When the Z-score is 0 then that corresponds to the upper limit of the 50th percentile (mean), so the 35th percentile is lower than this mean and Z will be negative. Since the distribution is symmetrical we can find Z for 65% and then negate it. We need to define X where Z=(X-μ)/σ (μ=270 and σ=25). So 35% corresponds to Z=0.3853, making -0.3853=(X-270)/25, and X=270-9.63=260.37 approx.
However, the 35th percentile is a range, not a single value. We also need to know the Z-score for 34%, and, following the same method as for 35%, this is -0.4125. X=270-10.31=259.69 approx. The 35th percentile is the range 259.7≤X≤260.4. X=260 is a whole number within this range, so we could say that 260 is the 35th percentile.