-
Target Test Prep 20% Off Flash Sale is on! Code: FLASH20
Redeem
Manhattan GMAT Challenge Problem of the Week – 27 July 2010
Here is a new Challenge Problem! If you want to win prizes, try entering our Challenge Problem Showdown. The more people enter our challenge, the better the prizes.
As always, the problem and solution below were written by one of our fantastic instructors. Each challenge problem represents a 700+ level question. If you are up for the challenge, however, set your timer for 2 mins and go!
Question
The harmonic mean of two numbers x and y, symbolized as h(x, y), is defined as 2 divided by the sum of the reciprocals of x and y, whereas the geometric mean g(x, y) is defined as the square root of the product of x and y (when this square root exists), and the arithmetic mean m(x, y) is defined as (x + y)/2. For which of the following pairs of values for x and y is g(x, y) equal to the arithmetic mean of h(x, y) and m(x, y)?A. x = -2, y = -1
B. x = -1, y = 2
C. x = 2, y = 8
D. x = 8, y = 8
E. x = 8, y = 64
Answer
We should be organized as we try to make sense of all the given definitions. First, translate the definitions into algebraic symbols:
[pmath]h(x, y) = 2/(1/x + 1/y)[/pmath]
[pmath]g(x, y) = sqrt(xy)[/pmath]
[pmath]m(x, y) [/pmath]is the normal arithmetic mean, [pmath](x + y)/2[/pmath]
Now, we are asked for a special pair of values for which the following is true: once we calculate these three means, we'll find that g is the normal average (arithmetic mean) of h and m. This seems like a lot of work, so we should look for a shortcut. One way is to look among the answer choices for "easy" pairs, for which h, g, and m are easy to calculate. We should also recognize that the question's statement can only be true for one pair; it must be different from the others, so if we spot two easy pairs, we should first compute h, g, and m for the "more different-looking" of the two candidate pairs. Scanning the answer choices, looking for an easy pair to calculate, our eye should be drawn to (D), since the two values are equal. If both x and y equal 8, then m is super easy to calculate: m also equals 8. Let's now figure out g and h. Since g is defined as the square root of xy, in this case g equals the square root of 64, so g = 8 as well. Finally, h equals 2/(1/8 + 1/8) = 2/(2/8) = 8. The arithmetic mean of h (= 8 ) and m (= 8 ) is also 8, which equals g. We can stop right now: there can only be one right answer.
The correct answer is (D).
Special Announcement: If you want to win prizes for answering our Challenge Problems, try entering our Challenge Problem Showdown. Each week, we draw a winner from all the correct answers. The winner receives a number of our our Strategy Guides. The more people enter, the better the prize.
Recent Articles
Archive
- March 2024
- February 2024
- January 2024
- December 2023
- November 2023
- October 2023
- September 2023
- July 2023
- June 2023
- May 2023
- April 2023
- March 2023
- February 2023
- January 2023
- December 2022
- November 2022
- October 2022
- September 2022
- August 2022
- July 2022
- June 2022
- May 2022
- April 2022
- March 2022
- February 2022
- January 2022
- December 2021
- November 2021
- October 2021
- September 2021
- August 2021
- July 2021
- June 2021
- May 2021
- April 2021
- March 2021
- February 2021
- January 2021
- December 2020
- November 2020
- October 2020
- September 2020
- August 2020
- July 2020
- June 2020
- May 2020
- April 2020
- March 2020
- February 2020
- January 2020
- December 2019
- November 2019
- October 2019
- September 2019
- August 2019
- July 2019
- June 2019
- May 2019
- April 2019
- March 2019
- February 2019
- January 2019
- December 2018
- November 2018
- October 2018
- September 2018
- August 2018
- July 2018
- June 2018
- May 2018
- April 2018
- March 2018
- February 2018
- January 2018
- December 2017
- November 2017
- October 2017
- September 2017
- August 2017
- July 2017
- June 2017
- May 2017
- April 2017
- March 2017
- February 2017
- January 2017
- December 2016
- November 2016
- October 2016
- September 2016
- August 2016
- July 2016
- June 2016
- May 2016
- April 2016
- March 2016
- February 2016
- January 2016
- December 2015
- November 2015
- October 2015
- September 2015
- August 2015
- July 2015
- June 2015
- May 2015
- April 2015
- March 2015
- February 2015
- January 2015
- December 2014
- November 2014
- October 2014
- September 2014
- August 2014
- July 2014
- June 2014
- May 2014
- April 2014
- March 2014
- February 2014
- January 2014
- December 2013
- November 2013
- October 2013
- September 2013
- August 2013
- July 2013
- June 2013
- May 2013
- April 2013
- March 2013
- February 2013
- January 2013
- December 2012
- November 2012
- October 2012
- September 2012
- August 2012
- July 2012
- June 2012
- May 2012
- April 2012
- March 2012
- February 2012
- January 2012
- December 2011
- November 2011
- October 2011
- September 2011
- August 2011
- July 2011
- June 2011
- May 2011
- April 2011
- March 2011
- February 2011
- January 2011
- December 2010
- November 2010
- October 2010
- September 2010
- August 2010
- July 2010
- June 2010
- May 2010
- April 2010
- March 2010
- February 2010
- January 2010
- December 2009
- November 2009
- October 2009
- September 2009
- August 2009