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A certain investment earned a fixed rate of 4 percent interest per yea

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Re: A certain investment earned a fixed rate of 4 percent interest per yea [#permalink]
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Let A=Investment amount in dollars
Rate of interest = 4 percent per year compouned annually
Time = 5 years
Let In be the interest earned for nth year
I3= A(1.04)^3 - A(1.04)^2
I1= A(1.04)^1 - A

We need to find I3-I1

1.
A(1.04)^1= 4160
=>A = 4000
Sufficient

2.
A(1.04)^2= 4326.40
=>A=4000
Sufficient
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Re: A certain investment earned a fixed rate of 4 percent interest per yea [#permalink]
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Bunuel wrote:
A certain investment earned a fixed rate of 4 percent interest per year, compounded annually, for five years. The interest earned for the third year of the investment was how many dollars greater than that for the first year?

(1) The amount of the investment at the beginning of the second year was $4,160.00. (2) The amount of the investment at the beginning of the third year was$4,326.40.

Kudos for a correct solution.

Let x be the amount at the beginning of year 1.

Statement 1)
1.04*x = 4'160.00, therefore you can calculate x and thus the interest for the first year as well as the interest for the 3rd year. In this example x will be 4'000. Interest on the first year will be 160.
To calculate the interest on the 3rd year, go step by step to get 1.04^3*x. But you do not really have to do this, all you need to know, is that you can do it. Sufficient.

Statement 2)
The same is true for this. Here you can calculate x too by 1.04^2*x=4'326.4 -->> if you have this formula for interest in mind, you do not have to go further. You know that you can calculate any given year. Sufficient.

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Re: A certain investment earned a fixed rate of 4 percent interest per yea [#permalink]
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A certain investment earned a fixed rate of 4 percent interest per year, compounded annually, for five years. The interest earned for the third year of the investment was how many dollars greater than that for the first year?

Basically we need to know initial principal since rate of interest and time are mentioned

(1) The amount of the investment at the beginning of the second year was $4,160.00. A= p ( 1 + r/100)^n We have A, r and n . Can calculate p. After that interest earned during 2nd and 3rd year can be found and subtracted. SUFFICIENT (2) The amount of the investment at the beginning of the third year was$4,326.40.

A= p ( 1 + r/100)^n . Again Can find p and repeat steps

HENCE D

Originally posted by rishi02 on 12 Jun 2016, 06:01.
Last edited by rishi02 on 23 Jun 2016, 02:16, edited 1 time in total.
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Re: A certain investment earned a fixed rate of 4 percent interest per yea [#permalink]
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assumption : x = initial investment

after 1 year = x(1+4/100)
after 2 years = x(1+4/100)^2
after 3 years = x(1+4/100)^3, etc...

Question : x(1+4/100)^3 - x = x [(1+4/100)^3 - 1], we need x

(1) The amount of the investment at the beginning of the second year was $4,160.00. x(1+4/100)=4,160, we can calculate x, Sufficient (2) The amount of the investment at the beginning of the third year was$4,326.40.

x(1+4/100)^2=4,326.40, we can calculate x, Sufficient
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Re: A certain investment earned a fixed rate of 4 percent interest per yea [#permalink]
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Bunuel wrote:
A certain investment earned a fixed rate of 4 percent interest per year, compounded annually, for five years. The interest earned for the third year of the investment was how many dollars greater than that for the first year?

(1) The amount of the investment at the beginning of the second year was $4,160.00. (2) The amount of the investment at the beginning of the third year was$4,326.40.

Kudos for a correct solution.

Target question: The interest earned for the third year of the investment was how many dollars greater than that for the first year?

Given: A certain investment earned a fixed rate of 4 percent interest per year, compounded annually, for five years.
So, we have:
Let P = the initial investment
After 1 year, the value of the investment = P(1.04)
After 2 years, the value of the investment = P(1.04)^2
After 3 years, the value of the investment = P(1.04)^3
After 4 years, the value of the investment = P(1.04)^4
After 5 years, the value of the investment = P(1.04)^5

Statement 1: The amount of the investment at the beginning of the second year was $4,160.00 The value of the investment at the BEGINNING of the second year is the same as value of the investment at the END of the first year So, we can write: P(1.04) =$4,160.00
Since we COULD solve this question for P, we COULD determine the value of the investment for each of the 5 years, which means we COULD answer the target question with certainty.
As such, statement 1 is SUFFICIENT

Statement 2: The amount of the investment at the beginning of the third year was $4,326.40 The value of the investment at the BEGINNING of the third year is the same as value of the investment at the END of the second year So, we can write: P(1.04)^2 =$4,326.40
Since we COULD solve this question for P, we COULD determine the value of the investment for each of the 5 years, which means we COULD answer the target question with certainty.
As such, statement 2 is SUFFICIENT

Cheers,
Brent
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Re: A certain investment earned a fixed rate of 4 percent interest per yea [#permalink]
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Bunuel wrote:
A certain investment earned a fixed rate of 4 percent interest per year, compounded annually, for five years. The interest earned for the third year of the investment was how many dollars greater than that for the first year?

