Question

question 3 you are in charge of calculating the growth of your companys revenue over time. in the first year, revenue grew 15%. the following year, revenue grew 20%, then 15%, then 10%, then 4%. what is the approximate per year growth of the revenue of your company? give as a percentage, without the percent symbol, accurate to one decimal place (for example, if your answer is 14.2759%, you should input 14.3) moving to another question will save this response

Explanation:

Step1: Define the formula for compound growth

To find the equivalent annual growth rate, we first consider the total growth factor over the number of years. If the growth rates are \( r_1, r_2, \dots, r_n \) over \( n \) years, the total growth factor \( G \) is \( (1 + r_1)(1 + r_2)\dots(1 + r_n) \). Then the equivalent annual growth rate \( g \) is found by \( G = (1 + g)^n \), so \( g=(G)^{\frac{1}{n}}-1 \). Here, the growth rates are \( 0.15, 0.20, 0.15, 0.10, 0.05 \) (converted from percentages) and \( n = 5 \) years.

Step2: Calculate the total growth factor

First, calculate the product of \( (1 + r_i) \) for each \( r_i \):
\( (1 + 0.15)(1 + 0.20)(1 + 0.15)(1 + 0.10)(1 + 0.05) \)
\( = 1.15\times1.20\times1.15\times1.10\times1.05 \)
First, \( 1.15\times1.20 = 1.38 \)
Then, \( 1.38\times1.15 = 1.587 \)
Then, \( 1.587\times1.10 = 1.7457 \)
Then, \( 1.7457\times1.05 = 1.832985 \)

Step3: Calculate the equivalent annual growth rate

Now, we have \( n = 5 \) years, so we need to find \( g \) such that \( (1 + g)^5=1.832985 \). Taking the fifth root of \( 1.832985 \):
\( g = 1.832985^{\frac{1}{5}}-1 \)
Calculate \( 1.832985^{\frac{1}{5}} \). We know that \( 1.13^5=1.13\times1.13\times1.13\times1.13\times1.13 = 1.13^2 = 1.2769, 1.2769\times1.13 = 1.442897, 1.442897\times1.13 = 1.63047361, 1.63047361\times1.13 = 1.8424351793 \) which is close to \( 1.832985 \). Let's use a calculator - like approach. Let \( x = 1.832985^{\frac{1}{5}} \), so \( \ln(x)=\frac{\ln(1.832985)}{5} \). \( \ln(1.832985)\approx0.605 \), so \( \ln(x)=\frac{0.605}{5}=0.121 \), then \( x = e^{0.121}\approx1.1286 \). So \( g = 1.1286 - 1 = 0.1286 \), which is \( 12.9 \) when rounded to one decimal place. (More accurately, using a calculator for \( 1.832985^{0.2} \): \( 1.832985^{0.2}=\text{exp}(0.2\times\ln(1.832985))=\text{exp}(0.2\times0.60503)=\text{exp}(0.121006)\approx1.1286 \), so \( g = 0.1286 \) or \( 12.9\% \) when rounded to one decimal place.)

Answer:

12.9