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A bank offers an interest of 5% per annum compounded annually on all its deposits. If $10,000 is deposited, what will be the ratio of the interest earned in the 4th year to the interest earned in the 5th year? A. 1:5 B. 625:3125 C. 100:105 D. 100^4:105^4 E. 725:3225 _________________ Kudos will encourage many others, like me. Good Questions also deserve few KUDOS. Originally posted by emmak on 11 Jun 2013, 03:58. Last edited by Bunuel on 28 Jan 2015, 07:43, edited 2 times in total. Edited the question. ##### Most Helpful Expert Reply Math Expert V Joined: 02 Sep 2009 Posts: 55273 Re: A bank offers an interest of 5% per annum compounded annua [#permalink] ### Show Tags 15 17 emmak wrote: A bank offers an interest of 5% per annum compounded annually on all its deposits. If$10,000 is deposited, what will be the ratio of the interest earned in the 4th year to the interest earned in the 5th year?

A. 1:5
B. 625:3125
C. 100:105
D. 1004:1054
E. 725:3225

The interest earned in the 1st year = $500 The interest earned in the 2nd year =$500*1.05
The interest earned in the 3rd year = $500*1.05^2 The interest earned in the 4th year =$500*1.05^3
The interest earned in the 5th year = $500*1.05^4 (500*1.05^3)/(500*1.05^4) = 1/1.05=100/105. Answer: C. _________________ ##### Most Helpful Community Reply Intern  Joined: 29 Sep 2013 Posts: 47 Re: A bank offers an interest of 5% per annum compounded annua [#permalink] ### Show Tags 5 3 emmak wrote: A bank offers an interest of 5% per annum compounded annually on all its deposits. If$10,000 is deposited, what will be the ratio of the interest earned in the 4th year to the interest earned in the 5th year?

A. 1:5
B. 625:3125
C. 100:105
D. 1004:1054
E. 725:3225

Thirty seconds approach, regardless of what the figure is at the 4th year it will at act as a base figure (100) for the next years 5% increase (to 105). So the ratio is 100:105 or option C
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Re: A bank offers an interest of 5% per annum compounded annua  [#permalink]

emmak wrote:
A bank offers an interest of 5% per annum compounded annually on all its deposits. If $10,000 is deposited, what will be the ratio of the interest earned in the 4th year to the interest earned in the 5th year? A. 1:5 B. 625:3125 C. 100:105 D. 1004:1054 E. 725:3225 Hi Bunuel, Here is my approach: is this correct? Interest earned in 4 year= 10000(1+0.05)^4 Interest earned in 5 year= 10000(1+0.05)^5 Ratio= {10000(1.05)^4}/{10000(1.05^5)} =>1.05^4/1.05^5 =>1/1.05 Multiplied by 100 in both numerator and denominator gives 100:105 Hence Ans:C _________________ Do not forget to hit the Kudos button on your left if you find my post helpful. Math Expert V Joined: 02 Sep 2009 Posts: 55273 Re: A bank offers an interest of 5% per annum compounded annua [#permalink] ### Show Tags atalpanditgmat wrote: emmak wrote: A bank offers an interest of 5% per annum compounded annually on all its deposits. If$10,000 is deposited, what will be the ratio of the interest earned in the 4th year to the interest earned in the 5th year?

A. 1:5
B. 625:3125
C. 100:105
D. 1004:1054
E. 725:3225

Hi Bunuel,
Here is my approach: is this correct?

