At the beginning of year 1, an investor puts p dollars into an investm

Question

At the beginning of year 1, an investor puts $p$ dollars into an investment whose value increases at a variable rate of $x_n%$ per year, where $n$ is an integer ranging from 1 to 3 indicating the year. If $85 \lt x_n \lt 110$ for all $n$ between 1 and 3, inclusive, then at the end of 3 years, the value of the investment must be between

A. $$p$ and $$2p$
B. $$2p$ and $$5p$
C. $$5p$ and $$10p$
D. $$10p$ and $$25p$
E. $$25p$ and $$75p$

Show Answer

Official Answer

C

A. $ p  and $ 2p 
B. $ 2p  and $ 5p 
C. $ 5p  and $ 10p 
D. $ 10p  and $ 25p 
E. $ 25p  and $ 75p

Bunuel · Accepted Answer

Official Solution:

At the beginning of year 1, an investor puts $p$ dollars into an investment whose value increases at a variable rate of $x_n%$ per year, where $n$ is an integer ranging from 1 to 3 indicating the year. If $85 \lt x_n \lt 110$ for all $n$ between 1 and 3, inclusive, then at the end of 3 years, the value of the investment must be between

A. $$p$ and $$2p$
B. $$2p$ and $$5p$
C. $$5p$ and $$10p$
D. $$10p$ and $$25p$
E. $$25p$ and $$75p$

Show Answer

Official Answer

C

A. $ p  and $ 2p 
B. $ 2p  and $ 5p 
C. $ 5p  and $ 10p 
D. $ 10p  and $ 25p 
E. $ 25p  and $ 75p

The quick solution to this problem is to pick a convenient number in the allowed range of growth rates. If  x_n  is always between 85 and 110, then the most convenient growth rate to pick is 100. An annual growth rate of 100% is exactly equivalent to doubling one's money. Doubling one's money three times is equivalent to multiplying one's investment by a factor of 8 ( = 2^3 ). The only range that includes $ 8p  is the third range ($ 5p  to $ 10p ).
 
Computing the exact outer limits of the allowed range is much more cumbersome. We would have to cube 1.85 for the lower limit and 2.1 for the upper limit. The cube of 1.85 is 6.331625, and the cube of 2.1 is 9.261. These values fall between 5 and 10.
 
Answer: C.

akumar5 · Answer

for finding the range :

Max limit = p(1+110/100)^3=10p
Min limit = p(1+85/100)^3= 5p

so Answer is C

82vkgmat · Answer

Nice problem! I think the whole xn business is just a tricky way of saying this is a compound interest problem.

Since the min rate of interest is 86% the min value of investment will be (1+0.86)^3*p = (1.86^3)*p. Similarly max value will be (2.1)^3*p.

2.1^3 is 9.5 or you can guess little over 8 so 10p may be good estimate.
1.86^3 is trickier to estimate. I knew root(3) = 1.73. So 1.86^2 is close to 3 and 3*1.8 is 5.4. So 5p is a close lower bound.

So range is 5p to 10p.
Answer C.

Enbee11 · Answer

At the beginning of year 1, an investor puts $p$ dollars into an investment whose value increases at a variable rate of $x_n%$ per year, where $n$ is an integer ranging from 1 to 3 indicating the year. If $85 \lt x_n \lt 110$ for all $n$ between 1 and 3, inclusive, then at the end of 3 years, the value of the investment must be between

A. $$p$ and $$2p$
B. $$2p$ and $$5p$
C. $$5p$ and $$10p$
D. $$10p$ and $$25p$
E. $$25p$ and $$75p$

Show Answer

Official Answer

C

A. $p and $2p
B. $2p and $5p
C. $5p and $10p
D. $10p and $25p
E. $25p and $75p

The min. value for Xn for 1, 2 & 3 year can be 86%, 87% & 88% ( They can be as close to 85 as possible but for simplicity i am taking these rate and carefull about variable so cannot take all value same) 
similarly Max value can be 109%, 108% & 107% 
Now consider p 100 
than solve by increasing it by saubsequent rate for both max and min for 
 for min rate it will be around 546 and for max it will be around 816 that is the range of final value and by puting p=100 in option u will get option C as the range.

PareshGmat · Answer

Answer = C = $5p and $10p

Min limit  = (1 + \frac{86}{100})^3 = (1 + 0.86)^3 = 1.86^3 =  Below 8

Max limit  = (1 + \frac{109}{100})^3 = (1 + 1.09)^3 = 2.09^3 =  Above 8

ShashankDave · Answer

One question...How do we know that they are talking about compound interest and not simple interest?

Shrivathsan · Answer

Even i have the same query as  ShashankDave

Bunuel  ,can you please clarify the below query

NikitaVChoudhary · Answer

How to know whether they are talking about simple interests or compound interests?
Is it because the problem says, "an investment whose value increases at a variable rate of Xn"?

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