Herky and Elaina want to compare their investment accounts to see how much they will have in the accounts after eight years. They substitute their values shown below into the compound interest formula.
| Name | Principal | Interest Rate | Number of Years | Compounded |
| :----- | :-------- | :------------ | :-------------- | :------------ |
| Herky | $500 | 5% | 8 | Once a year |
| Elaina | $400 | 6% | 8 | Once a year |
| | | | | $A=P(1+r)^t$ |
Which pair of equations would correctly calculate their compound interests?
A. Herky: $A=500(1+0.08)^5$, Elaina: $A=400(1+0.08)^6$
B. Herky: $A=(500 \times 1)+(0.05)^8$, Elaina: $A=(400 \times 1)+(0.06)^8$
C. Herky: $A=500(1+0.05)^8$, Elaina: $A=400(1+0.06)^8$
D. Herky: $A=(500 \times 1)+(0.05)^8$, Elaina: $A=(400 \times 1)+(0.05)^8
Answer (2)
Substitute Herky's values ( P = 500 , r = 0.05 , t = 8 ) into the compound interest formula: A = 500 ( 1 + 0.05 ) 8 .
Substitute Elaina's values ( P = 400 , r = 0.06 , t = 8 ) into the compound interest formula: A = 400 ( 1 + 0.06 ) 8 .
Identify the correct pair of equations based on the substitutions.
The correct equations are: Herky: A = 500 ( 1 + 0.05 ) 8 , Elaina: A = 400 ( 1 + 0.06 ) 8 .
Explanation
Understanding the Compound Interest Formula We are given the compound interest formula A = P ( 1 + r ) t , where:
A is the final amount
P is the principal amount
r is the interest rate (as a decimal)
t is the number of years
We need to find the correct equations for Herky and Elaina based on the given information.
Calculating Herky's Investment For Herky:
Principal, $P = $500
Interest rate, r = 5% = 0.05
Number of years, t = 8
Substituting these values into the formula, we get: A = 500 ( 1 + 0.05 ) 8
Calculating Elaina's Investment For Elaina:
Principal, $P = $400
Interest rate, r = 6% = 0.06
Number of years, t = 8
Substituting these values into the formula, we get: A = 400 ( 1 + 0.06 ) 8
Identifying the Correct Equations Comparing our calculated equations with the given options, we find that the correct pair of equations is:
Herky: A = 500 ( 1 + 0.05 ) 8 Elaina: A = 400 ( 1 + 0.06 ) 8
Final Answer Therefore, the correct answer is:
Herky: A = 500 ( 1 + 0.05 ) 8 , Elaina: A = 400 ( 1 + 0.06 ) 8
Examples
Compound interest is a powerful concept used in many real-life financial situations. For example, when you deposit money into a savings account, the bank pays you interest on your initial deposit (principal) and also on the accumulated interest from previous periods. This is how investments grow over time, and understanding the compound interest formula helps you project the future value of your investments. Another application is in loans, where interest is charged on the outstanding balance, and the compounding effect can significantly increase the total repayment amount. Understanding compound interest is crucial for making informed decisions about savings, investments, and loans.
The correct pair of equations to calculate the compound interest for Herky and Elaina are: Herky: A = 500 ( 1 + 0.05 ) 8 and Elaina: A = 400 ( 1 + 0.06 ) 8 . Therefore, the chosen answer is option C. This reflects the accurate application of the compound interest formula for both individuals.
;