Interest earned in 4 year= 10000(1+0.05)^4

Interest earned in 5 year= 10000(1+0.05)^5

Ratio= {10000(1.05)^4}/{10000(1.05^5)} =>1.05^4/1.05^5 =>1/1.05 Multiplied by 100 in both numerator and denominator gives 100:105

Hence Ans:C

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Re: A bank offers an interest of 5% per annum compounded annua  [#permalink]

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atalpanditgmat wrote:
emmak wrote:
A bank offers an interest of 5% per annum compounded annually on all its deposits. If $10,000 is deposited, what will be the ratio of the interest earned in the 4th year to the interest earned in the 5th year? A. 1:5 B. 625:3125 C. 100:105 D. 1004:1054 E. 725:3225 Hi Bunuel, Here is my approach: is this correct? Interest earned in 4 year= 10000(1+0.05)^4 Interest earned in 5 year= 10000(1+0.05)^5 Ratio= {10000(1.05)^4}/{10000(1.05^5)} =>1.05^4/1.05^5 =>1/1.05 Multiplied by 100 in both numerator and denominator gives 100:105 Hence Ans:C This formula is to calculate the total amount, not the compound interest You require to subtract the Principal to get the resultant compound interest We require to calculate ratio of interest earned in 4th & 5th year This method you're using is calculating ratio of 4 yr deposit to 5 yr deposit _________________ Kindly press "+1 Kudos" to appreciate Director  G Joined: 23 Jan 2013 Posts: 549 Schools: Cambridge'16 Re: A bank offers an interest of 5% per annum compounded annua [#permalink] ### Show Tags question ask what is: 10000*1.05^4/10000*1.05^5 we get 10000/10000*1.05=10000/10500=100/105 Retired Moderator Joined: 16 Jun 2012 Posts: 1006 Location: United States Re: A bank offers an interest of 5% per annum compounded annua [#permalink] ### Show Tags Temurkhon wrote: question ask what is: 10000*1.05^4/10000*1.05^5 we get 10000/10000*1.05=10000/10500=100/105 Hello. You're correct for choosing C but wrong for interest formula buddy. The question asks you to calculate ration of the interest earned in 4th year to the interest earned in 5th year. Your formula is to calculate Total value in 4th year and 5th year NOT interests. In order to calculate INTEREST in 4th and 5th year, you have to calculate INTEREST in 1st year. interest in 1st year = 10,000*0.05 = 500 interest in 2nd year = 500*1.05 interest in 3rd year = 500*1.05^2 interest in 4th year = 500*1.05^3 interest in 5th year = 500*1.05^4 Ratio = 1/1.05 = 100/105 Hope it helps. _________________ Please +1 KUDO if my post helps. Thank you. "Designing cars consumes you; it has a hold on your spirit which is incredibly powerful. It's not something you can do part time, you have do it with all your heart and soul or you're going to get it wrong." Chris Bangle - Former BMW Chief of Design. Intern  Joined: 16 Mar 2014 Posts: 4 Re: A bank offers an interest of 5% per annum compounded annua [#permalink] ### Show Tags Could somebody please explain, how interest could be calculated this way - The interest earned in the 1st year =$500
The interest earned in the 2nd year = $500*1.05 The interest earned in the 3rd year =$500*1.05^2
The interest earned in the 4th year = $500*1.05^3 The interest earned in the 5th year =$500*1.05^4

Since we are compounding, the interest for the second year should be 500 + 500*1.05
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Re: A bank offers an interest of 5% per annum compounded annua  [#permalink]

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emmak wrote:
A bank offers an interest of 5% per annum compounded annually on all its deposits. If $10,000 is deposited, what will be the ratio of the interest earned in the 4th year to the interest earned in the 5th year? A. 1:5 B. 625:3125 C. 100:105 D. 100^4:105^4 E. 725:3225 Interest earned in the first year =$10,000 *(5/100) = $500 i.e. The interest earned in the 1st year =$500

