On a trip, a motorist drove 150 miles in the morning and 50 miles in the afternoon. His average rate in the morning was twice his average rate in the afternoon. He spent 5 hours driving. Find his average rate on each part of the trip.

If three numbers have a ratio of 7:3:2, we can set up two equations from that:

a/b = 7/3

a/c = 7/2

and if the sum of the smallest and largest is 30 more than twice the second:

a + c = 2b + 30

We now have three equations and three unknowns.

I'll solve the first two equations for "a", then use them to create a new equation with two variables, and use one of them to change the third equation into a new equation with the same two variables:

a/b = 7/3

3a = 7b

a = 7b/3

a/c = 7/2

2a = 7c

a = 7c/2

7b/3 = 7c/2

14b = 21c

2b = 3c < - - first new equation

a + c = 2b + 30

(7c/2) + c = 2b + 30

7c + 2c = 4b + 60

9c = 4b + 60 < - - second new equation

Now that we have a new system of two equations and two unknowns, this is easilly solved.

2b = 3c

b = 3c/2

9c = 4b + 60

9c = 4 (3c/2) + 60

9c = 6c + 60

3c = 60

c = 20

Now that we have c, we can find a and b:

a = 7c/2

a = 7 (20) / 2

a = 70

a = 7b/3

70 = 7b/3

210 = 7b

b = 30

The solution is:

70, 30, and 20