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07 Jul 2025, 08:00
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E
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Difficulty:
95%
(hard)
Question Stats:
40% (01:58) correct
60%
(02:01)
wrong
based on 355
sessions
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An old photograph shows a group of people standing in a row. If 4 of the 10 people from the left are wearing hats and all remaining people in the row are wearing hats, how many people are in the row?
(1) 5 of the 6 people from the right are wearing hats.
(2) 3 of the 8 people from the left are wearing hats.
(1) 5 of the 6 people from the right are wearing hats.
(2) 3 of the 8 people from the left are wearing hats.
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07 Jul 2025, 08:30
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An old photograph shows a group of people standing in a row. If 4 of the 10 people from the left are wearing hats and all remaining people in the row are wearing hats, how many people are in the row?
From the stem, it’s clear that exactly 6 people in the row are not wearing hats: 4 hats among the first 10 people, and everyone else wears a hat. So, we always have 6 people without hats, and the goal is to see how these can be distributed in the lineup to satisfy the statements.
(1) 5 of the 6 people from the right are wearing hats.
Since only 4 of the first 10 wear hats, that means there must be at least 10 + 1 = 11 people in the row to fit the remaining 1 hat. A minimal valid lineup of 11 people could be:
X - X - X - X - X - X - H - H - H - H | H
This satisfies the condition: 4 hats in the first 10, and 5 hats in the last 6.
However, we could have more than 11 people as well. For example, 14:
H - H - H - X - X - X - X - X - X - H | H - H - H - H
Still 4 hats in the first 10, and still 5 hats in the last 6.
Not sufficient.
(2) 3 of the 8 people from the left are wearing hats.
This only tells us that 3 of the 4 hat-wearers in the first 10 are among the first 8. That still leaves one more hat somewhere in positions 9 or 10. So, the last two people in that group must be either X - H or H - X. However, we have no information about people beyond these 10. Not sufficient.
(1)+(2) Since the 14-person lineup considered in (1) also satisfies (2), it's still valid:
H - H - H - X - X - X - X - X - X - H | H - H - H - H
However, we can also have 15 people if we swap X and H in the last positions (remember, from (2) we know that one hat must be in position 9 or 10):
H - H - H - X - X - X - X - X - H - X | H - H - H - H - H
Still 4 hats in the first 10, still 5 hats in the last 6, and 3 hats in the first 8.
So even together, the total number of people is not uniquely determined.
Answer: E.
GMAT Club Official Explanation:
An old photograph shows a group of people standing in a row. If 4 of the 10 people from the left are wearing hats and all remaining people in the row are wearing hats, how many people are in the row?
From the stem, it’s clear that exactly 6 people in the row are not wearing hats: 4 hats among the first 10 people, and everyone else wears a hat. So, we always have 6 people without hats, and the goal is to see how these can be distributed in the lineup to satisfy the statements.
(1) 5 of the 6 people from the right are wearing hats.
Since only 4 of the first 10 wear hats, that means there must be at least 10 + 1 = 11 people in the row to fit the remaining 1 hat. A minimal valid lineup of 11 people could be:
X - X - X - X - X - X - H - H - H - H | H
This satisfies the condition: 4 hats in the first 10, and 5 hats in the last 6.
However, we could have more than 11 people as well. For example, 14:
H - H - H - X - X - X - X - X - X - H | H - H - H - H
Still 4 hats in the first 10, and still 5 hats in the last 6.
Not sufficient.
(2) 3 of the 8 people from the left are wearing hats.
This only tells us that 3 of the 4 hat-wearers in the first 10 are among the first 8. That still leaves one more hat somewhere in positions 9 or 10. So, the last two people in that group must be either X - H or H - X. However, we have no information about people beyond these 10. Not sufficient.
(1)+(2) Since the 14-person lineup considered in (1) also satisfies (2), it's still valid:
H - H - H - X - X - X - X - X - X - H | H - H - H - H
However, we can also have 15 people if we swap X and H in the last positions (remember, from (2) we know that one hat must be in position 9 or 10):
H - H - H - X - X - X - X - X - H - X | H - H - H - H - H
Still 4 hats in the first 10, still 5 hats in the last 6, and 3 hats in the first 8.
So even together, the total number of people is not uniquely determined.
Answer: E.
General Discussion
07 Jul 2025, 08:33
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Information given:
- There's a row of people in an old photograph
- Of the first 10 people from the left, 4 wear hats
- Everyone else in the row (beyond those first 10) wears hats
Question:
- How many people are in the row?
Solution:
- Statement 1: 5 of the 6 people from the right are wearing hats
- Does not directly connect to the known 4 of 10 from the left, or the rest
- Without overlap, we can not figure out the total row length
- Insufficient
- Statement 2: 3 of the 8 people from the left are wearing hats
- We know that from the first 10 people from the left, 4 wear hats
- So we can deduce that positions 9-10 add 1 more hat
- However, there's no link to the right end, no way to find the total length
- Statement 1 and 2 together
- From the left: 1 hat among people 9-10
- After position 10, everyone must be wearing hats
- From the right: last 6 people, 5 hats
- If we take total = 14, last 6 people would be positions 9-14
- Positions 9-10: 1 hat, positions 11-14: all hats (4 people), total 5 hats, matches statement (1)
- Larger or smaller total would break this balance
- So, only 14 fits both
- Sufficient
Answer: C, Both statements together are sufficient, but neither alone is sufficient
- There's a row of people in an old photograph
- Of the first 10 people from the left, 4 wear hats
- Everyone else in the row (beyond those first 10) wears hats
Question:
- How many people are in the row?
Solution:
- Statement 1: 5 of the 6 people from the right are wearing hats
- Does not directly connect to the known 4 of 10 from the left, or the rest
- Without overlap, we can not figure out the total row length
- Insufficient
- Statement 2: 3 of the 8 people from the left are wearing hats
- We know that from the first 10 people from the left, 4 wear hats
- So we can deduce that positions 9-10 add 1 more hat
- However, there's no link to the right end, no way to find the total length
- Statement 1 and 2 together
- From the left: 1 hat among people 9-10
- After position 10, everyone must be wearing hats
- From the right: last 6 people, 5 hats
- If we take total = 14, last 6 people would be positions 9-14
- Positions 9-10: 1 hat, positions 11-14: all hats (4 people), total 5 hats, matches statement (1)
- Larger or smaller total would break this balance
- So, only 14 fits both
- Sufficient
Answer: C, Both statements together are sufficient, but neither alone is sufficient
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