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Question
If Dorbish-Bowley's and Fisher's Price Index Numbers are 5 and 4, respectively, then find Laspeyre's and Paasche's Price Index Numbers.
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Solution
Let Laspeyre’s Price Index Number P01(L) = x
and Paasche’s Price Index Number P01(P) = y
Dorbish-Bowley’s Price Index Number P01(D-B) = 5
Fisher’s Price Index Number P01(F) = 4
`("P"_01("L") "P"_01("P"))/(2) = "P"_(01)("D"–"B")`
∴ `(x + y)/(2)` = 5
∴ x + y = 10 ...(i)
`sqrt("P"_01("L") xx "P"_01("P"))= "P"_01("F")`
∴ `sqrt(xy)` = 4
∴ xy = 16
∴ y = `(16)/x`
∴ `x + (16)/x` = 10 ...[From (i)]
∴ x2 + 16 = 10x
∴ x2 – 10x + 16 = 0
∴ x – 8x – 2x + 16 = 0
∴ x(x – 8) – 2(x – 8) = 0
∴ (x – 2) (x – 8)
∴ x = 2 or x = 8
If x = 2, then from equation (i), y = 8
If x = 8, then from equation (i), y = 2
∴ P01(L) = 8 and P01(P) = 2
or P01(P) = 8 and P01(L) = 2
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