The interest earned in the Second year = $10,000 *(5/100) +$500 *(5/100) = $500 + (5/100)*$500 = $500*1.05 i.e. The interest earned in the 2nd year =$500*1.05
Similarly,
The interest earned in the 3rd year = $500*1.05^2 The interest earned in the 4th year =$500*1.05^3
The interest earned in the 5th year = $500*1.05^4 (500*1.05^3)/(500*1.05^4) = 1/1.05=100/105. NOTE: Writing every step here is not a great idea as we must understand that Coumpound interest is a form of Geometric Progression in which the ratio of two consecutive terms remain constant hence 1st year interest / 2nd year interest = 2nd year interest / 3rd year interest = 3rd year interest / 4th year interest = 4th year interest / 5th year interest = 1/1.05 Answer: C. _________________ Prosper!!! GMATinsight Bhoopendra Singh and Dr.Sushma Jha e-mail: info@GMATinsight.com I Call us : +91-9999687183 / 9891333772 Online One-on-One Skype based classes and Classroom Coaching in South and West Delhi http://www.GMATinsight.com/testimonials.html ACCESS FREE GMAT TESTS HERE:22 ONLINE FREE (FULL LENGTH) GMAT CAT (PRACTICE TESTS) LINK COLLECTION Intern  Joined: 16 Aug 2013 Posts: 11 Re: A bank offers an interest of 5% per annum compounded annua [#permalink] ### Show Tags shankar245 wrote: Buneul, I have a doubt. Quote: The interest earned in the 1st year =$50
The interest earned in the 2nd year = $50*1.05 The interest earned in the 3rd year =$50*1.05^2
The interest earned in the 4th year = $50*1.05^3 The interest earned in the 5th year =$50*1.05^4

So we are just calculating the interest from interest.Are we not supposed to calculate the interest from the principle amount every year?

Hi Bunuel
I'm agree with shankar245
the interest earned in the 4th years is= 10000(1+0.05)^4-10000=10000((1+0.05)^4-1)
the interest earned in the 5th years is= 10000(1+0.05)^5-10000=10000((1+0.05)^5-1)
the ratio is ((1+0.05)^4-1)/((1+0.05)^5-1)=0.78
can you please clarify?
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A bank offers an interest of 5% per annum compounded annua  [#permalink]

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amirzohrevand wrote:
shankar245 wrote:
Buneul,

I have a doubt.

Quote:
The interest earned in the 1st year = $50 The interest earned in the 2nd year =$50*1.05
The interest earned in the 3rd year = $50*1.05^2 The interest earned in the 4th year =$50*1.05^3
The interest earned in the 5th year = $50*1.05^4 So we are just calculating the interest from interest.Are we not supposed to calculate the interest from the principle amount every year? Hi Bunuel I'm agree with shankar245 the interest earned in the 4th years is= 10000(1+0.05)^4-10000=10000((1+0.05)^4-1) the interest earned in the 5th years is= 10000(1+0.05)^5-10000=10000((1+0.05)^5-1) the ratio is ((1+0.05)^4-1)/((1+0.05)^5-1)=0.78 can you please clarify?$500*1.05 = 500 + (5/100)*500

Where 500 is interest earned on principle
And

(5/100)*500 is interest earned on previous interest.

So the expressions include both.

I hope this helps!
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Originally posted by GMATinsight on 17 Sep 2015, 23:47.
Last edited by GMATinsight on 18 Sep 2015, 04:06, edited 1 time in total.
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Hi Bunuel
I'm agree with shankar245
the interest earned in the 4th years is= 10000(1+0.05)^4-10000=10000((1+0.05)^4-1)
the interest earned in the 5th years is= 10000(1+0.05)^5-10000=10000((1+0.05)^5-1)
the ratio is ((1+0.05)^4-1)/((1+0.05)^5-1)=0.78
can you please clarify?[/quote]

$50*1.05 = 50 + (5/100)*50 Where 50 is interest earned on principle And (5/100)*50 is interest earned on previous interest. So the expressions include both. I hope this helps![/quote] Hi Dear GMATinsight I'm not agree with your approach cuz you dismissed the principle , however the principle must be seen. your approach yields 0.92 but my approach yields 0.78 I'm still confused could you elaborate more? plz can you please let me know what is wrong with my approach? tnx CEO  D Status: GMATINSIGHT Tutor Joined: 08 Jul 2010 Posts: 2931 Location: India GMAT: INSIGHT Schools: Darden '21 WE: Education (Education) Re: A bank offers an interest of 5% per annum compounded annua [#permalink] ### Show Tags 1 amirzohrevand wrote: Hi Dear GMATinsight I'm not agree with your approach cuz you dismissed the principle , however the principle must be seen. your approach yields 0.92 but my approach yields 0.78 I'm still confused could you elaborate more? plz can you please let me know what is wrong with my approach? tnx Principle =$10,000
Rate of Interest = 5%

