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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).
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