Measures of Center Question 11, 3.1.39 HW Score: \( 81.18 \% \), 13.8 of 17 points Part 1 of 4 Points: 0 of 1 The geometric mean is often used in business and economics for finding average rates of change, average rates of growth, or average ratios. Given \( n \) values (all of which are positive), the geometric mean nth root of their product. The average growth factor for money compounded at annual interest rates of \( 8.5 \%, 4.7 \% \), and \( 2.7 \% \) can be found by computing the geometric mean of \( 1.085,1.047 \), and 1.027 . Fir average growth factor, or geometric mean. What single percentage growth rate would be the same as having three successive growth rates of \( 8.5 \%, 4.7 \% \), and \( 2.7 \% \) ? Is that result the same as the mean o \( 4.7 \% \), and \( 27 \% \) ?

To determine the single equivalent growth rate that represents three successive annual growth rates of 8.5%, 4.7%, and 2.7%, we'll use the **geometric mean**. The geometric mean is particularly useful for average rates of growth because it accounts for the compounding effect over multiple periods. ### Step 1: Convert Percentage Rates to Growth Factors First, convert each percentage growth rate to its corresponding growth factor by adding 1 to each rate expressed as a decimal: - **8.5% growth rate**: \( 1 + 0.085 = 1.085 \) - **4.7% growth rate**: \( 1 + 0.047 = 1.047 \) - **2.7% growth rate**: \( 1 + 0.027 = 1.027 \) ### Step 2: Calculate the Geometric Mean The geometric mean \( G \) of these three growth factors is the cube root of their product: \[ G = \sqrt[3]{1.085 \times 1.047 \times 1.027} \] Let's compute the product first: \[ 1.085 \times 1.047 = 1.137795 \\ 1.137795 \times 1.027 \approx 1.168 \] Now, take the cube root: \[ G = \sqrt[3]{1.168} \approx 1.054 \] ### Step 3: Convert Back to a Percentage Growth Rate Subtract 1 and convert back to a percentage: \[ 1.054 - 1 = 0.054 \\ 0.054 \times 100 = 5.4\% \] **So, the single equivalent annual growth rate is approximately 5.4%.** ### Comparing to the Arithmetic Mean The **arithmetic mean** of the three original growth rates is calculated as: \[ \text{Arithmetic Mean} = \frac{8.5\% + 4.7\% + 2.7\%}{3} = \frac{15.9\%}{3} = 5.3\% \] Alternatively, if considering only two rates (assuming a typo in the original question): \[ \text{Arithmetic Mean} = \frac{4.7\% + 2.7\%}{2} = \frac{7.4\%}{2} = 3.7\% \] **Comparison:** - **Geometric Mean (Equivalent Growth Rate):** ≈ 5.4% - **Arithmetic Mean (All Three Rates):** 5.3% - **Arithmetic Mean (Two Rates):** 3.7% **Conclusion:** - The geometric mean (≈ 5.4%) is **very close** to the arithmetic mean of all three rates (5.3%) but **not** the same as the arithmetic mean of just two rates (3.7%). - In contexts involving growth rates over multiple periods, the **geometric mean is the preferred method** because it accurately reflects the compounding nature of growth. ### Key Takeaways - **Geometric Mean** is ideal for averaging rates of growth over multiple periods. - **Arithmetic Mean** can be used for simple average calculations but doesn't account for compounding, making it less accurate for growth rates. - Always ensure you're comparing the appropriate means based on the context of the problem. If you have any further questions or need clarification on any of these steps, feel free to ask!