Page 70 - Mathematics of Business and Finance

50 Chapter 1 | Review of Basic Arithmetic

Example 1.5(i) Determining the Geometric Mean
Determine the geometric mean of the following five terms: 3, 4, 6, 9, 14.

Solution 1 n
Using, G = ex ∙ x ∙ x ∙ x ∙ x ∙ ... ∙ x o
2
1
3

n
4
5
1
5
G = (3 × 4 × 6 × 9 × 14) = 6.187861... = 6.19
Therefore, the geometric mean is 6.19.
Example 1.5(j) Determining the Average Annual Growth Rate
If the growth rate of an investment for the past four years was 15%, 20%,–30%, and 10%, calculate the
average annual growth rate using the geometric mean.
st
Solution The value at the end of the 1 year = 115% or (1 + 0.15) = 1.15
The value at the end of the 2 year = 120% or (1 + 0.20) = 1.20
nd
The value at the end of the 3 year = 70% or (1 – 0.30) = 0.70
rd
th
The value at the end of the 4 year = 110% or (1 + 0.10) = 1.10
The geometric mean of the above growth rate is calculated using the Formula 1.5(c) as follows:
1
G = (1.50 × 1.20 × 0.70 × 1.10) 4 = 1.01529548.... or about 1.0153 or 101.53%
Therefore, the average growth rate over the 4-year period is 1.53%. This means, if we multiplied the
initial investment by 1.01529548… or 101.529…% each year, then we would get the same amount as if
we had multiplied it by 1.15 × 1.20 × 0.70 × 1.10. That is:

(1.01529548…) (1.01529548…) (1.01529548…) (1.01529548…) = 1.15 × 1.20 × 0.70 × 1.10.

1.5 Exercises Answers to the odd-numbered problems are available at the end of the textbook.

SIMPLE ARITHMETIC AVERAGE (ARITHMETIC MEAN)
1. Calculate the arithmetic average of the following numbers:
a. 50, 45, 72, and 86 b. 120.70, 118.25, 110.75, and 95.50
2. Calculate the arithmetic average of the following numbers:

a. 12, 18, 26, and 32 b. 35.50, 86.25, 75.75, and 12.50
3. It took Alexis the following times to commute to her office on four different days: 45 minutes, 55 minutes, 30 minutes,
and 47 minutes. What is the average time it took her to commute to work?
4. The number of hours Yara worked each of the last four weeks were: 40 hours, 35 hours, 37.5 hours, and 36 hours.
Determine the average number of hours she worked per week.
5. The average of four rates is 5.5%. If three of the rates are 6.25%, 4%, and 6.5%, what is the fourth rate?
6. The average of five numbers is 120. If four of the numbers are 118, 130, 125, and 110, what is the fifth number?
7. The mean of 5 numbers is 27. If one of the numbers is excluded, the mean reduces by 2. Determine the excluded
number.
8. The mean of 6 numbers is 50. If one of the numbers is excluded, the mean reduces by 5. Determine the excluded
number.
9. The average mark for a class of 40 students is 74. The 15 male students in the class have an average mark of 70. Calculate
the average mark of the female students in the class.
10. The average monthly salary of 50 people is $4200. The average monthly salary of 42 of them is $3500. Calculate the
average monthly salary of the remaining 8 people.