessboard into sixteen $2\times2$ squares.

In each of this $2\times2$ , it is evident that we can't fill in all 4 pawns. Otherwise there will be pawns attacking each other.
Similarly, it is also true that we can't fit in 3 pawns in otherwise there will be one pawn attacking another pawn.
This leaves us with 2 pawns: We arrange these 2 pawns in such a way that only 2 they are in the same row or the same column so that they are not attacking each other.

Hence, we can fill in a maximum of 2 pawns for each $2\times2$ squares. And so, we can fill in a maximum of $16\times2=\boxed{32}$ pawns in a $8\times8$ chessboard. The only thing left to do is to show that it is possible. A possible configuration is to place all the pawns in the rows 1, 3, 5 and 7. And we're done.

Relevant wiki: Chess Puzzles

NOTE: There are lots of ways to get to the same solution, but I will just look at one.

Let's take a look at the movement of a pawn. A pawn only moves forward and captures diagonally . Therefore, any pawn cannot be put diagonally next to another pawn since that pawn would attack it.

We can put a whole row of pawns on the first rank:

but now all the pawns control the squares in front of it:

so we cannot put any pawns on the 2nd rank so we'll have to skip a rank and place a row of pawns on the 3rd.

Now we'll have to skip a row and place pawns on the 5th rank (because of similar reasons above)

We'll have to skip the 6th rank as well and place pawns on the 7th rank.

We cannot place any pawns on the 8th rank because of similar reasons stated above

Now if we count the amount of pawns on the board, we will see that we have 32 pawns . The problem asked for the maximum percent of pawns you can put on the chessboard. The percent is calculated by: $\frac{32}{315} \times 100 \approx 10.16$ which is the answer.