Math / Algebra

QuestionCalculate the compound amount from the given data. principal $=\$ 500$ , compounded monthly, 9 years, annual rate $=4 \%$

Studdy Solution

STEP 1

1. The principal amount is $P = \$500$ .
2. The interest rate is compounded monthly.
3. The time period is $t = 9$ years.
4. The annual interest rate is $r = 4\%$ .

STEP 2

1. Determine the monthly interest rate from the annual rate.
2. Calculate the total number of compounding periods.
3. Use the compound interest formula to find the compound amount.

STEP 3

Convert the annual interest rate to a monthly interest rate. The annual rate is $r = 4\% = 0.04$ .
$r_{\text{monthly}} = \frac{r}{12} = \frac{0.04}{12} = 0.003333\ldots$

STEP 4

Calculate the total number of compounding periods. There are 12 months in a year, so for 9 years:
$n = 12 \times 9 = 108$

STEP 5

Use the compound interest formula to find the compound amount. The formula is:
$A = P \left(1 + \frac{r}{n}\right)^{nt}$
Here, $P = 500$ , $r_{\text{monthly}} = 0.003333\ldots$ , and $n = 108$ .
$A = 500 \left(1 + 0.003333\ldots\right)^{108}$

STEP 6

Simplify the expression inside the parentheses:
$1 + 0.003333\ldots = 1.003333\ldots$

STEP 7

Raise the base to the power of the total number of compounding periods:
$A = 500 \left(1.003333\ldots\right)^{108}$

STEP 8

Calculate the value of $\left(1.003333\ldots\right)^{108}$ :
$\left(1.003333\ldots\right)^{108} \approx 1.431292$

STEP 9

Multiply the principal by the calculated factor to find the compound amount:
$A = 500 \times 1.431292 \approx 715.646$
The compound amount after 9 years is approximately \$715.65.

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QuestionCalculate the compound amount from the given data. principal $=\$ 500$ , compounded monthly, 9 years, annual rate $=4 \%$

Studdy Solution

STEP 1

1. The principal amount is $P = \$500$ .
2. The interest rate is compounded monthly.
3. The time period is $t = 9$ years.
4. The annual interest rate is $r = 4\%$ .

STEP 2

1. Determine the monthly interest rate from the annual rate.
2. Calculate the total number of compounding periods.
3. Use the compound interest formula to find the compound amount.

STEP 3

Convert the annual interest rate to a monthly interest rate. The annual rate is $r = 4\% = 0.04$ .
$r_{\text{monthly}} = \frac{r}{12} = \frac{0.04}{12} = 0.003333\ldots$

STEP 4

Calculate the total number of compounding periods. There are 12 months in a year, so for 9 years:
$n = 12 \times 9 = 108$

STEP 5

Use the compound interest formula to find the compound amount. The formula is:
$A = P \left(1 + \frac{r}{n}\right)^{nt}$
Here, $P = 500$ , $r_{\text{monthly}} = 0.003333\ldots$ , and $n = 108$ .
$A = 500 \left(1 + 0.003333\ldots\right)^{108}$

STEP 6

Simplify the expression inside the parentheses:
$1 + 0.003333\ldots = 1.003333\ldots$

STEP 7

Raise the base to the power of the total number of compounding periods:
$A = 500 \left(1.003333\ldots\right)^{108}$

STEP 8

Calculate the value of $\left(1.003333\ldots\right)^{108}$ :
$\left(1.003333\ldots\right)^{108} \approx 1.431292$

STEP 9

Multiply the principal by the calculated factor to find the compound amount:
$A = 500 \times 1.431292 \approx 715.646$
The compound amount after 9 years is approximately \$715.65.

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QuestionCalculate the compound amount from the given data. principal =$500=\$ 500=$500, compounded monthly, 9 years, annual rate =4%=4 \%=4%

Studdy Solution

STEP 1

1. The principal amount is P=$500P = \$500P=$500.2. The interest rate is compounded monthly.3. The time period is t=9t = 9t=9 years.4. The annual interest rate is r=4%r = 4\%r=4%.

STEP 2

1. Determine the monthly interest rate from the annual rate.2. Calculate the total number of compounding periods.3. Use the compound interest formula to find the compound amount.

STEP 3

Convert the annual interest rate to a monthly interest rate. The annual rate is r=4%=0.04r = 4\% = 0.04r=4%=0.04.rmonthly=r12=0.0412=0.003333… r_{\text{monthly}} = \frac{r}{12} = \frac{0.04}{12} = 0.003333\ldots rmonthly​=12r​=120.04​=0.003333…

STEP 4

Calculate the total number of compounding periods. There are 12 months in a year, so for 9 years:n=12×9=108 n = 12 \times 9 = 108 n=12×9=108

STEP 5

STEP 6

Simplify the expression inside the parentheses:1+0.003333…=1.003333… 1 + 0.003333\ldots = 1.003333\ldots 1+0.003333…=1.003333…

STEP 7

Raise the base to the power of the total number of compounding periods:A=500(1.003333…)108 A = 500 \left(1.003333\ldots\right)^{108} A=500(1.003333…)108

STEP 8

Calculate the value of (1.003333…)108 \left(1.003333\ldots\right)^{108} (1.003333…)108:(1.003333…)108≈1.431292 \left(1.003333\ldots\right)^{108} \approx 1.431292 (1.003333…)108≈1.431292

STEP 9

Multiply the principal by the calculated factor to find the compound amount:A=500×1.431292≈715.646 A = 500 \times 1.431292 \approx 715.646 A=500×1.431292≈715.646The compound amount after 9 years is approximately \$715.65.

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QuestionCalculate the compound amount from the given data. principal $=\$ 500$ , compounded monthly, 9 years, annual rate $=4 \%$

1. The principal amount is $P = \$500$ .
2. The interest rate is compounded monthly.
3. The time period is $t = 9$ years.
4. The annual interest rate is $r = 4\%$ .

1. Determine the monthly interest rate from the annual rate.
2. Calculate the total number of compounding periods.
3. Use the compound interest formula to find the compound amount.

Convert the annual interest rate to a monthly interest rate. The annual rate is $r = 4\% = 0.04$ .
$r_{\text{monthly}} = \frac{r}{12} = \frac{0.04}{12} = 0.003333\ldots$

Calculate the total number of compounding periods. There are 12 months in a year, so for 9 years:
$n = 12 \times 9 = 108$

Simplify the expression inside the parentheses:
$1 + 0.003333\ldots = 1.003333\ldots$

Raise the base to the power of the total number of compounding periods:
$A = 500 \left(1.003333\ldots\right)^{108}$

Calculate the value of $\left(1.003333\ldots\right)^{108}$ :
$\left(1.003333\ldots\right)^{108} \approx 1.431292$

Multiply the principal by the calculated factor to find the compound amount:
$A = 500 \times 1.431292 \approx 715.646$
The compound amount after 9 years is approximately \$715.65.