Interest earned in the first year = $10,000 *(5/100) =$500
i.e. The interest earned in the 1st year = $500 The interest earned in the Second year =$10,000 *(5/100) + $500 *(5/100) =$500 + (5/100)*$500 =$500*1.05
i.e. The interest earned in the 2nd year = $500*1.05 Similarly, The interest earned in the 3rd year =$500*1.05^2
The interest earned in the 4th year = $500*1.05^3 The interest earned in the 5th year =$500*1.05^4

(500*1.05^3)/(500*1.05^4) = 1/1.05=100/105.
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A bank offers an interest of 5% per annum compounded annua  [#permalink]

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Quote:
amirzohrevand wrote:
Hi Bunuel
I'm agree with shankar245
the interest earned in the 4th years is= 10000(1+0.05)^4-10000=10000((1+0.05)^4-1)
the interest earned in the 5th years is= 10000(1+0.05)^5-10000=10000((1+0.05)^5-1)
the ratio is ((1+0.05)^4-1)/((1+0.05)^5-1)=0.78
can you please clarify?

$50*1.05 = 50 + (5/100)*50 Where 50 is interest earned on principle And (5/100)*50 is interest earned on previous interest. So the expressions include both. I hope this helps! Hi Dear GMATinsight I'm not agree with your approach cuz you dismissed the principle , however the principle must be seen. your approach yields 0.92 but my approach yields 0.78 I'm still confused could you elaborate more? plz can you please let me know what is wrong with my approach? tnx Hi, you are calculating total interest earned in 4 years i.e interest of 1year + 2year + 3year + 4year by that formula Correct way to calculate 4th year interest is- 10,000(1 + .05)^4 - 10,000(1+ .05)^3 hope this helps! Intern  B Joined: 07 Jan 2017 Posts: 19 Location: India Concentration: Other, Other Schools: UConn"19 WE: Medicine and Health (Pharmaceuticals and Biotech) Re: A bank offers an interest of 5% per annum compounded annua [#permalink] ### Show Tags C.I=P(1+r)^n Interest earned in 4 th year = P(1+0.05)^3 Interest earned in 5 th year = P(1+0.05)^4 Ratio of Interest earned in 4 th year : Ratio of Interest earned in 5 th year = P(1+0.05)(1+0.05)(1+0.05)/P(1+0.05)(1+0.05)(1+0.05)(1+0.05) =1/(1+0.05) =100/105{Multiplied by 100} Ans C Manager  G Joined: 30 May 2012 Posts: 201 Location: United States (TX) Concentration: Finance, Marketing GPA: 3.3 WE: Information Technology (Consulting) A bank offers an interest of 5% per annum compounded annua [#permalink] ### Show Tags 1 Bunuel or any mods, Please help understand this. Compound Interest (CI) for 1st year is $$\frac{5}{100}$$*10000 + 10000 = 10,500 Compound Interest (CI) for 2nd year is $$\frac{5}{100}$$*10500 + 10500 = 11,025 Compound Interest (CI) for 3rd year is $$\frac{5}{100}$$*11,025+ 11,025 ----- And this becomes an ugly math. Why? Where am I going wrong? Compound Interest (CI) for 4th year is $$\frac{5}{100}$$*? + ? ------ By this point, I was heavily estimating the calculation. And I am still left with calculating for 5th year. Compound Interest (CI) for 5th year is $$\frac{5}{100}$$*?+ ? I somehow arrived at the correct answer but I see that there is inherently some mistake in my approach. Could someone help? TIA. Math Expert V Joined: 02 Sep 2009 Posts: 55273 Re: A bank offers an interest of 5% per annum compounded annua [#permalink] ### Show Tags emmak wrote: A bank offers an interest of 5% per annum compounded annually on all its deposits. If$10,000 is deposited, what will be the ratio of the interest earned in the 4th year to the interest earned in the 5th year?

