Janelys is going to invest in an account paying an interest rate of \( 5.9 \% \) compounded daily. How much would Janelys need to invest, to the nearest ten dollars, for the value of the account to reach \( \$ 1,890 \) in 12 years?

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.