DS - Difficult integer one....

[karthikpandian19]

The average of 5 distinct single digit integers is 5. If two of the integers are discarded, the new average is 4. What is the largest of the 5 integers?

(1) Exactly 3 of the integers are consecutive primes.

(2) The smallest integer is 3


My answer is coming to C, but it's not OA?

[eagleeye]

Hi karthikpandian19:

I got the answer as D. Let me explain:

We are given that:
Average of 5 distinct single digit integers is 5. First off for single digit integers, I assume single digit refers to them being positive, correct me if I am wrong. Then we have 0,1,2,3,4,5,6,7,8,9 as the set to choose from. Let the integer digits be a,b,c,d,e where all are distinct.
Now we have (a+b+c+d+e)/5 = 5 then a+b+c+d+e = 5*5 = 25.
We are also told that when two are removed average of the rest is 4. Let the removed ones be a and b. Then (c+d+e)/3 = 4 or c+d+e = 12.
Then from these two we get that a+b = 13. Since sum of a and b is greater than the sum of c,d,e the largest number must be among a,b.
Now since the digits are distinct, a+b combination can be
a. 9+4
b. 8+5
c. 7+6

Hence from the options we have to check which of 7,8,9 is the greatest.

1) We have 3 of the integers as consecutive primes. From our set of 0 to 9, those 3 can either be,
a. 2,3,5 or
b. 3,5,7

Case 1a) 2,3,5: if this is the case, since 2+3+5 = 10; the other two numbers must equal 25-10 = 15, which means they can be either 9+6 or 8+7. However, it can't be 8+7 since then we get 4 consecutive integers (2,3,5,7) which violates exactly 3 consecutive primes.
It also can't be 9,6 because then there is no combination with sum of 13. Hence 2,3,5 doesn't work. Before editing this part, i had chosen the wrong option because i overlooked the word exactly. Now lets look at 3,5,7
2a) 3,5,7 : the other 2 must equal 25-(3+5+7)=10, that leaves us with only 4,6 as the other 2 (this is because 8,2 can't be an option as it gives 4 primes) Also it works because
3+4+5=12 and 6+7=13, and 3,5,7 are exactly 3 consecutive integers.
So we have the largest number as 7. Sufficient.

2) We are given that smallest integer is 3, since
our original groups (a+b) were:
a. 9+4
b. 8+5
c. 7+6
with no 3 as an option, we must have that 3 belongs to one of the numbers of c,d,e. let 3 = c, then c+d+e=12 => d+e = 12- c = 12-3 = 9.
d+e = 9, therefore we can have only 4,5 are the option for 4+5=9, since all numbers are distinct and 3 is the smallest number. therefore we have c,d,e = 3,4,5. Since 4 and 5 occur in c,d,e, this leaves only one option for a+b which is 6+7. So we get that the largest number is 7. Hence, this is sufficient also and D is the correct answer.
Thanks to Anurag and Gmatguruny for their inputs.
Let me know if it helps

[karthikpandian19]

@eagleeye....I found the explanation in the Princeton website like this :

A. Yes. Since the average of the five numbers is 5, their sum is 25. For statement (1), if exactly three of the integers are consecutive primes, then the only possibilities for the primes are (2, 3, 5) or (3, 5, 7). Neither of these sets of numbers has an average of 4, so at least one prime must be discarded. Now, try excluding numbers and working with the averages. For example, in the only case that works, exclude the 7 from the set of (X, Y, 3, 5, 7). So X = 4 so that when we remove Y the average of the three remaining numbers is 4. Finally, the fifth number Y must then be 6 so that the sum of all five of the numbers is 25 - (3, 4, 5, 6, 7). In all other cases, the required numbers are either not single digit, not distinct or give a fourth prime. Hence, AD. Statement (2) is insufficient information to determine all the numbers. The correct answer is A.

B. No. Since the average of the five numbers is 5, their sum is 25. For statement (1), if exactly three of the integers are consecutive primes, then the only possibilities for the primes are (2, 3, 5) or (3, 5, 7). Neither of thesesets of numbers has an average of 4, so at least one prime must be discarded. Now, try excluding numbers and working with the averages. For example, in the only case that works, exclude the 7 from the set of (X, Y, 3, 5, 7). So X = 4 so that when we remove Y the average of the three remaining numbers is 4. Finally, the fifth number Y must then be 6 so that the sum of all five of the numbers is 25 - (3, 4, 5, 6, 7). In all other cases, the required numbers are either not single digit, not distinct or give a fourth prime. Hence, AD. Statement (2) is insufficient information to determine all the numbers. The correct answer is A.

C. No. Since the average of the five numbers is 5, their sum is 25. For statement (1), if exactly three of the integers are consecutive primes, then the only possibilities for the primes are (2, 3, 5) or (3, 5, 7). Neither of these sets of numbers has an average of 4, so at least one prime must be discarded. Now, try excluding numbers and working with the averages. For example, in the only case that works, exclude the 7 from the set of (X, Y, 3, 5, 7). So X = 4 so that when we remove Y the average of the three remaining numbers is 4. Finally, the fifth number Y must then be 6 so that the sum of all five of the numbers is 25 - (3, 4, 5, 6, 7). In all other cases, the required numbers are either not single digit, not distinct or give a fourth prime. Hence, AD. Statement (2) is insufficient information to determine all the numbers. The correct answer is A.

