At a community center, 30 volunteers who participated in gardening gave the center an average feedback rating of 72. The 24 volunteers who participated in tutoring gave the center an average feedback rating of 68. Six volunteers participated in both activities, and those volunteers gave the center an average feedback rating of 80. Each volunteer gave exactly one feedback rating.
What is the mean feedback rating of the 48 distinct volunteers who participated in at least one of the two activities?
For overlapping groups, do not simply combine the two means or add the group totals without checking for shared members. First convert each mean into a total. Then subtract the shared group's total once because it was counted twice. Likewise, subtract the number of shared members once when finding the total number of distinct data values.
Hints
Use the meaning of a mean
Multiply each group's mean rating by its number of volunteers to find that group's total rating.
Watch for double-counting
The volunteers who participated in both activities appear in both group totals. Their ratings should be included only once in the total for all distinct volunteers.
Find the correct divisor
Use the number of distinct volunteers, not the sum of the two group sizes, when calculating the final mean.
Desmos Guide
Use a calculation expression
Algebraic calculation is quickest. Enter in Desmos. The subtraction in both the numerator and denominator accounts for the 6 volunteers counted in both groups.
Step-by-step Explanation
Count the distinct volunteers
Adding the group sizes counts the 6 volunteers in both groups twice. Therefore, the number of distinct volunteers is .
Find the total feedback rating
Convert each mean to a total rating, then remove the repeated contribution from the volunteers in both groups.
Find the overall mean
Divide the total rating by the number of distinct volunteers: