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

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kairoshan
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A certain investment grows at an annual interest rate of 8%, [#permalink] New post   Updated on: 04 Feb 2014, 05:30
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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)
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B

Originally posted by kairoshan on 18 Nov 2009, 21:41.
Last edited by Bunuel on 04 Feb 2014, 05:30, edited 1 time in total.
Edited the question.
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IanStewart
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Re: compound interest [#permalink] New post   06 Sep 2011, 20:01
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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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Re: A certain investment grows at an annual interest rate of 8%, [#permalink] New post   19 Sep 2018, 18:11
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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?

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

Answer: B
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Re: A certain investment grows at an annual interest rate of 8%, [#permalink] New post   18 Nov 2009, 23:52
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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
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Re: A certain investment grows at an annual interest rate of 8%, [#permalink] New post   19 Nov 2009, 06:51
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If P is the principle:

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

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

B is the answer...

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

Thanks!
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compound interest [#permalink] New post   06 Sep 2011, 19:49
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
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Re: compound interest [#permalink] New post   06 Sep 2011, 20:10
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?
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Re: compound interest [#permalink] New post   07 Sep 2011, 09:18
yup.. but usually gmac specifies whether the rate is annual or compunded quarterly.. so hopefully we should not ee any ambiguous content
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Re: A certain investment grows at an annual interest rate of 8%, [#permalink] New post   03 Aug 2014, 05:39
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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?[/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!
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Re: A certain investment grows at an annual interest rate of 8%, [#permalink] New post   12 Aug 2014, 09:22
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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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Re: A certain investment grows at an annual interest rate of 8%, [#permalink] New post   19 Oct 2014, 06:19
81= (1.04)^4x
3^4= [(1.04)^x]^4
3=(1.04)^x.

Answer..
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Re: A certain investment grows at an annual interest rate of 8%, [#permalink] New post   02 Sep 2016, 14:41
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!
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Re: A certain investment grows at an annual interest rate of 8%, [#permalink] New post   09 Oct 2018, 07:30
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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] New post   07 Mar 2020, 09:09
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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 --> correct: investment of i, an annual interest rate of 8%, compounded quarterly, so interest per quarter = 8%/4=2%, so 16*i= i*(1+2/100)^4*x => 2^4 = (1.02^x)^4 => 2= 1.02^x
C. 16 = 1.08^(4x)
D. 2 = 1.02^(x/4)
E. 1/16 = 1.02^(4x)
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A certain investment grows at an annual interest rate of 8%, [#permalink] New post   05 Jun 2022, 10:49
Given: A certain investment grows at an annual interest rate of 8%, compounded quarterly.

Asked: 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?

Let the principal amount be P
\(16P = P(1.02)^{4x}\)
\(16 = 1.02^{4x}\)
\(16^{\frac{1}{4}} = 2 = 1.02^x\)

IMO B
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Re: A certain investment grows at an annual interest rate of 8%, [#permalink] New post   13 Jul 2022, 23:06
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)



language is definitely not accurate

if we keep that aside (since options are designed with that inaccurate language), then

16 = 1(1+R/4)^4x (where R = 8% and X is number of years ) (since compounded quarterly hence rate will be take as R/4 = 2% per quarter and number of duration will become 4x quarters)

16 = (1+2%)^4x

2^4 = (1.02)^4x

2 = (1.02)^x
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Re: A certain investment grows at an annual interest rate of 8%, [#permalink] New post   15 Jul 2023, 01:27
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Re: A certain investment grows at an annual interest rate of 8%, [#permalink]
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