A sum of $6600 was taken as loan. This is to be repaid in two annual

Hi could someone kindly clarify which formula we would use in this question and why?

When doing questions based on equal installments compounded annually, then

\(P = \frac{x}{(1 + \frac{r}{100})^1} + \frac{x}{(1 + \frac{r}{100})^2} + ... + \frac{x}{(1 + \frac{r}{100})^n}\)


Where P = Principal Amount borrowed

x = Value of each installment

n = Number of years.


In the above question, P = 6600, r = 20% and n = 2

\(6600 = \frac{x}{(1 + \frac{20}{100})^1} + \frac{x}{(1 + \frac{20}{100})^2}\)

\(6600 = \frac{x}{(\frac{12}{10})^1} + \frac{x}{(\frac{12}{10})^2}\)

\(6600 = \frac{10x}{12} + \frac{100x}{144}\)

\(6600 = \frac{120x + 100x}{144} = \frac{220x}{144}\)

\(x = \frac{6600 * 144}{220} = 4320\)


Option A

Arun Kumar

(1st) Chart the sequence of Events


We will receive $6,600 the 1st year

throughout the year 20% Interest will accrue such that at this point, by the END of the 1st Year, we will have to pay back:

$6,600 * (1.2) = $6,600 * (6/5) -----> in Total

However, we are breaking up the Payments we pay back Each Year in 2 Equal Installments. Let the amount we pay back at the End of Year 1 = $X


Beg of Year 1: $6,600

Throughout Year 1: $6,600 * (6/5) is the amount we have to pay back so far

END of Year 1 we pay Back $X Installment so the amount Remaining:

$6,600 * (6/5) - $X



Beginning of Year 2 the Amount we still owe from the Loan is:
$6,600 * (6/5) - $X

Throughout Year 2 we will have to pay back 20% Interest on top of the Remaining Money we still owe:

[ $6,600 * (6/5) - $X ] * (6/5)

END of Year 2:

we want the Installment we pay back to EQUAL the Same Amount we paid back at the end of Year 1 (the Question says 2 Installments)

Therefore, the Amount we pay back at the End of Year 2 should be:

[ $6,600 * (6/5) - $X ] * (6/5) = $X

and now we have paid back the loan in 2 Equal Installments = $X



Q: What is the amount of the Installment?
or
What is $X = ?

Use the Equation Set up at the End of Year 2 to Solve for $X


[ (6,600) * (6/5) - X ] * (6/5) = X

----Distribute (6/5) through the Brackets-----

(6,600) * (6/5) * (6/5) - X * (6/5) = X



----Add: X * (6/5) to Each Side of the Equation----

(6,600) * (6/5) * (6/5) = (11/5) * X


----Cancel a 5 in the Denominator on Each Side of the Equation (i.e., Multiply Both Sides by *5)-----

(6,600) * (6/5) * (6) = (11) * X



---Cancel a Factor of 11 in the Numerator on Each Side of the Equation (i.e., Divide Both Sides by /11) -----

(600) * (6/5) * (6) = X


Solving For X ----->

(120) * (6) * (6) = (720) * (6) = 4,320


The Installment Payments Each Year on the Loan = $4,320

-A-

Tricky Question unless you know the Formula for Installment Payments on a Loan....

P(1+r/100)^n=Total amount. So P = Total amount/(1+r/100)^n where P=6600. Let total amount=x.
{x/(1+20/100)}+{x/(1+20/100)^2}=6600 or, x=4320(A).

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