Welcome to Westonci.ca, the place where your questions find answers from a community of knowledgeable experts. Join our platform to connect with experts ready to provide precise answers to your questions in various areas. Discover in-depth answers to your questions from a wide network of professionals on our user-friendly Q&A platform.
Sagot :
To determine which statements about the growth are true, let's analyze the table provided:
[tex]\[ \begin{tabular}{|c|c|c|} \hline Year & \text{Simple Interest} & \text{Compound Interest} \\ \hline 0 & 100.00 & 100.00 \\ \hline 1 & 107.00 & 105.09 \\ \hline 2 & 114.00 & 110.45 \\ \hline 3 & 121.00 & 116.08 \\ \hline 4 & 128.00 & 121.99 \\ \hline 5 & 135.00 & 128.20 \\ \hline \end{tabular} \][/tex]
### Simple Interest Analysis:
- Calculation:
- Initially, Beth has [tex]$\$[/tex]100[tex]$. - In a simple interest account, interest is calculated on the initial principal only. - With a 7% annual rate, each year Beth earns \(7\% \times 100 = 7\) dollars in interest. - Observation from the Table: - Year 1: $[/tex]100 + 7 = 107[tex]$ - Year 2: $[/tex]107 + 7 = 114[tex]$ - Year 3: $[/tex]114 + 7 = 121[tex]$ - Year 4: $[/tex]121 + 7 = 128[tex]$ - Year 5: $[/tex]128 + 7 = 135[tex]$ - Conclusion: - Each year, a constant amount of \$[/tex]7 is added, resulting in a linear growth pattern. Therefore, the simple interest shows linear growth because it is adding a constant amount each year.
### Compound Interest Analysis:
- Calculation:
- Initially, Beth has [tex]$\$[/tex]100[tex]$. - With compound interest, interest is calculated on the new total after each compounding period. - With a 5% annual rate compounded quarterly, the interest rate per quarter is \( \frac{5\%}{4} = 1.25\% \). - Observation from the Table: - The amounts in the table for compound interest do not indicate a constant increase. - Instead, each year the amount grows by varying amounts: - Year 1: \$[/tex]105.09 ([tex]$5.09$[/tex] interest)
- Year 2: \[tex]$110.45 ($[/tex]5.36[tex]$ interest) - Year 3: \$[/tex]116.08 ([tex]$5.63$[/tex] interest)
- Year 4: \[tex]$121.99 ($[/tex]5.91[tex]$ interest) - Year 5: \$[/tex]128.20 ([tex]$6.21$[/tex] interest)
- Conclusion:
- The compound interest shows increasing increments. The growth depends on the account balance, making it not linear but rather exponential growth because the amount grows at a compounding rate.
- Therefore, the compound interest does not show linear growth.
### Statement Verification:
Statements Provided:
1. The simple interest shows linear growth because it is adding a constant amount.
2. The simple interest shows exponential growth because it is adding a constant amount.
3. The compound interest shows linear growth.
Evaluations:
1. True: The simple interest shows linear growth because it is adding a constant amount.
2. False: The simple interest does not show exponential growth; adding a constant amount each year is indicative of linear growth.
3. False: The compound interest does not show linear growth; it shows exponential growth due to the nature of compounding.
Correct answer:
- The first statement is true.
- The second and third statements are false.
So, the correct assessment based on the analysis is:
- (True, False, False).
[tex]\[ \begin{tabular}{|c|c|c|} \hline Year & \text{Simple Interest} & \text{Compound Interest} \\ \hline 0 & 100.00 & 100.00 \\ \hline 1 & 107.00 & 105.09 \\ \hline 2 & 114.00 & 110.45 \\ \hline 3 & 121.00 & 116.08 \\ \hline 4 & 128.00 & 121.99 \\ \hline 5 & 135.00 & 128.20 \\ \hline \end{tabular} \][/tex]
### Simple Interest Analysis:
- Calculation:
- Initially, Beth has [tex]$\$[/tex]100[tex]$. - In a simple interest account, interest is calculated on the initial principal only. - With a 7% annual rate, each year Beth earns \(7\% \times 100 = 7\) dollars in interest. - Observation from the Table: - Year 1: $[/tex]100 + 7 = 107[tex]$ - Year 2: $[/tex]107 + 7 = 114[tex]$ - Year 3: $[/tex]114 + 7 = 121[tex]$ - Year 4: $[/tex]121 + 7 = 128[tex]$ - Year 5: $[/tex]128 + 7 = 135[tex]$ - Conclusion: - Each year, a constant amount of \$[/tex]7 is added, resulting in a linear growth pattern. Therefore, the simple interest shows linear growth because it is adding a constant amount each year.
### Compound Interest Analysis:
- Calculation:
- Initially, Beth has [tex]$\$[/tex]100[tex]$. - With compound interest, interest is calculated on the new total after each compounding period. - With a 5% annual rate compounded quarterly, the interest rate per quarter is \( \frac{5\%}{4} = 1.25\% \). - Observation from the Table: - The amounts in the table for compound interest do not indicate a constant increase. - Instead, each year the amount grows by varying amounts: - Year 1: \$[/tex]105.09 ([tex]$5.09$[/tex] interest)
- Year 2: \[tex]$110.45 ($[/tex]5.36[tex]$ interest) - Year 3: \$[/tex]116.08 ([tex]$5.63$[/tex] interest)
- Year 4: \[tex]$121.99 ($[/tex]5.91[tex]$ interest) - Year 5: \$[/tex]128.20 ([tex]$6.21$[/tex] interest)
- Conclusion:
- The compound interest shows increasing increments. The growth depends on the account balance, making it not linear but rather exponential growth because the amount grows at a compounding rate.
- Therefore, the compound interest does not show linear growth.
### Statement Verification:
Statements Provided:
1. The simple interest shows linear growth because it is adding a constant amount.
2. The simple interest shows exponential growth because it is adding a constant amount.
3. The compound interest shows linear growth.
Evaluations:
1. True: The simple interest shows linear growth because it is adding a constant amount.
2. False: The simple interest does not show exponential growth; adding a constant amount each year is indicative of linear growth.
3. False: The compound interest does not show linear growth; it shows exponential growth due to the nature of compounding.
Correct answer:
- The first statement is true.
- The second and third statements are false.
So, the correct assessment based on the analysis is:
- (True, False, False).
Thank you for visiting our platform. We hope you found the answers you were looking for. Come back anytime you need more information. Thanks for stopping by. We strive to provide the best answers for all your questions. See you again soon. Your questions are important to us at Westonci.ca. Visit again for expert answers and reliable information.