D. No. Since the average of the five numbers is 5, their sum is 25. For statement (1), if exactly three of the integers are consecutive primes, then the only possibilities for the primes are (2, 3, 5) or (3, 5, 7). Neither of these sets of numbers has an average of 4, so at least one prime must be discarded. Now, try excluding numbers and working with the averages. For example, in the only case that works, exclude the 7 from the set of (X, Y, 3, 5, 7). So X = 4 so that when we remove Y the average of the three remaining numbers is 4. Finally, the fifth number Y must then be 6 so that the sum of all five of the numbers is 25 - (3, 4, 5, 6, 7). In all other cases, the required numbers are either not single digit, not distinct or give a fourth prime. Hence, AD. Statement (2) is insufficient information to determine all the numbers. The correct answer is A.

E. No. Since the average of the five numbers is 5, their sum is 25. For statement (1), if exactly three of the integers are consecutive primes, then the only possibilities for the primes are (2, 3, 5) or (3, 5, 7). Neither of these sets of numbers has an average of 4, so at least one prime must be discarded. Now, try excluding numbers and working with the averages. For example, in the only case that works, exclude the 7 from the set of (X, Y, 3, 5, 7). So X = 4 so that when we remove Y the average of the three remaining numbers is 4. Finally, the fifth number Y must then be 6 so that the sum of all five of the numbers is 25 - (3, 4, 5, 6, 7). In all other cases, the required numbers are either not single digit, not distinct or give a fourth prime. Hence, AD. Statement (2) is insufficient information to determine all the numbers. The correct answer is A.


Now i am more confused, Can anyone clarify the doubt?

Hi karthikpandian19:

I got the answer as B. Let me explain:

We are given that:
Average of 5 distinct single digit integers is 5. First off for single digit integers, I assume single digit refers to them being positive, correct me if I am wrong. Then we have 0,1,2,3,4,5,6,7,8,9 as the set to choose from. Let the integer digits be a,b,c,d,e where all are distinct.
Now we have (a+b+c+d+e)/5 = 5 then a+b+c+d+e = 5*5 = 25.
We are also told that when two are removed average of the rest is 4. Let the removed ones be a and b. Then (c+d+e)/3 = 4 or c+d+e = 12.
Then from these two we get that a+b = 13. Since sum of a and b is greater than the sum of c,d,e the largest number must be among a,b.
Now since the digits are distinct, a+b combination can be
a. 9+4
b. 8+5
c. 7+6

Hence from the options we have to check which of 7,8,9 is the greatest.

1) We have 3 of the integers as consecutive primes. From our set of 0 to 9, those 3 can either be,
a. 2,3,5 or
b. 3,5,7

Case 1a) 2,3,5: if this is the case, since 2+3+5 = 10; the other two numbers must equal 25-10 = 15, which means they can be either 9+6 or 8+7. clearly we can't find the largest from this information.
2a) 3,5,7 : we can do similar analysis, but it isn't necessary. since we already know that the information is insufficient.

2) We are given that smallest integer is 3, since
our original groups (a+b) were:
a. 9+4
b. 8+5
c. 7+6
with no 3 as an option, we must have that 3 belongs to one of the numbers of c,d,e. let 3 = c, then c+d+e=12 => d+e = 12- c = 12-3 = 9.
d+e = 9, therefore we can have only 4,5 are the option for 4+5=9, since all numbers are distinct and 3 is the smallest number. therefore we have c,d,e = 3,4,5. Since 4 and 5 occur in c,d,e, this leaves only one option for a+b which is 6+7. So we get that the largest number is 7. Hence, this is sufficient and B is the correct answer.

Let me know if I got it right and if it helps

[Anurag@Gurome]

The average of 5 distinct single digit integers is 5. If two of the integers are discarded, the new average is 4. What is the largest of the 5 integers?

(1) Exactly 3 of the integers are consecutive primes.
(2) The smallest integer is 3

Sum of five integers = 5*5 = 25
Sum of three of the five integers = 3*4 = 12
Sum of other two of the five integers = (25 - 12) = 13

Statement 1: Three of the five integers are either (2, 3, 5) or (3, 5, 7)

• Analysis for (2, 3, 5)
  Other two integers are either (7 and 8) or (6 and 9)
  1. If other two integers are 7 and 8, then we end up with four consecutive primes
  2. If other two integers are 6 and 9, then we cannot get a sum of 12 or 13 as required
  
  Analysis for (3, 5, 7)
  Other two integers are either (1 and 9) or (2 and 8) or (3 and 7) or (4 and 6) or (5 and 5)
  1. The red underlined pairs are discarded as they violate the criteria 'distinct' integers
  2. If other two integers are 1 and 9, then we cannot get 12 as the sum of 3 integers and 13 as the sum of other two integers
  3. If other two integers are 2 and 8, then we end up with four consecutive primes
  4. If other two integers are 4 and 6, then (3 + 4 + 5) = 12 and (6 + 7) = 13 --> Possible

Hence, only possible set of five integers is (3, 4, 5, 6, and 7)

Sufficient

Statement 2: We can make 13 by selecting (4 and 9) or (5 and 8) or (6 and 7)
As 3 is the smallest of all five, 3 must be one of the remaining three integers. Hence, sum of other two integers is (12 - 3) = 9
Hence, other two integers can be (0 and 9) or (1 and 8) or (2 and 7) or (3 and 6) or (4 and 5). But the first three red underlined ones violates the criteria "Smallest integer is 3" and the fourth one violates the criteria "distinct"

Hence, only possible set of five integers is (3, 4, 5, 6, and 7)

Sufficient

The correct answer is D.

