How much more interest will maria receive if she invests 1000$ for one

Question

How much more interest will maria receive if she invests 1000$ for one year at x % annual interest, compounded semianually, than if she invest 1000$ for one year at x percent annual interest, compounded annually?

A. 5x

B. 10x

C.  \frac{x^2}{20}

D.  \frac{x^2}{40}

E.  (10x+\frac{x^2}{40})

Here is how i solved only to get it wrong:

let x=100%

annually :

SI = 1000*100/100*1 = 1000
total =1000+1000=2000

semiannually:

Si (for 1st 6 months) = 1000
total after 6 months =2000

Si(for last 6 months) = 2000
total after 6 months= 4000

more interest maria will receive = 2000

====looking for the answer choices

100^2/40 != 2000!!!

where am I wrong? Today it seems I am back to square 1

chetan2u · Accepted Answer

Hi,
 the correct way is--

whenever amount is compounded other than annually, reduce the interest that many time and increase the time period that many times.. 
 lets see in this Q.. 
it is semi annually that is twice in a year...
so our time becomes 2n and rate of interest = r/2..

1) when annually  
Amount = 1000( 1+\frac{r}{100})^1 = 1000(1+\frac{r}{100})

2) when semi annually 
 1000( 1+\frac{r}{2*100})^2 = 1000(1+(\frac{r}{200})^2+2*1*\frac{r}{200})

subtract 1 from 2..
 1000(1+(\frac{r}{200})^2+\frac{r}{100})- 1000( 1+\frac{r}{100})^1 ..
 1000(1+(\frac{r}{200})^2+\frac{r}{100})-(1+\frac{r}{100}) ..
 1000(\frac{r}{200})^2  =  \frac{r^2}{40} 
D

aniketm.87@gmail.com · Answer

simple soln :

take r = 10%

for annual interest

interest = 1000*10/100 = 100

for semi annual

it will be 5/100 * 1000 = 50 ------> after 6 months interest

after 12 months ========> 5/100 * 1050 + 50 = 52.5 + 50 = 102.5 --------> final interest

SO now gain in interest = 102.5 - 100 = 2.5

A), B) and E) are out since all them contain 10x + ...... (10*10 = 100 + x -----> definitely the increase is not in this range)

test c) x^2/20 = 100/20 = 5 ------> close but wrong

so ans is D) x^2/40

check -----> 100/40 = 10/4 = 5/2 = 2.5 -----> bingo

FightToSurvive · Answer

How much more interest will maria receive if she invests 1000$ for one year at x % annual interest, compounded semianually, than if she invest 1000$ for one year at x percent annual interest, compounded annually?

a> 5x
b> 10x
c> x^2/20
d> x^2/40
e>10x+ (x^2/40)

I dont from what source these questions are from.
See from simplification sake, take x =10%
(i) investing 1000$ for one year at x % annual interest, compounded semiannually
1000*1.1*1.1=1210
Interest amount = 1210-1000=210
(ii)investing 1000$ for one year at x percent annual interest, compounded annually
1000*1.1=1100
Interest amount = 1100-1000=100
Difference in two cases = 210-100 = 110

Putting x=10 in the above option , none of them give 110 as an answer. Could kindly check the options you have posted as it cannot be D as you given the OA.

email2vm · Answer

Gotcha!

But can you please take some value such as 5% or 10% and try to solve it.

And what did I miss in my calculations?

Ravi

FightToSurvive · Answer

My bad, i also missed it.

See Ravi,
if you take x = 10%
Annually,
Its simple 1000*10*1/100=100
Semi annually,
1000(1+10/2*100)^2 = 1102.5
Interest amount = 1102.5-1000 = 102.5

Difference in interest = 102.5-100 = 2.5

Put x = 10 and see option D gives you 2.5. Hence D is the answer.

FightToSurvive · Answer

Yes I understood. For quarterly compounded, rate of interest =r/4 and time = 4n.

