The four largest numbers in a set of seven numbers have a mean of 10.

Question

The four largest numbers in a set of seven numbers have a mean of 10. The four smallest numbers in the same set have a mean of 5. What is the least possible sum of the seven numbers?

(A) 30
(B) 40
(C) 50
(D) 60
(E) 70

Are You Up For the Challenge: 700 Level Questions

gmatophobia · Answer

Lets assume the numbers arranged in the ascending order are as follows

a .. b .. c .. d .. e .. f .. g

We know that

a + b + c + d = 20

d + e + f + g = 40

Adding both

a + b + c + d + d + e + f + g = 60

Taking one of the 'd' to the other side

a + b + c + d + e + f + g = 60 - d

Thus to minimize the sum, we have to maximize the value of d

We know that the sum of last four digits is 40, so the value of d will be maximum when d = 10

Therefore,

a + b + c + d + e + f + g = 60 - 10

= 50

IMO C

IanStewart · Answer

The four largest numbers sum to 40, and the four smallest sum to 20. So if we add those eight numbers, the sum is 60, but we've now added the median twice. So the sum of the seven numbers is 60 - m, where m is the median. Now the median can't be larger than 10, because then the four largest numbers would all be greater than 10 and could never have an average of 10. So m is at most 10, and the sum 60-m is at least 50.

If we wanted to construct a list with this sum, it could be: 10/3, 10/3, 10/3, 10, 10, 10, 10

If we wanted to construct a list with this sum, it could be: 10/3, 10/3, 10/3, 10, 10, 10, 10

sujoykrdatta · Answer

Let the numbers (smallest to largest) are: a, b, c, d, e, f, g

d + e + f + g = 40
a + b + c + d = 20

Add: a + b + c + d + d + e + f + g = 60

=> a + b + c + d + e + f + g = 60 - d

We need to figure out the possible values of d

To minimize the sum of the 7 numbers, we need to maximize d: 
Maximum value of d = 10 (since the largest 4 numbers have a mean 10, and d is the least of those 4 numbers)

Thus, the minimum sum of the 7 numbers = 60 - 10 = 50
 Answer C

Note:
To maximize the sum of the 7 numbers, we need to minimize d: 
Minimum value of d = 5 (since the smallest 4 numbers have a mean 5, and d is the largest of those 4 numbers)
Thus, the maximum sum of the 7 numbers = 60 - 5 = 55

Gaudapada · Answer

I've never seen a more logical solution in my life.

Posted from my mobile device

bumpbot · Answer

Automated notice from GMAT Club BumpBot: 

A member just gave Kudos to this thread, showing it’s still useful. I’ve bumped it to the top so more people can benefit. Feel free to add your own questions or solutions.

 This post was generated automatically.

The four largest numbers in a set of seven numbers have a mean of 10.

Prep Toolkit

Top 5 GMAT Debrief Videos

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards

GMAT Problem Solving (PS) Questions

The four largest numbers in a set of seven numbers have a mean of 10.

Prep Toolkit

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards