Tom and Antonio both want to open savings accounts earning equal compounded interest today. Tom wants to have $1,000 in his savings account six years from now. Antonio wants to have $1,000 in his savings account three years from now. Tom will need to deposit twice the amount of money today as Antonio.


Is it true or false?

To determine whether Tom will need to deposit twice the amount of money today as Antonio, we need to calculate the present value (PV) of their savings goals, using the compound interest formula. We assume the interest rate is the same for both since the problem states that the accounts earn equal compounded interest.

### Formula for Present Value

The present value needed to reach a future value ($FV$) is calculated using the formula:

\[
PV = \frac{FV}{(1 + r)^n}
\]

Where:
- $FV$ is the future value desired ($1,000 in this case).
- $r$ is the annual interest rate.
- $n$ is the number of years until the future value is reached.

### Present Value for Tom

For Tom, who wants $1,000 in six years:

\[
PV_{\text{Tom}} = \frac{1000}{(1 + r)^6}
\]

### Present Value for Antonio

For Antonio, who wants $1,000 in three years:

\[
PV_{\text{Antonio}} = \frac{1000}{(1 + r)^3}
\]

### Relationship Between Tom's and Antonio's Deposits

According to the problem, Tom's deposit is twice Antonio's deposit:

\[
PV_{\text{Tom}} = 2 \times PV_{\text{Antonio}}
\]

Substituting the expressions for the present values, we get:

\[
\frac{1000}{(1 + r)^6} = 2 \times \frac{1000}{(1 + r)^3}
\]

Simplifying the equation:

\[
\frac{1}{(1 + r)^6} = \frac{2}{(1 + r)^3}
\]

Multiplying both sides by $(1 + r)^6$:

\[
1 = 2 \times (1 + r)^3
\]

Dividing both sides by 2:

\[
(1 + r)^3 = \frac{1}{2}
\]

Taking the cube root of both sides:

\[
1 + r = \left(\frac{1}{2}\right)^{1/3}
\]

Subtracting 1 from both sides gives us:

\[
r = \left(\frac{1}{2}\right)^{1/3} - 1
\]

This shows a valid interest rate $ r $ would exist under these conditions, making it theoretically possible for Tom to deposit twice what Antonio does, assuming this interest rate is achievable.

### Conclusion

The statement that Tom will need to deposit twice the amount of money today as Antonio is **true**, given that a feasible interest rate could exist under these specific conditions.

### Would you like to know more?

Feel free to ask if you have further questions or need additional clarification.

### Relative Questions

1. How do you determine the interest rate needed for specific future values?
2. What happens to the present value if the interest rate increases?
3. How does the duration of the investment period affect the amount needed to invest?
4. How is the formula for present value derived from compound interest?
5. What is the effect of compound interest frequency on the present value?
6. How do present and future value calculations apply to real-life financial planning?
7. What is the impact of varying interest rates on savings and investments?
8. How can you determine the future value from a given present value and interest rate?

### Tip

When comparing investments with different time frames, always account for the time value of money, as longer periods allow more interest to accrue.

Explore the comparison of savings deposits with equal compounded interest. Calculate present values for different time frames to achieve $1,000 savings goals. Determine if Tom needs to deposit twice the amount Antonio does today.

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