Calculation of Weighted Price Index Number

Given data:
- Increase of 10% and 20% in the prices of 2 commodities respectively.
- Decrease of 30% in the price of another commodity.
- Relative importance of 3 commodities in the ratio 3:3:1.

Let us assume the initial prices of the 3 commodities to be P1, P2, and P3 respectively.

Calculation of New Prices

After the given increase and decrease in prices, the new prices of the 3 commodities will be:
- New price of P1 = 1.1P1
- New price of P2 = 1.2P2
- New price of P3 = 0.7P3

Calculation of Weighted Mean

The weighted mean is calculated by multiplying the new prices with their respective relative importance ratios and then adding them up. This sum is then divided by the total relative importance ratio.

Weighted mean = [(3/7) x 1.1P1] + [(3/7) x 1.2P2] + [(1/7) x 0.7P3]
Weighted mean = (3.3/7)P1 + (3.6/7)P2 + (0.1/7)P3

Calculation of Weighted Price Index Number

The weighted price index number is the ratio of the weighted mean to the initial price index, multiplied by 100.

Weighted price index number = (weighted mean / initial price index) x 100

The initial price index can be calculated using the formula:
Initial price index = (3/7)P1 + (3/7)P2 + (1/7)P3

Initial price index = (3/7) x P1 + (3/7) x P2 + (1/7) x P3

Substituting the values of new prices in the above equation, we get:

Initial price index = (3/7) x 1.1P1 + (3/7) x 1.2P2 + (1/7) x 0.7P3

Initial price index = (3.3/7)P1 + (3.6/7)P2 + (0.1/7)P3

Substituting the values of weighted mean and initial price index in the formula of weighted price index number, we get:

Weighted price index number = [(3.3/7)P1 + (3.6/7)P2 + (0.1/7)P3 / (3.3/7)P1 + (3.6/7)P2 + (0.1/7)P3] x 100

Weighted price index number = 100

Therefore, the correct answer is option (a) 80.

Explanation

The weighted price index number is a measure of the average price changes of a basket of goods or commodities, taking into account their relative importance. In this question, we are given the initial prices of 3 commodities and their relative importance ratios. We are also given the percentage increase and decrease in prices of these commodities. Using this information, we can calculate the new prices of the commodities and then calculate the weighted mean and the initial price index. Finally, we can use these values

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