If you need $53,000 15 years from now, what is the minimum amount of money you need to deposit into a bank account that pays 2% annual interest, compounded:

a) Annually
b) monthly
c) daily (365 days per year)

This is a compound interest problem where we are looking for the **present value (PV)** of an amount that will grow to $53,000 after 15 years. The formula for compound interest is:

\[
A = P \left(1 + \frac{r}{n}\right)^{nt}
\]

Where:
- $ A $ is the future value ($53,000),
- $ P $ is the present value (the amount we want to find),
- $ r $ is the annual interest rate (2%, or 0.02),
- $ n $ is the number of times the interest is compounded per year (depends on the compounding frequency),
- $ t $ is the number of years (15).

We can rearrange this formula to solve for $ P $ (the present value):

\[
P = \frac{A}{\left(1 + \frac{r}{n}\right)^{nt}}
\]

Now let's calculate for each case:

### a) Compounded Annually ($ n = 1 $):

\[
P = \frac{53,000}{\left(1 + \frac{0.02}{1}\right)^{1 \times 15}} = \frac{53,000}{(1.02)^{15}}
\]

### b) Compounded Monthly ($ n = 12 $):

\[
P = \frac{53,000}{\left(1 + \frac{0.02}{12}\right)^{12 \times 15}} = \frac{53,000}{\left(1 + \frac{0.02}{12}\right)^{180}}
\]

### c) Compounded Daily ($ n = 365 $):

\[
P = \frac{53,000}{\left(1 + \frac{0.02}{365}\right)^{365 \times 15}} = \frac{53,000}{\left(1 + \frac{0.02}{365}\right)^{5475}}
\]

I will now compute these values.The minimum amount of money you need to deposit to reach $53,000 in 15 years, given a 2% annual interest rate, is:

a) **Compounded Annually:** $39,379.78  
b) **Compounded Monthly:** $39,273.17  
c) **Compounded Daily:** $39,263.69  

As you can see, more frequent compounding slightly reduces the initial deposit needed due to the compounding effect.

Would you like more details on the calculation, or do you have any other questions?

Here are 5 related questions:
1. What would the present value be if the interest rate was 3% instead of 2%?
2. How does the compounding frequency affect the final amount over different periods?
3. What happens to the present value if the time is reduced to 10 years instead of 15?
4. How is continuous compounding different from daily compounding in this context?
5. How does the future value change if you deposit a fixed amount monthly instead of a lump sum?

**Tip:** Increasing the compounding frequency will slightly decrease the required present value for a fixed future sum due to more frequent application of interest.

If you need $53,000 15 years from now, what is the minimum amount of money you need to deposit into a bank account that pays 2% annual interest, compounded: a) Annually b) Monthly c) Daily (365 days per year)?

Discover how to calculate the minimum deposit required to reach $53,000 in 15 years with a 2% interest rate, compounded annually, monthly, and daily. This problem explains key financial math concepts such as compound interest and present value calculations, suitable for Grades 10-12.

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