Question

A survey of 80 students found that 24 students both play in the band and play a sport. But 22 students are not in band and do not play a sport. There are 48 students in the band. If being in band is the row variable and playing sports is the column variable, fill in the labels in the table.

A 4-column table with 3 rows. Column 1 has entries in band, not in band, total. Column 2 is labeled play a sport with entries a, d, g. Column 3 is labeled do not play a sport with entries b, e, h. Column 4 is labeled total with entries c, f, i.

Which of the following correctly represents the given data in the problem?
a = 24, g = 48, h = 22, i = 80
a = 22, c = 80, d = 24, i = 48
a = 24, b = 48, c = 22, i = 48
a = 24, c = 48, e = 22, i = 80

Answer 1

Let's break down the information given in the problem and create a structure for the table.

We know:
- There are 80 students total.
- 24 students are both in the band and play a sport (this will be represented as 'a' in the table).
- 22 students are neither in the band nor play a sport (this will represent the total students in the last cell as 'i').
- There are 48 students in the band.

From this information, we can calculate additional entries in the table:

1. **Total in Band**: 48
- Students who are in band and play a sport (a) = 24
- Let's denote students who are in band and do not play a sport as 'b': \( b + a = 48 \)
- Therefore, \( b = 48 - 24 = 24 \).

2. **Total not in Band**: Total students - Students in Band = \( 80 - 48 = 32 \)
- Of these 32, let's denote the students who do not play a sport (e): we know that 22 students are not in band and do not play a sport.
- So, the students who are not in band and play a sport (d) can be calculated as: \( d + e = 32 \)
- With \( e = 22 \), \( d = 32 - 22 = 10 \).

So we can fill in the cells a, b, d, and e as follows:

- a = 24 (in band, play a sport)
- b = 24 (in band, do not play a sport)
- d = 10 (not in band, play a sport)
- e = 22 (not in band, do not play a sport)

Now we can summarize the totals:
- For the column "play a sport", the total (c): \( a + d = 24 + 10 = 34 \).
- For the column "do not play a sport", the total (f): \( b + e = 24 + 22 = 46 \).

We can now summarize everything in the table format:

| | Play a sport | Do not play a sport | Total |
|------------------|--------------|----------------------|-------|
| In band | 24 (a) | 24 (b) | 48 (c)|
| Not in band | 10 (d) | 22 (e) | 32 (f)|
| Total | 34 | 46 | 80 (i)|

Now let's match the results to the correct option. The correct data derived from the calculations based on the problem statement is:

- a = 24,
- b is unknown but should be calculated (24),
- d = 10,
- e = 22,
- c = 48,
- i = 80.

Based on this reasoning, the option that correctly represents the given data in the problem is:

**None of the provided options are completely correct**. However, based on filling the variables accurately from data, let’s assert the closest one for potential understanding:

In your problem entry, if referring to a = 24 match-ups, focus on b vs e, and match further down.

But based on calculations and final conclusions attached with counting, please check oversight if the facts were misaligned initially.

None of them is 100% aligned, check solution and verify back count either way.