Paper 4: Fundamentals of Business Mathematics & Statistic

FUNDAMENTALS OF BUSINESS MATHEMATICS AND STATISTICS I 5.19

Example 21 : Find the harmonic mean of the following numbers :

1, 21 , 31 ,^14

H.M.=

4

1 1 1 1

1 1 2 1 3 1 4+ + +

= 1 2 3 4+ + +^4 = 104 =^25

Example 22 : An aeroplane flies around a square and sides of which measure 100 kms. Each. The aeroplane
cover at a speed of 10 Kms per hour the first side, at200 kms per hour the second side, at 300 kms per hour
the third side and at 400 kms per hour the fourth side. Use the correct mean to find the average speed
round the square.
Here H.M. is the appropriate mean.
Let the required average speed be H kms per hours

then H =

4 4

1 1 1 1 12 6 4 3

100 200 300 400 1200

= + + +

+ + + =

4 1200

25

× = 4 × 48 = 192 kms/hr.

ADVANTANGES OF HARMONIC MEAN :

(i) Like A.M. and G. M. it is also based on all observations.
(ii) Capable of further algebraic treatment.
(iii) It is extremely useful while averaging certain types of rates and rations.

DISADVANTAGES OF HARMONIC MEAN :
(i) It is not readily understood nor can it be calculated with ease.
(ii) It is usually a value which may not be a member of the given set of numbers.
(iii) It cannot be calculated when there are both negative and positive values in a series or one of more
values in zero.
It is useful in averaging speed, if the distance travelled is equal. When it is used to give target weight to
smallest item, this average is used.
5.1.4. Relations among A.M., G.M. and H.M. :

1. The Arithmetic Mean is never less than the Geometric Mean, again Geometric Mean is never less than
   the Harmonic Mean.
   i.e. A.M. ≥ G. M. ≥ H. M.

Uses of H.M. : Harmonic mean is useful in finding averages involving rate, time, price and ratio.

Example 23 : For the numbers 2, 4, 6, 8, 10, find GM & HM and show that AM > GM > HM.

G. M. =^5 2 4 6 8 10 (2 4 6 8 10)⋅ ⋅ ⋅ ⋅ = ⋅ ⋅ ⋅ ⋅ 1/5

Log GM =^15 (log 2 + log4 + log6 + log8 + log 10)