Pay attention
If some of the audience fell asleep for some of this talk, what was the average proportion of the talk that people heard?
Age
14 to 16
Challenge level
Problem
When a speaker gave a talk, 6% of the audience slept through the whole thing.
22% of the audience stayed awake and heard the entire talk.
Of the rest of the audience, half of them heard $\frac{2}{3}$ of the talk, and half of them heard $\frac{1}{3}$ of the talk.
What was the average proportion of the talk that people heard?
This problem is adapted from the World Mathematics Championships
Student Solutions
Using numbers to make the proportions easier to work with:
Let's imagine that there were 100 people in the audience and the talk was 60 minutes long.
Then 6 people heard 0 minutes, 22 people heard 60 minutes.
100 $-$ 6 $-$ 22 = 72, and 72$\div$2 = 36, so 36 people heard 40 minutes ($\frac{2}{3}$ of an hour) and 36 people heard 20 minutes ($\frac{1}{3}$ of an hour).
To find the average number of minutes heard, we need the total number of minutes heard divided by the number of people (which is 100).
The 6 people who heard 0 minutes did not contribute to the total number of minutes heard.
The 22 people who each heard 60 minutes heard a total of 22$\times$60 = 1320 minutes.
The 36 people who each heard 40 minutes heard a total of 36$\times$40 = 1440 minutes.
The 36 people who each heard 20 minutes heard a total of 36$\times$20 = 720 minutes.
That is a total of 1320 + 1440 + 720 = 3480 minutes.
So the average number of minutes heard was 3480$\div$100 = 34.8.
So the average proportion of the talk heard was $\frac{34.8}{60}=\frac{348}{600}=\frac{29}{50}=58\%$
Using 100 people and proportions of the talk:
Let's imagine that there were 100 people in the audience.
To find the average proportion of the talk heard, we need the total proportion of the talk heard divided by 100.
The 6 people who slept for the whole talk heard none of the talk.
The 22 people who heard all of the talk heard 22$\times$ the whole talk.
100 $-$ 6 $-$ 22 = 72, and 72$\div$2 = 36, so 36 people heard $\frac{2}{3}$ of the talk and 36 people heard $\frac{1}{3}$ of the talk.
The 36 people who heard $\frac{2}{3}$ of the talk heard a total of 36$\times\frac{2}{3}$= 24$\times$ the whole talk.
The 36 people who heard $\frac{1}{3}$ of the talk heard a total of 36$\times\frac{1}{3}$= 12$\times$ the whole talk.
That is a total of 22 + 24 + 12 = 58 $\times$ the whole talk.
So on average, $\frac{58}{100}=58\%$ of the talk was heard.
Using percentages and proportions:
If everyone in the audience had heard all of the talk, then the proportion of the talk heard by the audience would be 1. If half of the audience had slept through the talk and half had heard half of it, then the proportion of the talk heard by the audience would be half of a half, which is a quarter. That would also be the average proportion of the talk heard by the audience.
6% of the audience heard none of the talk, so they did not contribute to the proportion of the talk that was heard by the audience.
22% of the audience heard all of the talk, so they contribute 22% to the proportion that was heard.
100 $-$ 6 $-$ 22 = 72, and 72$\div$2 = 36, so 36% of the audience heard $\frac{2}{3}$ of the talk and 36% heard $\frac{1}{3}$ of the talk.
The 36% who heard $\frac{2}{3}$ of the talk contribute 36%$\times\frac{2}{3}$=24% to the proportion of the talk that was heard by the audience.
The 36% who heard $\frac{1}{3}$ of the talk contribute 36%$\times\frac{1}{3}$=12% to the proportion of the talk that was heard.
So the total proportion of the talk that was heard by the audience was:
22% + 24% + 12% = 58%.
So the average proportion of the talk that was heard was 58%, or $\frac{29}{50}$.