Page 64 - Mathematics of Business and Finance

44 Chapter 1 | Review of Basic Arithmetic

Example 1.5(a) Calculating the Arithmetic Average of Numbers
Calculate the arithmetic average of 45, 65, 85, and 90.

45 + 65 + 85 + 90 285
Solution Arithmetic average = = = 71.25
4 4
Note: The answer is greater than the least value of 45 and less than the greatest value of 90.

Summation Notation (or Sigma Notation)

A simple method for indicating the sum of a number of terms in a sequence is the summation
notation. This involves the Greek letter sigma, Σ. When using the sigma notation, the variable
defined below the Σ is called the index of summation. The lower number is the lower limit of the
index (the term where the summation starts), and the upper number is the upper limit of the
summation (the term where the summation ends)
n
/ x
i This expression is read as the sum of x as x goes from x to x .
i = 1 i i 1 n
n
The arithmetic average formula can be written using sigma notation as shown below.

Formula 1.5(a) Arithmetic Average
Sum of all values of terms x + x + x + ... + x n
Arithmetic Average = = 1 2 3 =
Number of terms n

x refers to the 1 term, x refers to the 2 term, etc., and x refers to the n term, where n is the
th
st
nd
1
n
2
total number of terms.
Example 1.5(b) Calculating the Average Earnings per Day
The following are a worker's earnings for the last five days: $150, $225, $350, $270, and $325. Determine
his average earnings per day.
n
Solution / x
Average earnings per day = i = n i = 150 + 225 + 350 + 270 + 325 = 1320 = $264.00
1
5
5
Therefore, his average earning per day is $264.00.

Example 1.5(c) Calculating an Unknown Number Given the Other Numbers and the Average of All Numbers
The average of six numbers is 50. If five of the numbers are 40, 25, 75, 30, and 50, what is the sixth number?

Solution Let x represent the sixth number.

40 + 25 + 75 + 30 + 50 + x The average of six numbers is 50
= 50
6 Therefore, the sum of six numbers is 50 × 6 = 300
40 + 25 + 75 + 30 + 50 + x = 50 × 6 OR The sum of five numbers is 40 + 25 + 75 + 30 + 50 = 220
220 + x = 300 Therefore, the sixth number is 300 – 220 = 80
x = 300 – 220

x = 80
Therefore, the sixth number is 80.