Suppose you want to save $7,500 to go on a dream vacation in three years.

You can save for your vacation by either depositing one single lump of money into a savings account and letting it accumulate interest or by making regular monthly payments.

How much money, as a one-time deposit,  PV
 , would you need to deposit into an account that earns 1.5% compounded monthly to earn a future value,  FV
 , of $7,500 in three years?

This amount can be found by using algebra to rearrange the function,    FV=PV(1+rn)^(nY)
 , so that  FV
  becomes an input variable and  PV
  becomes the output variable.

PV=FV(1+rn)^(−nY)
 

Use this function to find the amount needed as a one-time deposit to earn $7,500 in 3 years.


To solve this problem, we will use the formula for the present value (PV) of a lump sum, given the future value (FV), interest rate ($r$), number of compounding periods per year ($n$), and the number of years ($Y$).

### The formula:
\[
PV = \frac{FV}{\left(1 + \frac{r}{n}\right)^{nY}}
\]
Where:
- $FV = 7500$ (future value you want in 3 years)
- $r = 0.015$ (1.5% annual interest rate)
- $n = 12$ (compounded monthly)
- $Y = 3$ (number of years)

### Step-by-step calculation:

1. First, calculate $\frac{r}{n}$:
\[
\frac{0.015}{12} = 0.00125
\]

2. Then calculate $nY$, which is the total number of compounding periods:
\[
nY = 12 \times 3 = 36
\]

3. Now compute $\left(1 + \frac{r}{n}\right)^{nY}$:
\[
\left(1 + 0.00125\right)^{36} = 1.046011
\]

4. Finally, calculate $PV$:
\[
PV = \frac{7500}{1.046011} \approx 7171.28
\]

### Result:
You would need to deposit **$7,171.28** as a one-time deposit to accumulate $7,500 in 3 years at 1.5% interest compounded monthly.

Would you like any further details or clarifications?

### Related questions:
1. How much interest will you have earned after 3 years?
2. What if the interest rate was compounded quarterly instead of monthly?
3. How would the required deposit change if you had only 2 years to save?
4. What if the interest rate was 2% instead of 1.5%?
5. How much would you need to save monthly to reach $7,500 in 3 years at 1.5% interest?

### Tip:
The more frequently interest compounds, the less you need to deposit initially to reach a target future value!

Suppose you want to save $7,500 to go on a dream vacation in three years. You can save for your vacation by either depositing one single lump of money into a savings account and letting it accumulate interest or by making regular monthly payments. How much money, as a one-time deposit, would you need to deposit into an account that earns 1.5% compounded monthly to earn a future value of $7,500 in three years? Use the function FV=PV(1+rn)^(nY) to find the amount needed as a one-time deposit.

Learn how to calculate the present value (PV) required to save $7,500 for a vacation in 3 years using a 1.5% interest rate compounded monthly. This example uses the compound interest formula FV = PV(1 + r/n)^(nY) and is suitable for high school students studying algebra and finance.

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