set

Question

Set R contains five numbers that have an average value of 55. If the median of the set is equal to the mean, and the largest number in the set is equal to 20 more than three times the smallest number, what is the largest possible range for the numbers in the set?

a) 78
b) 77 1/5
c) 66 1/7
d) 55 1/7
5) 52

thephoenix · Answer

daretodream wrote:Set R contains five numbers that have an average value of 55. If the median of the set is equal to the mean, and the largest number in the set is equal to 20 more than three times the smallest number, what is the largest possible range for the numbers in the set?

a) 78
b) 77 1/5
c) 66 1/7
d) 55 1/7
5) 52

If the median of the set is equal to the mean,it means that the numbers in the set are equally spaced.
Let the numbers be N1,N2,N3,N4,N5 in the ascending order.
So,N2 - N1 = N3 - N2= N4 - N3 = N5 - N4 = x(lets say)
largest number in the set is equal to 20 more than three times the smallest number-that means N5 = 3N1 +20

Now, N1+N2+N3+N4+N5 = 55 x 5 =275

Now, for largest possible range we should have the difference between the no.s to be the largest,
Now, N2 = N1 +x,
N3 = N2 + x = N1 +2x
Like this we have ,N5 = N1 + 4x = 3N1 + 20
=>2N1 = 4x -20
=>N1 = 2x -10

I am stuck at this point.
Can someone help has to how to proceed further.

harsh.champ · Answer

daretodream wrote:Set R contains five numbers that have an average value of 55. If the median of the set is equal to the mean, and the largest number in the set is equal to 20 more than three times the smallest number, what is the largest possible range for the numbers in the set?

a) 78
b) 77 1/5
c) 66 1/7
d) 55 1/7
5) 52

If the median of the set is equal to the mean,it means that the numbers in the set are equally spaced.
Let the numbers be N1,N2,N3,N4,N5 in the ascending order.
So,N2 - N1 = N3 - N2= N4 - N3 = N5 - N4 = x(lets say)
largest number in the set is equal to 20 more than three times the smallest number-that means N5 = 3N1 +20

Now, N1+N2+N3+N4+N5 = 55 x 5 =275

Now, for largest possible range we should have the difference between the no.s to be the largest,
Now, N2 = N1 +x,
N3 = N2 + x = N1 +2x
Like this we have ,N5 = N1 + 4x = 3N1 + 20
=>2N1 = 4x -20
=>N1 = 2x -10

I am stuck at this point.
Can someone help has to how to proceed further.

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shashank.ism · Answer

thephoenix wrote:daretodream wrote:Set R contains five numbers that have an average value of 55. If the median of the set is equal to the mean, and the largest number in the set is equal to 20 more than three times the smallest number, what is the largest possible range for the numbers in the set? a) 78 b) 77 1/5 c) 66 1/7 d) 55 1/7 5) 52 Mean = 55 Total = 55 x 5 = 275 The five numbers are: a, b, m, x, y y = 3a + 20 in oreder to widen the gap b/n a nad y We can re-phrase 5 numbers: a, a, m, m, y hence they are: a, a, 55, 55, 3a+20 because we can widen the gap between a and y only when we assign minimum possible values for b and x. The minimum possible value for b is a and of x is m. sum = 275 = a+a+55+55+3a+20 a = 29 y = 107 range = 107 - 29 = 78 phoenix that's a very nice approach but I am not able to understand "The minimum possible value for b is a and of x is m. "

shubhamkumar · Answer

shashank.ism wrote:thephoenix wrote:daretodream wrote:Set R contains five numbers that have an average value of 55. If the median of the set is equal to the mean, and the largest number in the set is equal to 20 more than three times the smallest number, what is the largest possible range for the numbers in the set? a) 78 b) 77 1/5 c) 66 1/7 d) 55 1/7 5) 52 Mean = 55 Total = 55 x 5 = 275 The five numbers are: a, b, m, x, y y = 3a + 20 in oreder to widen the gap b/n a nad y We can re-phrase 5 numbers: a, a, m, m, y hence they are: a, a, 55, 55, 3a+20 because we can widen the gap between a and y only when we assign minimum possible values for b and x. The minimum possible value for b is a and of x is m. sum = 275 = a+a+55+55+3a+20 a = 29 y = 107 range = 107 - 29 = 78 phoenix that's a very nice approach but I am not able to understand "The minimum possible value for b is a and of x is m. " Let the 5 numbers be a,b,c,d,e.Arranging the numbers in ascending order {a,b,c,d,e} We know that 55 is the median.The numbers now look like {a,b,55,d,e} We know that the largest number is 20 more than 3 times the number {a,b,55,d,3a+20} Now to maximize the range we need to have a maximum value of a and 3a+20.For this we need to minimize b and d.Now b and d can either be more than the preceding number or equal to it. For ex. series3,3,3,4,5,7,7,7,8. Therefore to have b and d the minimum value let them be equal to the previous number. {a,a,55,55,3a+20} We just have one unknown in this series,which we can now find out since we know the average of these numbers. OA A.Source mgmat

Anurag@Gurome · Answer

daretodream wrote:Set R contains five numbers that have an average value of 55. If the median of the set is equal to the mean, and the largest number in the set is equal to 20 more than three times the smallest number, what is the largest possible range for the numbers in the set?

a) 78
b) 77 1/5
c) 66 1/7
d) 55 1/7
5) 52

Let us assume that set R = {A, B, C, D, E}
The average of 5 numbers in set R = 55 implies the sum of these 5 numbers = 5 * 55 = 275
It is given that median of the set is equal to the mean implies median = C = 55
The largest number in the set is equal to 20 more than three times the smallest number implies E = 20 + 3A

So, the set R = {A, B, 55, D, 20 + 3C}
So, A + B + 55 + D + 20 + 3A = 275

Range of a set of data is the difference between the highest and lowest values in the set.
Here, range = E - A = 20 + 3A - A = 20 + 2A
Now for the range to be maximum, the value of A should be maximum. For A to be maximum, we should minimize all other terms, which means B and D.
Minimum value of B = A and minimum value of D = median, C = 55

Now the set R = {A, A, 55, 55, 3A + 20}
So, A + A + 55 + 55 + 3A + 20 = 275
5A + 130 = 275
5A = 145
A = 29

Therefore, range = 20 + 2A = 20 + (2 * 29) = 20 + 58 = 78

The correct answer is A.

Anurag Mairal, Ph.D., MBA
GMAT Expert, Admissions and Career Guidance
Gurome, Inc.
1-800-566-4043 (USA)

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Scott@TargetTestPrep · Answer

daretodream wrote: ↑Fri Feb 19, 2010 3:31 am Set R contains five numbers that have an average value of 55. If the median of the set is equal to the mean, and the largest number in the set is equal to 20 more than three times the smallest number, what is the largest possible range for the numbers in the set? a) 78 b) 77 1/5 c) 66 1/7 d) 55 1/7 5) 52 Solution: The sum of the five numbers is 55 * 5 = 275. If we let x be the smallest number, then the largest number is 3x + 20. Since we want to find the largest possible range, we also want the second smallest number to be the same as the smallest number and the second largest number the same as the median. Therefore, we can create the equation. x + x + 55 + 55 + 3x + 20 = 275 5x + 130 = 275 5x = 145 x = 29 Since the largest number is 3(29) + 20 = 107 and the smallest number is 29, the largest possible range is 107 - 29 = 78. Answer: A

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