On the first of the year, James invested x dollars at Proudstar bank

Question

On the first of the year, James invested x dollars at Proudstar bank in an account that yields 2% in interest every quarter year. At the end of the year, during which he made no additional deposits or withdrawals, he had y dollars in the account. If James had invested the same amount in an account which pays interest on a yearly basis, what must the interest rate be for James to have y dollars at the end of the year?

A. 2.04%
B. 6.12%
C. 8%
D. 8.25%
E. 10%

A. 2.04%
B. 6.12%
C. 8%
D. 8.25%
E. 10%

Bunuel · Accepted Answer

If the interest were compounded annually instead of quarterly then in one year the interest would be 2*4=8%. Now, since the interest is compounded quarterly then there would be interest earned on interest (very small amount) thus the actual interest would be a little bit more than 8%, only answer choice D fits. Answer: D. Similar questions to practice: jolene-entered-an-18-month-investment-contract-that-127308.html marcus-deposited-8-000-to-open-a-new-savings-account-that-128395.html Hope it helps.

Bunuel · Answer

The balance in one year would be:

Final \ balance = principal*(1 + \frac{yearly \ interest}{the \ number \ of \ times \ it's \ compounded \ annually})^{(the \ number \ of \ times \ it's \ compounded \ annually)}=

=x*(1 + \frac{0.08}{4})^4=x*(1.02)^4=x*1.08243216

So, the  exact  answer is 8.243216%, not 8.25%. In that the question is not precise (real GMAT question won't be worded this way), it should ask "  If James had invested the same amount in an account which pays interest on a yearly basis,   approximately   what must the interest rate be for James to have y dollars at the end of the year? "

Plus, I'd also like to see the word  compounded  here:

On the first of the year, James invested x dollars at Proudstar bank in an account that yields   compounded interest of   2% in interest every quarter year.

Because in current wording it's not 100% clear that  2% every quarter year  is compounded or not (we all above assumed that it is but it's not explicitly mentioned). So, theoretically one can make a case for C, arguing that we are not told that 2% is compounded quarterly. If it's not then the answer would simply be 8%.

mikemcgarry · Answer

Hi, there. I'm happy to help with this.

Essentially, this question is asking for the effective interest rate . So, every increase of 2% means we multiply x by the multiplier 1.02. The initial amount x gets multiplied by this multiply four times, one for each quarter, so . . . y = x*(1.02)^4 = (1.08243216)*x ====> effective interest = 8.2432% That's how you'd get the exact answer with a calculator, but of course you don't have a calculator available on GMAT PS questions. Think about it this way. With simple interest, 2% a quarter would add up to 8% annual. With compound interest, where you get interest on your interest, you will do a little better than you would with simple interest, so the answer should be something slightly above 8%. That leads us to . . . Answer = D Does that make sense? Please let me know if you have any additional questions on what I've said there. Mike

romeokillsu · Answer

But according to compound interest formula y= x*( 1+ 0.02/4(quarterly)^time(1yr)*4(quarterly) -> so it comes to x* (1.02015). Please correct me where I'm going wrong.
Thanks

mikemcgarry · Answer

Dear romeokillsu , My friend, in most standard problems, the interest rate given is an ANNUAL interest rate, and for compounding quarterly, we have to divide it by four, as that formula does. BUT, in this problem we are told: On the first of the year, James invested x dollars at Proudstar bank in an account that yields 2% in interest every quarter year . So, this problem is following a different pattern --- it is not giving us an ANNUAL interest rate that needs to be divided by four. It is giving us a QUARTERLY interest rate. That formulas you cited is only used when the problem gives us an annual interest rate. DO NOT blindly apply formulas! That is a recipe for failure! You must understand WHY the formula is true. You have to be able to recreate the argument whereby you derive the formula from scratch. That is really understanding, and that is what the GMAT is testing. Knowing just the formula and not where it comes from, not why it is true, is precisely the kind of superficial knowledge that the GMAT loves to exploit and punish. For more on the compounding interest formula, see this blog: https://magoosh.com/gmat/2014/compound-i ... -the-gmat/ Does all this make sense? Mike :-)

PareshGmat · Answer

Per quarter, interest = 2%

So for a year, interest = 8%

Due to quarter cummulation, effective yield (YTM) would be slight higher than 8%

Answer = 8.25% = D

Note: I've not made any calculations here; solved conceptually

gmatchile · Answer

El ejercicio solicita comparar interés compuesto con interés simple:

Recordemos que:

1 trimestres = 3 meses, entonces 1 año = 4 trimestres = 12 meses.

