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We define the harmonic mean of a set of numbers as the recip

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We define the harmonic mean of a set of numbers as the recip

by Max@Math Revolution » Wed Feb 06, 2019 6:56 am

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[GMAT math practice question]

We define the harmonic mean of a set of numbers as the reciprocal of the average (arithmetic mean) of the reciprocals of the numbers. What is the harmonic mean of 20 and 30?

A. 22
B. 24
C. 25
D. 26
E. 28
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by Brent@GMATPrepNow » Wed Feb 06, 2019 7:21 am
Max@Math Revolution wrote:[GMAT math practice question]

We define the harmonic mean of a set of numbers as the reciprocal of the average (arithmetic mean) of the reciprocals of the numbers. What is the harmonic mean of 20 and 30?

A. 22
B. 24
C. 25
D. 26
E. 28
What is the harmonic mean of 20 and 30?
NOTE: 20 = 20/1
So, the reciprocal of 20 = 1/20
And the reciprocal of 30 = 1/30

The average of 1/20 and 1/30 = (1/20 + 1/30 )/2
= (3/60 + 2/60 )/2
= (5/60 )/2
= (5/60 )/(2/1)
= (5/60 )(1/2)
= 5/120

The reciprocal of 5/120 = 120/5 = 24

Answer: B

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by fskilnik@GMATH » Wed Feb 06, 2019 7:24 am
Max@Math Revolution wrote:[GMAT math practice question]

We define the harmonic mean of a set of numbers as the reciprocal of the average (arithmetic mean) of the reciprocals of the numbers. What is the harmonic mean of 20 and 30?

A. 22
B. 24
C. 25
D. 26
E. 28
$$? = H\left( {20,30} \right)$$
$$?\,\,\, = \,\,\,{1 \over {\mu \left( {{1 \over {20}} + {1 \over {30}}} \right)}}\,\,\,\mathop = \limits^{\left( * \right)} \,\,\,{1 \over {\,\,{1 \over {24}}\,\,}}\,\,\, = \,\,\,24$$
$$\left( * \right)\,\,\,\,\,\mu \left( {{1 \over {20}} + {1 \over {30}}} \right) = {1 \over 2}\left( {{{1 \cdot 3} \over {20 \cdot 3}} + {{1 \cdot 2} \over {30 \cdot 2}}} \right) = {1 \over 2}\left( {{1 \over {3 \cdot 4}}} \right) = {1 \over {24}}$$

The correct answer is therefore (B).


We follow the notations and rationale taught in the GMATH method.

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Fabio.
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by Max@Math Revolution » Fri Feb 08, 2019 1:22 am
=>

1 / { ( 1/20 + 1/30 ) / 2 } = 1 / { ( 3/60 + 2/60 ) / 2 } = 1 / { (5/60) / 2 } = 1 / { 5 / 120 } = 120 / 5 = 24.

Therefore, the answer is B.
Answer: B
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