To analyze the claims made by Tara and Jim regarding the possible ratios derived from the survey data, let's first clarify the context of their claims.
The survey data tells us:
- 35 people exercise in the morning
- 45 people exercise in the afternoon
- 20 people exercise at night
Tara's claim consists of the following ratios:
- 35:55
- 45:80
- 20:65
Jim's claim consists of the following ratios:
- 35:100
- 45:100
- 20:100
Evaluating Tara's Ratios:
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35:55: This can be interpreted as the ratio of people who exercise in the morning to those who do not exercise in the morning. The total number of people is 100, so the number of people who do not exercise in the morning is \(100 - 35 = 65\). Thus, the ratio 35:65 can potentially be reduced to 35:55, but this is not feasible as 55 is not the count for non-morning exercisers. Hence, this ratio does not relate to the survey results correctly.
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45:80: This appears to represent the ratio of people who exercise in the afternoon to those who do not exercise in the afternoon. The number of people who do not exercise in the afternoon is \(100 - 45 = 55\). Again, 80 cannot accurately represent the number of people not exercising in the afternoon.
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20:65: This suggests a ratio of night exercisers (20) to those who do not exercise at night (which should be \(100 - 20 = 80\)). This ratio cannot be established since 65 is not a valid number in this context.
Therefore, none of Tara's ratios are valid based on the survey data.
Evaluating Jim's Ratios:
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35:100: This ratio indicates that 35 out of 100 people exercise in the morning. This is a valid ratio.
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45:100: This shows that 45 out of 100 people exercise in the afternoon. This is also a valid ratio.
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20:100: This indicates that 20 out of 100 people exercise at night. This is valid as well.
Since all ratios in Jim's claim are properly based on the survey data, Jim is correct.
In summary, Jim's claim of 35:100, 45:100, and 20:100 are accurate representations of the proportions of individuals exercising at different times of the day from the survey, while Tara's ratios do not correctly reflect the available data.