Cuestión planteada por el ejercicio:

Cierta cantidad a un interes trimestral del 2%, pasado un año, debe ser igual a la misma cierta cantidad a la cuál se le aplica un interes desconocido (lo que se pregunta) anual. ¿ cuál es este último interés anual?

El interés del 2% trimestral, ya que involucra más de un período en el año, corresponde a interés compuesto, modelo
x(1 + 2/100)^4 = aproximadamente pero mayor que x(1 + o,o8).

Es decir la misma cierta cantidad a un interés anual cercano pero ligeramente mayor al 8% (interes simple), equivale a la cierta cantidad con un interés del 2% trimestral al finalizar una año (interés compuesto).

Respuesta Correcta D.
claudio hurtado
Tutor GMAT GRE MATH in Chile
www.gmatchile.cl

law258 · Answer

Quarterly interest rate is not the same as annual interest rate compounded quarterly. We are given quarterly interest rate, therefore we do not need to divide by 4 in the compound interest formula.

ACCT A: x(1.02)^4=y 
ACCT B: x(1+r)=y

x(1.02)^4=x(1+r)
(1.02)^4=(1+r)
0.0824=r

D.

er.arun88 · Answer

Hi Mike,
I have a confusion here.Where in the question it is mentioned that we need to check t he COMPOUND INTEREST?

In QUESTION STEM
 If James had invested the same amount in an account which pays interest on a yearly basis, what must the interest rate be for James to have y dollars at the end of the year?
 
Here it says "interest rate /per year" Nowhere it us mentioned "Compounded annually"??

IN my opinion it should be Simple interest?No?

mikemcgarry · Answer

Dear er.arun88 , I'm happy to respond. :-)

There are two reasons why we can be 100% sure that compound interest in intended here. 1) The GMAT prepares you for the real business world and reflects the values of the real business world. Simple interest does not exist outside of kiddie math text books. No bank or credit card or financial institution on the planet would ever use simple interest: they all use compound interest. Thus, whenever the GMAT talks about interest and doesn't specify, we can be 100% sure that they mean compound interest. The GMAT would have to specify very clearly that they were using simple interest, although I don't know if I have ever even seen an official question about simple interest. It's true, the official GMAT is usually exceptionally clear, and probably would have said " compounds " rather than " yields " in the first sentence. 2) If there were no compounding, then in simple interest having the 2% interest accrue quarterly would make absolutely no sense, because by the end of the year, one would have exactly the same amount as an 8% annual account That essentially would be a question: do you know how to multiply by four ? That is not the level at which the GMAT tests. The entire setup for the question is about measuring the compounding difference, and the subtle reasoning about how different the compound effects would be from the all-at-once interest payment is precisely what raises this question to the level of the conceptual complexity of the GMAT Quant section. Does all this make sense? Mike :-)

AMsac123 · Answer

But here they are asking for rate in compund interest anually right , soit should be less than compounded monthly rate right? Please help me!!

mikemcgarry · Answer

Dear Anazeer , I'm happy to respond. :-)

When we have a quarterly interest rate, and these quarter compound, then over the course of a full year, the account makes a total amount of interested in the year that is more than it makes in any one quarter. Since all percentages are compared to the principal, which remains the same, the annual amount must be a larger percentage. Does this make sense? Mike :-)

JeffTargetTestPrep · Answer

We can let James’ initial investment of x be $100. That is, we can say that James invested 100 dollars at Proudstar bank in an account that yields 2% in interest every quarter year. Recall the compound interest formula: 
 
A = P(1 + i)^n 
 
In which P = principal, i = interest rate per period, n = number of periods, and A = amount at the end of n periods.
 
