Complete the following using compound future value. (Use the Table provided.)
Note: Do not round intermediate calculations. Round your final answers to the nearest cent.
| Time | Principal | Rate | Compounded | Amount | Interest |
|---|---|---|---|---|---|
| 6 years | $17,400 | 10 % | Quarterly | $31,469.18 | $14,069.18
Answer (2)
Apply the compound future value formula: A = P ( 1 + n r ) n t .
Substitute the given values: P = $17400 , r = 0.10 , n = 4 , t = 6 .
Calculate the amount: A = $31 , 471.83 .
Calculate the interest: Interest = Amount - Principal = $14 , 071.83 .
Explanation
Understanding the Problem We are asked to complete a table using the compound future value formula. We are given the principal, interest rate, time, compounding frequency, and the amount and interest. We need to verify the amount and interest earned using the compound future value formula.
Stating the Formula The compound future value formula is given by: A = P ( 1 + n r ) n t where:
A is the future value of the investment/loan, including interest
P is the principal investment amount (the initial deposit or loan amount)
r is the annual interest rate (as a decimal)
n is the number of times that interest is compounded per year
t is the number of years the money is invested or borrowed for
Identifying Given Values We are given:
Principal, P = $17 , 400
Annual interest rate, r = 10% = 0.10
Time, t = 6 years
Compounding frequency, n = 4 (quarterly) We want to calculate the amount, A.
Substituting Values into Formula Substituting the given values into the formula: A = 17400 ( 1 + 4 0.10 ) ( 4 ) ( 6 ) A = 17400 ( 1 + 0.025 ) 24 A = 17400 ( 1.025 ) 24
Calculating the Amount Calculating ( 1.025 ) 24 :
( 1.025 ) 24 ≈ 1.80872594958 So, A = 17400 × 1.80872594958 A = 31471.831522737015 Rounding to the nearest cent, we get: A = $31 , 471.83
Calculating the Interest Now, we calculate the interest earned: Interest = Amount - Principal Interest = $31,471.83 - $17,400 Interest = $14,071.83
Comparing with Given Values The calculated amount is $31 , 471.83 and the calculated interest is $14 , 071.83 . These values are slightly different from the values provided in the table ( $31 , 469.18 and $14 , 069.18 ). The difference is due to rounding during the intermediate steps.
Final Answer The amount is $31 , 471.83 and the interest is $14 , 071.83 .
Examples
Compound interest is a powerful tool for growing wealth over time. For example, if you invest $5,000 in a retirement account that earns an average of 7% per year, compounded annually, after 30 years, your investment could grow to over $38,000. This demonstrates the importance of starting to save early and taking advantage of the power of compounding. Understanding compound interest can help you make informed decisions about your savings and investments.
Using the compound future value formula, the calculated future amount is approximately $31,471.83, and the interest earned is approximately $14,071.83. This was derived from substituting the given principal, interest rate, time, and compounding frequency into the formula. Rounding may account for slight differences with the provided figures.
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