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A bank account earned 2% annual interest, compounded daily, for as lon

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A bank account earned 2% annual interest, compounded daily, for as lon [#permalink] A bank account earned 2% annual interest, compounded daily, for as lon   21 Sep 2019, 19:02
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A bank account earned 2% annual interest, compounded daily, for as long as the balance was under $1,000, starting when the account was opened. Once the balance reached $1,000, the account earned 2.5% annual interest, compounded daily until the account was closed. No deposits or withdrawals were made. Was the total amount of interest earned at the 2% rate greater than the total amount earned at the 2.5% rate?

(1) The account earned exactly $25 in interest at the 2.5% rate.
(2) The account was open for exactly three years.


DS29931.01
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C
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A bank account earned 2% annual interest, compounded daily, for as lon [#permalink] A bank account earned 2% annual interest, compounded daily, for as lon   Updated on: 20 Oct 2019, 17:08
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Suppose initial Principal amount=P
Assume, after x days, balance would be 1000

\(P(1+\frac{2}{365*100})^x=1000\)......(1)

Assume account was opened for N days
Balance after N days= B

\(1000(1+\frac{2.5}{365*100})^{N-x}= B\).....(2)

We need to figure out Both P and B.

Statement 1
Statement 1 gives us the value of B. We can figure out N-x, but P is still unknown.

Statement 2-
Statement gives us the value of N, but both P and B are unknown

Combining both Statements
From statement 1
we have value of B and with the help of equation (2), we can find out N-x

From statement 2 we have value of N; Hence, we can find the value of x. With the help of equation 1, we can find P.

Sufficient







gmatt1476
A bank account earned 2% annual interest, compounded daily, for as long as the balance was under $1,000, starting when the account was opened. Once the balance reached $1,000, the account earned 2.5% annual interest, compounded daily until the account was closed. No deposits or withdrawals were made. Was the total amount of interest earned at the 2% rate greater than the total amount earned at the 2.5% rate?

(1) The account earned exactly $25 in interest at the 2.5% rate.
(2) The account was open for exactly three years.


DS29931.01

Originally posted by nick1816 on 30 Sep 2019, 06:18.
Last edited by nick1816 on 20 Oct 2019, 17:08, edited 2 times in total.
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Re: A bank account earned 2% annual interest, compounded daily, for as lon [#permalink] A bank account earned 2% annual interest, compounded daily, for as lon   05 Sep 2020, 12:46
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gmatt1476
A bank account earned 2% annual interest, compounded daily, for as long as the balance was under $1,000, starting when the account was opened. Once the balance reached $1,000, the account earned 2.5% annual interest, compounded daily until the account was closed. No deposits or withdrawals were made. Was the total amount of interest earned at the 2% rate greater than the total amount earned at the 2.5% rate?

(1) The account earned exactly $25 in interest at the 2.5% rate.
(2) The account was open for exactly three years.


DS29931.01
Solution:

Statement One Only:
The account earned exactly $25 in interest at the 2.5% rate.

Since we don’t know the original amount deposited in the account, i.e., the principal, we can’t determine whether the total amount of interest earned at the 2% rate is greater than the total amount earned at the 2.5% rate. For example, if the principal is $970, then the interest earned at 2% will be $30, which is greater than the $25 interest earned at the 2.5% rate. However, if the principal is $980, then the interest earned at 2% will be $20, which is less than the $25 interest earned at the 2.5% rate.

Statement Two Only:
The account was open for exactly three years.

Since we don’t know the length of time it takes the principal to reach the $1000 benchmark, we can’t determine whether the total amount of interest earned at the 2% rate is greater than the total amount earned at the 2.5% rate. For example, if it takes 2 years for the principal to reach $1000, then certainly the interest earned at 2% (during the first 2 years) will be greater than the $25 interest earned at the 2.5% rate (during the last year). However, if it takes 1 year for the principal to reach $1000, then certainly the interest earned at 2% (during the first year) will be less than the $25 interest earned at the 2.5% rate (during the last 2 years).

