Janelys is going to invest in an account paying an interest rate of compounded daily. How much would Janelys
need to invest, to the nearest ten dollars, for the value of the account to reach in 12 years?

Question

Janelys is going to invest in an account paying an interest rate of

compounded daily. How much would Janelys
need to invest, to the nearest ten dollars, for the value of the account to reach

in 12 years?

UpStudy AI · Accepted Answer

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.
Janelys needs to invest approximately \$930 to reach \$1,890 in 12 years with a 5.9% daily compounded interest rate.

Janelys is going to invest in an account paying an interest rate of compounded daily. How much would Janelys
need to invest, to the nearest ten dollars, for the value of the account to reach in 12 years?

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Janelys is going to invest in an account paying an interest rate of compounded daily. How much would Janelys need to invest, to the nearest ten dollars, for the value of the account to reach in 12 years?

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Janelys is going to invest in an account paying an interest rate of compounded daily. How much would Janelys
need to invest, to the nearest ten dollars, for the value of the account to reach in 12 years?