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Manager  Joined: 11 Aug 2009
Posts: 65
A certain investment grows at an annual interest rate of 8%,  [#permalink]

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23 00:00

Difficulty:   95% (hard)

Question Stats: 41% (02:11) correct 59% (01:57) wrong based on 331 sessions

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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)

Originally posted by kairoshan on 18 Nov 2009, 20:41.
Last edited by Bunuel on 04 Feb 2014, 04:30, edited 1 time in total.
Edited the question.
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Joined: 24 Jun 2008
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ssruthi 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?
16 = (1.02)x/4
2 = (1.02)x
16 = (1.08)4x
2 = (1.02)x/4
1/16 = (1.02)4x

If we apply 8% annual interest, compounded quarterly, then we apply one quarter of the interest (or 2% interest) four times per year. That is, in one year, we will multiply the value of our investment by 1.02 four times, or in other words, by (1.02)^4. So, if we invest for x years, we will apply 2% interest 4x times, so will multiply the value of our initial investment by (1.02)^(4x). Now, we know that the value has increased by a factor of 16, so

(1.02)^(4x) = 16
(1.02^x)^4 = 2^4
1.02^x = 2
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##### General Discussion
Manager  Joined: 30 Aug 2009
Posts: 213
Location: India
Concentration: General Management
Re: A certain investment grows at an annual interest rate of 8%,  [#permalink]

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kairoshan 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?
16 = (1.02)x/4
2 = (1.02)x
16 = (1.08)4x
2 = (1.02)x/4
1/16 = (1.02)4x

B -> 2 = (1.02)^x

PS: this should be in Problem Solving forum
Manager  Joined: 13 Aug 2009
Posts: 119
Schools: Sloan '14 (S)
Re: A certain investment grows at an annual interest rate of 8%,  [#permalink]

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2
1
If P is the principle:

16*P = P*(1+8/400)^(4x)

which can be reduced to:2 = (1.02)^x

BUT... can we really simplify x^4=y^(4*z) to x=y^z?

Thanks!
Intern  Joined: 06 Sep 2011
Posts: 7

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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?
16 = (1.02)x/4
2 = (1.02)x
16 = (1.08)4x
2 = (1.02)x/4
1/16 = (1.02)4x
Intern  Joined: 06 Sep 2011
Posts: 7

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IanStewart wrote:
ssruthi 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?
16 = (1.02)x/4
2 = (1.02)x
16 = (1.08)4x
2 = (1.02)x/4
1/16 = (1.02)4x

If we apply 8% annual interest, compounded quarterly, then we apply one quarter of the interest (or 2% interest) four times per year. That is, in one year, we will multiply the value of our investment by 1.02 four times, or in other words, by (1.02)^4. So, if we invest for x years, we will apply 2% interest 4x times, so will multiply the value of our initial investment by (1.02)^(4x). Now, we know that the value has increased by a factor of 16, so

(1.02)^(4x) = 16
(1.02^x)^4 = 2^4
1.02^x = 2

Thanks, i was confused with 8% annual rate.
If the question was like 8% interest rate means we will take 1.08 only rgt?
Manager  Joined: 04 Jun 2011
Posts: 114

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yup.. but usually gmac specifies whether the rate is annual or compunded quarterly.. so hopefully we should not ee any ambiguous content
Intern  Joined: 05 Nov 2013
Posts: 20
Re: A certain investment grows at an annual interest rate of 8%,  [#permalink]

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1
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?[/b]

I am 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 "increase by a factor of 16" means [b]x + 16x?

Hope an expert clarifies this doubt!
Math Expert V
Joined: 02 Sep 2009
Posts: 61524
Re: A certain investment grows at an annual interest rate of 8%,  [#permalink]

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2
1
pratikshr 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?[/b]

I am 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 "increase by a factor of 16" means [b]x + 16x?

Hope an expert clarifies this doubt!

Increasing something by a factor of x means multiplying by x.
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Intern  Joined: 26 May 2013
Posts: 1
Re: A certain investment grows at an annual interest rate of 8%,  [#permalink]

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81= (1.04)^4x
3^4= [(1.04)^x]^4
3=(1.04)^x.

Intern  B
Joined: 11 Jan 2015
Posts: 32
Re: A certain investment grows at an annual interest rate of 8%,  [#permalink]

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IanStewart wrote:
ssruthi 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?
16 = (1.02)x/4
2 = (1.02)x
16 = (1.08)4x
2 = (1.02)x/4
1/16 = (1.02)4x

If we apply 8% annual interest, compounded quarterly, then we apply one quarter of the interest (or 2% interest) four times per year. That is, in one year, we will multiply the value of our investment by 1.02 four times, or in other words, by (1.02)^4. So, if we invest for x years, we will apply 2% interest 4x times, so will multiply the value of our initial investment by (1.02)^(4x). Now, we know that the value has increased by a factor of 16, so

(1.02)^(4x) = 16
(1.02^x)^4 = 2^4
1.02^x = 2

Could you please elaborate on the yellow part?

Thanks!
Target Test Prep Representative V
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Joined: 14 Oct 2015
Posts: 9514
Location: United States (CA)
Re: A certain investment grows at an annual interest rate of 8%,  [#permalink]

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2
2
kairoshan 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^(¼) = [1.02^(4x)]^(¼)

2 = 1.02^x

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Intern  B
Joined: 04 Aug 2018
Posts: 7
Re: A certain investment grows at an annual interest rate of 8%,  [#permalink]

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Bunuel wrote:
pratikshr 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?[/b]

I am 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 "increase by a factor of 16" means [b]x + 16x?

Hope an expert clarifies this doubt!

Increasing something by a factor of x means multiplying by x.

I have a question about ]"Increasing something by a factor of x means multiplying by x"
why it is not x+x*16?
there is a similar expression on OG2019
"From 2000 to 2003,the number of employees at a certain company increased by a factor of 1/4 . From 2003 to 2006,the number of employees at this company decreased by a factor of 1/3. If there were 100 employees at the company in 2006, how many employees were there at the company in 2000?"
here x (1+1/4)(1-1/3)=100; x=120
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Re: A certain investment grows at an annual interest rate of 8%,  [#permalink]

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