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Taking 1965 as the base year, with an index number of $100$, calculate an index number for 1972, based on weighted average of price relatives derived from the following:
Commodities A B C Weights $34$ $26$ $40$ Price per unit in 1965 $16$ $20$ $32$ Price per unit in 1972 $24$ $19$ $36$ - The weights are now changed so that the weight for A is $40$ and the total weight is $100$. If the value of the index number for 1972 is now $120.5$, calculate the weights applied to B and C.
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Let base price per unit $= P_o$
Let current price per unit $= P_i$
Commodities $P_o \; \left(1965\right)$ $P_i \; \left(1972\right)$ Price relatives for 1972 $x = \dfrac{P_i}{P_o} \times 100$ Weights $\left(w\right)$ $w \cdot x$ A $16$ $24$ $150$ $34$ $5100$ B $20$ $19$ $95$ $26$ $2470$ C $32$ $36$ $112.5$ $40$ $4500$
$\Sigma w = 100$, $\;$ $\Sigma \left(w \cdot x\right) = 12070$
Index number for 1972 $= \dfrac{\Sigma \left(w \cdot x\right)}{\Sigma w} = \dfrac{12070}{100} = 120.70$ -
New index number for 1972 $= 120.5$
New weight for A $= 40$
Total weight $= \Sigma w = 100$
Let new weight for B $= x$
New weight for C $= 100 - 40 - x = 60 - x$
Commodities $P_o \; \left(1965\right)$ $P_i \; \left(1972\right)$ Price relatives for 1972 $x = \dfrac{P_i}{P_o} \times 100$ Weights $\left(w\right)$ $w \cdot x$ A $16$ $24$ $150$ $40$ $6000$ B $20$ $19$ $95$ $x$ $95 x$ C $32$ $36$ $112.5$ $60 - x$ $112.5 \left(60 - x\right)$
$\begin{aligned} \Sigma \left(w \cdot x\right) & = 6000 + 95 x + 112.5 \left(60 - x\right) \\\\ & = 6000 + 95 x + 6750 - 112.5 x \\\\ & = 12750 - 17.5 x \end{aligned}$
$\therefore \;$ New index number for 1972 $= \dfrac{\Sigma \left(w \cdot x\right)}{\Sigma w} = \dfrac{12750 - 17.5 x}{100} = 120.5$
i.e. $\;$ $12750 - 17.5 x = 12050$
i.e. $\;$ $17.5 x = 700$ $\implies$ $x = 40$
$\therefore \;$ New weight for B $= 40$
New weight for C $= 60 - x = 60 - 40 = 20$