Question

The price relatives and weights of a set of commodities are given as:

Commodity	P	Q	R
Price Relative	100	130	180
Weight	x	2x	y

If the sum of weights is 54 and index for the set is 130, then the values of x and y are

• A. x=12, y=18
• B. x=15, y=9
• C. x=16, y=6
• D. x=9, y=15

Answer 1

The Correct Option is B

To solve the problem, we need to determine the values of x and y using the given conditions. We have a table of commodities with their price relatives and weights:

Commodity	P	Q	R
Price Relative	100	130	180
Weight	x	2x	y

The sum of the weights is given as:

$$x + 2x + y = 3x + y = 54$$

We are also given that the price index for the set is 130. The formula for calculating the weighted price index I is:

$$I = \frac{\sum (P_i \cdot W_i)}{\sum W_i}$$

Using the provided data:

$$130 = \frac{100x + 130(2x) + 180y}{54}$$

Let's simplify and solve:

$$130 = \frac{100x + 260x + 180y}{54}$$

$$130 = \frac{360x + 180y}{54}$$

Multiplying both sides by 54 to clear the denominator:

$$130 \times 54 = 360x + 180y$$

$$7020 = 360x + 180y$$

Now, we have two equations:

• 1. \(3x + y = 54\)
• 2. \(360x + 180y = 7020\)

First, simplify equation (2) by dividing through by 180:

$$2x + y = 39$$

Now, solve the system of equations:

1. Equation (1): \(3x + y = 54\)
2. Equation (3): \(2x + y = 39\)

Subtract equation (3) from equation (1) to eliminate \(y\):

$$3x + y - (2x + y) = 54 - 39$$

$$x = 15$$

Substitute \(x = 15\) into equation (3):

$$2(15) + y = 39$$

$$30 + y = 39$$

$$y = 9$$

Thus, the values are \(x = 15\) and \(y = 9\). Therefore, the correct answer is:

x=15, y=9