A family has two cars. During one particular week, the first car consumed 30 gallons of gas and the second consumed 35 gallons of gas. The two cars drove a combined total of 1800 miles, and the sum of their fuel efficiencies was 55miles per gallon. What were the fuel efficiencies of each of the cars that week?

The first car gets 25 miles per gallon, and the second car gets 30 miles per gallon.

Fuel efficiency is calculated by dividing the number of miles driven by the number of gallons of fuel used.  Let x be the number of miles the first car drives.  Its fuel efficiency is given by x/30.

1800-x will be the number of miles driven by the second car.  Its fuel efficiency is given by (1800-x)/35.  

Together the fuel efficiency is 55:
x/30 + (1800-x)/35 = 55

Multiply everything by 30 to cancel it:
(x/30)*30 + (1800-x)/35*30 = 55*30
x + 30(1800-x)/35 = 1650

Multiply everything by 35 to cancel it:
x*35 + 30(1800-x)/35*35 = 1650*35
35x + 30(1800-x) = 57750

Using the distributive property, we have:
35x + 30*1800 - 30*x = 57750
35x + 54000 - 30x = 57750

Combining like terms, we have:
5x + 54000 = 57750

Subtract 54000 from both sides:
5x + 54000 - 54000 = 57750 - 54000
5x = 3750

Divide both sides by 5:
5x/5 = 3750/5
x = 750

The first car drives 750 miles.  Its fuel efficiency is 750/30 = 25 miles per gallon.
The second car drives 1800-750 = 1050.  Its fuel efficiency is 1050/35 = 30 miles per gallon.