The price of a certain commodity increased at a rate of

Hi there! I'm happy to help with this!

Let's say, the price in 2000 is A. That the original amount. Each year it increases X%. To represent a percent increase (a) write the percent as a fraction/decimal, here X/100; (b) add one ---> 1 + X/100; (c) that's the multiplier -- multiplying a number by that multiplier results in a X% increase.

price in 2000 = A
price in 2001 = A*(1 + X/100)
price in 2002 = A*(1 + X/100)^2
price in 2003 = A*(1 + X/100)^3
price in 2004 = A*(1 + X/100)^4

For simplicity, I am going to define r = (1 + X/100). Then these equations become:

price in 2000 = A
price in 2001 = A*r
price in 2002 = A*r^2
price in 2003 = A*r^3
price in 2004 = A*r^4

Now, suppose we have M = 2001 price = A*r and N = 2003 price = A*r^3. How do we represent the 2002 prince (A*r^2) in terms of M and N?

There are two methods.

Method One: express r in terms of M and N

This is more a crank-it-out algebraic solution approach. We notice that N/M = (A*r^3)/(A*r) = r^2, so r = sqrt(N/M). Well,

2002 price = (2001 price)*(r) = M * sqrt(N/M) = [sqrt(M)*sqrt(M)]*[sqrt(N)/sqrt(M)] = sqrt(M)*sqrt(N) = sqrt(NM).

Through some fast-and-loose manipulation of the laws of squareroots, we arrive at answer .

Method Two: a more elegant solution for a more civilized age . . .

When you have an arithmetic sequence --- that is, adding the same number to get new terms (e.g. 8, 11, 14, 17, 20, 23, . . . ), when you take any three numbers in a row, the middle number is the mean, the arithmetic average, of the outer two. For example ---11, 14, 17 --- (11 + 17)/2 = 14. The arithmetic average is the ordinary average --- add the two numbers, and divide by two.

When you have a geometric sequence -- that is, multiplying the same ratio to get new terms (e.g. 2, 6, 18, 54, 162, . . . ), when you take any three numbers in a row, the middle number is the geometric mean of the outer two. The geometric mean of two numbers means multiply the two numbers and take the squareroot. For example --- 6, 18, 54 --- 6*54 = 324, and sqrt(324) = 18.

When you apply a fixed percentage increase from one term to the next, as we have in this problem, that's a geometric sequence. Thus, to find the 2002 price, all you have to do is take the geometric mean of the 2001 price and the 2003 price. 2002 price = sqrt(MN). Bam. Done. Again, answer = .

The ideas about arithmetic & geometric sequences, and the associated means, are good tricks to have up your sleeve for the more challenging GMAT math problems.

Does all this make sense? Please let me know if you have any questions.

Mike
_________________

The price in 2001 is M
The price in 2002 is M(1+\(\frac{X}{100}\)) ----------------------(1)

The price in 2003 is M(1+\(\frac{X}{100}\))(1+\(\frac{X}{100}\)) = N ---------(2)

Solving equation (2) we get
(1+\(\frac{X}{100}\)) = \(\sqrt{\frac{N}{M}}\)

Putting this value in equation (1) to get the desired answer
The price in 2002 is M\(\sqrt{\frac{N}{M}}\) = \(\sqrt{MN}\)

Hence the option A

I think that plug-in method is easiest for this problem.

Let the price in 2001 be 100 and the annual rate be 10%. Then:
2001 = 100 = M;
2002 = 110;
2003 = 121 = N;

Now, plug 100 and 121 in the answer choices to see which one gives 110:
A. \(\sqrt{MN}=\sqrt{100*121}=10*11=110\), correct answer right away.

Answer: A.

P.S. For plug-in method it might happen that for some particular numbers more than one option may give "correct" answer. In this case just pick some other numbers and check again these "correct" options only.

Hope it helps.
_________________

The price of a certain commodity increased at a rate of X % per year between 2000 and 2004. If the price was M dollars in 2001 and N dollars in 2003, what was the price in 2002 in terms of M and N ?

Given:
Price in 2001 : M
Price in 2003 : N
Rate per year: X %

Price in 2002 \(= M ( 1 + X/100)\)........................................(1)
And, Price in 2003: \(N = M( 1 + X/100)^2\)

Or, \((1 + x/100) =\sqrt{(N/M)}\)........(2)

Substituting the value of (2) in (1) above,
Price in 2002 \(= M \sqrt{(N/M)} = \sqrt{MN}\)

Answer: (A)

When no absolute values(or variables pertaining to them) are given, number plugging is the way to go..

Let the Value initially be = 1
and let X% = 100%(in other words..value doubles every year)

values are..
M = 2
N = 8

and we're looking for the value in 2002..which is equal to 4.

Answer (A).

I think this is the easiest approach here:

2001 - M

2002 - ? (Let it be 'x')

2003 - N

Now according to ques, since the rate of increase is equal both year and the rates should be calculated on the previous year's price,

\(\frac{x-M}{M} * 100= \frac{N-x}{x}*100\)

\(x^2 - Mx = MN - Mx\)

\(x = \sqrt{MN}\)

Answer: A

Hope it helps
+1 Kudos would be nice

Let the original price be $100 in the year 2000.

Let the X% increase be 100% increase which is a multiplier of 2 times.

Year 2000 = 100

Year 2001 = (100) (2) = M

Year 2002 = (100) (2) (2)

Year 2003 = (100) (2) (2) (2) = N

We are looking for the price in the year 2002 in terms of M and N

Target value: (100) (2)^2

If we multiply (M) (N) we get:

(100) (2) (100) (2) (2) (2) =

(100)^2 (2)^4

Then if we take the square root, we get the value in 2002: (100) (2)^2

Answer: *A*

Sqrt(MN)

Posted from my mobile device

Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email.
_________________