Concept explainers
Geometric
The geometric mean of 4 and 16 is
The geometric mean of 1, 3, and 9 is
The geometric mean is useful in finding the average of percentages, ratios, indexes, or growth rates. For example, if a person receives a 20% raise after 1 year of service and a 10% raise after the second year of service, the average percentage raise per year is not 15 but 14.89%, as shown.
Or
His salary is 120% at the end of the first year and 110% at the end of the second year. This is equivalent to an average of 14.89%, since 114.89% − 100% = 14.89%.
This answer can also be shown by assuming that the person makes $10,000 to start and receives two raises of 20% and 10%.
Raise 1 = 10, 000 ⋅ 20% = $2000
Raise 2 = 12, 000 ⋅ 10% = $1200
His total salary raise is $3200. This total is equivalent to
Find the geometric mean of each of these.
a. The growth rates of the Living Life Insurance Corporation for the past 3 years were 35, 24, and 18%.
b. A person received these percentage raises in salary over a 4-year period: 8, 6, 4, and 5%.
c. A stock increased each year for 5 years at these percentages: 10, 8, 12, 9, and 3%.
d. The price increases, in percentages, for the cost of food in a specific geographic region for the past 3 years were 1, 3, and 5.5%.
a.
The geometric mean of growth rate.
Answer to Problem 37EC
The geometric mean of growth rate is 24.73%.
Explanation of Solution
Given Info:
The given total 3 values are 35, 24 and 18%.
Calculation:
Formula to calculate the geometric mean is,
Where
Substitute
The growth rate of the living life insurance corporation is 24.73%.
b.
The geometric mean of the data.
Answer to Problem 37EC
The geometric mean for the salary raise percentages is 5.57%
Explanation of Solution
Given info:
The salary raise percentages of a person over a 4 year period are 8, 6, 4 and 5%.
Calculation:
Substitute
Thus, the geometric mean for the salary raise percentages is 5.57%
c.
The geometric mean of the data.
Answer to Problem 37EC
The geometric mean for the increase in stock over a 5 year period is 7.63%
Explanation of Solution
Given Info:
The percentage of stock increases each year over a period of 5 years and the percentages are 10, 8, 12, 9 and 3%
Calculation:
Substitute
Thus, the geometric mean for the increase in stock over a 5 year period is 7.63%
d.
The geometric mean of the data.
Answer to Problem 37EC
The geometric mean for the percentage increase in the cost of food for a specific geographical location over the past 3 years is 2.55%
Explanation of Solution
Given Info:
The percentages of increase in cost of food for a specific geographical location over the past 3 years are 1, 3 and 5%.
Calculation:
Substitute
Determine product of 1, 3 and 5.5, then find cube root of product.
Thus, the geometric mean for the percentage increase in the cost of food for a specific geographical location over the past 3 years is 2.55%.
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