The consumer price index for June, 2005 was 125. The food index was 120 and that of other items. What is the percentage of the total weight given to food?

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Solution

$\begin{matrix} Item & Index & Weights & WI Food & 120 & (W_{1}) & 120 (W_{1}) Other Items & 135 & (W_{2}) & 135 (W_{2}) \sum W = 100 & \sum W I = 120 W_{1} + 135 W_{2} \end{matrix}$

Let the total weight = 100

$W_{1}$ denotes weight of food

$W_{2}$ denotes weight of other items

So, $W_{1} + W_{2} = 100 . . . .$ Eqn (i)

CPI = 125

CPI = $\frac{\sum W R}{\sum W}$

125 = $\frac{120 W_{1} + 135 W_{2}}{100}$

12500 = $120 W_{1} + 135 W_{2} . . . .$ Eqn (ii)

Multiplying both sides of Eq. (i) by 135 and subtracting Eq. (ii) From (i) we get

13500 = $135 W_{1} + 135 W_{2}$

12500 = $120 W_{1} + 135 W_{2}$

13500 = $135 W_{1} + 135 W_{2}$

$\frac{(-) (-) (-)}{1000 = 15 W_{1} + 0}$

So, $W_{1} = \frac{1000}{15} = 66.67$

Substituting the value of $W_{1}$ in the Eq. (i) , we get

$W_{1} + W_{2} = 100$

or 66.67+ $W_{2} = 100$

$W_{2} = 33.33$
Therefore, percentage of total weight given to food is $66.67 %$ and other items $33.33 %$

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