(1) The amount of the investment at the beginning of the second year was $4,160.00. (2) The amount of the investment at the beginning of the third year was$4,326.40.

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Re: A certain investment earned a fixed rate of 4 percent interest per yea [#permalink]
Bunuel wrote:
A certain investment earned a fixed rate of 4 percent interest per year, compounded annually, for five years. The interest earned for the third year of the investment was how many dollars greater than that for the first year?

(1) The amount of the investment at the beginning of the second year was $4,160.00. (2) The amount of the investment at the beginning of the third year was$4,326.40.

Solution:

Question Stem Analysis:

We need to determine the interest earned for the third year of the investment was how many dollars greater than that for the first year. Notice that we are given the interest rate, the time, and how the interest is accrued. Therefore, if we can determine the original principal, then we can determine the interest earned for any particular year.

Statement One Alone:

From statement one, we see that the interest has accrued for one full year. So, we can create the equation where P is the original principal:

P(1 + 0.04)^1 = 4,160

From the equation above, we see that we can determine a value for P. Thus, we can determine how much more the interest earned for the third year of the investment was than the interest earned in the first year. Statement one alone is sufficient.

Statement Two Alone:

From statement two, we see that the interest has accrued for two full years. So, we can create the equation where P is the principal:

P(1 + 0.04)^2 = 4,326.40

From the equation above, we see that we can determine a value for P. Thus, we can determine how much more the interest earned for the third year of the investment was than the interest earned in the first year. Statement two alone is sufficient.

