Determine the fraction of rice remaining after Day 1.

The school used $\frac{1}{6}$ of the total rice on the first day.
Fraction remaining after Day 1 is:

$$1 - \frac{1}{6} = \frac{6}{6} - \frac{1}{6} = \frac{5}{6}$$

On the second day, $\frac{1}{5}$ of the remaining rice was used.
Fraction used on Day 2 = $\frac{1}{5} \times (\text{fraction remaining after Day 1})$

$$\frac{1}{5} \times \frac{5}{6} = \frac{1}{6}$$

Total fraction used = (Fraction used on Day 1) + (Fraction used on Day 2)

$$\frac{1}{6} + \frac{1}{6} = \frac{2}{6} = \frac{1}{3}$$

a) Step 4: Determine the fraction of rice that was shared equally for the last 4 days.
The "rest of the rice" was shared equally for the last 4 days. This is the fraction remaining after Day 2.
Fraction remaining after Day 2 = $1 - (\text{Total fraction used after Day 2})$

$$1 - \frac{1}{3} = \frac{3}{3} - \frac{1}{3} = \frac{2}{3}$$

This is the fraction of rice shared equally for the last 4 days.
The fraction of rice that was shared equally for the last 4 days is $\boxed{\frac{2}{3}}$.

b) Step 5: Determine the fraction of the total rice used on the last day.
The fraction $\frac{2}{3}$ of the total rice was shared equally over 4 days (Day 3, Day 4, Day 5, Day 6).
The amount used on each of these 4 days is $\frac{1}{4}$ of this remaining fraction.
Fraction used on the last day (Day 6) = $\frac{1}{4} \times \frac{2}{3}$

$$\frac{1}{4} \times \frac{2}{3} = \frac{2}{12} = \frac{1}{6}$$

We know that the mass used on the last day was 37.5 kg, which corresponds to $\frac{1}{6}$ of the total mass.
Let $T$ be the total mass of rice bought.

$$\frac{1}{6} T = 37.5 \text{ kg}$$

To find $T$, multiply both sides by 6:

$$T = 37.5 \times 6$$ $$T = 225 \text{ kg}$$

The mass of rice that the school had bought is $\boxed{225 \text{ kg}}$.