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A bank offers an interest of 5% per annum compounded annua

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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
[Reveal] Spoiler: OA

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Last edited by Bunuel on 28 Jan 2015, 07:43, edited 2 times in total.
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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. 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.
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Re: A bank offers an interest of 5% per annum compounded annua [#permalink]

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New post 14 Jun 2013, 03:22
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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New post 14 Jun 2013, 03:28
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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


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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. 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]

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New post 15 Apr 2014, 20:39
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?
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Re: A bank offers an interest of 5% per annum compounded annua [#permalink]

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New post 15 Apr 2014, 22:00
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
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Re: A bank offers an interest of 5% per annum compounded annua [#permalink]

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New post 15 Apr 2014, 22:06
Bunuel 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


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

(50*1.05^3)/(50*1.05^4) = 1/1.05=100/105.

Answer: C.



Bunuel, can you please correct this?

It should be 500

\(\frac{10000 * 5 * 1}{100} = 500\)
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New post 23 Jul 2014, 09:42
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 : 100^5
e) 725 : 3225


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∗21/20+1/20∗(10,000∗21/20)=21/20(10,000∗21/20)=(21/20)2∗10,000
Third year: (21/20)2(10,000)+1/20∗(21/20)2(10,000)=21/20∗(21/20)2(10,000)=(21/20)3∗10,000)
If you follow the pattern, the total value at the end of each year will simply be (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:

1/20∗(21/20)3(10,000)
And the amount of interest earned in the 5th year will be:

1/20∗(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:

(21/20)3/(21/20)4
Factoring out the common (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.


MY CONFUSION : IT WAS A LONG WORDY EXPLANATION, AND I GOT LOST IN IT, SO I NEED A MORE CONCISE EXPLANATION IF POSSIBLE, ALSO WHERE DOES THE 21/ 20 COME FROM? IN SECOND YEAR HOW COME WE ARE ADDING 21/20 AND 1/20 AND THEN MULTIPLYING BY 10,000 AND 1/20 WHY NOT MULTIPLY BY 21/20?
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New post 23 Jul 2014, 09:47
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sagnik2422 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 : 100^5
e) 725 : 3225


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∗21/20+1/20∗(10,000∗21/20)=21/20(10,000∗21/20)=(21/20)2∗10,000
Third year: (21/20)2(10,000)+1/20∗(21/20)2(10,000)=21/20∗(21/20)2(10,000)=(21/20)3∗10,000)
If you follow the pattern, the total value at the end of each year will simply be (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:

1/20∗(21/20)3(10,000)
And the amount of interest earned in the 5th year will be:

1/20∗(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:

(21/20)3/(21/20)4
Factoring out the common (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.


MY CONFUSION : IT WAS A LONG WORDY EXPLANATION, AND I GOT LOST IN IT, SO I NEED A MORE CONCISE EXPLANATION IF POSSIBLE, ALSO WHERE DOES THE 21/ 20 COME FROM? IN SECOND YEAR HOW COME WE ARE ADDING 21/20 AND 1/20 AND THEN MULTIPLYING BY 10,000 AND 1/20 WHY NOT MULTIPLY BY 21/20?


Merging topics. Please refer to the discussion above.

P.S. Please read carefully and follow: rules-for-posting-please-read-this-before-posting-133935.html Pay attention to rules 1 and 3. Thank you.


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

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New post 03 Sep 2014, 12:11
Hi Bunel, I like your explanation but have 2 questions.

You say (500*1.05^3)/(500*1.05^4) = 1/1.05. I understand the 1, but when you divide exponents, I thought you subtract (i.e. 1.05^3/1.05^4 = 1.05^-1).

Also, what would the equation look like if they wanted the ratio of the total amounts (principal + interest) for years 4 and 5?
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New post 03 Sep 2014, 21:59
question ask what is:

10000*1.05^4/10000*1.05^5

we get 10000/10000*1.05=10000/10500=100/105
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Re: A bank offers an interest of 5% per annum compounded annua [#permalink]

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New post 03 Sep 2014, 23:12
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.
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Re: A bank offers an interest of 5% per annum compounded annua [#permalink]

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New post 07 Jul 2015, 11:40
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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New post 07 Jul 2015, 14:20
Expert's post
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.
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Re: A bank offers an interest of 5% per annum compounded annua [#permalink]

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New post 17 Sep 2015, 21:26
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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New post 17 Sep 2015, 23:47
Expert's post
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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Last edited by GMATinsight on 18 Sep 2015, 04:06, edited 1 time in total.
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A bank offers an interest of 5% per annum compounded annua [#permalink]

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New post 18 Sep 2015, 01:06
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
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Re: A bank offers an interest of 5% per annum compounded annua [#permalink]

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New post 18 Sep 2015, 04:04
Expert's post
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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Re: A bank offers an interest of 5% per annum compounded annua [#permalink]

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New post 28 Sep 2015, 08:42
suk1234 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


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


I am literally ashamed that I didn't see this :cry:
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Re: A bank offers an interest of 5% per annum compounded annua   [#permalink] 28 Sep 2015, 08:42

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