Maths problems: Take two

Math problems: Basic Technical Mathematics with Calculus 5th Ed
Ch 1. Exercise 1-13
Q3. Three meshed spur gears have a total of 107 teeth. If the second gear has 13 more cogs than the first and the third has 15 more than the second, how many gears have each?

Preamble -- I'm assuming this is another set of simultaneous equations
( this is more of a scratch pad than an explanation, hopefully previous take will give general idea)
Three unknowns, totalling 107
a + b + c = 107
second is 13 more than first
b = a + 13
third is 15 more than second
c = b + 15
we can substitute in stages:
a + b + (b + 15) = 107
a + (a + 13) + (b + 15) = 107
a + (a + 13) + ((a + 13)) + 15) = 107

quick check: 13 + 13 + 15 = 41

a + a + a + 13 + 13 + 15 = 107

3a + 41 = 107
3a = 107 -41
3a = 66
a = 66 / 3
a = 22
b = a + 13 = 35
c = b + 15 = 50
22 + 35 + 50 = 107
3a = 107 - 41
3a = 66
a = 66 /3 = 22
b = a + 13 = 35
c = b + 15 = 50
22 + 35 + 50 = 107