How long must someone invest $400 to have it grow to a total amount of $1000, if it is put in an account earning 7% interest compounded quarterly? 
N =
I%=
PV=
PMT=
FV=
P/Y=
C/Y= 


To solve for how long it takes to grow an investment of $400 to $1000 at an interest rate of 7% compounded quarterly, we can use the formula for compound interest:

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

Where:
- $ FV $ = future value ($1000)
- $ PV $ = present value ($400)
- $ r $ = annual interest rate (0.07)
- $ n $ = number of times interest is compounded per year (4 for quarterly)
- $ t $ = number of years

Given the values:
- $ FV = 1000 $
- $ PV = 400 $
- $ r = 0.07 $
- $ n = 4 $

### Step 1: Set up the equation
\[
1000 = 400 \times \left(1 + \frac{0.07}{4}\right)^{4t}
\]

### Step 2: Simplify the equation
Calculate the term $ \frac{0.07}{4} = 0.0175 $.

Thus, the equation becomes:
\[
1000 = 400 \times \left(1.0175\right)^{4t}
\]

### Step 3: Divide both sides by 400
\[
2.5 = \left(1.0175\right)^{4t}
\]

### Step 4: Take the natural logarithm of both sides
\[
\ln(2.5) = \ln\left(\left(1.0175\right)^{4t}\right)
\]
\[
\ln(2.5) = 4t \cdot \ln(1.0175)
\]

### Step 5: Solve for $ t $
\[
t = \frac{\ln(2.5)}{4 \cdot \ln(1.0175)}
\]

### Step 6: Calculate the values
Calculating:
\[
\ln(2.5) \approx 0.9163
\]
\[
\ln(1.0175) \approx 0.0173
\]
Substituting these into the equation:
\[
t = \frac{0.9163}{4 \cdot 0.0173} \approx \frac{0.9163}{0.0692} \approx 13.24
\]

Thus, it will take approximately **13.24 years** for the investment to grow from $400 to $1000 at a 7% interest rate compounded quarterly.

### Summary of the Values:
- $ N $ = total number of periods (years)
- $ I\% $ = annual interest rate (7%)
- $ PV $ = present value ($400)
- $ PMT $ = payment per period (not applicable here since it's a single investment)
- $ FV $ = future value ($1000)
- $ P/Y $ = payments per year (1 since no additional payments)
- $ C/Y $ = compounding periods per year (4)

Would you like further details or have any questions? Here are some related questions to consider:

1. How would the time change if the interest rate increased to 8%?
2. What if the interest were compounded annually instead of quarterly?
3. How much total interest would be earned over the investment period?
4. What would be the future value if $400 were invested for 15 years at the same interest rate?
5. How does the frequency of compounding affect the total amount earned? 

**Tip:** Always consider different compounding frequencies when planning investments, as they can significantly impact growth over time.

How long must someone invest $400 to have it grow to a total amount of $1000, if it is put in an account earning 7% interest compounded quarterly?

Discover how to calculate the time needed for a $400 investment to grow to $1000 at a 7% interest rate compounded quarterly. This guide explains the compound interest formula and provides a detailed solution, making it suitable for high school students.

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