Question

The harmonic mean of three numbers $3,5,7$ is
A. $2.43$
B. $3.43$
C. $4.43$
D. $5.43$

Answer 1

We know that harmonic progression (HP) is defined as a sequence of real numbers which is determined by dividing the number of terms by the reciprocals of the terms that does not contain zero. In order to solve this question we should know that if $a,b,c$ are in H.P, then $\dfrac{1}{a},\dfrac{1}{b},\dfrac{1}{c}$ should be in arithmetic progression.

Formula used:
If we have $a,b,c$ in H.P then we can calculate the H.P by the formula:
$H.P = \dfrac{3}{{\dfrac{1}{a} + \dfrac{1}{b} + \dfrac{1}{c}}}$

Complete step by step answer:
We have been given that $3,5,7$ is in Harmonic Mean. Here we have three terms, so the number of terms is $3$. Now we apply the formula and by comparing we have :
$a = 3,b = 5,c = 7$
So we can write:
$\dfrac{3}{{\dfrac{1}{3} + \dfrac{1}{5} + \dfrac{1}{7}}}$
We will solve it now,
$\dfrac{3}{{\dfrac{{35 + 21 + 15}}{{105}}}}=\dfrac{3}{{\dfrac{{71}}{{105}}}}$
The above expression can also be written as:
$3 \times \dfrac{{71}}{{105}}$
On multiplication it gives the value
$\dfrac{{315}}{{71}} = 4.436$

Hence the correct option is C.

Note: We should note that Harmonic Mean has the least value among all the three means. The relationship between arithmetic mean, geometric mean and harmonic mean is that: The product of arithmetic mean and harmonic mean of any two numbers $a,b$ in such a way that $a > b > 0$ is equal to the share of their geometric mean. We can write this in expression as: $AM \times HM = G{M^2}$.