A bank offers an interest of 5% per annum compounded

[RBBmba@2014]

A bank offers an interest of 5% per annum compounded annually on all of 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 : 1005

(E) 725 : 3225

OA : C

@ Experts - What could be the fastest and smartest way to solve this sort of problem under time constraint?

[GMATGuruNY]

A bank offers an interest of 5% per annum compounded annually on all of 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 : 1005

(E) 725 : 3225

OA : C

@ Experts - What could be the fastest and smartest way to solve this sort of problem under time constraint?

The interest ratio from one year to the next is always THE SAME.
Since the amount in the account increases by 5% each year, the amount of interest also increases by 5% each year.
Thus, if $100 interest is earned one year, then $105 interest is earned the next year.
Resulting ratio from one year to the next = 100:105.

The correct answer is C.

To verify the line of reasoning above, we can test cases.

Case 1: Amount in the account in the 4th year = 2000.
4th-year interest = 5% of 2000 = 100.
Amount in the account in the 5th year = 2000 + 100 = 2100.
5th-year interest = 5% of 2100 = 105.
(4th-year interest) : (5th-year interest) = 100:105.

Case 2: Amount in the account in the 4th year = 4000.
4th-year interest = 5% of 4000 = 200.
Amount in the account in the 5th year = 4000 + 200 = 4200.
5th-year interest = 5% of 4200 = 210.
(4th-year interest) : (5th-year interest) = 200:210 = 100:105.

In each case, the resulting interest ratio is THE SAME.

[RBBmba@2014]

A bank offers an interest of 5% per annum compounded annually on all of 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 : 1005

(E) 725 : 3225

OA : C

@ Experts - What could be the fastest and smartest way to solve this sort of problem under time constraint?

The interest ratio from one year to the next is always THE SAME.
Since the amount in the account increases by 5% each year, the amount of interest also increases by 5% each year.
Thus, if $100 interest is earned one year, then $105 interest is earned the next year.
Resulting ratio from one year to the next = 100:105.

The correct answer is C.

To verify the line of reasoning above, we can test cases.

Case 1: Amount in the account in the 4th year = 2000.
4th-year interest = 5% of 2000 = 100.
Amount in the account in the 5th year = 2000 + 100 = 2100.
5th-year interest = 5% of 2100 = 105.
(4th-year interest) : (5th-year interest) = 100:105.

Case 2: Amount in the account in the 4th year = 4000.
4th-year interest = 5% of 4000 = 200.
Amount in the account in the 5th year = 4000 + 200 = 4200.
5th-year interest = 5% of 4200 = 210.
(4th-year interest) : (5th-year interest) = 200:210 = 100:105.

In each case, the resulting interest ratio is THE SAME.

Mitch - just a quick clarification that the test cases you've demonstrated (i.e Case 1: Amount in the account in the 4th year = 2000 or Case 2: Amount in the account in the 4th year = 4000) are arbitrary, I think. Right?

I mean, this amount can hold any value, I believe. Correct me please if wrong!

[coolhabhi]

[quote="RBBmba@2014"][color=blue][b]A bank offers an interest of 5% per annum compounded annually on all of 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?[/b]

(A) 1:5

(B) 625 : 3125

(C) 100 : 105

(D) 1004 : 1005

(E) 725 : 3225[/color]

OA : [spoiler]C[/spoiler]

@ Experts - What could be the fastest and smartest way to solve this sort of problem [u]under time constraint[/u]?[/quote]

I might probably help you out here.
There is a formula for this type of problems. It is

Interest earned I = P[1 + (r/100)]^n
where P is the Principle Amount I.e., 10000
r is the Rate of interest I.e., 5
n is the number of years

Applying the formula
Interest at the end if 4th year P4 = 10000[1 + (5/100)]^4

Interest at the end if 5th year P5 = 10000[1 + (5/100)]^5

=>ratio of the interest earned in the 4th year to the interest earned in the 5th year
10000[1 + (5/100)]^4 : 10000[1 + (5/100)]^5
=>1 : [1 + (5/100)]
=>1 : [105/100]
=>105:100
=>C

[GMATGuruNY]

Mitch - just a quick clarification that the test cases you've demonstrated (i.e Case 1: Amount in the account in the 4th year = 2000 or Case 2: Amount in the account in the 4th year = 4000) are arbitrary, I think. Right?

I mean, this amount can hold any value, I believe. Correct me please if wrong!

Correct.
The amount in the account at the beginning of the fourth year is irrelevant.
It can be ANY VALUE.
Whether there is $1 in the account at the beginning of the fourth year -- or $1,000,000 -- the amount of interest accrued in the fifth year will be 5% greater than the amount of interest accrued in the fourth year.