Math / Algebra

QuestionUse the compound interest formula to compute the total amount accumulated and the interest earned. $\$ 2500$ for 5 years at $3.2 \%$ compounded monthly
The total amount accumulated after 5 years is $\$$ $\square$ (Round to the nearest cent as needed.) The amount of interest earned is $\$$ $\square$ (Round to the nearest cent as needed.)

Studdy Solution

STEP 1

What is this asking? How much money will we have after putting $\$2500$ in a savings account for 5 years with a monthly interest rate of 3.2%, and how much of that is pure profit? Watch out! The interest rate is given annually, but it's compounded monthly, so we need to adjust it accordingly.
Don't forget that the question asks for two values: the total amount and the interest earned.

STEP 2

1. Calculate the monthly interest rate.
2. Calculate the total number of times the interest is compounded.
3. Calculate the total amount accumulated.
4. Calculate the interest earned.

STEP 3

We're given an annual interest rate of $3.2\%$ , but since the interest is compounded monthly, we need to find the monthly interest rate.
To do this, we divide the annual rate by 12, the number of months in a year.

STEP 4

So, the monthly interest rate is $\frac{3.2\%}{12} = \frac{0.032}{12} \approx 0.00266667$ .
We'll keep a few extra decimal places for accuracy and round at the very end.

STEP 5

The problem states that the money is invested for 5 years and compounded monthly.
Since there are 12 months in a year, the interest is compounded $5 \cdot 12 = \textbf{60}$ times.

STEP 6

Now, we can use the compound interest formula: $A = P(1 + r)^n$ Where: $A$ is the total amount accumulated. $P$ is the principal amount (initial investment), which is $\$2500$ . $r$ is the monthly interest rate, which we calculated to be approximately $0.00266667$ . $n$ is the number of times the interest is compounded, which is 60.

STEP 7

Let's plug in the values: $A = 2500(1 + 0.00266667)^{60}$ $A \approx 2500(1.00266667)^{60}$ $A \approx 2500(1.173035)$ $A \approx 2932.59$

STEP 8

So, the total amount accumulated after 5 years is approximately $\$\textbf{2932.59}$ .

STEP 9

To find the interest earned, we subtract the principal amount from the total amount accumulated: $Interest = A - P$ $Interest = 2932.59 - 2500$ $Interest = 432.59$

STEP 10

Therefore, the interest earned is $\$\textbf{432.59}$ .

STEP 11

The total amount accumulated after 5 years is $\$2932.59$ . The amount of interest earned is $\$432.59$ .

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Studdy Solution

STEP 1

STEP 2

1. Calculate the monthly interest rate.
2. Calculate the total number of times the interest is compounded.
3. Calculate the total amount accumulated.
4. Calculate the interest earned.

STEP 3

We're given an annual interest rate of $3.2\%$ , but since the interest is compounded monthly, we need to find the monthly interest rate.
To do this, we divide the annual rate by 12, the number of months in a year.

STEP 4

So, the monthly interest rate is $\frac{3.2\%}{12} = \frac{0.032}{12} \approx 0.00266667$ .
We'll keep a few extra decimal places for accuracy and round at the very end.

STEP 5

The problem states that the money is invested for 5 years and compounded monthly.
Since there are 12 months in a year, the interest is compounded $5 \cdot 12 = \textbf{60}$ times.

STEP 6

STEP 7

Let's plug in the values: $A = 2500(1 + 0.00266667)^{60}$ $A \approx 2500(1.00266667)^{60}$ $A \approx 2500(1.173035)$ $A \approx 2932.59$

STEP 8

So, the total amount accumulated after 5 years is approximately $\$\textbf{2932.59}$ .

STEP 9

To find the interest earned, we subtract the principal amount from the total amount accumulated: $Interest = A - P$ $Interest = 2932.59 - 2500$ $Interest = 432.59$

STEP 10

Therefore, the interest earned is $\$\textbf{432.59}$ .

STEP 11

The total amount accumulated after 5 years is $\$2932.59$ . The amount of interest earned is $\$432.59$ .

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Studdy Solution

STEP 1

STEP 2

1. Calculate the monthly interest rate.2. Calculate the total number of times the interest is compounded.3. Calculate the total amount accumulated.4. Calculate the interest earned.

STEP 3

We're given an **annual** interest rate of 3.2%3.2\%3.2%, but since the interest is compounded **monthly**, we need to find the **monthly** interest rate.To do this, we **divide** the annual rate by **12**, the number of months in a year.

STEP 4

So, the monthly interest rate is 3.2%12=0.03212≈0.00266667\frac{3.2\%}{12} = \frac{0.032}{12} \approx 0.00266667123.2%​=120.032​≈0.00266667.We'll keep a few extra decimal places for accuracy and round at the very end.

STEP 5

The problem states that the money is invested for **5 years** and compounded **monthly**.Since there are **12** months in a year, the interest is compounded 5⋅12=605 \cdot 12 = \textbf{60}5⋅12=60 times.

STEP 6

STEP 7

Let's plug in the values: A=2500(1+0.00266667)60A = 2500(1 + 0.00266667)^{60}A=2500(1+0.00266667)60 A≈2500(1.00266667)60A \approx 2500(1.00266667)^{60}A≈2500(1.00266667)60A≈2500(1.173035)A \approx 2500(1.173035)A≈2500(1.173035)A≈2932.59A \approx 2932.59A≈2932.59

STEP 8

So, the total amount accumulated after 5 years is approximately $2932.59\$\textbf{2932.59}$2932.59.

STEP 9

To find the interest earned, we **subtract** the **principal amount** from the **total amount accumulated**: Interest=A−PInterest = A - PInterest=A−P Interest=2932.59−2500Interest = 2932.59 - 2500Interest=2932.59−2500Interest=432.59Interest = 432.59Interest=432.59

STEP 10

Therefore, the interest earned is $432.59\$\textbf{432.59}$432.59.

STEP 11

The total amount accumulated after 5 years is $2932.59\$2932.59$2932.59. The amount of interest earned is $432.59\$432.59$432.59.

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1. Calculate the monthly interest rate.
2. Calculate the total number of times the interest is compounded.
3. Calculate the total amount accumulated.
4. Calculate the interest earned.

We're given an annual interest rate of $3.2\%$ , but since the interest is compounded monthly, we need to find the monthly interest rate.
To do this, we divide the annual rate by 12, the number of months in a year.

So, the monthly interest rate is $\frac{3.2\%}{12} = \frac{0.032}{12} \approx 0.00266667$ .
We'll keep a few extra decimal places for accuracy and round at the very end.

The problem states that the money is invested for 5 years and compounded monthly.
Since there are 12 months in a year, the interest is compounded $5 \cdot 12 = \textbf{60}$ times.

Let's plug in the values: $A = 2500(1 + 0.00266667)^{60}$ $A \approx 2500(1.00266667)^{60}$ $A \approx 2500(1.173035)$ $A \approx 2932.59$

So, the total amount accumulated after 5 years is approximately $\$\textbf{2932.59}$ .

To find the interest earned, we subtract the principal amount from the total amount accumulated: $Interest = A - P$ $Interest = 2932.59 - 2500$ $Interest = 432.59$

Therefore, the interest earned is $\$\textbf{432.59}$ .

The total amount accumulated after 5 years is $\$2932.59$ . The amount of interest earned is $\$432.59$ .