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A certain investment earned a fixed rate of 4 percent

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by jjjinapinch » Thu Jul 27, 2017 12:06 pm

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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

Official Guide question
Answer: D
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Source: — Data Sufficiency |

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by Brent@GMATPrepNow » Fri Jul 28, 2017 5:44 am
jjjinapinch 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

Official Guide question
Answer: D
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
Brent Hanneson - Creator of GMATPrepNow.com
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by [email protected] » Fri Jul 28, 2017 11:20 am
Hi jjjinapinch,

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.

Final Answer: D

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Rich
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Re: A certain investment earned a fixed rate of 4 percent

by iammohakjain » Wed Jul 01, 2020 9:53 am
There is no need to solve, Just form the eqns and see if you they are solvable.
1. We know that P at beginning of second year is x(some dollars).
-- Amount at the end of 1st year = P + P(rate(4%) *time(1year)) ---- Through this we know we can get first years interest.. and also if we solve the same way we can get second and third year interest.

Hence this is sufficient.

2. The amount at the end of 2nd year - P + P(rate(4%) *time(1year)) + [( P + P(rate(4%) *time(1year))) +( P + P(rate(4%) *time(1year)))*((rate(4%) *time(1year))] .. Through this we know we can get first years interest.. and also if we solve the same way we can get third year interest.


Hence this is sufficient.


Hence D
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Re: A certain investment earned a fixed rate of 4 percent

by Scott@TargetTestPrep » Sat May 08, 2021 10:12 am
jjjinapinch wrote: ↑
Thu Jul 27, 2017 12:06 pm
 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

Official Guide question
Answer: D
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

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