Since a year has 4 quarters, n = 4 and we have:
 
A = 100(1 + 0.02)^4 
 
A = 100(1.02)^4
 
A = 100(1.0824)
 
A = 108.24
 
We can see that he will get back $108.24 when the interest is compounded quarterly, and this has to be the value of y (if x = 100). So the question becomes: if James had invested $100 in an account that pays interest on a yearly basis, what must the interest rate be for James to have $108.24 at the end of the year?
 
We can use the same formula, except now we need to solve for i when n = 1:
 
108.24 = 100(1 + i)^1
 
1.0824 = 1 + i
 
0.0824 = i
 
8.24% = i 
 
We see that the closest answer choice is D: 8.25%.
 
Alternate solution:
 
In this problem, we are given that a certain amount of money is being compounded quarterly with an interest rate of 2%. That is like paying an annual interest rate of 8% (4 x 0.02 = 0.08), but SOMEWHAT BETTER, since the interest is compounded (i.e., interest added on principal plus interest) every quarter of the year. So we are looking for an annual interest rate of slightly more than 8%. The only two answer choices that are more than 8% are 8.25% and 10%. Recall, we’ve said “somewhat better,” so it can’t be 10%. This leaves 8.25% as the most reasonable answer choice and the correct answer choice. 
 
Answer: D

uzumakinaruto444 · Answer

Hello,

i have a doubt...shouldn't the "i" in the formula be divided by "n" as well where "n" is the number of times interest is compounded annually...and the answer comes out to be different in that case...please help me out

thanks in advance.

Fdambro294 · Answer

JeffTargetTestPrep  alternative solution is definitely the way to go. Thinking about the problem like that can get you in and out in about 40 seconds.

To calculate, you might be better of estimating and doing the calculations year by year. Let X = $100 to make it easier.

2% each quarter

To calculate the 2% interests each quarter: first, find 1% of the principal by moving the decimal 2 places to the left. Then double that result to get the 2% compound interest.

Start with X = $100:

Quarter 1 interest: 1% = $1 ——-> 2% = $2

$102 at end of Q1

Q2: 1% = $1.02 ——-> 2% = $2.04

$104.04 at end of Q2 ~ $104

Q3: 1% = $1.04 ———> 2% = $2.08

$106.08 at end of Q3 ~ $106

Q4: 1% = $1.06 ———> 2% = $2.12

$108.12 is the approximate amount at the end of Q4

So Y = $108.12 (the actual amount will be a little more because this is an UNDER estimate....we cutoff some of the decimal portion above to make the calculation easier)

We are asked to find the simple interest rate to get from X = $100 to ———-> Y = $108.12 at the end of 1 year

We are going to need a little more than 8% of $100 to get us to $108.12 (an under-estimate)

%8.25 is the only answer that is possible

Posted from my mobile device

Nr967 · Answer

Hi,

Why don't we divide by 4 since it is compounding on a quarterly basis? Shouldn't it be like this:

x(1+(2/(100)(4)))^4

woohoo921 · Answer

I realize that this question is discussed in great depth. However, I just wanted to clarify what this would look like if you use the formal compounding interest formula with let's say a starting principal amount of $100: I did =100*(1+0.08/4)^4 Is this correct? I converted the 2% interest every quarter to 8% interest in a year because the compounding interest formula is --> Final Amount=Principal*(1+(annual interest rate as a decimal/# of compoundings per year))^(# of years*# of compoundings per year) Many thanks

Bunuel · Answer

Check similar question I've just created: https://gmatclub.com/forum/warren-inves ... 00610.html

darsh.shah · Answer

Hi Mike, Is there any way to arrive at numbers like y = x*(1.02)^4 = (1.08243216)*x within 2 minutes without any calculator? Because this will take some more time to calculate and it would be difficult to estimate if options are near.

On the first of the year, James invested x dollars at Proudstar bank

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On the first of the year, James invested x dollars at Proudstar bank

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