[karthikpandian19]

@Anurag.....Please find my query below in BOLD Red

The average of 5 distinct single digit integers is 5. If two of the integers are discarded, the new average is 4. What is the largest of the 5 integers?

(1) Exactly 3 of the integers are consecutive primes.
(2) The smallest integer is 3

Sum of five integers = 5*5 = 25
Sum of three of the five integers = 3*4 = 12
Sum of other two of the five integers = (25 - 12) = 13

Statement 1: Three of the five integers are either (2, 3, 5) or (3, 5, 7)

• Analysis for (2, 3, 5)
  Other two integers are either (7 and 8) or (6 and 9)
  1. If other two integers are 7 and 8, then we end up with four consecutive primes
  2. If other two integers are 6 and 9, then we cannot get a sum of 12 or 13 as required
  
  Analysis for (3, 5, 7)
  Other two integers are either (1 and 9) or (2 and 8) or (3 and 7) or (4 and 6) or (5 and 5)
  1. The red underlined pairs are discarded as they violate the criteria 'distinct' integers
  2. If other two integers are 1 and 9, then we end up with four consecutive primes---where is 4 cons. primes ???[/color][/b]
  3. If other two integers are 2 and 8, then we end up with four consecutive primes
  4. If other two integers are 4 and 6, then (3 + 4 + 5) = 12 and (6 + 7) = 13 --> Possible

Hence, only possible set of five integers is (3, 4, 5, 6, and 7)

Sufficient

Statement 2: We can make 13 by selecting (4 and 9) or (5 and 8) or (6 and 7)
As 3 is the smallest of all five, 3 must be one of the remaining three integers. Hence, sum of other two integers is (12 - 3) = 9
Hence, other two integers can be (0 and 9) or (1 and 8) or (2 and 7) or (3 and 6) or (4 and 5). But the first three red underlined ones violates the criteria "Smallest integer is 3" and the fourth one violates the criteria "distinct"

Hence, only possible set of five integers is (3, 4, 5, 6, and 7)

Sufficient

The correct answer is D.

[GMATGuruNY]

The average of 5 distinct single digit integers is 5. If two of the integers are discarded, the new average is 4. What is the largest of the 5 integers?

(1) Exactly 3 of the integers are consecutive primes.

(2) The smallest integer is 3

Since the average of the 5 integers is 5, the sum of all 5 integers = 5*5 = 25.
After 2 integers are discarded, the average of the remaining 3 integers is 4, implying that the sum of the remaining 3 integers = 3*4 = 12.
Thus, the sum of the 2 discarded integers = 25-12 = 13.
Thus, the correct set of 5 integers must exhibit the following characteristics:
The integers are distinct and between 0 and 9, inclusive.
The sum of all 5 integers is 25.
The sum of 3 of the integers is 12.
The sum of the other 2 integers is 13.

Statement 1: Exactly 3 of the integers are consecutive primes.
Case 1: 2,3,5
Thus, the options for the remaining 2 integers are 0,1,4,6,8,9.
Since 2+3+5 = 10, the sum of the remaining 2 integers = 25-10 = 15.
Only one combination works: 6+9.
Thus, the 5 integers would be 2,3,5,6,9.
This list does not include two integers whose sum is 13.

Case 2: 3,5,7
Thus, the options for the remaining 2 integers are 0,1,4,6,8,9.
Since 3+5+7 = 15, the sum of the remaining 2 integers = 25-15 = 10.
Only two combinations work: 1+9 and 4+6.
Thus, the 5 integers are either 1,3,5,7,9 or 3,4,5,6,7.
Only the second option includes two integers with a sum of 13 (6+7=13).
Thus, the 5 integers are 3,4,5,6,7.
SUFFICIENT.

Statement 2: The smallest integer is 3.
We know from statement 1 that 3,4,5,6,7 works.
For the smallest integer to be 3, the sum of the remaining 4 integers must be 25-3 = 22.
Given that these remaining 4 integers must each be greater than 3, there are no options aside from 4+5+6+7 = 22.
Thus, the 5 integers are 3,4,5,6,7.
SUFFICIENT.

The correct answer is D.

[Anurag@Gurome]

@Anurag.....Please find my query below in BOLD Red

2. If other two integers are 1 and 9, then we end up with four consecutive primes---where is 4 cons. primes ???

Thanks for pointing out.
That was a silly copy-paste error.
Edited the reply.