A class of 12 scored an average of 8. Another class of 18 scored an average of 13. What is the average for all 30 students together?
Hints
Take them one at a time. The first gives nothing away.
1A nudge
Turn both averages into totals.
2The strategy
An average multiplied by its group size gives that group's total. Add the two totals, then divide by the number of students altogether. Never average two averages of different-sized groups.
3The full solution
The totals are 12 x 8 = 96 and 18 x 13 = 234, adding to 330. Then 330 divided by 30 = 11.
Solution
The reliable way
Turn each average into a total. The first class scored 12 x 8 = 96 and the second 18 x 13 = 234, so together 330 across 30 students. The combined average is 330 divided by 30 = 11. Note it is not 10.5, the average of the two averages - the larger class pulls the result towards 13. The transferable idea: averages of different-sized groups combine through their totals, never by averaging the averages.
The elegant way
The classes split 2 to 3, so the answer sits two fifths of the way from 13 to 8: 13 - 2 = 11.
Why this is on the test: Averaging the averages is the defining error of weighted-average problems and it produces an answer that looks completely reasonable.