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27% (02:20) correct
73%
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In a group of 145 interns, 115 have a car, 120 have a motorcycle, and 140 have a bicycle. Which of the following statements must be true?
I. Most of the interns have both a motorcycle and a bicycle.
II. More interns have both a motorcycle and a bicycle than have both a motorcycle and a car.
III. Most interns who do not have a car have both a bicycle and a motorcycle.
(A) I only
(B) II only
(C) III only
(D) I and II only
(E) I and III only
I. Most of the interns have both a motorcycle and a bicycle.
II. More interns have both a motorcycle and a bicycle than have both a motorcycle and a car.
III. Most interns who do not have a car have both a bicycle and a motorcycle.
(A) I only
(B) II only
(C) III only
(D) I and II only
(E) I and III only
12 Feb 2026, 12:54
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This is a classic three-set overlapping problem that tests your understanding of maximum and minimum constraints. The key is to figure out what *must* be true versus what *could* be true.
**Step 1: Find the minimum overlap**
Total interns = 145
When we add up all three groups: 115 + 120 + 140 = 375
Since we only have 145 interns total, the "excess" is 375 - 145 = 230. This excess represents people who were counted multiple times because they belong to multiple categories.
**Step 2: Evaluate Statement I**
"Most of the interns have both a motorcycle and a bicycle."
Since 120 have motorcycles and 140 have bicycles, and we only have 145 people total, the minimum number who must have both is: 120 + 140 - 145 = 115.
So at least 115 out of 145 interns have both a motorcycle and a bicycle. That's 115/145 = roughly 79%, which is definitely "most." Statement I must be true.
**Step 3: Evaluate Statement II**
"More interns have both a motorcycle and a bicycle than have both a motorcycle and a car."
From Step 2, we know at least 115 have both motorcycle and bicycle.
For motorcycle and car: 120 + 115 - 145 = 90 minimum.
So yes, at least 115 > at least 90. Statement II must be true.
**Step 4: Evaluate Statement III**
"Most interns who do not have a car have both a bicycle and a motorcycle."
Interns without a car = 145 - 115 = 30
Now here's the trap: we know at least 115 people have both M and B. But could all 115 of these people be the same people who have cars? Yes! If all 115 car owners also have both M and B, then among the 30 non-car people, we'd only have 5 motorcycles (120 - 115) and 25 bicycles (140 - 115) to distribute. At most 5 of the 30 non-car people would have both.
So Statement III is NOT necessarily true.
**Answer: (D) I and II only**
**Common trap:** Students often assume that the minimum overlap must be evenly distributed across all categories. With three overlapping sets, you need to consider the most extreme distributions to test what MUST be true versus what COULD be true.
**Takeaway:** When you see "must be true" questions with overlapping sets, always calculate minimum overlaps using the principle: Min(A ∩ B) = |A| + |B| - Total. Then test whether those overlaps can be concentrated in specific subgroups.
**Step 1: Find the minimum overlap**
Total interns = 145
When we add up all three groups: 115 + 120 + 140 = 375
Since we only have 145 interns total, the "excess" is 375 - 145 = 230. This excess represents people who were counted multiple times because they belong to multiple categories.
**Step 2: Evaluate Statement I**
"Most of the interns have both a motorcycle and a bicycle."
Since 120 have motorcycles and 140 have bicycles, and we only have 145 people total, the minimum number who must have both is: 120 + 140 - 145 = 115.
So at least 115 out of 145 interns have both a motorcycle and a bicycle. That's 115/145 = roughly 79%, which is definitely "most." Statement I must be true.
**Step 3: Evaluate Statement II**
"More interns have both a motorcycle and a bicycle than have both a motorcycle and a car."
From Step 2, we know at least 115 have both motorcycle and bicycle.
For motorcycle and car: 120 + 115 - 145 = 90 minimum.
So yes, at least 115 > at least 90. Statement II must be true.
**Step 4: Evaluate Statement III**
"Most interns who do not have a car have both a bicycle and a motorcycle."
Interns without a car = 145 - 115 = 30
Now here's the trap: we know at least 115 people have both M and B. But could all 115 of these people be the same people who have cars? Yes! If all 115 car owners also have both M and B, then among the 30 non-car people, we'd only have 5 motorcycles (120 - 115) and 25 bicycles (140 - 115) to distribute. At most 5 of the 30 non-car people would have both.
So Statement III is NOT necessarily true.
**Answer: (D) I and II only**
**Common trap:** Students often assume that the minimum overlap must be evenly distributed across all categories. With three overlapping sets, you need to consider the most extreme distributions to test what MUST be true versus what COULD be true.
**Takeaway:** When you see "must be true" questions with overlapping sets, always calculate minimum overlaps using the principle: Min(A ∩ B) = |A| + |B| - Total. Then test whether those overlaps can be concentrated in specific subgroups.
12 Feb 2026, 19:33
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I am having trouble understanding your logic for statement two. Since we are evaluating what must be true shouldn't we compare the minimum number of motorcycle and bike owners 115 with the maximum number of motorcycle and car owners which would also be 115. If both of the numbers could be 115 then this statement could be false in exactly one situation. So it isnt always true right? Or am I missing something? Confused on why we compare both minimums
Edskore
