First, we will determine the success rate for each player. The success rate is the number of goals divided by the number of shots.
1. Player J:
- Goals: 28
- Shots: 36
- Success Rate: [tex]\(\frac{28}{36} \approx 0.7778\)[/tex]
2. Player K:
- Goals: 22
- Shots: 40
- Success Rate: [tex]\(\frac{22}{40} = 0.55\)[/tex]
3. Player L:
- Goals: 16
- Shots: 30
- Success Rate: [tex]\(\frac{16}{30} \approx 0.5333\)[/tex]
Next, we need to convert these success rates to a common scale for comparison and express them as integers.
The lowest common multiple (LCM) of the shot numbers (36, 40, 30) is determined, and we use it to scale the success rates.
The LCM of 36, 40, and 30 is 360.
Now, we scale the success rates to this common multiple:
1. Scaling Player J's Success Rate:
- Original Success Rate: [tex]\(0.7778\)[/tex]
- Scaled: [tex]\(0.7778 \times 360 = 280\)[/tex]
2. Scaling Player K's Success Rate:
- Original Success Rate: 0.55
- Scaled: [tex]\(0.55 \times 360 = 198\)[/tex]
3. Scaling Player L's Success Rate:
- Original Success Rate: 0.5333
- Scaled: [tex]\(0.5333 \times 360 = 192\)[/tex]
So, Player J's scaled success rate is 280, Player K's is 198, and Player L's is 192.
Finally, we simplify these numbers to their simplest integer form by finding the greatest common divisor (GCD) of these three numbers and dividing each by the GCD.
The GCD of 280, 198, and 192 is 2.
1. Simplifying Player J's Scaled Rate:
- [tex]\( \frac{280}{2} = 140 \)[/tex]
2. Simplifying Player K's Scaled Rate:
- [tex]\( \frac{198}{2} = 99 \)[/tex]
3. Simplifying Player L's Scaled Rate:
- [tex]\( \frac{192}{2} = 96 \)[/tex]
Therefore, the simplest integer form of the success rates for Player J, Player K, and Player L in ratio form is:
[tex]\[ J : K : L = 140 : 99 : 96 \][/tex]
