Question

The H.M. of the numbers \[\dfrac{1}{5},\dfrac{1}{{10}},\dfrac{1}{{15}},\dfrac{1}{{20}},\dfrac{1}{{25}},\dfrac{1}{{30}},\dfrac{1}{{35}}\] is:
A) $\dfrac{1}{{20}}$
B) $\dfrac{1}{{16}}$
C) $\dfrac{1}{{15}}$
D) $\dfrac{1}{{13}}$

Answer 1

Hint: In simple words, Harmonic mean is defined as the type of numerical average and it is calculated by dividing the number of observations by the reciprocal of each of the numbers given in the series. Here we will use the formula and place the given data and simplify for the required value.

Complete step by step solution:
Harmonic Mean of \[\{ {a_1},{a_2},{a_{3,......,}}{a_n}\} \] is represented by $ = \dfrac{n}{{\dfrac{1}{{{a_1}}} + \dfrac{1}{{{a_2}}} + ....\dfrac{1}{{{a_n}}}}}$
Place the given values in the above expression –
Harmonic Mean \[ = \dfrac{7}{{\dfrac{1}{{\dfrac{1}{5}}} + \dfrac{1}{{\dfrac{1}{{10}}}} + .... + \dfrac{1}{{\dfrac{1}{{35}}}}}}\]
Denominator’s denominator goes to the numerator.
Harmonic Mean $ = \dfrac{7}{{5 + 10 + 15 + ... + 35}}$
Simplify the above expression, finding the sum of the terms in the denominators.
Harmonic Mean $ = \dfrac{7}{{140}}$
Find the factors for the term in the denominator.
Harmonic Mean $ = \dfrac{7}{{7 \times 20}}$
Common factors from the numerator and the denominator cancels each other. Therefore, remove from the numerator and the denominator from the above expression.
Harmonic Mean $ = \dfrac{1}{{20}}$
Therefore, option (A) is the correct answer.

Note:
Always remember that the denominator’s denominator goes to the numerator. Common factors from the numerator and the denominator always cancels each other which gives the equivalent value for the fraction. Be good in multiples to find the factorization and to get the reduced fraction or the value.