Statements One and Two Together:

Using statement one, we can calculate the time necessary to earn an interest of $25 from a principal of $1,000 at 2.5% compounded daily. Using statement two, we can subtract the time we calculated in the previous step from three years to determine the amount of time the principal acquired interest at the 2% rate. Since we know the account was worth $1,000 at the end of this time, we can calculate the principal and the interest earned in this period. Finally, we can compare this interest to $25 and determine whether the account acquired more interest in the 2% period or the 2.5% period.

Recall that in a DS question, we only need to determine whether we have enough information to answer the question; we don’t have to perform the steps and find the actual answer. As we can see from above, statement one and statement two together provide sufficient information to answer the question.

Answer: C
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A bank account earned 2% annual interest, compounded daily, for as lon [#permalink] A bank account earned 2% annual interest, compounded daily, for as lon   01 Oct 2019, 16:01
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gmatt1476
A bank account earned 2% annual interest, compounded daily, for as long as the balance was under $1,000, starting when the account was opened. Once the balance reached $1,000, the account earned 2.5% annual interest, compounded daily until the account was closed. No deposits or withdrawals were made. Was the total amount of interest earned at the 2% rate greater than the total amount earned at the 2.5% rate?

(1) The account earned exactly $25 in interest at the 2.5% rate.
(2) The account was open for exactly three years.


DS29931.01

The question is asking if the Interest earned at 2% > than the Interest earned at 2.5%
Let's say that:
Interest Earned at 2% = x
Interest Earned at 2.5 = Y

Here's my approach, which requires little math and more reasoning/logic:

1.) Interested Earned at 2.5% = $25
If the initial investment is $1, we would know that the interested earned from 2% would be greater than Interest earned from 2.5% ($25) X>Y
If the initial investment is $999, we would know that the interest earned from 2% would be smaller than the Interest earned from 2.5% ($25) Y>X
Insufficient

2.) Time = 3 years
We don't know the starting values, if the initial investment is $1, in three years X>Y. If the initial investment is $999, then Y>X. Insufficient

1) + 2)
We know that time is 3 years, and that the interest earned at 2.5% is $25. This basically "locks" in the value of Y with a specific amount of time, so then we can directly solve for how much of the 3 years was the investment under $1000, and hence X. Sufficient

C is the Answer!
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Re: A bank account earned 2% annual interest, compounded daily, for as lon [#permalink] A bank account earned 2% annual interest, compounded daily, for as lon   20 Oct 2019, 13:38
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nick1816
Suppose initial Principal amount=P
Assume, after x days, balance would be 1000

\(P(1+\frac{2}{365})^x=1000\)......(1)

Assume account was opened for N days
Balance after N days= B

\(1000(1+\frac{2.5}{365})^{N-x}= B\).....(2)

We need to figure out Both P and B.

Statement 1
Statement 1 gives us the value of B. We can figure out N-x, but P is still unknown.

Statement 2-
Statement gives us the value of N, but both P and B are unknown

Combining both Statements
From statement 1
we have value of B and with the help of equation (2), we can find out N-x

From statement 2 we have value of N; Hence, we can find the value of x. With the help of equation 1, we can find P.

Sufficient







gmatt1476
A bank account earned 2% annual interest, compounded daily, for as long as the balance was under $1,000, starting when the account was opened. Once the balance reached $1,000, the account earned 2.5% annual interest, compounded daily until the account was closed. No deposits or withdrawals were made. Was the total amount of interest earned at the 2% rate greater than the total amount earned at the 2.5% rate?

(1) The account earned exactly $25 in interest at the 2.5% rate.
(2) The account was open for exactly three years.


DS29931.01
Hey nick1816,
Which formula have u used for the C.I. equation?
I remember 1 formula - C.I. = P(1 + (r/n))^(n*t)
Where - n- no. of times/ year
t - no. of years
r - rate
But what you've used isn't analogous to this one...
Please help me understand your formula...
Thanks in advance... :)
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Re: A bank account earned 2% annual interest, compounded daily, for as lon [#permalink] A bank account earned 2% annual interest, compounded daily, for as lon   20 Oct 2019, 18:54
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Brother if you're considering interest rate annually, convert the duration in years.
if you're considering interest rate monthly, convert the duration in months.
if you're considering interest rate daily, convert the duration in days.

