A summer camp has several campers whose ages are all different

Question

A summer camp has several campers whose ages are all different, and the combined age of all the campers is 215 years. The 4 oldest campers are assigned to Cabin A, the 3 youngest campers are assigned to Cabin B, and the remaining campers are assigned to Cabin C. If the combined age of the 4 oldest campers is 110 years, how many campers are assigned to Cabin C?

(1) The combined age of the campers assigned to Cabin C is 60 years.

(2) The combined age of the campers assigned to Cabin B is 45 years.

M47-218

Bunuel · Accepted Answer

GMAT Club Official Solution:

A summer camp has several campers whose ages are all different, and the combined age of all the campers is 215 years. The 4 oldest campers are assigned to Cabin A, the 3 youngest campers are assigned to Cabin B, and the remaining campers are assigned to Cabin C. If the combined age of the 4 oldest campers is 110 years, how many campers are assigned to Cabin C?

(1) The combined age of the campers assigned to Cabin C is 60 years.

Since the total combined age is 215 years and the campers in Cabin A have a combined age of 110 years, the campers in Cabin B have a combined age of:

215 - 110 - 60 = 45 years.

The average age in Cabin A is:

110/4 = 27.5

The average age in Cabin B is:

45/3 = 15

Every camper in Cabin C is older than each camper in Cabin B and younger than each camper in Cabin A. Therefore, the average age in Cabin C must be between 15 and 27.5.

If there are n campers in Cabin C, then their average age is 60/n. So:

15 < 60/n < 27.5

The only positive integer value of n that satisfies this inequality is 3.

Sufficient.

(2) The combined age of the campers assigned to Cabin B is 45 years.

Since the total combined age is 215 years and the campers in Cabin A have a combined age of 110 years, the campers in Cabin C have a combined age of:

215 - 110 - 45 = 60 years.

The average age in Cabin A is:

110/4 = 27.5

The average age in Cabin B is:

45/3 = 15

From here, the same logic as in statement (1) applies: the average age in Cabin C must be between 15 and 27.5, and since the combined age in Cabin C is 60, Cabin C must have exactly 3 campers.

Sufficient.

Answer: D.

paragw · Answer

Let total age = 215.
4 oldest (Cabin A) = 110.
So, ages of Cabin B + Cabin C = 215 − 110 = 105.
We need the number of campers in Cabin C.

(1) Cabin C total age = 60.
Then Cabin B total age = 105 − 60 = 45.
Knowing only the sums (60 and 45) does not determine how many campers make up the 60 years in Cabin C (different numbers of distinct ages are possible).
 Not sufficient.

(2) Cabin B total age = 45.
Cabin B has exactly 3 youngest campers.
Then Cabin C total age = 105 − 45 = 60.
Again, knowing only the total age of Cabin C does not determine how many campers are in Cabin C.
 Not sufficient.

Together: Still only know Cabin B = 45 and Cabin C = 60, which does not fix the number of campers in Cabin C.

Answer:  E  (Statements (1) and (2) together are not sufficient).

HarishChaitanya · Answer

Total age of all campers = 215.
Sum of ages of the 4 oldest campers (Cabin A) = 110.
So, remaining age (Cabin B + Cabin C) = 215−110=105215 - 110 = 105215−110=105.
Cabin B contains 3 youngest campers, and Cabin C contains the remaining campers.
From here:
Cabin B sum + Cabin C sum = 105.
Using statement (1):
Cabin C total age = 60 ⇒ Cabin B total = 105−60=45105 - 60 = 45105−60=45.
But knowing only the total age of Cabin C does not tell how many campers are in Cabin C, since multiple distinct-age combinations can give sum = 60. Hence, statement (1) is not sufficient.
Using statement (2):
Cabin B total age = 45 ⇒ Cabin C total = 105−45=60105 - 45 = 60105−45=60.
Again, we only get the sum of Cabin C, but not the number of campers. Multiple possibilities exist, so statement (2) is also not sufficient.
Using both statements together:
We still only know Cabin B total = 45 (3 campers) and Cabin C total = 60. Even with distinct ages, there is no unique way to determine the number of campers in Cabin C. Different combinations of distinct ages can still satisfy the same total.
Therefore, the number of campers in Cabin C cannot be uniquely determined.
Final Answer: (E) — Not sufficient even when both statements are used together.

vasu1104 · Answer

group A= 4 people ( oldest) 
 group B= 3 people (youngest) 
 group C = remaining, a-7

combined age of A= x, 
 combined age of B=y 
 combined age of C =z

we know x= 110 
 x+y+z= 215 
 y+z= 105 
 we need to find X-7.

now here everyone has different age.

S-1  
 z= 60 
 y+60= 105 
 y = 45

now we know there are 3 people in B and they are youngest in entire group. 
 lets try to find the age range of this group. 
 since each one is of different age, it cant be that each is 15 years. 
 we can try to make age as close to each other. 
 we get 14,15,16 
 so the oldest young person is 116 years old.

nowA has all the oldest people and the sum is 110 years old with 4 people. 
 the avg would be 27.5 and there are only 4 people. also try to make ages as close to each other. 
 so 2nd and 3rd person would be 27,28. so 1st would be 26 and 4th would be 29. 
 sum easily leads upto 110.

now youngest old person is 26 years in group A. 
 we know oldest oldest young is 16 years.

that means each person in C must be between 16 and 26

we know their combined age is 60. 
 its not possible to have 2 people in this age range and get the sum to 60.  
 its also not possible to have 4 people. even if we consider the the first four num in this range, it will lead us to 74 total and its not possible.

so now it has to be 3 people in the group.  
 from this age range, only 19,20,21 is possible to get us 60.

so there must be 3 people in C. 
 sufficient.

