M08-01

Question

The price of a certain commodity increased at a constant rate of  x\%  per year between 2000 and 2004, where  x > 0 . If the commodity was priced at  m  dollars in 2001 and  n  dollars in 2003, where  mn 
eq 0 , then, in terms of  m  and  n , what was the price of the commodity in 2002?

A.  \sqrt{mn} 
B.  n\sqrt{\frac{n}{m} } 
C.  n\sqrt{m} 
D.  \frac{nm}{\sqrt{n} } 
E.  nm^{\frac{3}{2} }

Bunuel · Answer

Official Solution:

The price of a certain commodity increased at a constant rate of  x\%  per year between 2000 and 2004, where  x > 0 . If the commodity was priced at  m  dollars in 2001 and  n  dollars in 2003, where  mn 
eq 0 , then, in terms of  m  and  n , what was the price of the commodity in 2002?

A.  \sqrt{mn} 
B.  n\sqrt{\frac{n}{m} } 
C.  n\sqrt{m} 
D.  \frac{nm}{\sqrt{n} } 
E.  nm^{\frac{3}{2} }

The price in 2001 =  m ;
 
The price in 2002 =  m*(1+\frac{x}{100}) . Our goal is to express this in terms of  m  and  n , so we need to express  1+\frac{x}{100}  using  m  and  n .
 
The price in 2003 =  m*(1+\frac{x}{100})*(1+\frac{x}{100})=n , which leads to  m*(1+\frac{x}{100})^2=n .
 
From the above equation:  (1+\frac{x}{100}) = \sqrt{\frac{n}{m} } 
 
Therefore, the price in 2002 =  m*(1+\frac{x}{100}) = m*\sqrt{\frac{n}{m} }=\sqrt{mn}

Alternatively, we can use the plug-in method.
 
Suppose the price in 2001 is 100 and the annual rate is 10%. Then:
 
 2001 = 100 = m 
 
 2002 = 110 
 
 2003 = 121 = n 
 
Now, plug 100 and 121 into the answer choices to see which one yields 110:
 
A.  \sqrt{mn}=\sqrt{100*121}=10*11=110 , which is the correct answer.
 
P.S. For the plug-in method, it might happen that for some specific numbers, more than one option may give the "correct" answer. In this case, simply pick other numbers and check those "correct" options again.

Answer: A

Bunuel · Answer

I have edited the question and the solution by adding more details to enhance its clarity. I hope it is now easier to understand.

BottomJee · Answer

I think this is a  high-quality  question and I agree with explanation.

jack5397 · Answer

Hi  Bunuel ,

I understood the solution algebraically. I have one doubt as it is stated in the question time from 2000 to 2004 then in the compounding will be as per below in my understanding

2000 - a
2001 - a * (1+x/100) = m 
2002 - a * (1+x/100)^2
2003 - a * (1+x/100)^3 = n

Then why we are considering directly from 2001 instead of 2000 while finding the value of 2002.

Bunuel · Answer

Why do you need year 2000 at all? Why adding extra variable  a ? However, if you you continue the way you are doing you'd still get the same:

2002 - a * (1+x/100)^2 =  m * (1+x/100)
2003 - a * (1+x/100)^3 = n = m * (1+x/100)^2

renzosm93 · Answer

Hi Bunuel, can you tell me if this approach is acceptable?

I did:
Price in 01: M
Price in 02: X
Price in 03: N

Since the % change is constant, then (X-M)/M = (N - X)/X
 so: X^2 = MN

Bunuel · Answer

_______________________________
Yes.

AaryanGupta1200 · Answer

I think question is above the 705 level.

bondi123 · Answer

I like the solution - it’s helpful.

XLmafia21 · Answer

its a gp. so the middle term is the geometric mean of the remaining two. so 2002 price is sqrt(mn)

Advait01 · Answer

I like the solution - it’s helpful. I think such questions are easier to solve by assuming a value of x like 10%, and then moving towards options are elimination

M08-01

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