At a school, of the students participate in music programs. Of the students who do not participate in music programs, participate in athletics. What percent of all students participate in neither music programs nor athletics?
The cue in a nested percent problem is a phrase like “of the students who do not”: the second percent applies to a smaller group, not to everyone. Find the share outside the first group, then the share outside the second. You can check the result in Desmos by imagining 100 students. The tempting mistake is treating the second share as a percent of all students.
Hints
- Hint 1
A complement is the part left out of a group. All students make up , so what share is left after the students in music programs are removed?
- Hint 2
The base of a percent is the group it is taken of. Here, the athletics percent is taken only from students outside music. What share of that smaller group does not play athletics?
- Hint 3
To find a percent of a percent, take the second share of the group left after the first. Imagine 100 students so that a count out of 100 also gives the percent of all students.
Step-by-step
Find each group left
Step 1Find the share outside music
All students make up . Subtract the in music to find its complement, the students not in music:
- Step 2
Find the share outside athletics
Of the students outside music, play athletics. Subtract to find the complement within that group:
These are the students in neither activity. The describes students who do play athletics.
- Step 3
Apply the second share to the right group
The is taken of the outside music, not of the whole school. That gives the nested percent relationship:
Each percent applies to the group named after “of.”
- Step 4
Calculate the percent of all students
Imagine 100 students, so a count out of 100 is also a percent of all students. Type in Desmos. It shows : in this imagined school, 45 of 100 students participate in neither music nor athletics. So of all students do. Choice C.