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Question: The harmonic mean of two numbersxandy, symbolized ash(x, y), is defined as 2 divided by the sum of the reciprocals ofxandy, whereas the geometric meang(x, y) is defined as the square root of the product ofxandy(when this square root exists), and the arithmetic meanm(x, y) is defined as. For which of the following pairs of values forxandyisg(x, y) equal to the arithmetic mean ofh(x, y) andm(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
Approach Solution (1)
We should be organized as we try to make sense of all the given definitions. First, translate the definitions into algebraic symbols:
h(x, y) =
g(x, y) = xy
m(x, y) is the normal arithmetic mean,
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 thatgis the normal average (arithmetic mean) ofhandm. 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 whichh,g, andmare 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 computeh,g, andmfor 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 bothxandyequal 8, thenmis super easy to calculate:malso equals 8.
Let's now figure outgandh. Sincegis defined as the xy, in this case g equals the square root of 64, sog= 8 as well. Finally,hequals. The arithmetic mean ofh(= 8) andm(= 8) is also 8, which equalsg. We can stop right now: there can only be one right answer.
Correct option: D
h(x, y) =
g(x, y) = xy
m(x, y) is the normal arithmetic mean,
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 thatgis the normal average (arithmetic mean) ofhandm. 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 whichh,g, andmare 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 computeh,g, andmfor 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 bothxandyequal 8, thenmis super easy to calculate:malso equals 8.
Let's now figure outgandh. Sincegis defined as the xy, in this case g equals the square root of 64, sog= 8 as well. Finally,hequals. The arithmetic mean ofh(= 8) andm(= 8) is also 8, which equalsg. We can stop right now: there can only be one right answer.
Correct option: D
Approach Solution (2)
A: g(x, y) =
h(x, y) =
So, h(x, y) =
m(x, y) =
Average of h(x, y) and m(x, y) =
Not equal to the g(x, y)
Tried them all except B, where g(x, y) would have been which doesn’t exist.
D: g(x, y) = 8 (Take only positive root)
h(x, y) = 218+18=228=162=8
m(x, y) = 162=8
Average of h(x, y) and m(x, y) = (8+8)2=162=8
Correct option: D
h(x, y) =
So, h(x, y) =
m(x, y) =
Average of h(x, y) and m(x, y) =
Not equal to the g(x, y)
Tried them all except B, where g(x, y) would have been which doesn’t exist.
D: g(x, y) = 8 (Take only positive root)
h(x, y) = 218+18=228=162=8
m(x, y) = 162=8
Average of h(x, y) and m(x, y) = (8+8)2=162=8
Correct option: D
Approach Solution (3)
We require to satisfy below condition:
Just observe the above formed equation
RHS >> Addition of 2 fractions would result in another fraction or an integer, but ot a square root
LHS >> to get a proper integer, only options C, D & E can be considered as they will produce a perfect square root
Option C fails: will produce fraction
Option D success
Correct option: D
Just observe the above formed equation
RHS >> Addition of 2 fractions would result in another fraction or an integer, but ot a square root
LHS >> to get a proper integer, only options C, D & E can be considered as they will produce a perfect square root
Option C fails: will produce fraction
Option D success
Correct option: D
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