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Algebra P1B (sp24ALGB) #5979-Activities
Module 1 Comprehension Check
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Algebra P 1 B (sp24ALGB) #5979/Module 1: Exponents
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"You have \($2000 to Invest in an account and need to have $\)2500 in five years. What annual interest rate would you need to have in order to have this if the amount is compounde
3.77%
2%
5.5%
4.47%

Verified Answer :

GinnyAnswer

Sure! Let's walk through the steps needed to solve this problem for finding the annual interest rate:

1. Identify the Given Values:
- Initial principal amount (P): \($2000 - Final amount (A): $\)2500
- Time period (t): 5 years

2. Understand the Problem:
We need to find the annual interest rate (r) for the amount to grow from \($2000 to $\)2500 in 5 years with compounded interest.

3. Use the Compound Interest Formula:
For annually compounded interest, the formula is:
\(\[ A = P(1 + r)^t \]\)
where \(\(A\)\) is the amount of money accumulated after n years, including interest, \(\(P\)\) is the principal amount (the initial amount of money), \(\(r\)\) is the annual interest rate (in decimal form), and \(\(t\)\) is the time the money is invested for in years.

Substitute the given values into the formula:
\(\[ 2500 = 2000(1 + r)^5 \]\)

4. Solve for \(\(r\)\):
To isolate \(\(r\)\), perform the following steps:
- Divide both sides by 2000:
\(\[ \frac{2500}{2000} = (1 + r)^5 \]\)
\(\[ 1.25 = (1 + r)^5 \]\)

- Take the fifth root of both sides to remove the exponent:
\(\[ (1.25)^{1/5} = 1 + r \]\)
\(\[ 1.04563955259127317 \approx 1 + r \]\)

- Subtract 1 from both sides to solve for \(\(r\)\):
\(\[ r \approx 0.04563955259127317 \]\)

5. Convert \(\(r\)\) to a Percentage:
To express the interest rate as a percentage, multiply the decimal by 100:
\(\[ r \approx 0.04563955259127317 \times 100 \]\)
\(\[ r \approx 4.563955259127317\% \]\)

6. Round the Rate (if necessary):
Matching closely to one of the provided answer choices, the annual interest rate would be approximately:
\(\[ 4.47\% \]\)

Therefore, in order to grow your \($2000 investment to $\)2500 in 5 years with annually compounded interest, you would need an annual interest rate of approximately 4.47%.

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