Students analyze a circle graph to answer questions about favorite films.

Let’s solve each question step by step using the circle graph (also called a pie chart). The whole circle = 100% of surveyed people.

We are told:
- “The Lion King” = 35%
- “Toy Story” = 25%
- “Finding Nemo” = 20%
- “Shrek” = 10%
- “Cinderella” = 10%

Also, we’re told that 40 people said their favorite film was *The Lion King*. That’s our key to find total number of people surveyed.

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Step 1: Find total number of people surveyed

If 35% = 40 people, then:

Total people = 40 ÷ 0.35
= 40 ÷ (35/100)
= 40 × (100/35)
= 4000 ÷ 35
≈ 114.2857...

Wait — that’s not a whole number. But number of people must be whole. Let’s check if 35% of some nice number gives 40.

Try: 35% of 114 = 0.35 × 114 = 39.9 → too low
35% of 115 = 0.35 × 115 = 40.25 → too high

Hmm… maybe the percentages are rounded? Or perhaps we should use exact fractions.

Actually, let’s think differently. Maybe the 40 people correspond exactly to 35%, so total = 40 / 0.35 = 800/7 ≈ 114.2857 — still not integer.

But wait — look at the other percentages: 25%, 20%, 10%, 10%. All multiples of 5%. So likely total is divisible by 20 (since 5% = 1/20).

Let’s suppose total = T.

Then 35% of T = 40 → T = 40 / 0.35 = 800/7 ≈ 114.2857 — not integer.

This suggests maybe the 40 is approximate? But in school problems, numbers usually work out nicely.

Wait — perhaps I misread. Let me double-check the image description.

Actually, looking back: the problem says “40 people said their favorite film was ‘The Lion King’”, and the pie chart shows “The Lion King” as 35%.

So unless there’s a typo, we have to go with it.

But 35% = 7/20, so if 7 parts = 40, then 1 part = 40/7, total = 20 parts = 800/7 ≈ 114.2857 — not possible for people.

Alternative idea: maybe the 35% is rounded, and actual value is such that 40 people is exact.

Suppose total people = T.

Then 35% of T = 40 → T = 40 / 0.35 = 114.2857... — still same.

Perhaps the problem expects us to use 40 ÷ 0.35 and round? But that’s messy.

Wait — let’s check if 35% corresponds to 40, what is 1%? 40/35 = 8/7 ≈ 1.142857 per percent.

Then total = 100% = 100 × (8/7) = 800/7 ≈ 114.2857 — again same.

Maybe the problem has a mistake? Or perhaps I need to proceed with fraction.

Let’s keep it as fraction: T = 800/7

But that’s ugly. Let’s see if other questions can be answered without total.

Question 1: What percentage of surveyed people liked Shrek the most?

From pie chart: Shrek = 10% → Answer: 10%

No calculation needed.

Question 2: What percentage liked Finding Nemo? → 20%

Question 3: What fraction liked either Finding Nemo or Toy Story?

Finding Nemo = 20%, Toy Story = 25% → together = 45%

Fraction = 45/100 = 9/20

Question 4: Ratio of people who liked Shrek to Cinderella?

Both are 10% each → ratio = 10:10 = 1:1

Question 5: Ratio of people who liked Toy Story to total?

Toy Story = 25% → ratio = 25:100 = 1:4

Question 6: How many people were asked?

Here we need total. Given that 35% = 40 people.

So total = 40 / 0.35 = 4000 / 35 = 800 / 7 ≈ 114.2857

But since number of people must be integer, perhaps the 35% is approximate, and actual number is 40 out of 114 or 115.

Let’s calculate 35% of 114 = 0.35*114 = 39.9 ≈ 40 — close enough for rounding.

35% of 115 = 40.25 — also close.

But 40.25 is closer to 40 than 39.9? No, 39.9 is 0.1 away, 40.25 is 0.25 away — so 114 is better.

But 35% of 114.2857 is exactly 40.

