QuestionBill wants to save for a café by depositing monthly into an annuity at $3.6\%$ interest. How much to deposit monthly for \$25,000 in 9 years? Round to the nearest cent.

Studdy Solution

STEP 1

Assumptions1. The total value of the annuity after9 years is $25,000. The interest rate is3.6% per year, compounded monthly3. Deposits are made at the end of each month4. The time for saving is9 years5. We use the formula for the future value of an ordinary annuity

STEP 2

First, we need to convert the annual interest rate to a monthly interest rate. We do this by dividing the annual interest rate by the number of months in a year.
$r = Annual\, interest\, rate / Number\, of\, months\, in\, a\, year$

STEP 3

Now, plug in the given values for the annual interest rate and the number of months in a year to calculate the monthly interest rate.
$r =3.6\% /12$

STEP 4

Convert the percentage to a decimal value.
$3.6\% =0.036$ $r =0.036 /12$

STEP 5

Calculate the monthly interest rate.
$r =0.036 /12 =0.003$

STEP 6

Next, we need to find the total number of deposits made. We do this by multiplying the number of years by the number of months in a year.
$n = Number\, of\, years \\times Number\, of\, months\, in\, a\, year$

STEP 7

Now, plug in the given values for the number of years and the number of months in a year to calculate the total number of deposits.
$n =9 \\times12$

STEP 8

Calculate the total number of deposits.
$n = \\times12 =108$

STEP 9

Now, we can use the formula for the future value of an ordinary annuity to find the monthly deposit. The formula is $MT = FV / [( + r)^n -] / r$ Where- PM is the monthly deposit- FV is the future value of the annuity- r is the monthly interest rate- n is the total number of deposits

STEP 10

Now, plug in the given values for the future value, the monthly interest rate, and the total number of deposits to calculate the monthly deposit.
$MT = \$25,000 / [( +0.003)^{108} -] /0.003$

STEP 11

Calculate the monthly deposit.
$MT = \$25,000 / [( +0.003)^{108} -] /0.003 = \$198.79$ Bill needs to deposit $198.79 each month.

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QuestionBill wants to save for a café by depositing monthly into an annuity at $3.6\%$ interest. How much to deposit monthly for \$25,000 in 9 years? Round to the nearest cent.

Studdy Solution

STEP 1

Assumptions1. The total value of the annuity after9 years is $25,000. The interest rate is3.6% per year, compounded monthly3. Deposits are made at the end of each month4. The time for saving is9 years5. We use the formula for the future value of an ordinary annuity

STEP 2

First, we need to convert the annual interest rate to a monthly interest rate. We do this by dividing the annual interest rate by the number of months in a year.
$r = Annual\, interest\, rate / Number\, of\, months\, in\, a\, year$

STEP 3

Now, plug in the given values for the annual interest rate and the number of months in a year to calculate the monthly interest rate.
$r =3.6\% /12$

STEP 4

Convert the percentage to a decimal value.
$3.6\% =0.036$ $r =0.036 /12$

STEP 5

Calculate the monthly interest rate.
$r =0.036 /12 =0.003$

STEP 6

Next, we need to find the total number of deposits made. We do this by multiplying the number of years by the number of months in a year.
$n = Number\, of\, years \\times Number\, of\, months\, in\, a\, year$

STEP 7

Now, plug in the given values for the number of years and the number of months in a year to calculate the total number of deposits.
$n =9 \\times12$

STEP 8

Calculate the total number of deposits.
$n = \\times12 =108$

STEP 9

Now, we can use the formula for the future value of an ordinary annuity to find the monthly deposit. The formula is $MT = FV / [( + r)^n -] / r$ Where- PM is the monthly deposit- FV is the future value of the annuity- r is the monthly interest rate- n is the total number of deposits

STEP 10

Now, plug in the given values for the future value, the monthly interest rate, and the total number of deposits to calculate the monthly deposit.
$MT = \$25,000 / [( +0.003)^{108} -] /0.003$

STEP 11

Calculate the monthly deposit.
$MT = \$25,000 / [( +0.003)^{108} -] /0.003 = \$198.79$ Bill needs to deposit $198.79 each month.

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QuestionBill wants to save for a café by depositing monthly into an annuity at 3.6%3.6\%3.6% interest. How much to deposit monthly for \$25,000 in 9 years? Round to the nearest cent.

Studdy Solution

STEP 1

Assumptions1. The total value of the annuity after9 years is $25,000. The interest rate is3.6% per year, compounded monthly3. Deposits are made at the end of each month4. The time for saving is9 years5. We use the formula for the future value of an ordinary annuity

STEP 2

STEP 3

Now, plug in the given values for the annual interest rate and the number of months in a year to calculate the monthly interest rate.r=3.6%/12r =3.6\% /12r=3.6%/12

STEP 4

Convert the percentage to a decimal value.3.6%=0.0363.6\% =0.0363.6%=0.036r=0.036/12r =0.036 /12r=0.036/12

STEP 5

Calculate the monthly interest rate.r=0.036/12=0.003r =0.036 /12 =0.003r=0.036/12=0.003

STEP 6

Next, we need to find the total number of deposits made. We do this by multiplying the number of years by the number of months in a year.n=Number of yearstimesNumber of months in a yearn = Number\, of\, years \\times Number\, of\, months\, in\, a\, yearn=NumberofyearstimesNumberofmonthsinayear

STEP 7

Now, plug in the given values for the number of years and the number of months in a year to calculate the total number of deposits.n=9times12n =9 \\times12n=9times12

STEP 8

Calculate the total number of deposits.n=times12=108n = \\times12 =108n=times12=108

STEP 9

STEP 10

Now, plug in the given values for the future value, the monthly interest rate, and the total number of deposits to calculate the monthly deposit.MT=$25,000/[(+0.003)108−]/0.003MT = \$25,000 / [( +0.003)^{108} -] /0.003MT=$25,000/[(+0.003)108−]/0.003

STEP 11

Calculate the monthly deposit.MT=$25,000/[(+0.003)108−]/0.003=$198.79MT = \$25,000 / [( +0.003)^{108} -] /0.003 = \$198.79MT=$25,000/[(+0.003)108−]/0.003=$198.79Bill needs to deposit $198.79 each month.

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QuestionBill wants to save for a café by depositing monthly into an annuity at $3.6\%$ interest. How much to deposit monthly for \$25,000 in 9 years? Round to the nearest cent.

Now, plug in the given values for the annual interest rate and the number of months in a year to calculate the monthly interest rate.
$r =3.6\% /12$

Convert the percentage to a decimal value.
$3.6\% =0.036$ $r =0.036 /12$

Calculate the monthly interest rate.
$r =0.036 /12 =0.003$

Next, we need to find the total number of deposits made. We do this by multiplying the number of years by the number of months in a year.
$n = Number\, of\, years \\times Number\, of\, months\, in\, a\, year$

Now, plug in the given values for the number of years and the number of months in a year to calculate the total number of deposits.
$n =9 \\times12$

Calculate the total number of deposits.
$n = \\times12 =108$

Now, plug in the given values for the future value, the monthly interest rate, and the total number of deposits to calculate the monthly deposit.
$MT = \$25,000 / [( +0.003)^{108} -] /0.003$

Calculate the monthly deposit.
$MT = \$25,000 / [( +0.003)^{108} -] /0.003 = \$198.79$ Bill needs to deposit $198.79 each month.