Why it works.
Consider interest rate annually=r
time duration in years= 0.5 years

A=\(P(1+r/100)^{0.5}\)= \(P(1+\frac{0.5r}{100}+(0.5)(-0.5)(\frac{r}{100})^2/2!..........)\)=P(1+r/200)
or
A=\(P(1+r/1200)^6\)= \(P(1+\frac{6r}{1200}+6*5(\frac{r}{1200})^2/2!.........)\)= P(1+r/200)

As r/100 is very small, higher power of r/100 is almost negligible. Hence, you can neglect 3rd and higher terms.



RK007
nick1816
Suppose initial Principal amount=P
Assume, after x days, balance would be 1000

\(P(1+\frac{2}{365})^x=1000\)......(1)

Assume account was opened for N days
Balance after N days= B

\(1000(1+\frac{2.5}{365})^{N-x}= B\).....(2)

We need to figure out Both P and B.

Statement 1
Statement 1 gives us the value of B. We can figure out N-x, but P is still unknown.

Statement 2-
Statement gives us the value of N, but both P and B are unknown

Combining both Statements
From statement 1
we have value of B and with the help of equation (2), we can find out N-x

From statement 2 we have value of N; Hence, we can find the value of x. With the help of equation 1, we can find P.

Sufficient







gmatt1476
A bank account earned 2% annual interest, compounded daily, for as long as the balance was under $1,000, starting when the account was opened. Once the balance reached $1,000, the account earned 2.5% annual interest, compounded daily until the account was closed. No deposits or withdrawals were made. Was the total amount of interest earned at the 2% rate greater than the total amount earned at the 2.5% rate?

(1) The account earned exactly $25 in interest at the 2.5% rate.
(2) The account was open for exactly three years.


DS29931.01
Hey nick1816,
Which formula have u used for the C.I. equation?
I remember 1 formula - C.I. = P(1 + (r/n))^(n*t)
Where - n- no. of times/ year
t - no. of years
r - rate
But what you've used isn't analogous to this one...
Please help me understand your formula...
Thanks in advance... :)
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Re: A bank account earned 2% annual interest, compounded daily, for as lon [#permalink] A bank account earned 2% annual interest, compounded daily, for as lon   25 Jan 2020, 14:13
Hello,

In this question we established that the account earned 2.5% only for 1 year since it Staement 1 tells us that it earned 25$ therefore only for 1year out of 3 did it have $1000 savings; this means that for 2 years the account had a balance of under $1000.

How do we know that this balance was not $1, meaning 2% interest/year on a $1 balance would mean that the account earned less interest at the 2% rate vs 2.5%?

Thanks!
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Re: A bank account earned 2% annual interest, compounded daily, for as lon [#permalink] A bank account earned 2% annual interest, compounded daily, for as lon   31 Jan 2020, 03:20
jsound996
gmatt1476
A bank account earned 2% annual interest, compounded daily, for as long as the balance was under $1,000, starting when the account was opened. Once the balance reached $1,000, the account earned 2.5% annual interest, compounded daily until the account was closed. No deposits or withdrawals were made. Was the total amount of interest earned at the 2% rate greater than the total amount earned at the 2.5% rate?

(1) The account earned exactly $25 in interest at the 2.5% rate.
(2) The account was open for exactly three years.


DS29931.01

The question is asking if the Interest earned at 2% > than the Interest earned at 2.5%
Let's say that:
Interest Earned at 2% = x
Interest Earned at 2.5 = Y

Here's my approach, which requires little math and more reasoning/logic:

1.) Interested Earned at 2.5% = $25
If the initial investment is $1, we would know that the interested earned from 2% would be greater than Interest earned from 2.5% ($25) X>Y
If the initial investment is $999, we would know that the interest earned from 2% would be smaller than the Interest earned from 2.5% ($25) Y>X
Insufficient

2.) Time = 3 years
We don't know the starting values, if the initial investment is $1, in three years X>Y. If the initial investment is $999, then Y>X. Insufficient

1) + 2)
We know that time is 3 years, and that the interest earned at 2.5% is $25. This basically "locks" in the value of Y with a specific amount of time, so then we can directly solve for how much of the 3 years was the investment under $1000, and hence X. Sufficient

C is the Answer!