In real surveys, they might report rounded percentages.

For example, if total is 114, then:

- Lion King: 40 people → 40/114 ≈ 35.087% → rounded to 35%

- Toy Story: 25% of 114 = 28.5 → probably 28 or 29 people

But the problem says "the following graph shows these results", implying the percentages are given, and 40 is exact for Lion King.

Perhaps we should use the exact value.

Another thought: maybe the 40 is for Lion King, and 35% is exact, so total is 800/7, but that's not integer.

Unless the problem allows fractional people? No.

Perhaps I miscalculated the percentages.

Let me add up the percentages: 35 + 25 + 20 + 10 + 10 = 100% — good.

So only issue is question 6.

Perhaps in such cases, we accept the fractional total for calculation, but for final answer, round to nearest integer.

Or maybe the 40 is not exact, but the problem states it as fact.

Let’s look for another way.

Suppose total people = T.

Lion King: 0.35T = 40 → T = 40/0.35 = 114.2857...

Now, for other films:

Toy Story: 0.25 * 114.2857 = 28.5714 — not integer

Finding Nemo: 0.20 * 114.2857 = 22.8571 — not integer

Shrek: 0.10 * 114.2857 = 11.4286 — not integer

Cinderella: same as Shrek.

None are integers, which is impossible.

Therefore, the only logical conclusion is that the 35% is rounded, and the actual number of people for Lion King is 40, so we need to find T such that 40/T is approximately 35%, and all other counts are integers.

Let’s assume total T, and 40 people for Lion King.

Then percentage for Lion King = (40/T)*100% = 35% approximately.

So 40/T = 0.35 → T = 40/0.35 = 114.2857, as before.

But perhaps T is 114, then Lion King % = 40/114 * 100 ≈ 35.087% — rounds to 35%.

Toy Story: 25% of 114 = 28.5 — not integer. If we take 25% as exact, it should be 28.5, but can't have half person.

Perhaps the percentages are based on actual counts.

Let me denote:

Let L = Lion King = 40

T_s = Toy Story

F = Finding Nemo

S = Shrek

C = Cinderella

Total = L + T_s + F + S + C = T

And from pie chart, L/T = 35/100 = 7/20

So 40/T = 7/20 → T = 40 * 20 / 7 = 800/7 ≈ 114.2857 — same as before.

This is frustrating.

Perhaps the "40 people" is for a different film? No, the problem says "40 people said their favorite film was 'The Lion King'".

Another idea: maybe the pie chart percentages are not exact, but the 40 is exact, and we should use that to find total, and then calculate others based on percentages, even if not integer, but for question 6, we report the calculated total.

But that doesn't make sense for people count.

Perhaps in the context of the problem, we are to use the percentage to find total, and accept that it's not integer, but for practical purposes, we can use it for ratios.

For question 6, "how many people were asked", we can say approximately 114, but let's see if there's a better way.

Let's calculate the number for each film using the percentages and the total from Lion King.

From Lion King: 35% = 40 people, so 1% = 40/35 = 8/7 people

Then:

- Toy Story: 25% = 25 * 8/7 = 200/7 ≈ 28.571 people

- Finding Nemo: 20% = 20 * 8/7 = 160/7 ≈ 22.857 people

- Shrek: 10% = 10 * 8/7 = 80/7 ≈ 11.429 people

- Cinderella: 10% = 80/7 ≈ 11.429 people

Sum: 40 + 200/7 + 160/7 + 80/7 + 80/7 = 40 + (200+160+80+80)/7 = 40 + 520/7 = 40 + 74.2857 = 114.2857 — consistent.

But none are integers except Lion King.

This suggests that the percentages are rounded, and the actual counts are integers.

For example, suppose total T = 114

Then Lion King: 40 people → 40/114 ≈ 35.087% — rounds to 35%

Toy Story: 25% of 114 = 28.5 — so perhaps 28 or 29 people. If 28, then 28/114 ≈ 24.56% — rounds to 25%? 24.56% is closer to 25% than to 24%, yes.