ShashankDave · Answer

To simplify calculations..take x = 100%..

keats · Answer

I got this as Q3 on my GMATPrep. Is it possible to receive 700 level questions this early on the GMAT?

abhimahna · Answer

There are two things I wanna say : 1. I don't think this question is of 700 level. It's an easy question. I will count it as 600-650 level. It could be solved in hardly 1 min if you solve with a proper approach. 2. Yes, On GMAT sometimes you may get 700 level questions too early. And if you excel in those, you may head towards your dream score very easily. :-D

keats · Answer

1. I see what you have to say  abhimahna . I quoted 700 level based on GMATClub stats. Moreover, what might be easy for you (or probably me) might not be easy for the other one. 
2. Since I did hit a Q 51 on this GMAT Prep Exam 5 (EP2) with one mistake on Q 30, I genuinely wanted to see if we get high level questions this early (in case of a Q 51). I googled my questions starting Q 2 to Q 7 and except Q 2 (600-700), all were rated as 700 level per GMATClub stats.

abhimahna · Answer

Yes brother, I understand your point. But as I said all you need to get through this question is a systematic approach. If you follow the same, you will appreciate my point that its an easy and straightforward question. If someone is considering this question a tougher one, then the person either lacks the basic skills or not able to recollect those at the time of answering this question.

Regarding Q51, I also felt the same when I got the same score in Mocks. Its all about how you are gonna play with GMAT algorithm. If you beat it at the beginning only, GMAT will throw a score of 51 on you.

Revant · Answer

Hi Chetan,

Could you please tell me how did you deduce the second condition where it says semi annually.. It seems you have seperated r/100. I couldn't figure out how ?

syahasa2 · Answer

1,000 compounded Anually(x/100 rate): 1,000*x/100=10x (1)

1,000 compounded Semi-anually (x/200 rate): 1,000*x/200=5x  1st 6-month period

(1,000+5x)*x/200=5x +x^2/40  2nd 6-month period . So total 5x+5x+x^2/40=10x+x^2/40(2)

So  (2)-(1) =10x+x^2/40-10x=  x^2/40 Ans. (D)

Cez005 · Answer

Your figure for the annual interest is incorrect. Is should be 20%. This gives 200 interest on a principle of 1000 versus interest of 210 when compounded semi-annually. D gives 10.

llamsivel · Answer

I can't quite wrap my head around the semi annual and quarterly compounding interest.

I tried the following, but ended up with "A" as my answer:

Let X percent be 200

Semi-Annual = 1000(1+200/2*100)^2 == > 1000(2)^2 ==> 4,000

Then interest for semi-annual should equal $3,000

Annual = 1000(1+200/100)^1

Then interest for annual should equal $2,000

Difference in interest is $1,000. Given X = 200, why isn't A correct?

lpardess · Answer

I have a question about gmat intrests- this question doesnt mention simple intrest or not. should i assume that?
because, lets say 10% intrest annualy- isnt 5% semi annualy as the question implies but 4.88 %. 1000*1.0488=1048.8 for the first half a year and 1048.8*1.048 for the second equels= 1100. which equals to 10%. 
so actually on the first 30 seconds I tried to look if there is an answer that says- 0. or after few calculation says it, and look for a sentence that would clear that issue.

so, are we suppose to assume simple intrest?

abhimahna · Answer

Hi  lpardess ,

Welcome to GMATClub :)

First of all, notice that this question contains the word "compounded". So, whenever you see "compounded" word in your question, it has to be compound interest.

If not, take it as simple interest.

Does that make sense?

lpardess · Answer

Hi
thank for the answer, I thought i responded earlier but I guess my post wasnt sent. so Ill post again.

I probobly didnt explain well what I mean by simple interest, while yes compounded means not simple, but still that doesnt resolve my problem with the question.

as i wrote on my example, lets say 10% interest annually - isnt 5% semi annually but 4.88 %. 1000*1.0488=1048.8 for the first half a year and 1048.8*1.048 for the second equels= 1100. meaning 10% annual interest. 
the question asks about X% annually . but the actual annual interest is different by the answer.

X percent annual interest is X percent. doesnt mattar if its compounded annually , semiannually or every month. 
so the correct answer should be 0. at least in real life. are the gmat rules different for interest?

rajat0627 · Answer

I believe this question can be handled in max 20 sec without getting into much formulas..

If you think about it, the difference is going to be nothing but the interest for last 6 months on the interest of first 6 months..

Interest of first 6 months = ((1000/100)*(x/2)) = 5x

Difference = ((5x/100)*(x/2)) = x^2/40

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