Louie takes out a three-month loan of $1000. The lender charges him 10

\(CI = P(1+\frac{r}{100})^t\)

Assume he pays off entire amount in 3rd month or interest is accrued for 2 months. Find the amount at end of 3 months and divide by 3 to know monthly EMI

I Couldnt get why interest would be paid for 2 months, as per me

1. 1st month at the end monthly interest would be Accrued and monthly installment would be deducted from that amount.
2. For 2nd month start amount would be remaining amt of 1st month and at the end of 2nd month, monthly interest would be Accrued and thereafter again monthly installment would be deducted
3. For the 3rd month start amt would again be the remaning amt of 2nd month and at the end of 3rd month monthly interest would be accrued which should be equal to monthly installment.

So as per this interest was paid thrice ..request you to please clarify !!

this is because you are paying off in the third and last months. This is assuming the interest rate is calculated at the end of the month. So it is assumed you paid off the balance at the end of third month so 0 balance. Like CC statements - if you didnt pay off your statement by end of month you get charged interest - you dont get charged interest throughout.

Actually what I wrote above assumes that Louie doesn't start repaying the loan immediately, instead he waits until all the interest has compounded. This is not necessarily correct since usually you start paying off loans as soon as you take them out. This means that you pay off some principle each month and therefore your interest is lower.

Here's the calculation for that case, assume monthly payment is X.

After 1st month: (1000)(1.1)-X = 1100-X
After 2nd month: (1100-X)(1.1)-X = 1210-2.21X
After 3rd month: (1210-2.21X)(1.1)-X = 1331-3.31X

Now, the amount after the last payment in 3rd month must bring the total to 0. Hence:

1331-3.31X = 0
X = 1331/3.31 = 402.11

The answer is C. However, I think this is a poorly worded question and on the real GMAT, they would specify that the payment is to be started immediately after loan inception.

The interest has to be calculated on a reducing balance.
If monthly repayment = x
At the end of the 3 month period,
1.1*[1.1*{1.1*(1000)-x}-x]-x = 0
=> 3.31x = 1331
=> x ~ 402

Why are we assuming he pays from the 3rd month? The question does not specify that, it just says he has to pay in 3 installments.

Why not this way?
Total Loan disbursed in 3 months = 1.1 * 1.1* 1.1* 1000 = 1331
Repaid in 3 months, hence per month = 1331/3 = 443

In case of CI ,Repayment in equal installments (X) can be given as:


X =P*r/ [1-(100/100+r)^n]

where X :each installment
r: rate
n: number of installments
P: Principal amount borrowed by borrower.


So in this case it would be 1000*10/[1-(10/11)^3] = 133100/331 = 402

I get a different answer by using the Compound Interest formula, i.e- P[1 +(r)/100n]^nt

Since this formula uses annualized figures, so:
r = 10% per month = 120% per year
n = 12 (as interest is compounded monthly)
t = 3 months = 3/12 years

Using the formula for compound interest, I get:
P + C.I = 1000(1.1)^3 = 1331

So, EMI = 1331/3 = 443.66 which is ~ $444

What's wrong with this approach?

Thanks,
Ishan

Since he pays after each month, then after the firs month (after the first payment) the interest is calculated on reduced balance.

Does this make sense?

Since the loan = 1000, and this amount were to be charged 10% interest per month, compounded monthly ...

1000/3 = 333

1 --- 333 * 1.1 = 366
2 --- 366 * 1.1 = 402
3 --- 402 * 1.1 = 442

Sum the 3 months: 366 + 402 + 442 = 1210 total owed

3 equal installments: 1210/3 = roughly 403

The closest answer (sans the slight rounding): Choice C.

\(Amount = 1000( 1 + 10/100)^3\)

Amount is 1331

Since repayment is to be made in 3 equal installments , EMI is 1331/3 = 433 , Answer must be (D)

The solution will be as given by Bunuel in the comment on first page.

Why not this? Interest is always calculated on the outstanding amount only. If I pay say $400 at the end of month 1 out of the $1100 (P + I) till then, my interest in the second month will be charged on $700 only.
This calculation assumes no payment till month 3 to get the amount of $1331. That is not correct.

Bunuel KarishmaB @Martymurrat, I thought the would be 1331/3. Could you please explain why $x to be paid each month? Thanks in advance.
­

­
Have you reviewed the following posts addressing the same doubt?
https://gmatclub.com/forum/louie-takes- ... l#p1265916
https://gmatclub.com/forum/louie-takes- ... l#p2529835 

­
I missed that. Thanks!­

The lender charges him 10% interest per month compunded monthly. The terms of the loan state that Louie must repay the loan in three equal monthly payments. KarishmaB  GMATinsight chetan2u

Let the monthly installment be E. Basically EMI for a particular month means the total amount which is the principal plus the compound interest levied so far.

For the first month ,
E = P1 ( 1 + 10/100)^1 = P1 * 1.1 ( The standard formulae for the compound interest i.e A or AMount = P *( 1 + r/n)^nt )

For the second month,
E = P2 * (1 + 10/100)^2 = P2 * 1.1^2 = 1.21 * P2

For the third month,
E = P3 * ( 1 + 10/100)^3 = 1.331*P3

P1 + P2 + P3 = 1000
E/ 1.1 + E/1.21 + E/1.331 = 1000
E(.909 + .826 + .751) = 1000
E * 2.486 = 1000
E= 402

C is the answer.­

P = (L x C x r) / (C – 1) where P is the instalment amount, L = amount of the loan, r = annual interest rate, and C = compounding factor = (1 + r)^N where N = number of annual payments.

Here, N=3 , r = 0.1, L=1000

C= (1+0.1)^3 = 1.331

=> P = (1000*1.331*0.1)/(1.331 - 1)
=> P = (133.1)/(0.331)
=> P ~ 402