The consumer price index for June, 2005 was 125. The food index was 120 and that of other items 135. What is the percentage of the total weight given to food?
The percentage of the total weight given to food is 66.67% (or 66⅔%).
Explanation
This question applies the weighted aggregative price index formula shown in the context: CPI = ΣWR / ΣW. Since there are only two categories (food and other items), we can set up an equation where the weights sum to 100. The overall CPI of 125 is the weighted average of the food index (120) and other items index (135). Solving this linear equation gives us the weight of food as 66.67%.
Solution Steps
Step 1: Let the weight given to food be W₁ and the weight given to other items be W₂.
Step 2: Since these are the only two categories, W₁ + W₂ = 100.
Step 3: Using the weighted index formula: CPI = (W₁ × Food Index + W₂ × Other Items Index) / 100
Step 4: Substituting values: 125 = (W₁ × 120 + W₂ × 135) / 100
Step 5: Simplifying: 12500 = 120W₁ + 135W₂
Step 6: Since W₂ = 100 - W₁, substitute: 12500 = 120W₁ + 135(100 - W₁)
Step 7: Solving: 12500 = 120W₁ + 13500 - 135W₁
Step 8: Rearranging: 15W₁ = 13500 - 12500 = 1000
Step 9: Therefore, W₁ = 1000/15 = 66.67%