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

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A certain investment earned a fixed rate of 4 percent interest per yea  [#permalink]

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New post 21 Oct 2015, 20:37
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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?

(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.

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Re: A certain investment earned a fixed rate of 4 percent interest per yea  [#permalink]

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New post 21 Oct 2015, 21:40
2
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
Answer D
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Re: A certain investment earned a fixed rate of 4 percent interest per yea  [#permalink]

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New post 23 Oct 2015, 00:05
2
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.

Answer D.
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A certain investment earned a fixed rate of 4 percent interest per yea  [#permalink]

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New post Updated on: 23 Jun 2016, 01:16
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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
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Originally posted by rishi02 on 12 Jun 2016, 05:01.
Last edited by rishi02 on 23 Jun 2016, 01:16, edited 1 time in total.
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A certain investment earned a fixed rate of 4 percent interest per yea  [#permalink]

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New post 22 Jun 2016, 08:39
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1
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
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A certain investment earned a fixed rate of 4 percent interest per yea  [#permalink]

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New post 22 Jun 2016, 12:11
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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New post 15 Dec 2017, 17:34
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Hi All,

This is a great 'concept' question - meaning that if you recognize the concept(s) involved, you don't have to do much (if any) math to get to the correct answer.

In this prompt, we're told that an investment earns a fixed 4% interest, compounded annually for 5 years. We're asked for the difference, in interest earnings, between the 3rd year and the 1st year.

To start, we don't know the initial investment. However, since we're compounding the interest each year, then we know that the interest will be 4% of whatever total amount is already there. Thus, we have a constant 'multiplier' (1.04) that we'll just keep multiplying year-after-year. This means that if we know ANY dollar figure at any point in the 5 year history, then we can figure out ALL of the other values (using either division - to move backwards, or multiplication - to move forwards).

1) The amount of the investment at the beginning of the second year was $4,160.00

With this Fact, we can move 'backwards' and 'forwards' as described above and get the necessary information to answer the question.
Fact 1 is SUFFICIENT.

2) The amount of the investment at the beginning of the third year was $4,326.40

Just as in Fact 1 (above), we can move 'backwards' and 'forwards' as described and get the necessary information to answer the question.
Fact 2 is SUFFICIENT.

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Re: A certain investment earned a fixed rate of 4 percent interest per yea  [#permalink]

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New post 13 Jan 2018, 06:37
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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

Answer: D

Cheers,
Brent
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