Question
Abraham will deposit $1,760 in an account that earns 6% simple interest every year. His brother, Mahammed, will deposit $1,700 in an account that earns 8% interest compounded annually. The deposits will be made on the same day, and no additional money will be deposited or withdrawn from the accounts.
Answers
GPT-4o mini
To compare Abraham's and Mahammed's accounts, we will calculate the total amount in each account after a certain number of years using the formulas for simple interest and compound interest.
### Abraham's Account (Simple Interest)
The formula for simple interest is:
\[
A = P(1 + rt)
\]
Where:
- \(A\) is the total 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 (decimal).
- \(t\) is the time the money is invested or borrowed for in years.
For Abraham:
- \(P = 1760\)
- \(r = 0.06\) (6%)
- \(t = t\) (in years)
Plugging in the values, we get:
\[
A = 1760(1 + 0.06t) = 1760(1 + 0.06t)
\]
### Mahammed's Account (Compound Interest)
The formula for compound interest is:
\[
A = P(1 + r)^t
\]
Where:
- \(A\) is the amount of money accumulated after n years, including interest.
- \(P\) is the principal amount.
- \(r\) is the annual interest rate (decimal).
- \(t\) is the number of years the money is invested or borrowed for.
For Mahammed:
- \(P = 1700\)
- \(r = 0.08\) (8%)
- \(t = t\)
Plugging in the values, we get:
\[
A = 1700(1 + 0.08)^t = 1700(1.08)^t
\]
### Example Calculation
Let's calculate and compare the total amount after 5 years (you can adjust the number of years according to your needs).
#### After 5 Years:
**Abraham's Amount:**
\[
A = 1760(1 + 0.06 \times 5) = 1760(1 + 0.30) = 1760 \times 1.30 = 2288
\]
**Mahammed's Amount:**
\[
A = 1700(1.08)^5
\]
Calculating \( (1.08)^5 \):
\[
(1.08)^5 \approx 1.4693
\]
Now calculate:
\[
A \approx 1700 \times 1.4693 \approx 2507.81
\]
### Summary After 5 Years
- Abraham's total amount: **$2,288**
- Mahammed's total amount: **$2,507.81**
Based on this calculation, Mahammed's account earns more money due to the higher interest rate and compounding effect over the 5-year period. You can apply the same formulas to compute for different years as needed!
### Abraham's Account (Simple Interest)
The formula for simple interest is:
\[
A = P(1 + rt)
\]
Where:
- \(A\) is the total 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 (decimal).
- \(t\) is the time the money is invested or borrowed for in years.
For Abraham:
- \(P = 1760\)
- \(r = 0.06\) (6%)
- \(t = t\) (in years)
Plugging in the values, we get:
\[
A = 1760(1 + 0.06t) = 1760(1 + 0.06t)
\]
### Mahammed's Account (Compound Interest)
The formula for compound interest is:
\[
A = P(1 + r)^t
\]
Where:
- \(A\) is the amount of money accumulated after n years, including interest.
- \(P\) is the principal amount.
- \(r\) is the annual interest rate (decimal).
- \(t\) is the number of years the money is invested or borrowed for.
For Mahammed:
- \(P = 1700\)
- \(r = 0.08\) (8%)
- \(t = t\)
Plugging in the values, we get:
\[
A = 1700(1 + 0.08)^t = 1700(1.08)^t
\]
### Example Calculation
Let's calculate and compare the total amount after 5 years (you can adjust the number of years according to your needs).
#### After 5 Years:
**Abraham's Amount:**
\[
A = 1760(1 + 0.06 \times 5) = 1760(1 + 0.30) = 1760 \times 1.30 = 2288
\]
**Mahammed's Amount:**
\[
A = 1700(1.08)^5
\]
Calculating \( (1.08)^5 \):
\[
(1.08)^5 \approx 1.4693
\]
Now calculate:
\[
A \approx 1700 \times 1.4693 \approx 2507.81
\]
### Summary After 5 Years
- Abraham's total amount: **$2,288**
- Mahammed's total amount: **$2,507.81**
Based on this calculation, Mahammed's account earns more money due to the higher interest rate and compounding effect over the 5-year period. You can apply the same formulas to compute for different years as needed!