Quantitative Aptitude · Index Numbers

Cost of Living Index and Consumer Price Index

Updated 1 October 2026 · Fact-checked

The cost of living index (consumer price index) measures how the price of a fixed basket of items a family buys changes between a base year and a current year. Use Σp1q0 ÷ Σp0q0 × 100 (aggregate expenditure) or ΣPV ÷ ΣV (family budget). Both give the same result for the same data.

Understand Cost of Living Index and Consumer Price Index

A cost of living index (also called the Consumer Price Index, CPI) tells you how much more or less a typical family must spend now to buy the same goods and services it bought in the base year. It is a weighted index number, because food matters far more to a family budget than, say, newspapers.

The base year index is 100. If the CLI is 135, the same basket costs 35% more than in the base year. It says nothing about the quality of life, only about the cost of the fixed basket.

There are two ways to build it. In the aggregate expenditure method, you are given base quantities q0 and both prices. You compute what the base basket costs at current prices and compare it with its cost at base prices. In the family budget method, you are given price relatives and the amount spent on each item (the weights). You take a weighted average of the price relatives.

The two methods are the same idea. The weight V = p0q0 is the base-year spending on an item. So Σ(P × V) ÷ ΣV works out to Σp1q0 ÷ Σp0q0 × 100. That is why the answers match when the data describe the same family.

The main use of CPI is to measure real wages, deflate money incomes, and decide dearness allowance. Real wage tells you what your pay can actually buy after price changes.

Key formulas to remember

Aggregate expenditure method: CLI = (Σp1q0 ÷ Σp0q0) × 100; p0, p1 are base and current prices; q0 is the base-year quantity. This is the Laspeyres form.
Family budget method: CLI = ΣPV ÷ ΣV; P = (p1 ÷ p0) × 100 is the price relative; V = p0q0 is the value (weight) of the item in the base year.
Real wage: Real wage = (Money wage ÷ CLI) × 100; Use the CLI of the year for which the money wage is given, with the base year as 100.
Purchasing power of money: Purchasing power = 100 ÷ CLI; Take it as a rupee value relative to the base year, which is ₹1.
Wage needed to keep the same standard: Required wage = Base wage × (Current CLI ÷ 100); Applies when the base index is 100. Otherwise use current CLI ÷ base CLI.

How to solve Cost of Living Index and Consumer Price Index questions

Pick the method from the data given, then keep the work in a small table so you do not mix columns.

1Read what is given. If you see p0, p1 and q0, use the aggregate expenditure method. If you see price relatives or indices and weights, use the family budget method.
2For aggregate expenditure, make columns for p0q0 and p1q0 for each item. Add each column.
3Divide Σp1q0 by Σp0q0 and multiply by 100.
4For family budget, find P = p1 ÷ p0 × 100 for each item if it is not given. Find V = p0q0 if it is not given directly.
5Compute PV for each item, then ΣPV and ΣV. The CLI is ΣPV ÷ ΣV.
6If the question is about wages, divide the money wage by the CLI and multiply by 100 to get the real wage. Or multiply the base wage by CLI ÷ 100 for the wage needed.
7Check the result: the CLI must lie between the smallest and largest price relative. Then match with the option.

Quickest way: Weighted average shortcut with option elimination

When to use it: Use in the MCQ paper whenever weights and price relatives are given, or when numbers are large.

Check the range first. In the family budget method, the answer lies between the lowest and highest P. This often removes two options at once.
If weights add up to 100, then ΣV = 100 and the answer is just ΣPV ÷ 100. Drop two zeros.
Shrink the weights. If the weights share a common factor, such as 10, 20, 30, divide them by it before multiplying.
Work with deviations from 100 for large indices. Weighted average of (P − 100) plus 100 gives the same answer with smaller numbers.
For real wage questions, estimate first. A wage of ₹30,000 at CLI 150 gives ₹20,000, so look for a nearby option before you work out decimals.
If the sums need more than about two minutes, mark it and move on. A wrong answer costs 0.25.

