Question
Deposit 8,000 in investment account
Investment p pays 4.25% simple interest
Investment q pays 3.95% interest compounded annually
He will not make any additional deposits or withdrawals
Which investment will have paid more in interest after 5 years and by how much
Investment p pays 4.25% simple interest
Investment q pays 3.95% interest compounded annually
He will not make any additional deposits or withdrawals
Which investment will have paid more in interest after 5 years and by how much
Answers
GPT-4o mini
To determine which investment will yield more interest after 5 years, we can calculate the total interest earned for both investment options.
### Investment P (Simple Interest)
The formula for simple interest is:
\[
I = P \times r \times t
\]
where:
- \(I\) is the interest
- \(P\) is the principal amount (initial deposit)
- \(r\) is the annual interest rate (as a decimal)
- \(t\) is the time in years
For Investment P:
- Principal \(P = 8,000\)
- Interest rate \(r = 4.25\% = 0.0425\)
- Time \(t = 5\) years
Calculating the interest:
\[
I_P = 8,000 \times 0.0425 \times 5 = 8,000 \times 0.2125 = 1,700
\]
### Investment Q (Compound Interest)
The formula for compound interest is:
\[
A = P \times (1 + r)^t
\]
where:
- \(A\) is the amount of money accumulated after n years, including interest.
- \(P\) is the principal amount.
- \(r\) is the annual interest rate (as a decimal).
- \(t\) is the time the money is invested for in years.
First, we'll calculate the total amount for Investment Q:
For Investment Q:
- Principal \(P = 8,000\)
- Interest rate \(r = 3.95\% = 0.0395\)
- Time \(t = 5\) years
Calculating the total amount:
\[
A_Q = 8,000 \times (1 + 0.0395)^5
\]
Calculating:
\[
(1 + 0.0395)^5 \approx 1.0395^5 \approx 1.2146 \quad \text{(using a calculator)}
\]
\[
A_Q \approx 8,000 \times 1.2146 \approx 9,717.09
\]
Now, to find the interest earned:
\[
I_Q = A_Q - P = 9,717.09 - 8,000 \approx 1,717.09
\]
### Summary of Interest Earned
- **Investment P (Simple Interest)**: \(I_P = 1,700\)
- **Investment Q (Compound Interest)**: \(I_Q \approx 1,717.09\)
### Conclusion
Investment Q pays more interest after 5 years. The difference in interest earned is:
\[
I_Q - I_P \approx 1,717.09 - 1,700 = 17.09
\]
So, Investment Q pays approximately **$17.09 more** than Investment P after 5 years.
### Investment P (Simple Interest)
The formula for simple interest is:
\[
I = P \times r \times t
\]
where:
- \(I\) is the interest
- \(P\) is the principal amount (initial deposit)
- \(r\) is the annual interest rate (as a decimal)
- \(t\) is the time in years
For Investment P:
- Principal \(P = 8,000\)
- Interest rate \(r = 4.25\% = 0.0425\)
- Time \(t = 5\) years
Calculating the interest:
\[
I_P = 8,000 \times 0.0425 \times 5 = 8,000 \times 0.2125 = 1,700
\]
### Investment Q (Compound Interest)
The formula for compound interest is:
\[
A = P \times (1 + r)^t
\]
where:
- \(A\) is the amount of money accumulated after n years, including interest.
- \(P\) is the principal amount.
- \(r\) is the annual interest rate (as a decimal).
- \(t\) is the time the money is invested for in years.
First, we'll calculate the total amount for Investment Q:
For Investment Q:
- Principal \(P = 8,000\)
- Interest rate \(r = 3.95\% = 0.0395\)
- Time \(t = 5\) years
Calculating the total amount:
\[
A_Q = 8,000 \times (1 + 0.0395)^5
\]
Calculating:
\[
(1 + 0.0395)^5 \approx 1.0395^5 \approx 1.2146 \quad \text{(using a calculator)}
\]
\[
A_Q \approx 8,000 \times 1.2146 \approx 9,717.09
\]
Now, to find the interest earned:
\[
I_Q = A_Q - P = 9,717.09 - 8,000 \approx 1,717.09
\]
### Summary of Interest Earned
- **Investment P (Simple Interest)**: \(I_P = 1,700\)
- **Investment Q (Compound Interest)**: \(I_Q \approx 1,717.09\)
### Conclusion
Investment Q pays more interest after 5 years. The difference in interest earned is:
\[
I_Q - I_P \approx 1,717.09 - 1,700 = 17.09
\]
So, Investment Q pays approximately **$17.09 more** than Investment P after 5 years.