Hey can you please help me understand your approach once again?
As per Statement 1, the account earned exactly $25 in interest at the 2.5% rate, how are we evaluating this value in context of 3 years? isn't this value primarily for 1 year? I think I am not able to understand this question.
Thanks.
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A bank account earned 2% annual interest, compounded daily, for as lon [#permalink] A bank account earned 2% annual interest, compounded daily, for as lon   03 Feb 2020, 03:55
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gmatt1476
A bank account earned 2% annual interest, compounded daily, for as long as the balance was under $1,000, starting when the account was opened. Once the balance reached $1,000, the account earned 2.5% annual interest, compounded daily until the account was closed. No deposits or withdrawals were made. Was the total amount of interest earned at the 2% rate greater than the total amount earned at the 2.5% rate?

(1) The account earned exactly $25 in interest at the 2.5% rate.
(2) The account was open for exactly three years.


DS29931.01

The question is asking if the Interest earned at 2% > than the Interest earned at 2.5%
Let's say that:
Interest Earned at 2% = x
Interest Earned at 2.5 = Y

Here's my approach, which requires little math and more reasoning/logic:

1.) Interested Earned at 2.5% = $25
If the initial investment is $1, we would know that the interested earned from 2% would be greater than Interest earned from 2.5% ($25) X>Y
If the initial investment is $999, we would know that the interest earned from 2% would be smaller than the Interest earned from 2.5% ($25) Y>X
Insufficient

2.) Time = 3 years
We don't know the starting values, if the initial investment is $1, in three years X>Y. If the initial investment is $999, then Y>X. Insufficient

1) + 2)
We know that time is 3 years, and that the interest earned at 2.5% is $25. This basically "locks" in the value of Y with a specific amount of time, so then we can directly solve for how much of the 3 years was the investment under $1000, and hence X. Sufficient

C is the Answer!

Hey can you please help me understand your approach once again?
As per Statement 1, the account earned exactly $25 in interest at the 2.5% rate, how are we evaluating this value in context of 3 years? isn't this value primarily for 1 year? I think I am not able to understand this question.
Thanks.

Hi, this question basically is pretty simple as the value isn't asked. so this makes our work easy.
clearly each alone isn't sufficient.

Let's see the formula for C.I.=P[(1+r/n)^nt -1] (the -1 is for finding interest)

so now for

1) C.I=1000[((1+2.5/365*100)^365*t)-1]=25
from this equation we can somehow find the value of t in terms of a fraction which converts into days. i.e, days for which the 2.5% interest is calculated.

now from

2) 3 years minus t found from (1) gives no.of years at 2% interest. (Say T)
this converts the known equation: 1000=P(1+2/365*100)^365T.

where T = 3-t.

now on solving you can find the interest required and do the remaining. Thus its option(C)

HOPE THIS HELPS YOU SOLVE YOUR CONFUSION. HIT KUDOS IF YOU UNDERSTOOD.
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Re: A bank account earned 2% annual interest, compounded daily, for as lon [#permalink] A bank account earned 2% annual interest, compounded daily, for as lon   12 Apr 2020, 03:34
Bunuel Can you please advise how to approach this problem.
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Re: A bank account earned 2% annual interest, compounded daily, for as lon [#permalink] A bank account earned 2% annual interest, compounded daily, for as lon   22 Feb 2022, 12:09
gmatt1476
A bank account earned 2% annual interest, compounded daily, for as long as the balance was under $1,000, starting when the account was opened. Once the balance reached $1,000, the account earned 2.5% annual interest, compounded daily until the account was closed. No deposits or withdrawals were made. Was the total amount of interest earned at the 2% rate greater than the total amount earned at the 2.5% rate?

(1) The account earned exactly $25 in interest at the 2.5% rate.
(2) The account was open for exactly three years.


DS29931.01

HI KarishmaB! Can you help me with this question? Thank you! :)
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Re: A bank account earned 2% annual interest, compounded daily, for as lon [#permalink] A bank account earned 2% annual interest, compounded daily, for as lon   22 Feb 2022, 19:22
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gmatt1476
A bank account earned 2% annual interest, compounded daily, for as long as the balance was under $1,000, starting when the account was opened. Once the balance reached $1,000, the account earned 2.5% annual interest, compounded daily until the account was closed. No deposits or withdrawals were made. Was the total amount of interest earned at the 2% rate greater than the total amount earned at the 2.5% rate?

