12 Days of Christmas GMAT Competition - Day 2: Jingle, an elf

Question

12 Days of Christmas 🎅 GMAT Competition with Lots of Questions & Fun

Jingle, an elf, traveled a total of 1,080 miles over the course of three days. Was his daily average (arithmetic mean) distance greater than the median of the distances he covered on these days?

(1) The range of the distances Jingle covered over these days was equal to his daily average (arithmetic mean) distance.
(2) On one of the days, Jingle covered 360 miles.

Bunuel · Accepted Answer

GMAT Club Official Explanation:

Jingle, an elf, traveled a total of 1,080 miles over the course of three days. Was his daily average (arithmetic mean) distance greater than the median of the distances he covered on these days?

Jingle's total travel of 1,080 miles over three days results in a daily average distance of 1,080/3 = 360 miles. The question then becomes: is the median distance less than 360 miles?

(1) The range of the distances Jingle covered over these days was equal to his daily average (arithmetic mean) distance.

This means the range = 360. Let's consider sets that could yield both a NO and a YES answers.

For distances of 200, 320, and 560 miles, the median of 320 is less than 360.
For distances of 150, 420, and 510 miles, the median of 420 is greater than 360.

Not sufficient.

(2) On one of the days, Jingle covered 360 miles.

Since Jingle covered 360 miles on one of the days, on one of the remaining two days he must have covered a distance less than or equal to 360 miles, and on the other, a distance greater than or equal to 360 miles. This leads to the three distances being {≤360, 360, ≥360}. Therefore, the median, as the middle value, would also be 360, matching the average and yielding a NO answer to the question. Sufficient.

Sufficient.

Answer: B.

Kinshook · Answer

Given: Jingle, an elf, traveled a total of 1,080 miles over the course of three days.
Asked: Was his daily average (arithmetic mean) distance greater than the median of the distances he covered on these days?

Let the distance travelled on three days be d1, d2 & d3 such that d1 <= d2 <= d3
D = d1 + d2 + d3 = 1080 miles

Is (d1 + d2 + d3)/3 > d2?
Is d2 < 1080/3 = 360 miles ?

(1) The range of the distances Jingle covered over these days was equal to his daily average (arithmetic mean) distance.
d3 - d1 = (d1 + d2 + d3)/3 = 360; d3 = 360+d1
d1, d2, 360+d1; 360 + 2d1 + d2 = 1080; 2d1 + d2 = 720;
Case 1: {d1, d2, d3} = {240, 240, 600}: d2 =240 < 360
Case 2: {d1, d2, d3} = {150, 420, 510}: d2 =420 > 360
NOT SUFFICIENT

(2) On one of the days, Jingle covered 360 miles.
d1=360-x; d2 = 360; d3=360+x; where x is non-negative number.
Median of distance = d2 = 360 = Average of the distance
SUFFICIENT

IMO B

missionmba2025 · Answer

Let's assume that the distances covered be x, y and z. x \leq y \leq z x + y + z = 1080 --- Equation 1 Ques: \frac{x + y + z }{3} > y x + y + z > 3y x + z > 2y From equation 1 1080 - y > 2y 3y < 1080 y < 360 Hence the second largest distance should be less 360. Statement 2 (2) On one of the days, Jingle covered 360 miles. Let's assume that the smallest distance covered is 360. In that case, the other distances should also be 360 for the sum to be 1080. In this case the mean = median. The answer is No. If the second largest distance is 360, then the other two distances must sum to 720, and the mean is equal to median. The answer is No. The largest distance is 360, then the other distances should also be 360 for the sum to be 1080. In this case the mean = median. The answer is No. The statement is sufficient. Statement 1 (1) The range of the distances Jingle covered over these days was equal to his daily average (arithmetic mean) distance. z - x = \frac{x + y + z }{3} 3z - 3x = x + y + z 2z - 4x = y With this equation we cannot determine anything useful. We can multiple possible values depending on the values chosen for x, y and z. Hence this statement alone is not sufficient to answer the question. IMO B

KarishmaB · Answer

Mean is fixed at 1080/3 = 360 miles. 
We need to find the values that median can take.

Median can be less than/more than or at 360. For example:
358, 359, 363
357, 361, 362
359, 360, 361

(1) The range of the distances Jingle covered over these days was equal to his daily average (arithmetic mean) distance.

The range is 360 too.This is irrelevant. Note in our two cases above, the range is the same

358, 359, 363 (range 5)
357, 361, 362 (range 5)

So we can get two cases - one with median > mean and other with median < mean with range 360.
Not sufficient

(2) On one of the days, Jingle covered 360 miles.

If 360 miles was covered on a day and it is the mean, it means 360 must be the middle value. The other two values can be one greater than 360 and one smaller. Or all values may be 360.
But it is not possible to have the smallest value as 360 and then two greater values. Then the mean will be more than 360.

Hence median is 360 and we know that it is equal to mean.
Sufficient alone

Answer (B)

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12 Days of Christmas GMAT Competition - Day 2: Jingle, an elf

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