Page 123 - DCOM203_DMGT204_QUANTITATIVE_TECHNIQUES_I

Quantitative Techniques – I

Notes 10log90.4 25log125.5
GM = antilog
35
10 1.9562 25 2.0986
= antilog = antilog 2.0579 = 114.27
35
To determine the average rate of change of price for the entire period when the rate
of change of prices for different periods are given

Let P be the price of a commodity in the beginning of the first year. If it increases by k % in the
0 1
first year, the price at the end of 1st year (or beginning of second year) is given by

k 1 k 1 k 1
P P 1
P = P + 0 = 0 = P (1 + r ), where r = denotes the rate of increase per rupee
1 0 100 100 0 1 1 100
in first year. Similarly, if the price changes by k % in second year, the price at the end of second
2
year is given by
k k
P 2 P 1 2
P = P + 1 = 1 = P (1 + r )
2 1 100 100 1 2

Replacing the value of P as P (1 + r ) we can write
1 0 1
P = P (1 + r )(1 + r )
2 0 1 2
Proceeding in this way, if 100r % is the rate of change of price in the i th year, the price at the end
i
of nth period, P , is given by
n
P = P (1 + r )(1 + r ) ...... (1 + r ) .... (1)
n 0 1 2 n
Further, let 100r % per year be the average rate of increase of price that gives the price P at the
n
end of n years. Therefore, we can write
P = P (1 + r)(1 + r) ...... (1 + r) = P (1 + r) n .... (2)
n 0 0
Equating (1) and (2), we can write
(1 + r) = (1 + r )(1 + r ) ...... (1 + r )
n
1 2 n
1
or (1 + r) = 1 r 1 1 r  1 r n n .... (3)
2
This shows that (1 + r) is geometric mean of (1 + r ), (1 + r ), ...... and (1 + r ).
1 2 n
From (3), we get

1
r = 1 r 1 1 r  1 r n n – 1 .... (4)
2
Note: Here r denotes the per unit rate of change. This rate is termed as the rate of increase or the
rate of growth if positive and the rate of decrease or the rate of decay if negative.

6.6.4 Average Rate of Growth of Population

The average rate of growth of price, denoted by r in the above section, can also be interpreted as
the average rate of growth of population. If P denotes the population in the beginning of the
0

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