Question

Find x if the cost of living index is 150 : 

Hint:

This is essentially a weighted average problem. The formula \(\text{Average} = \frac{\sum (\text{Value} \times \text{Weight})}{\sum \text{Weight}}\) is fundamental. Just plug in the knowns and solve the resulting linear equation for the unknown weight.

Answer 1

The Cost of Living Index Number (CLI) is calculated using the formula for the weighted aggregate method: \[ \text{CLI} = \frac{\sum IW}{\sum W} \] Where \(I\) represents the price index for each group and \(W\) represents the weight.
Given:
CLI = 150
We first calculate \(\sum IW\) and \(\sum W\) from the table.
\(\sum IW = (180 \times 4) + (120 \times 5) + (300 \times 6) + (100 \times x) + (160 \times 3)\) \[ \sum IW = 720 + 600 + 1800 + 100x + 480 = 3600 + 100x \] \(\sum W = 4 + 5 + 6 + x + 3 = 18 + x\)
Now, substitute these into the CLI formula: \[ 150 = \frac{3600 + 100x}{18 + x} \] Multiply both sides by \((18 + x)\): \[ 150(18 + x) = 3600 + 100x \] \[ 2700 + 150x = 3600 + 100x \] Rearrange the terms to solve for \(x\): \[ 150x - 100x = 3600 - 2700 \] \[ 50x = 900 \] \[ x = \frac{900}{50} = 18 \] The value of x is 18.