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Ajay analyzed two companies and formulated tables to model each company's growth, where [tex]\( x \)[/tex] represents the years and [tex]\( y \)[/tex] represents the number of employees.

Company 1
[tex]\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
0 & 0 \\
\hline
1 & 6 \\
\hline
2 & 12 \\
\hline
3 & 18 \\
\hline
4 & 24 \\
\hline
\end{array}
\][/tex]

Company 2
[tex]\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
0 & 1 \\
\hline
1 & 6 \\
\hline
2 & 36 \\
\hline
3 & 216 \\
\hline
4 & 1,296 \\
\hline
\end{array}
\][/tex]

If the number of employees is an indicator of a successful business, in which company should Ajay invest?

A. Company 1 because it is adding six employees each year while Company 2 is multiplying its number of employees by six each year.
B. Company 1 because it is multiplying its number of employees by six each year while Company 2 is adding six employees each year.
C. Company 2 because it is adding six employees each year while Company 1 is multiplying its number of employees.

Select the best option.



Answer :

GinnyAnswer
To determine which company Ajay should invest in, we need to analyze the growth rates of the two companies over the given period.

### Company 1 Analysis
The data for Company 1 is summarized in the following table:
[tex]\[ \begin{array}{|c|c|} \hline x & y \\ \hline 0 & 0 \\ \hline 1 & 6 \\ \hline 2 & 12 \\ \hline 3 & 18 \\ \hline 4 & 24 \\ \hline \end{array} \][/tex]
To find the growth rate of Company 1, we notice the number of employees is increasing linearly. Specifically, the difference in the number of employees between successive years is consistently 6 (i.e., 6 - 0, 12 - 6, 18 - 12, 24 - 18). Therefore, Company 1 is adding 6 employees each year. The overall growth rate can be calculated as the change in the number of employees over the change in time:

[tex]\[ \text{Company 1 Growth Rate} = \frac{\text{Final number of employees} - \text{Initial number of employees}}{\text{Final year} - \text{Initial year}} \][/tex]
[tex]\[ = \frac{24 - 0}{4 - 0} = \frac{24}{4} = 6 \][/tex]

### Company 2 Analysis
The data for Company 2 is summarized in the following table:
[tex]\[ \begin{array}{|c|c|} \hline x & y \\ \hline 0 & 1 \\ \hline 1 & 6 \\ \hline 2 & 36 \\ \hline 3 & 216 \\ \hline 4 & 1296 \\ \hline \end{array} \][/tex]
For Company 2, the number of employees grows exponentially, as we can see:
[tex]\[ \text{From year 0 to 1:} \; 1 \rightarrow 6 \][/tex]
[tex]\[ \text{From year 1 to 2:} \; 6 \rightarrow 36 \][/tex]
[tex]\[ \text{From year 2 to 3:} \; 36 \rightarrow 216 \][/tex]
[tex]\[ \text{From year 3 to 4:} \; 216 \rightarrow 1296 \][/tex]

Company 2's data suggest a pattern where each year the number of employees is roughly multiplied by 6. To express the growth rate in terms of employees, we look at the ratio of the final number to the initial number:
[tex]\[ \text{Company 2 Growth Rate} = \frac{\text{Final number of employees}}{\text{Initial number of employees}} \][/tex]
[tex]\[ = \frac{1296}{1} = 1296 \][/tex]

### Conclusion
Comparing the growth rates:
- Company 1 adds 6 employees each year.
- Company 2 multiplies its number of employees by a factor leading to 1296 times the initial number in 4 years.

Since the growth rate of Company 2 (1296) is significantly higher than that of Company 1 (6), Ajay should invest in Company 2.

Therefore, the correct recommendation is:
Company 2 because it is multiplying its number of employees while Company 1 is adding.

The final investment recommendation is Company 2.

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