Let's solve the problem step-by-step:
### Part (a): Complete the table
First, we need to verify if the given relative frequencies sum up to 1. These frequencies are:
- Won: 0.4
- Lost: 0.15
- Draw: 0.45
To check this, sum the relative frequencies:
[tex]\[ 0.4 + 0.15 + 0.45 = 1.0 \][/tex]
Since the sum is indeed 1.0, the relative frequencies are valid, and the table is already complete.
### Part (b): How many games did the team play altogether?
We are provided with the following information:
- The team won 20 more games than they lost.
- The relative frequencies of winning, losing, and drawing a game.
Let's denote:
- [tex]\( L \)[/tex] as the number of games the team lost.
- [tex]\( W \)[/tex] as the number of games the team won.
- [tex]\( D \)[/tex] as the number of games the team drew.
- [tex]\( T \)[/tex] as the total number of games the team played.
From the relative frequencies, we know:
[tex]\[ \text{Relative frequency of won} = \frac{W}{T} = 0.4 \][/tex]
[tex]\[ \text{Relative frequency of lost} = \frac{L}{T} = 0.15 \][/tex]
[tex]\[ \text{Relative frequency of draw} = \frac{D}{T} = 0.45 \][/tex]
In addition, we are given that the team won 20 more games than they lost:
[tex]\[ W = L + 20 \][/tex]
Using the relative frequencies and this additional information, we can set up the following equation:
[tex]\[ W = 0.4T \][/tex]
[tex]\[ L = 0.15T \][/tex]
Since [tex]\( W = L + 20 \)[/tex], we can substitute [tex]\( W \)[/tex] and [tex]\( L \)[/tex] into the equation:
[tex]\[ 0.4T = 0.15T + 20 \][/tex]
Solving for [tex]\( T \)[/tex]:
[tex]\[ 0.4T - 0.15T = 20 \][/tex]
[tex]\[ 0.25T = 20 \][/tex]
[tex]\[ T = \frac{20}{0.25} \][/tex]
[tex]\[ T = 80 \][/tex]
Thus, the total number of games the team played is [tex]\( 80 \)[/tex].
To find the individual counts of wins, losses, and draws:
[tex]\[ W = 0.4T = 0.4 \times 80 = 32 \][/tex]
[tex]\[ L = 0.15T = 0.15 \times 80 = 12 \][/tex]
[tex]\[ D = 0.45T = 0.45 \times 80 = 36 \][/tex]
### Summary:
- The total number of games played: [tex]\( 80 \)[/tex]
- Number of games won: [tex]\( 32 \)[/tex]
- Number of games lost: [tex]\( 12 \)[/tex]
- Number of games drawn: [tex]\( 36 \)[/tex]
