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

Question

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

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

Bunuel · Accepted Answer

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.

suk1234 · Answer

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

atalpanditgmat · Answer

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

Bunuel · Answer

Check here: a-bank-offers-an-interest-of-5-per-annum-compounded-annua-154203.html#p1234708

PareshGmat · Answer

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

Temurkhon · Answer

question ask what is:

10000*1.05^4/10000*1.05^5

we get 10000/10000*1.05=10000/10500=100/105

pqhai · Answer

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.

Rashima · Answer

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

GMATinsight · Answer

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.

amirzohrevand · Answer

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?

GMATinsight · Answer

$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!

amirzohrevand · Answer

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

GMATinsight · Answer

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.

subhamgarg91 · Answer

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!

Sachin07 · Answer

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

Blackbox · Answer

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.

Bunuel · Answer

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.

EgmatQuantExpert · Answer

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, Saquib e-GMAT Quant Expert

ManishaPrepMinds · Answer

For compound Interest, the percent increase in the interest earned for every period/year is equal to the rate of interest per period or rate of interest per year 
Hence,
Interest earned in the First Year: $10000*0.05=500
Interest earned in the second Year: 500*1.05
Interest earned in the Third Year: 500*(1.05)^2
Interest earned in the Fourth Year: 500*(1.05)^3
Interest earned in the Fifth Year: 500* (1.05)^4
thus, ratio of interest earned in 4th year to the interest earned in 5th year=  \frac{500*(1.05)^3}{ 500* (1.05)^4} 
= \frac{1}{1.05} 
= \frac{100}{105}

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

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