S-2 
 y+z= 105 
 z= 60

its same calculation as above.

so sufficient.

option D

Edskore · Answer

This one is sneaky - I'd go with D, and here's why the age-range constraint matters.

Start with what we know: Cabin A has the 4 oldest with sum 110, Cabin B has the 3 youngest, Cabin B + Cabin C sums to 105. Statement 1 tells us Cabin C = 60, so Cabin B = 45. Statement 2 tells us Cabin B = 45, so Cabin C = 60. Both statements give us the same derived totals, so we only need to check if one is sufficient.

The key question is: knowing C has total age 60, can we determine how many people are in C?

First, establish bounds on C's age range. Since A has the 4 oldest, to minimize A's youngest member, pack the 4 A-members as close together as possible: if A_min is the youngest, then A_min + (A_min+1) + (A_min+2) + (A_min+3) <= 110, which gives A_min >= 26. So every Cabin C member has age <= 25.

Since B has the 3 youngest summing to 45, to maximize B's oldest member, pack B together too: B_max + (B_max-1) + (B_max-2) >= 45 means B_max >= 16. But the constraint B ages < C ages means C_min >= B_max + 1 >= 17.

So C members have distinct integer ages in the range [17, 25].

Now: how many distinct integers from {17,...,25} can sum to 60?
- 2 members: maximum is 24 + 25 = 49 < 60. Impossible.
- 3 members: minimum is 17+18+19 = 54, maximum is 23+24+25 = 72. 60 is in range - works (e.g., 19+20+21 = 60).
- 4 members: minimum is 17+18+19+20 = 74 > 60. Impossible.

So C must have exactly 3 people. Statement 1 is sufficient by itself, and by symmetry so is Statement 2.

Answer: D.

The E trap is real - it feels like knowing only the sum shouldn't tell you the count. But the age-ordering constraints box in the possible values so tightly that the count is forced.

SS990 · Answer

Given: A1+A2...An=215 1st 3 of lowest are in B; Last 4 of highest are in A..the rest Total N-(4+3) =C, total age of A=110 Age of C= 105yrs

To find: no. of ppl in group C

1) Total Age of people in C= 60yrs so B=45yrs.
min of A = 27.5yrs , B can go from 1 to 27.5 but if B=15; C would have been 1 or 2 or 3. and if B=20 C would be min 1 max 3.. so this is not sufficient

2) Age of B= 45. This is same as option above. Hence insufficient

Hence answer is E

CrazyBlender123 · Answer

Let N be the total number of campers. Let their distinct integer ages in descending order be A1>A2>A3...>A_N. 
 Total Sum=215 
 Cabin A Sum (4 oldest) = 110 
 Therefore, Sum (Cabin B + Cabin C): 215−110 = 105 
 We must find n, the number of campers in Cabin C. 
 First, establish the maximum possible age for anyone in Cabin C. We do this by minimizing the ages of the 4 oldest campers (Cabin A) by making them consecutive integers, ending with the youngest in that group (A4): 
 (A_4+3)+(A_4+2)+(A_4+1)+A_4≤110
4A_4+6≤110
4A_4≤104⟹A_4≤26

Because all ages are distinct, the oldest camper in Cabin C (A5) must be strictly younger than A_4. Every camper in Cabin C must be ≤25 years old. 
 S1: 
 Since Cabin B + Cabin C = 105, Cabin B must sum to 45. Cabin B holds the 3 youngest campers. We establish the minimum possible age for anyone in Cabin C by maximizing the ages in Cabin B: 
 A_N−2+(A_N−2−1)+(A_N−2−2)≥45
3A_N−2−3≥45
3A_N−2≥48⟹A_N−2≥16
Because all ages are distinct, the youngest camper in Cabin C (A_N−3) must be strictly older than A_N−2. Every camper in Cabin C must be ≥17 years old. 
 We now know Cabin C's sum is exactly 60, and every camper inside must be between 17 and 25 years old. Let's test the number of campers (n): 
 If n=2: The maximum possible sum is 25+24=49. (49<60, so n!=2) 
 If n=4: The minimum possible sum is 17+18+19+20=74. (74>60, so n!=4) 
 If n=3: The minimum sum is 17+18+19=54, and the maximum is 25+24+23=72. Since 60 falls perfectly within this range, 3 campers is the only mathematical possibility. 
 Statement (1) alone is sufficient.

S2: 
 Since Cabin B + Cabin C = 105, Cabin C must sum to 60. Because the sum of the 3 youngest campers (Cabin B) is 45, we run the exact same inequality as Statement 1 to find the lower bound: 
 A_N−2+(A_N−2−1)+(A_N−2−2)≥45
3A_N−2≥48⟹A_N−2≥16 
 Every camper in Cabin C must be ≥17 years old. 
 Cabin C's sum is 60, and camper ages are restricted between 17 and 25. Testing the possible values for n yields the identical result as in Statement 1. 
 Statement (2) alone is sufficient. 
 So IMO D.

A summer camp has several campers whose ages are all different

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