A. 1:5
B. 625:3125
C. 100:105
D. 100^4:105^4
E. 725:3225

VERITAS PREP OFFICIAL SOLUTION:

Solution: (C)

This is a great example of a problem that looks much more difficult than it really is. If we calculate the balance of this investment year-to-year, it would be:

First year: 10,000 + 5
NOTE: Using fractions is typically the easiest way to calculate, so we’ll represent 5% as 1/20 from here on out.

Second year: $$10,000∗\frac{21}{20}+\frac{1}{20}∗(10,000∗\frac{21}{20})=\frac{21}{20}(10,000∗\frac{21}{20})=(\frac{21}{20})^2∗10,000$$

Third year: $$(\frac{21}{20})^2(10,000)+\frac{1}{20}∗(\frac{21}{20})^2(10,000)=\frac{21}{20}∗(\frac{21}{20})^2(10,000)=(\frac{21}{20})^3∗10,000)$$

If you follow the pattern, the total value at the end of each year will simply be $$(\frac{21}{20})^n(10,000)$$ at the end of the nth year. The amount of interest each year is 1/20 of the previous year’s balance (that …+1/20 * the previous year). So, the amount of interest calculated in the 4th year will be: $$\frac{1}{20}∗(\frac{21}{20})^3(10,000)$$

And the amount of interest earned in the 5th year will be: $$\frac{1}{20}∗(\frac{21}{20})^4(10,000)$$

Putting those into ratio, you’ll see that the 1/20 and the 10,000 is common to both, so those terms divide out, leaving simply: $$\frac{(\frac{21}{20})^3}{(\frac{21}{20})^4}$$

Factoring out the common $$(\frac{21}{20})^3$$ term, we’re left with 1/(21/20). Dividing by a fraction is the same as multiplying by the reciprocal, so that can be expressed as 20/21, which is the same as 100/105. Therefore, the correct answer is C.
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Re: A bank offers an interest of 5% per annum compounded annua  [#permalink]

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Blackbox wrote:
Bunuel or any mods,

Please help understand this.

Compound Interest (CI) for 1st year is $$\frac{5}{100}$$*10000 + 10000 = 10,500

Compound Interest (CI) for 2nd year is $$\frac{5}{100}$$*10500 + 10500 = 11,025

Compound Interest (CI) for 3rd year is $$\frac{5}{100}$$*11,025+ 11,025 ----- And this becomes an ugly math. Why? Where am I going wrong?

Compound Interest (CI) for 4th year is $$\frac{5}{100}$$*? + ? ------ By this point, I was heavily estimating the calculation. And I am still left with calculating for 5th year.

Compound Interest (CI) for 5th year is $$\frac{5}{100}$$*?+ ?

I somehow arrived at the correct answer but I see that there is inherently some mistake in my approach. Could someone help? TIA.

I know it is an old post, but still thought of adding my 2 cents. The mistake in your approach is that you went on doing the calculation, when there was no need of any to be done.

I always try to do most of my calculations at the end. And by end, I mean when either I cannot move further, without doing the calculation or when the calculation would give me the final answer.

In this case, we need to find the ratio of two numbers and there would be a high chance that a lot of common terms will get canceled. So while calculating my amount for the 1st, 2nd or 3rd year, I would prefer to keep principal and interest in the actual form, instead of multiplying them.

This helps me because in the end I can cancel them out in the numerator and denominator.

As general rule, always write down what you need to find out and try to avoid the calculations as much as possible unless it really necessary for us to do so.

Regards,
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