QuestionRavi buys a laptop for R\$7000, deposits R\$1400, and pays 12\% annual interest compounded quarterly for 3 years. Find his monthly payment.

Studdy Solution

STEP 1

Assumptions1. The price of the laptop is R\$7000. The initial deposit is R\$14003. The interest rate is12% per annum, compounded quarterly4. The time for repayment is3 years5. He pays an insurance fee of R\$75 per month for the duration of the loan

STEP 2

First, we need to find the principal amount. This is the price of the laptop minus the initial deposit.
$Principal = Price\, of\, laptop - Initial\, deposit$

STEP 3

Now, plug in the given values for the price of the laptop and the initial deposit to calculate the principal.
$Principal = R\$7000 - R\$1400$

STEP 4

Calculate the principal amount.
$Principal = R\$7000 - R\$1400 = R\$5600$

STEP 5

Next, we need to find the total amount payable after3 years with the compound interest. The formula for compound interest is $A = \left(1 + \frac{r}{n}\right)^{nt}$ where- A is the amount of money accumulated after n years, including interest. - is the principal amount (the initial amount of money). - r is the annual interest rate (in decimal). - n is the number of times that interest is compounded per year. - t is the time the money is invested for in years.

STEP 6

Now, plug in the given values for the principal (), interest rate (r), number of times compounded per year (n), and time in years (t) to calculate the total amount payable.
$A = R\$5600 \left(1 + \frac{12\%}{4}\right)^{4 \times3}$

STEP 7

Convert the percentage to a decimal value.
$12\% =0.12$ $A = R\$5600 \left(1 + \frac{0.12}{4}\right)^{4 \times3}$

STEP 8

Calculate the total amount payable after3 years with the compound interest.
$A = R\$5600 \left(1 + \frac{0.12}{4}\right)^{4 \times3} = R\$7486.89$

STEP 9

Now, we need to calculate the total insurance fee over the3 years. This is the monthly insurance fee multiplied by the total number of months in3 years.
$Total\, insurance\, fee = Monthly\, insurance\, fee \times Number\, of\, months$

STEP 10

Plug in the values for the monthly insurance fee and the number of months to calculate the total insurance fee.
$Total\, insurance\, fee = R\$75 \times36$

STEP 11

Calculate the total insurance fee over the3 years.
$Total\, insurance\, fee = R\$75 \times36 = R\$2700$

STEP 12

Now, we need to calculate the total amount payable over the years. This is the total amount payable after years with the compound interest plus the total insurance fee.
$Total\, amount\, payable = Total\, amount\, with\, compound\, interest + Total\, insurance\, fee$

STEP 13

Plug in the values for the total amount with compound interest and the total insurance fee to calculate the total amount payable.
$Total\, amount\, payable = R\$7486.89 + R\$2700$

STEP 14

Calculate the total amount payable over the3 years.
$Total\, amount\, payable = R\$7486.89 + R\$2700 = R\$10186.89$

STEP 15

Now that we have the total amount payable over the3 years, we can divide this by the total number of months in3 years to find the monthly payment.
$Monthly\, payment = Total\, amount\, payable / Number\, of\, months$

STEP 16

Plug in the values for the total amount payable and the number of months to calculate the monthly payment.
$Monthly\, payment = R\$10186.89 /36$

STEP 17

Calculate the monthly payment amount.
$Monthly\, payment = R\$10186.89 /36 = R\$283.02$ Ravi needs to pay R\$283.02 each month.

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