(1) The account earned exactly $25 in interest at the 2.5% rate.
(2) The account was open for exactly three years.


DS29931.01

This is what we know about compounding:

\(Amount = P(1 + \frac{r}{100})^n\)

(1) The account earned exactly $25 in interest at the 2.5% rate.

So \(1025 = 1000(1 + \frac{2.5}{365*100})^{365n}\)
We can get the value of n i.e. the number of years for which 2.5% rate was applicable.
But we don't know how long the account earned 2% interest so not sufficient alone.

(2) The account was open for exactly three years.
We don't know how long each interest rate was applicable. Not sufficient.

Both together,
We know the number of years the 2.5% rate was applicable. We also know that total for 3 years the account was open. So 2% rate was applicable for (3 - n) years which we get because we know n.

\(1000 = P(1 + \frac{2}{365*100})^{365(3-n)}\)
Since we know n, we can get P and hence we know the exact interest amount earned at 2% too. This can help us compare the interest amounts on the two rates. Sufficient.

Answer (C)
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A bank account earned 2% annual interest, compounded daily, for as lon [#permalink] A bank account earned 2% annual interest, compounded daily, for as lon   27 Aug 2023, 23:58
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gmatt1476
A bank account earned 2% annual interest, compounded daily, for as long as the balance was under $1,000, starting when the account was opened. Once the balance reached $1,000, the account earned 2.5% annual interest, compounded daily until the account was closed. No deposits or withdrawals were made. Was the total amount of interest earned at the 2% rate greater than the total amount earned at the 2.5% rate?

(1) The account earned exactly $25 in interest at the 2.5% rate.
(2) The account was open for exactly three years.


DS29931.01
Solution:

Statement One Only:
The account earned exactly $25 in interest at the 2.5% rate.

Since we don’t know the original amount deposited in the account, i.e., the principal, we can’t determine whether the total amount of interest earned at the 2% rate is greater than the total amount earned at the 2.5% rate. For example, if the principal is $970, then the interest earned at 2% will be $30, which is greater than the $25 interest earned at the 2.5% rate. However, if the principal is $980, then the interest earned at 2% will be $20, which is less than the $25 interest earned at the 2.5% rate.

Statement Two Only:
The account was open for exactly three years.

Since we don’t know the length of time it takes the principal to reach the $1000 benchmark, we can’t determine whether the total amount of interest earned at the 2% rate is greater than the total amount earned at the 2.5% rate. For example, if it takes 2 years for the principal to reach $1000, then certainly the interest earned at 2% (during the first 2 years) will be greater than the $25 interest earned at the 2.5% rate (during the last year). However, if it takes 1 year for the principal to reach $1000, then certainly the interest earned at 2% (during the first year) will be less than the $25 interest earned at the 2.5% rate (during the last 2 years).

Statements One and Two Together:

Using statement one, we can calculate the time necessary to earn an interest of $25 from a principal of $1,000 at 2.5% compounded daily. Using statement two, we can subtract the time we calculated in the previous step from three years to determine the amount of time the principal acquired interest at the 2% rate. Since we know the account was worth $1,000 at the end of this time, we can calculate the principal and the interest earned in this period. Finally, we can compare this interest to $25 and determine whether the account acquired more interest in the 2% period or the 2.5% period.

Recall that in a DS question, we only need to determine whether we have enough information to answer the question; we don’t have to perform the steps and find the actual answer. As we can see from above, statement one and statement two together provide sufficient information to answer the question.

Answer: C

I used the same method as you did, and got the "right" answer. But I wasn't convinced because I (and you) am making an assumption here (the highlighted text above). How do we know that the starting balance for the 2nd period is 1000$ exactly? It could be 1000.01 or 1000.1 (you get my point). Because we are told that compounded happens daily and the daily could take it from just below 1000$ to just above 1000$. Of course, we are simplifying here, and in this case, it leads us to the right answer. But will it always? Why is it OK to ignore this rounding error?
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