Answer
Janelys needs to invest approximately \$930 to reach \$1,890 in 12 years with a 5.9% daily compounded interest rate.
Solution
To find out how much Janelys needs to invest today (the present value \( P \)) to reach a future value \( A \) of \$1,890 in 12 years with an interest rate of \( 5.9\% \) compounded daily, we can use the formula for compound interest:
\[
A = P \left(1 + \frac{r}{n}\right)^{nt}
\]
Where:
- \( A \) is the future value of the investment/loan, including interest.
- \( P \) is the principal investment amount (the initial deposit or loan amount).
- \( r \) is the annual interest rate (decimal).
- \( n \) is the number of times that interest is compounded per year.
- \( t \) is the number of years the money is invested or borrowed.
Given:
- \( A = 1890 \)
- \( r = 5.9\% = 0.059 \)
- \( n = 365 \) (since the interest is compounded daily)
- \( t = 12 \)
We need to rearrange the formula to solve for \( P \):
\[
P = \frac{A}{\left(1 + \frac{r}{n}\right)^{nt}}
\]
Now, let's calculate \( P \) step by step.
1. Calculate \( \frac{r}{n} \).
2. Calculate \( nt \).
3. Calculate \( \left(1 + \frac{r}{n}\right)^{nt} \).
4. Finally, calculate \( P \).
Let's perform these calculations.
### Step 1: Calculate \( \frac{r}{n} \)
\[
\frac{r}{n} = \frac{0.059}{365}
\]
### Step 2: Calculate \( nt \)
\[
nt = 365 \times 12
\]
### Step 3: Calculate \( \left(1 + \frac{r}{n}\right)^{nt} \)
Now we can calculate \( P \) using the values from the previous steps.
Let's perform these calculations.
Calculate the value by following steps:
- step0: Calculate:
\(365\times 12\)
- step1: Multiply the numbers:
\(4380\)
Calculate or simplify the expression \( 0.059/365 \).
Calculate the value by following steps:
- step0: Calculate:
\(\frac{0.059}{365}\)
- step1: Convert the expressions:
\(\frac{\frac{59}{1000}}{365}\)
- step2: Multiply by the reciprocal:
\(\frac{59}{1000}\times \frac{1}{365}\)
- step3: Multiply the fractions:
\(\frac{59}{1000\times 365}\)
- step4: Multiply:
\(\frac{59}{365000}\)
Calculate or simplify the expression \( (1 + 0.000162)^(4380) \).
Calculate the value by following steps:
- step0: Calculate:
\(\left(1+0.000162\right)^{4380}\)
- step1: Add the numbers:
\(1.000162^{4380}\)
- step2: Convert the expressions:
\(\left(\frac{500081}{500000}\right)^{4380}\)
- step3: Use the properties of exponents:
\(\frac{500081^{4380}}{500000^{4380}}\)
Calculate or simplify the expression \( 1890/(2.03298) \).
Calculate the value by following steps:
- step0: Calculate:
\(\frac{1890}{2.03298}\)
- step1: Convert the expressions:
\(\frac{1890}{\frac{101649}{50000}}\)
- step2: Multiply by the reciprocal:
\(1890\times \frac{50000}{101649}\)
- step3: Reduce the numbers:
\(630\times \frac{50000}{33883}\)
- step4: Multiply:
\(\frac{630\times 50000}{33883}\)
- step5: Multiply:
\(\frac{31500000}{33883}\)
Now that we have completed the calculations, let's summarize the steps and results:
1. **Calculate \( \frac{r}{n} \)**:
\[
\frac{r}{n} = \frac{0.059}{365} \approx 0.000162
\]
2. **Calculate \( nt \)**:
\[
nt = 365 \times 12 = 4380
\]
3. **Calculate \( \left(1 + \frac{r}{n}\right)^{nt} \)**:
\[
\left(1 + 0.000162\right)^{4380} \approx 2.03298
\]
4. **Calculate \( P \)**:
\[
P = \frac{1890}{2.03298} \approx 929.67
\]
To find the amount Janelys needs to invest, we round \( 929.67 \) to the nearest ten dollars:
\[
P \approx 930
\]
Thus, Janelys would need to invest approximately **\$930** to reach a value of \$1,890 in 12 years at an interest rate of \( 5.9\% \) compounded daily.
Reviewed and approved by the UpStudy tutoring team
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