Making sense of index numbers in Add Math

This is the chapter that often sits at the very end of the syllabus, and students meet it with least practice, yet it is one of the more forgiving topics once the core idea lands. An index number is simply a way of comparing a quantity at one time against the same quantity at an earlier, chosen time called the base.

To make comparisons easy, the base value is always rescaled to $100$. So an index of $125$ means the quantity is $25\%$ above the base; an index of $90$ means it has fallen $10\%$ below it.

That single reading, "how far above or below $100$", is the intuition to carry through the whole chapter.

The most common index in the syllabus is the price index, which compares the price of an item now, $P_{1}$, with its price in the base year, $P_{0}$:

$$I = \frac{P_{1}}{P_{0}} \times 100$$

Suppose a kilogram of rice cost RM4 in the base year and RM5 now. Its price index is $\dfrac{5}{4} \times 100 = 125$, telling you the price has risen by $25\%$.

The formula also runs backwards: if you know the index and the base price, you can recover the new price, or if you know the index and the new price, you can find what the base price must have been.

Everything else in the chapter is built on this one relationship. The words change, cost of living, production, quantity, but the structure is always "new over base, times one hundred".

A single price index describes one item. But real questions ask about a basket of items that do not all matter equally, a household spends far more on rice than on salt, so a change in the rice price should count for more.

This is what a weight does, and combining several indices with their weights gives the composite index:

$$\bar{I} = \frac{\sum I_{i}\, w_{i}}{\sum w_{i}}$$

It is just a weighted average of the individual indices. Say three building materials have price indices $120$, $110$ and $130$, with weights $3$, $2$ and $5$ reflecting how much of each is used.

Then:

$$\bar{I} = \frac{120(3) + 110(2) + 130(5)}{3 + 2 + 5} = \frac{1230}{10} = 123$$

The composite index is $123$, so the overall cost of the materials has risen about $23\%$. Notice the weight with the biggest value pulls the answer towards its own index, that is the whole purpose of weighting, and a good sanity check on your arithmetic.

Sometimes you are given an index from one base year and asked to link it to another. Two indices measured across a shared middle year can be chained together.

If $I_{2020/2018} = 110$ (year 2020 based on 2018) and $I_{2023/2020} = 120$ (2023 based on 2020), then the index of 2023 based on 2018 is:

$$I_{2023/2018} = \frac{I_{2023/2020} \times I_{2020/2018}}{100} = \frac{120 \times 110}{100} = 132$$

The division by $100$ is the step people forget, it is there because each index already carries a factor of $100$, and multiplying two of them would otherwise double it. Think of it as: a $10\%$ rise followed by a $20\%$ rise is not a $30\%$ rise but $1.10 \times 1.20 = 1.32$, a $32\%$ rise.

Index numbers reward tidy, labelled working more than almost any other chapter, because the arithmetic is easy but the bookkeeping is where things slip.

• Forgetting the $\times 100$, so an index of $1.25$ is left where $125$ belongs.
• Dividing the composite index by the number of items instead of by the sum of the weights $\sum w_{i}$.
• Forgetting to divide by $100$ when chaining two indices across base years.
• Reading "$25\%$ increase" as "index $25$", a $25\%$ rise is an index of $125$.

Index numbers are unusual: the maths is gentle, but because the chapter is short and often rushed at the end of the year, students arrive at the exam having barely practised it. That is a shame, because it can be one of the most reliable sets of marks on the paper.

One-to-one, a teacher can make sure the base-year direction, the weighting, and the base-change step are all solid, and can do it quickly, since the ideas are not deep, only easy to muddle. Our teachers are experienced; lessons are online and taught in English, while the SPM papers are set bilingually in Bahasa Melayu and English.

If index numbers were skimmed in class, you are welcome to begin with a one-hour paid trial class at the teacher's own rate (from RM50 per hour, depending on experience). We will not promise a grade, but this is exactly the sort of tidy, well-defined topic where focused practice tends to pay off.