Finding Nemo: 20% of 114 = 22.8 — so 23 people? 23/114 ≈ 20.175% — rounds to 20%

Shrek: 10% of 114 = 11.4 — so 11 people? 11/114 ≈ 9.65% — rounds to 10%

Cinderella: 11 people? 11/114 ≈ 9.65% — rounds to 10%

Sum: 40 + 28 + 23 + 11 + 11 = 113 — not 114. Missing one person.

If Toy Story is 29, then 40+29+23+11+11 = 114

Check percentages:

- Lion King: 40/114 ≈ 35.087% → 35%

- Toy Story: 29/114 ≈ 25.438% → 25% (since 25.438 is closer to 25 than to 26? 25.438 - 25 = 0.438, 26 - 25.438 = 0.562, so closer to 25% — yes)

- Finding Nemo: 23/114 ≈ 20.175% → 20%

- Shrek: 11/114 ≈ 9.649% → 10% (since 9.649 is closer to 10 than to 9? 10 - 9.649 = 0.351, 9.649 - 9 = 0.649, so closer to 10% — yes)

- Cinderella: 11/114 ≈ 9.649% → 10%

Sum of percentages: 35+25+20+10+10 = 100% — good.

And sum of people: 40+29+23+11+11 = 114 — perfect.

So total people surveyed = 114.

The percentages are rounded to nearest whole number.

For the purpose of this problem, we'll use total = 114.

Now, let's answer each question.

1) Percentage who liked Shrek: from pie chart, 10% — so answer is 10%

2) Percentage who liked Finding Nemo: 20%

3) Fraction who liked either Finding Nemo or Toy Story: 20% + 25% = 45% = 45/100 = 9/20

4) Ratio of Shrek to Cinderella: both 10%, so 10:10 = 1:1

5) Ratio of Toy Story to total: 25% : 100% = 25:100 = 1:4

6) Total people: as above, 114

To confirm, with total 114:

- Lion King: 40 people (given)

- Toy Story: 25% of 114 = 28.5, but we used 29 to make sum work, and 29/114 ≈ 25.438% which rounds to 25%

Similarly for others.

Since the problem likely intends for us to use the given percentage and the 40 to find total, and 40 / 0.35 = 114.2857, but since people must be integer, and 114 works with rounding, we'll go with 114.

In many textbooks, they would expect you to calculate total as 40 / 0.35 = 114.2857, and perhaps round to 114, or leave as fraction, but for "how many people", it should be integer.

Given that, and the sum works with 114, I think 114 is correct.

So final answers:

1) 10%

2) 20%

3) 9/20

4) 1:1

5) 1:4

6) 114

Now, for the response, since ExplainAnswer is false, etc., I just need to provide Final Answer.

But the instruction says: "Complete ALL reasoning and explanations BEFORE writing the Final Answer." and "If an option is false, completely omit that section."

Since all options are false, I should only write Final Answer, but after reasoning.

The format says: "Always follow the structure below." and "If ExplainAnswer is true: Explanation: ..." etc.

Since all are false, I should not include those sections, but I need to do reasoning before Final Answer.

Perhaps I should write the reasoning in my mind, and only output Final Answer.

But the instruction says: "Complete ALL reasoning and explanations BEFORE writing the Final Answer." — so I need to have the reasoning, but since no sections are enabled, maybe I can write it as part of the process.

To comply, I'll write the Final Answer directly, as per format.

But let me see the example or standard.

I think for this case, since no additional sections are requested, I'll just provide the Final Answer with the answers listed.

So:

Final Answer:
1) 10%
2) 20%
3) \frac{9}{20}
4) 1:1
5) 1:4
6) 114

To box it or something, but the format doesn't specify.

Just list them.

In the initial request, it says "provide only the final correct answer", so probably just the answers.

So I'll go with that.