A group of people were asked how much time they spent exercising yesterday. Their responses are shown in the table below.

What fraction of these people spent less than 20 minutes exercising yesterday? Give your answer in its simplest form.

\begin{tabular}{|c|c|}
\hline Time, [tex]$t$[/tex] (minutes) & Frequency \\
\hline [tex]$0 \leq t\ \textless \ 10$[/tex] & 4 \\
\hline [tex]$10 \leq t\ \textless \ 20$[/tex] & 11 \\
\hline [tex]$20 \leq t\ \textless \ 30$[/tex] & 9 \\
\hline [tex]$30 \leq t\ \textless \ 40$[/tex] & 8 \\
\hline [tex]$40 \leq t\ \textless \ 50$[/tex] & 3 \\
\hline
\end{tabular}



Answer :

In order to determine the fraction of people who spent less than 20 minutes exercising, we need to follow these steps:

1. Identify the relevant frequencies:
We need to know the number of people who spent time in the intervals \(0 \leq t < 10\) and \(10 \leq t < 20\).

- For \(0 \leq t < 10\), the frequency is 4.
- For \(10 \leq t < 20\), the frequency is 11.

2. Calculate the total number of people who spent less than 20 minutes exercising:
Add the frequencies from the two relevant intervals:
[tex]\[ 4 + 11 = 15 \][/tex]
So, 15 people spent less than 20 minutes exercising.

3. Calculate the total number of people surveyed:
Add up all the frequencies given in the table:
[tex]\[ 4 + 11 + 9 + 8 + 3 = 35 \][/tex]
This gives us the total number of people surveyed.

4. Determine the fraction of people who spent less than 20 minutes exercising:
We need to form a fraction with the number of people who spent less than 20 minutes exercising (numerator) and the total number of people surveyed (denominator):
[tex]\[ \frac{15}{35} \][/tex]

5. Simplify the fraction:
Simplify \(\frac{15}{35}\) by finding the greatest common divisor (GCD) of 15 and 35, which is 5. Divide both the numerator and the denominator by their GCD:
[tex]\[ \frac{15 \div 5}{35 \div 5} = \frac{3}{7} \][/tex]

Therefore, the fraction of people who spent less than 20 minutes exercising yesterday is [tex]\(\frac{3}{7}\)[/tex].