Key concept #1: Adding or subtracting the same number from every number in a set does not change the standard deviation
Key concept #2: If every number in a set is multiplied by k, then the new standard deviation = (k)(old standard deviation)

Let S = the standard deviation of ${a, b, c, d, e}$

By concept #2, the standard deviation of ${\frac{2}{3}a, \frac{2}{3}b, \frac{2}{3}c, \frac{2}{3}d, \frac{2}{3}e}$ $= \frac{2}{3}S$

By concept #1, the standard deviation of ${\frac{2}{3}a-6, \frac{2}{3}b-6, \frac{2}{3}c-6, \frac{2}{3}d-6, \frac{2}{3}e-6}$ $= \frac{2}{3}S$

If the standard deviation of Set B is 12.5, we can write: $\frac{2}{3}S = 12.5$
Multiply both sides of the equation by $\frac{3}{2}$ to get: $S = 18.75$

Since S represents the standard deviation of Set A, the answer is $18.75$