To determine which statement accurately describes the participation trends, we need to analyze the rate at which participation in the spring and fall leagues has increased over the four years.
First, we look at the number of participants each year for both leagues:
- Spring league participants: 68 (2010), 79 (2011), 90 (2012), 101 (2013)
- Fall league participants: 145 (2010), 148 (2011), 151 (2012), 154 (2013)
Next, we calculate the annual increase in the number of participants for each league:
Spring league:
- From 2010 to 2011: [tex]\( 79 - 68 = 11 \)[/tex]
- From 2011 to 2012: [tex]\( 90 - 79 = 11 \)[/tex]
- From 2012 to 2013: [tex]\( 101 - 90 = 11 \)[/tex]
So, the annual increases for the spring league are [tex]\( 11, 11, \)[/tex] and [tex]\( 11 \)[/tex].
Fall league:
- From 2010 to 2011: [tex]\( 148 - 145 = 3 \)[/tex]
- From 2011 to 2012: [tex]\( 151 - 148 = 3 \)[/tex]
- From 2012 to 2013: [tex]\( 154 - 151 = 3 \)[/tex]
So, the annual increases for the fall league are [tex]\( 3, 3, \)[/tex] and [tex]\( 3 \)[/tex].
Now, let's find the average annual increase for each league:
- Spring league average increase:
[tex]\[
\frac{11 + 11 + 11}{3} = \frac{33}{3} = 11
\][/tex]
- Fall league average increase:
[tex]\[
\frac{3 + 3 + 3}{3} = \frac{9}{3} = 3
\][/tex]
To compare the rate of increase between the two leagues, we check the ratio of their average increases:
[tex]\[
\frac{11}{3} \approx 3.67
\][/tex]
This ratio indicates that the average increase in spring league participation (11 participants per year) is more than three times the average increase in fall league participation (3 participants per year).
Given this information, the statement that best describes the programs is:
Participation in the spring league is increasing more than twice as fast as participation in the fall league.
