Question 5 OR (c)
Finding the rise in the price of food from a cost-of-living index
9 marks
- Time to write
- About 18 minutes
- Topic
- Index numbers
Question
Question 5 OR (c) · 9 marksThe cost-of-living index uses these weights for a basket of five commodities:
Food 35, Rent 20, Clothing 15, Fuel 10, Miscellaneous 20. In the period 2018-2023, the cost-of-living index goes up from 100 to 186. In the same period, the percentage rise in prices was:
Rent 40, Clothing 120, Fuel 90 and Miscellaneous 150. What was the percentage change in the price of Food?
Reworded from the university paper: names and figures differ, the task is the same. The original paper belongs to the University of Delhi. See DU's question bank.
Before you write
Read off the question and its marks
Write it in this order
- Read the data; note exactly what is asked 1 min
- 1 Method: weighted average of price relatives 2 marks 4 min
- 2 Price relatives and weighted relatives of the known groups 3 marks 5 min
- 3 Equation for the food relative and its solution 3 marks 5 min
- 4 Percentage rise in the price of food, with check 1 mark 2 min
- Check: arithmetic, units (₹, %), final answer underlined 1 min
Try it first. About 18 minutes for 9 marks. Write it out, then open the solution.
Solution
Hover or tap a marked figure, like 72,000, to see how it is made and where else it appears.
Method. The cost-of-living index is the weighted average of the price relatives of the groups in the basket, CLI = ΣWR / ΣW, where R is the current price as a percentage of the base price; such an index measures how the cost of living has changed over the period. A rise of k% in a group's prices means R = 100 + k. Let x be the price relative of food.
Working table (2023, base 2018 = 100)
| Group | Weight W | % rise in price | Price relative R | WR |
|---|---|---|---|---|
| Food | 35 | ? | x | 35x |
| Rent | 20 | 40 | 140 | 2,800 |
| Clothing | 15 | 120 | 220 | 3,300 |
| Fuel | 10 | 90 | 190 | 1,900 |
| Miscellaneous | 20 | 150 | 250 | 5,000 |
| Total | 100 | 35x + 13,000 |
CLI = ΣWR / ΣW, so 186 = (35x + 13,000) / 100.
35x + 13,000 = 18,600, so 35x = 5,600 and x = 160.
The price relative of food is 160: food prices rose by 60% between 2018 and 2023.
Check: (35 × 160 + 13,000) / 100 = (5,600 + 13,000) / 100 = 186.
Where the 9 marks sit
| # | Component | Marks |
|---|---|---|
| 1 | Method: weighted average of price relatives | 2 |
| 2 | Price relatives and weighted relatives of the known groups | 3 |
| 3 | Equation for the food relative and its solution | 3 |
| 4 | Percentage rise in the price of food, with check | 1 |
| Total | 9 |
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