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 The deposits will be made on the same day, and no additional money will be deposited or withdrawn from the accounts.

Answer 1

To compare the two accounts, we can calculate the total amount in each account after a certain number of years.

### For Abraham's Account (Simple Interest)

The formula for calculating the total amount with simple interest is:

\[
A = P(1 + rt)
\]

Where:
- \( A \) is the total amount after \( t \) years.
- \( P \) is the principal amount (initial deposit).
- \( r \) is the annual interest rate (as a decimal).
- \( t \) is the time in years.

For Abraham:
- \( P = 1760 \)
- \( r = 0.06 \)
- \( t \) (we will evaluate for any given number of years)

So,

\[
A = 1760(1 + 0.06t) = 1760 + 105.6t
\]

### For Mahammed's Account (Compound Interest)

The formula for calculating the total amount with compound interest is:

\[
A = P(1 + r)^t
\]

Where:
- \( A \) is the total amount after \( t \) years.
- \( P \) is the principal amount (initial deposit).
- \( r \) is the annual interest rate (as a decimal) for compound interest.
- \( t \) is the time in years.

For Mahammed:
- \( P = 1700 \)
- \( r = 0.08 \)

So,

\[
A = 1700(1 + 0.08)^t = 1700(1.08)^t
\]

### Total Amounts After \( t \) Years

Now we can calculate and compare the amounts for any number of years.

1. **After 1 year**:
- Abraham's Account:
\[
A = 1760(1 + 0.06 \cdot 1) = 1760 \cdot 1.06 = 1865.6
\]
- Mahammed's Account:
\[
A = 1700(1.08)^1 = 1700 \cdot 1.08 = 1836
\]

2. **After 2 years**:
- Abraham's Account:
\[
A = 1760(1 + 0.06 \cdot 2) = 1760 \cdot 1.12 = 1971.2
\]
- Mahammed's Account:
\[
A = 1700(1.08)^2 = 1700 \cdot 1.1664 = 1982.88
\]

3. **After 3 years**:
- Abraham's Account:
\[
A = 1760(1 + 0.06 \cdot 3) = 1760 \cdot 1.18 = 2076.8
\]
- Mahammed's Account:
\[
A = 1700(1.08)^3 = 1700 \cdot 1.259712 = 2131.5
\]

### Summary of Amounts After 1, 2, and 3 Years

- **After 1 year**:
- Abraham: $1865.60
- Mahammed: $1836.00

- **After 2 years**:
- Abraham: $1971.20
- Mahammed: $1982.88

- **After 3 years**:
- Abraham: $2076.80
- Mahammed: $2131.51

When looking at these calculations, we can see that Mahammed's account will start out lower but will grow faster due to the higher interest rate and the compounding effect. Over time, Mahammed's investment will surpass Abraham's account. You can continue this pattern to see the amounts for additional years if needed.