**Answer:** 72 buses.

They have enough room for 60 of these, expanding the 60 capacity by 40% will give them enough room for 84, which we know is 12 more spaces than they currently need.

If they have B buses and S spaces before the expansion, they have enough room for:

S = B - 12 [1]

After the expansion they have more spaces, and enough room for:

S + 0.4 x S = B + 12 [2]

Rewriting [2] as:

1.4S = B + 12

and again as

B = 1.4S - 12 [3]

We can rewrite [1] as:

B = S + 12 [4]

Making [3] = [4] we have:

1.4S - 12 = S + 12

Subtracting S from both sides gives:

0.4S - 12 = 12

Adding 12 to both sides gives:

0.4S = 24

Multiplying by 10 and dividing by 4 on both sides gives:

S = 60

Using S = 60 in [4] gives B = 72. QED.