Question

use the range and the mean to determine the missing numbers in each ordered data set.

1. _, 18, 20, _, 38, 42

range = 25 mean = 28

1. 3, _, 8.5, 10, 12.5, 14.5, _

range = 22 mean = 11

Explanation:

Step1: Solve for the first data set

Step1.1: Find the first missing number

The range is the difference between the largest and the smallest number. Let the first number be \(x\). Given the range \(R = 42 - x=25\).

$$
LATEXBLOCK0
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Step1.2: Find the second missing number

The mean \(\bar{x}=\frac{\sum_{i = 1}^{n}x_{i}}{n}\). Here \(n = 6\), \(\bar{x}=28\), and the sum of the known numbers is \(17+18 + 20+38+42=135\). Let the second - missing number be \(y\). Then \(\frac{135 + y}{6}=28\).

$$
LATEXBLOCK1
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Step2: Solve for the second data set

Step2.1: Find the last missing number

The range is the difference between the largest and the smallest number. Let the last number be \(z\). Given the range \(R=z - 3=22\).

$$
LATEXBLOCK2
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Step2.2: Find the second - missing number

The mean \(\bar{x}=\frac{\sum_{i = 1}^{n}x_{i}}{n}\). Here \(n = 7\), \(\bar{x}=11\), and the sum of the known numbers is \(3+8.5 + 10+12.5+14.5+25=73.5\). Let the second - missing number be \(w\). Then \(\frac{73.5+w}{7}=11\).

$$
LATEXBLOCK3
$$

Answer:

1. The missing numbers are \(17\) and \(33\).
2. The missing numbers are \(3.5\) and \(25\).