Examkin

Soru

Zorluk: ZorMean, Median, and Mode

A logistics company recorded the daily number of deliveries made by each of its 8 delivery vans on a given day. Each van completed a distinct positive integer number of deliveries. The arithmetic mean of the number of deliveries made by the 8 vans was 25, the median was 24, and the range was 18. If MM represents the maximum number of deliveries completed by any single van that day, what is the maximum possible value of MM?

Cevap: 37

Cevap

The maximum possible value of MM is 37.
To maximize the largest element M=x8M = x_8, we express MM in terms of the smallest element x1x_1 using the range: M=x1+18M = x_1 + 18. Thus, maximizing MM is equivalent to maximizing x1x_1. Testing x1=20x_1 = 20 forces the minimal possible sum of the 8 distinct terms to be 20+21+22+23+25+26+27+38=20220 + 21 + 22 + 23 + 25 + 26 + 27 + 38 = 202, which exceeds the required sum of 200. Testing x1=19x_1 = 19 allows a minimal sum of 198, which can be adjusted to 200 by setting the set to {19,20,21,23,25,26,29,37}\{19, 20, 21, 23, 25, 26, 29, 37\}. Thus, the maximum possible value of MM is 19+18=3719 + 18 = 37.

Adım Adım Çözüm

1
Formulate the algebraic equations from the statistical properties given.
Sum of 8 terms = 8×25=2008 \times 25 = 200; x4+x5=48x_4 + x_5 = 48; x8=x1+18=Mx_8 = x_1 + 18 = M.
Mean gives total sum, even number of items gives median as average of 4th and 5th terms, and range links the maximum and minimum values.
2
Relate maximizing the maximum term MM to maximizing the minimum term x1x_1.
Maximizing M=x1+18M = x_1 + 18 requires making x1x_1 as large as possible.
Since the range is fixed at 18, MM increases directly as x1x_1 increases.
3
Test x1=20x_1 = 20 to determine feasibility.
Minimum possible sum for x1=20x_1 = 20 is 20+21+22+23+25+26+27+38=202>20020 + 21 + 22 + 23 + 25 + 26 + 27 + 38 = 202 > 200, which is invalid.
Distinct integer constraints force x4≥23x_4 \ge 23; since x4+x5=48x_4 + x_5 = 48 and x4<x5x_4 < x_5, x4x_4 must be 23 and x5x_5 must be 25, forcing all lower bounds up.
4
Test x1=19x_1 = 19 to confirm feasibility and construct a valid set.
The valid set {19,20,21,23,25,26,29,37}\{19, 20, 21, 23, 25, 26, 29, 37\} meets all criteria with a sum of 200.
The minimal sum for x1=19x_1 = 19 is 198, leaving headroom to increase x7x_7 to 29 to reach the sum of 200.

Anahtar Kavram

Optimization of Extreme Values in Finite Ordered Sets using Mean, Median, and Range
Bu soruyu puanla

Konular

Mean, Median, and Mode

Bu konudan daha fazla soru