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

Question

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)

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)

IanStewart · Accepted Answer

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

ScottTargetTestPrep · Answer

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) =  ^(¼)

2 = 1.02^x

Answer: B

kp1811 · Answer

B -> 2 = (1.02)^x

PS: this should be in Problem Solving forum

h2polo · Answer

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!

ssruthi · Answer

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

ssruthi · Answer

Thanks, i was confused with 8% annual rate.
If the question was like 8% interest rate means we will take 1.08 only rgt?

viks4gmat · Answer

yup.. but usually gmac specifies whether the rate is annual or compunded quarterly.. so hopefully we should not ee any ambiguous content

pratikshr · Answer

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)

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  x + 16x?

Hope an expert clarifies this doubt!

Bunuel · Answer

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

adityaashish · Answer

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

Answer..

paddy41 · Answer

Could you please elaborate on the yellow part?

Thanks!

xiaoxueren · Answer

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

hiranmay · Answer

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)

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)

Kinshook · Answer

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

gauravjain0211 · Answer

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

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