Sure, let's solve this step-by-step.
1. Identify the total number of people surveyed:
To find the total number of people surveyed, we add up the frequencies of all the intervals listed:
[tex]\[
3 + 22 + 6 + 3 + 1 = 35
\][/tex]
So, the total number of people surveyed is 35.
2. Identify the number of people who spent less than 20 minutes exercising:
We look at the intervals that are less than 20 minutes. These intervals are [tex]\(0 \leq t < 10\)[/tex] and [tex]\(10 \leq t < 20\)[/tex].
- For the interval [tex]\(0 \leq t < 10\)[/tex], the frequency is 3.
- For the interval [tex]\(10 \leq t < 20\)[/tex], the frequency is 22.
Adding these frequencies together gives us:
[tex]\[
3 + 22 = 25
\][/tex]
So, 25 people spent less than 20 minutes exercising.
3. Determine the fraction of people who spent less than 20 minutes exercising:
The fraction of people who spent less than 20 minutes exercising is the number of people who spent less than 20 minutes divided by the total number of people surveyed:
[tex]\[
\frac{25}{35}
\][/tex]
4. Simplify the fraction:
To simplify the fraction [tex]\(\frac{25}{35}\)[/tex], we divide both the numerator and the denominator by their greatest common divisor (GCD). The GCD of 25 and 35 is 5.
[tex]\[
\frac{25 \div 5}{35 \div 5} = \frac{5}{7}
\][/tex]
Hence, the fraction of people who spent less than 20 minutes exercising yesterday is [tex]\(\frac{5}{7}\)[/tex] in its simplest form.
