A certain investment grows at an annual interest rate of 8%,

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BTGmoderatorDC: Moderator; Posts: 7187; Joined: Thu Sep 07, 2017 4:43 pm; Followed by:23 members

A certain investment grows at an annual interest rate of 8%,

by BTGmoderatorDC » Mon Sep 17, 2018 6:21 pm

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A certain investment grows at an annual interest rate of 8%, compounded quarterly. Which of the following equations can be solved to find the number of years, x, that it would take for the investment to increase by a factor of 16?

A. 16 = 1.02^(x/4)
B. 2 = 1.02^x
C. 16 = 1.08^(4x)
D. 2 = 1.02^(x/4)
E. 1/16 = 1.02^(4x)

OA B

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Source: Beat The GMAT — Problem Solving | Mon Sep 17, 2018 6:21 pm

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by GMATGuruNY » Mon Sep 17, 2018 7:35 pm

BTGmoderatorDC wrote:A certain investment grows at an annual interest rate of 8%, compounded quarterly. Which of the following equations can be solved to find the number of years, x, that it would take for the investment to increase by a factor of 16?

A. 16 = 1.02^(x/4)
B. 2 = 1.02^x
C. 16 = 1.08^(4x)
D. 2 = 1.02^(x/4)
E. 1/16 = 1.02^(4x)

When a value increases repeatedly by r%:

Final amount = Original amount * (1 + r/100)^number of changes.

Let original amount = 1.
Since the original amount increases by a factor of 16:
Final amount = 16.
Since the investment increases by 2% every quarter:
r = 2.
Since x = the number of years, and there are 4 changes every year:
Number of changes = 4x.

Plugging these values into the formula:

16 = 1*(1.02)^4x
16^(1/4) = 1.02^(4x*(1/4))
2 = 1.02^x

The correct answer is B.

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swerve: Legendary Member; Posts: 2499; Joined: Sun Oct 29, 2017 2:04 pm; Followed by:6 members

by swerve » Tue Sep 18, 2018 3:32 pm

I'm confused by the wording here:

It is implied in the OA that "increase by a factor of 16" means that the amount increased to 16 times its original amount.

Don't you think that "increased by a factor of 16" means x + 16x?

Hope someone clarifies this doubt! Thanks!

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Scott@TargetTestPrep: GMAT Instructor; Posts: 8085; Joined: Sat Apr 25, 2015 10:56 am; Location: Los Angeles, CA; Thanked: 43 times; Followed by:29 members

by Scott@TargetTestPrep » Wed Sep 19, 2018 5:09 pm

BTGmoderatorDC wrote:A certain investment grows at an annual interest rate of 8%, compounded quarterly. Which of the following equations can be solved to find the number of years, x, that it would take for the investment to increase by a factor of 16?

A. 16 = 1.02^(x/4)
B. 2 = 1.02^x
C. 16 = 1.08^(4x)
D. 2 = 1.02^(x/4)
E. 1/16 = 1.02^(4x)

The compound interest formula is: A = P(1 + r/n)^(nt)

where t = the number of years, A = the amount after t years, P = the principal or the initial amount, r = the annual interest rate, and n = the number of times compounded per year.

Here, r = 8% = 0.08, n = 4, t = x. We are not given a value of P but we are given that A = 16P. So we have:

16P = P(1 + 0.08/4)^(4x)

16 = (1 + 0.02)^(4x)

16 = 1.02^(4x)

Taking the fourth root of both sides of the equation, we have:

16^(Â¼) = [1.02^(4x)]^(Â¼)

2 = 1.02^x

Answer: B

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