Common mistakes in Cost of Living Index and Consumer Price Index

Using current quantities (q1) instead of base quantities (q0).
Students remember the Paasche index from the chapter and mix up the formulas.
Fix: For CLI, the basket is fixed at the base year. Use Σp1q0 and Σp0q0, not q1.
Forgetting to multiply by 100 in the aggregate method.
The ratio comes out as 1.21875 and looks like a finished answer.
Fix: The base is 100, so the CLI is always a number like 121.9. Multiply by 100 as the last step.
Using the given weights as prices or quantities in the family budget method.
The table has several columns, and students confuse weight W with q0.
Fix: In the family budget method, V is the money spent in the base year. Use ΣPV ÷ ΣV, and read the question to see if the weight is a value or a quantity.
Computing the price relative as p0 ÷ p1.
Students flip the ratio when in a hurry.
Fix: Price relative is always current over base: P = p1 ÷ p0 × 100. A price rise should give P above 100.
Dividing the CLI by the wage for real wage.
The formula is memorised without checking what it means.
Fix: Real wage = money wage ÷ CLI × 100. If prices rise, real wage is less than money wage.
Dividing by the number of items instead of ΣV.
Students treat it as a simple average.
Fix: Divide by the sum of the weights, not by the number of items.

Worked examples

Example 1

A family buys three items. Base price (₹), current price (₹) and base quantity are: A: 10, 12, 5; B: 20, 25, 3; C: 5, 6, 10. The cost of living index by the aggregate expenditure method is closest to: (a) 112.5 (b) 121.9 (c) 128.4 (d) 135.0

Show the solution

Find p0q0: A = 10 × 5 = 50; B = 20 × 3 = 60; C = 5 × 10 = 50. Σp0q0 = 160.
Find p1q0: A = 12 × 5 = 60; B = 25 × 3 = 75; C = 6 × 10 = 60. Σp1q0 = 195.
CLI = 195 ÷ 160 × 100 = 121.875.
Rounded to one decimal, 121.875 is 121.9.

Answer: (b) 121.9

Example 2

Five groups have price relatives (P) and base-year expenditure (V in ₹): Food: P = 150, V = 40; Rent: P = 120, V = 20; Clothing: P = 130, V = 10; Fuel: P = 140, V = 15; Others: P = 110, V = 15. The cost of living index by the family budget method is: (a) 130.5 (b) 134.5 (c) 138.0 (d) 145.0

Show the solution

Compute PV: Food = 150 × 40 = 6,000; Rent = 120 × 20 = 2,400; Clothing = 130 × 10 = 1,300; Fuel = 140 × 15 = 2,100; Others = 110 × 15 = 1,650.
ΣPV = 6,000 + 2,400 + 1,300 + 2,100 + 1,650 = 13,450.
ΣV = 40 + 20 + 10 + 15 + 15 = 100.
CLI = 13,450 ÷ 100 = 134.5.
Check: 134.5 lies between 110 and 150, so it is plausible.

Answer: (b) 134.5

Example 3

A worker earned ₹24,000 a month in the base year, when the CLI was 100. The CLI is now 160. What monthly wage keeps the worker's real wage unchanged? (a) ₹15,000 (b) ₹33,600 (c) ₹38,400 (d) ₹40,000

Show the solution

To keep the same real wage, the money wage must rise in the same proportion as the CLI.
Required wage = 24,000 × 160 ÷ 100.
24,000 × 1.6 = 38,400.
Check: real wage = 38,400 ÷ 160 × 100 = 24,000, which equals the base wage.
Option (a), ₹15,000, comes from dividing by 1.6, which is the wrong direction.

Answer: (c) ₹38,400

Exam tips

Read the question to see if it gives q0 with prices, or price relatives with weights. This decides the method in the first ten seconds.
Always do a range check on the family budget answer. It must lie between the smallest and largest price relative.
Real wage questions usually give two years of wages and indices. Compare real wages, not money wages, to say whether the worker is better off.
Option traps are common: the reciprocal answer, the answer without ×100, and the answer using q1. Check your result against these before marking.
If the question asks about purchasing power, use 100 ÷ CLI. Do not confuse it with real wage.

Practice questions from Index Numbers

Cost of Living Index and Consumer Price Index: frequently asked questions

What is the difference between the aggregate expenditure method and the family budget method?

The aggregate expenditure method uses actual prices and base quantities: Σp1q0 ÷ Σp0q0 × 100. The family budget method uses price relatives weighted by base-year spending: ΣPV ÷ ΣV. They give the same answer when V = p0q0.

Is the cost of living index the same as CPI?

At Foundation level, you can treat them as the same thing. Both measure the change in the cost of a fixed basket of consumer goods and services bought by a family. CPI is the name usually used for the published index.

How do I find the real wage from the CLI?

Divide the money wage by the CLI and multiply by 100. For example, ₹30,000 at a CLI of 150 gives a real wage of ₹20,000. It shows what the wage can buy at base-year prices.

Do the weights in the family budget method need to add up to 100?

No. The formula divides by ΣV, so any total works. When the weights do add up to 100, you can just divide ΣPV by 100, which saves time.