(Note: The value of P is $4,000 for both equations.) Answer: D Tutor Joined: 17 Jul 2019 Posts: 1304 Own Kudos [?]: 1745 [1] Given Kudos: 66 Location: Canada GMAT 1: 780 Q51 V45 GMAT 2: 780 Q50 V47 GMAT 3: 770 Q50 V45 Re: A certain investment earned a fixed rate of 4 percent interest per yea [#permalink] 1 Kudos Expert Reply Video solution from Quant Reasoning: Subscribe for more: https://www.youtube.com/QuantReasoning? ... irmation=1 Intern Joined: 31 Mar 2021 Posts: 21 Own Kudos [?]: 4 [0] Given Kudos: 290 Re: A certain investment earned a fixed rate of 4 percent interest per yea [#permalink] BrentGMATPrepNow wrote: Bunuel wrote: A certain investment earned a fixed rate of 4 percent interest per year, compounded annually, for five years. The interest earned for the third year of the investment was how many dollars greater than that for the first year? (1) The amount of the investment at the beginning of the second year was$4,160.00.
(2) The amount of the investment at the beginning of the third year was $4,326.40. Kudos for a correct solution. Target question: The interest earned for the third year of the investment was how many dollars greater than that for the first year? Given: A certain investment earned a fixed rate of 4 percent interest per year, compounded annually, for five years. So, we have: Let P = the initial investment After 1 year, the value of the investment = P(1.04) After 2 years, the value of the investment = P(1.04)^2 After 3 years, the value of the investment = P(1.04)^3 After 4 years, the value of the investment = P(1.04)^4 After 5 years, the value of the investment = P(1.04)^5 Statement 1: The amount of the investment at the beginning of the second year was$4,160.00
The value of the investment at the BEGINNING of the second year is the same as value of the investment at the END of the first year
So, we can write: P(1.04) = $4,160.00 Since we COULD solve this question for P, we COULD determine the value of the investment for each of the 5 years, which means we COULD answer the target question with certainty. As such, statement 1 is SUFFICIENT Statement 2: The amount of the investment at the beginning of the third year was$4,326.40
The value of the investment at the BEGINNING of the third year is the same as value of the investment at the END of the second year
So, we can write: P(1.04)^2 = $4,326.40 Since we COULD solve this question for P, we COULD determine the value of the investment for each of the 5 years, which means we COULD answer the target question with certainty. As such, statement 2 is SUFFICIENT Answer: D Cheers, Brent why haven’t be used Simple Interest formula… how to determine which one to use Posted from my mobile device GMAT Club Legend Joined: 12 Sep 2015 Posts: 6805 Own Kudos [?]: 30799 [1] Given Kudos: 799 Location: Canada Re: A certain investment earned a fixed rate of 4 percent interest per yea [#permalink] 1 Kudos Expert Reply Top Contributor iamvishnu wrote: BrentGMATPrepNow wrote: Bunuel wrote: A certain investment earned a fixed rate of 4 percent interest per year, compounded annually, for five years. The interest earned for the third year of the investment was how many dollars greater than that for the first year? (1) The amount of the investment at the beginning of the second year was$4,160.00.
(2) The amount of the investment at the beginning of the third year was $4,326.40. Kudos for a correct solution. Target question: The interest earned for the third year of the investment was how many dollars greater than that for the first year? Given: A certain investment earned a fixed rate of 4 percent interest per year, compounded annually, for five years. So, we have: Let P = the initial investment After 1 year, the value of the investment = P(1.04) After 2 years, the value of the investment = P(1.04)^2 After 3 years, the value of the investment = P(1.04)^3 After 4 years, the value of the investment = P(1.04)^4 After 5 years, the value of the investment = P(1.04)^5 Statement 1: The amount of the investment at the beginning of the second year was$4,160.00
The value of the investment at the BEGINNING of the second year is the same as value of the investment at the END of the first year
So, we can write: P(1.04) = $4,160.00 Since we COULD solve this question for P, we COULD determine the value of the investment for each of the 5 years, which means we COULD answer the target question with certainty. As such, statement 1 is SUFFICIENT Statement 2: The amount of the investment at the beginning of the third year was$4,326.40
The value of the investment at the BEGINNING of the third year is the same as value of the investment at the END of the second year
So, we can write: P(1.04)^2 = $4,326.40 Since we COULD solve this question for P, we COULD determine the value of the investment for each of the 5 years, which means we COULD answer the target question with certainty. As such, statement 2 is SUFFICIENT Answer: D Cheers, Brent why haven’t be used Simple Interest formula… how to determine which one to use Posted from my mobile device The question tells us the investment earned a fixed rate of 4 percent interest per year, compounded annually Manager Joined: 05 Mar 2024 Posts: 142 Own Kudos [?]: 27 [0] Given Kudos: 5 A certain investment earned a fixed rate of 4 percent interest per yea [#permalink] alejandrogaleas wrote: A certain investment earned a fixed rate of 4 percent interest per year, compounded annually, for five years. The interest earned for the third year of the investment was how many dollars greater than that for the first year? (1) The amount of the investment at the beginning of the second year was$4,160.00.
(2) The amount of the investment at the beginning of the third year was $4,326.40. Hope this helps someone: Using the formula for compound interest: $$FV=PV (1+r)^n$$ FV: future Value PV: present Value r: annual interest rate n: number of periods (1) Amount of the investment at the beggining of the second year (or initial investment + interest accumulated at the end of the first year) is 4160, which can be represented as: $$4160=PV (1+0.04$$)$$^1$$ $$4000=PV$$ PV is the same as the initial investment, therefore the interest earned after the first year is 160. once PV is obtained, the difference between the interest earned after the third year and after the first year is: $$[4000(1.04)^3-4000(1.04)^2]$$$$-160$$ further calculation isn´t needed since we are able to respond the question. (1) Sufficient (2) We are able to get PV by using the same formula, only FV and n values are changed: $$4326.40=PV (1+0.04)^2$$ $$4000=PV$$ once PV is obtained, the difference between the interest earned after the third year and after the first year is: $$[4000(1.04)^3-4000(1.04)^2]$$$$-[4000(1.04)^1-4000]$$ further calculation isn´t needed since we are able to respond the question. (2) Sufficient Answer is D ­I don't get why you subtracted 160 in 1)­ Math Expert Joined: 02 Sep 2009 Posts: 94342 Own Kudos [?]: 640863 [0] Given Kudos: 85011 Re: A certain investment earned a fixed rate of 4 percent interest per yea [#permalink] Expert Reply PrabhatKC wrote: alejandrogaleas wrote: A certain investment earned a fixed rate of 4 percent interest per year, compounded annually, for five years. The interest earned for the third year of the investment was how many dollars greater than that for the first year? (1) The amount of the investment at the beginning of the second year was$4,160.00.
(2) The amount of the investment at the beginning of the third year was \$4,326.40.

Hope this helps someone:

Using the formula for compound interest:

$$FV=PV (1+r)^n$$

FV: future Value
PV: present Value
r: annual interest rate
n: number of periods

(1)
Amount of the investment at the beggining of the second year (or initial investment + interest accumulated at the end of the first year) is 4160, which can be represented as:

$$4160=PV (1+0.04$$)$$^1$$
$$4000=PV$$

PV is the same as the initial investment, therefore the interest earned after the first year is 160.

once PV is obtained, the difference between the interest earned after the third year and after the first year is:

$$[4000(1.04)^3-4000(1.04)^2]$$$$-160$$

further calculation isn´t needed since we are able to respond the question.

(1) Sufficient

(2)
We are able to get PV by using the same formula, only FV and n values are changed:
$$4326.40=PV (1+0.04)^2$$
$$4000=PV$$

once PV is obtained, the difference between the interest earned after the third year and after the first year is:

$$[4000(1.04)^3-4000(1.04)^2]$$$$-[4000(1.04)^1-4000]$$

further calculation isn´t needed since we are able to respond the question.

(2) Sufficient