A composite index is one number that summarises several items, and it always compares with a base year whose index is 100. To compare two other years, divide their indices, do not subtract them.
This lesson belongs to SPM Additional Mathematics index numbers. If weights are still unfamiliar, read combining weighted indices first.
What does a composite index measure?
It measures the weighted average of the item indices, relative to one base year. The formula is I = ΣIw ÷ Σw, where I is each item’s index and w is its weight.
Here is a household cost index with base year 2020.
| Item | Index in 2024 | Weight |
|---|---|---|
| Food | 140 | 5 |
| Transport | 120 | 3 |
| Housing | 120 | 2 |
ΣIw = 140×5 + 120×3 + 120×2 = 700 + 360 + 240 = 1 300, and Σw = 10. So the composite index for 2024 is 1 300 ÷ 10 = 130.
That means household costs in 2024 were 30% higher than in 2020. It does not say they rose 30% in the last year.
Worked example: comparing two years that share a base
Suppose the composite index for 2022 is 104 and for 2024 is 130, both with base year 2020. The question asks for the percentage increase in costs from 2022 to 2024.
The tempting method. Subtract: 130 − 104 = 26, so costs rose by 26%. The working is short, and it is wrong.
The correct method. Re-base so that 2022 is the new base year:
Index of 2024 based on 2022 = 130 ÷ 104 × 100 = 125.
So costs rose by 25% from 2022 to 2024. As a check, take a basket priced at RM2 080 in 2022. Its 2020 price was 2 080 ÷ 104 × 100 = RM2 000, and its 2024 price is 2 000 × 130 ÷ 100 = RM2 600. Then 2 600 ÷ 2 080 = 1.25, which matches.
The mistake that costs marks
The slip is treating the gap between two indices as a percentage change. The gap is measured in points of the base-year value, and the percentage change between two later years is measured against the earlier year.
| Step | Wrong | Right |
|---|---|---|
| Base year assumed | Still 2020 | Earlier year becomes 100 |
| Operation | 130 − 104 | 130 ÷ 104 × 100 |
| Answer | 26% | 25% |
The gap looks small here, but it grows when the earlier index is large. With indices 200 and 260, subtraction gives 60% while the true increase is 30%.
Reading a question about base years
Underline which year is the base before doing anything. Then decide which year the question wants as its comparison.
- Note the base year given in the question, the one with index 100.
- Note the two years that are being compared.
- If neither of them is the base year, divide the later index by the earlier one and multiply by 100.
- State the answer in words, for example “costs were 25% higher in 2024 than in 2022”.
Try your own pairs in the percentage base and index comparison explorer, which shows how the percentage changes when the base changes.
Check yourself
The composite index for 2021 is 110 and for 2025 is 143, both with base year 2019. A basket was priced at RM2 200 in 2021. Find the index of 2025 based on 2021, and the price of the same basket in 2025.
Answer
Index of 2025 based on 2021 = 143 ÷ 110 × 100 = 130.
Price in 2025 = 2 200 × 143 ÷ 110. Since 2 200 ÷ 110 = 20, the answer is 20 × 143 = RM2 860.
Check: RM2 860 ÷ RM2 200 = 1.3, which is the 130 found above.
What to study next
Start from calculating a price index from two values if the formula I = Q₁ ÷ Q₀ × 100 still needs work. When you are ready, test the whole chapter with the index numbers practice set.
If you want a teacher to go through index questions with your own mistakes, see online one-to-one Additional Mathematics tuition. The mistake log helps you spot repeat slips first.