Index numbers, explained through everyday prices

You've almost certainly read a headline like "prices rose compared with last year" without thinking about the maths behind it. That sentence is describing an index number: today's price, compared against an earlier reference price, turned into a single figure that's easy to talk about.

Add Math simply gives you the tools to build and read that figure for yourself, rather than taking someone else's word for it.

Formally, an index number is a value that shows how a quantity, usually a price, has changed relative to a fixed reference point, called the base. The base itself is always set to 100, so an index of 100 means no change, an index above 100 means an increase, and an index below 100 means a decrease.

An index of 115, for instance, means the value has risen by 15% since the base period.

The simplest index number compares one item's price now against its price at the base period:

$$I = \dfrac{Q_{1}}{Q_{0}} \times 100$$

Here $Q_0$ is the price at the base period and $Q_1$ is the price at the period you're comparing it to. Say a kilogram of rice cost RM2.80 at the base period and RM3.15 now.

The price index is $I = \frac{3.15}{2.80}\times 100 \approx 112.5$, meaning the price has risen by about 12.5% since the base period.

The same formula runs in reverse just as often. If you're told the index is 120 and the base price was RM4.00, the current price is found by rearranging: $Q_1 = \frac{I \times Q_0}{100} = \frac{120\times 4.00}{100} = \text{RM}4.80$.

Recognising which of $I$, $Q_0$, or $Q_1$ is missing, and rearranging for it, is most of what this part of the chapter tests.

A single item's price index is useful, but a household doesn't spend equally on every item it buys, rice, transport, and rent don't carry the same weight in a monthly budget. A composite index combines several price indices into one overall figure, weighted by how much each item actually matters:

$$\bar{I} = \dfrac{\sum W_{i} I_{i}}{\sum W_{i}}$$

Each item gets an index $I_i$ and a weight $W_i$ reflecting its relative importance, often the quantity bought, or its share of total spending. Suppose three items have price indices of 110, 120, and 105, with weights 3, 2, and 5 respectively.

The composite index is $\bar{I} = \frac{(3\times 110)+(2\times 120)+(5\times 105)}{3+2+5} = \frac{330+240+525}{10} = 109.5$.

Notice that the composite index leans toward whichever item has the largest weight, here, the item weighted 5 pulls the average closer to its own index of 105 than a simple, unweighted average of 110, 120, and 105 (which would be $111.\overline{6}$) would suggest. That's the entire purpose of weighting: it stops one lightly-purchased item from swaying a figure meant to represent everyday spending.

The base period isn't fixed forever, it's simply the earlier point everything else is compared against, chosen because it's a stable, convenient reference. When prices are re-indexed against a more recent base period, that's called rebasing, and it resets the base period's own index back to exactly 100.

A composite index is interpreted the same way regardless of which base period was chosen: a value above 100 means overall prices have risen since that base period, and a value below 100 means they've fallen. What changes between two different base periods is only the reference point, not the underlying logic of the calculation.

Index-number questions tend to give you a table with some cells filled in and others left blank, a price, a quantity, an index, or a weight missing from an otherwise complete row. The habit worth building is treating each formula as an equation with three parts, and identifying which one is unknown before you touch a calculator:

• If the price index is missing, use $I=\frac{Q_1}{Q_0}\times 100$ directly.
• If a price is missing, rearrange for $Q_1$ or $Q_0$ first, then substitute.
• If the composite index is missing, total the products $W_iI_i$ before dividing by the total weight, don't divide item by item.
• If a weight is missing, use the composite-index formula with everything else known, and solve for that one weight algebraically.

Because marking is analytic, laying out which values are known, which are unknown, and which formula connects them, before calculating anything, earns method marks on its own, and makes it far easier to spot which rearrangement you actually need.

Index numbers rarely trip students up on the algebra, rearranging $I=\frac{Q_1}{Q_0}\times 100$ is straightforward once you've seen it done. What's harder to pick up alone is reading a real question: spotting which value is missing from a table, or seeing why a composite index leans toward a heavily-weighted item.

A teacher working through examples with you, live, closes that gap far faster than staring at a mark scheme after the fact. Our teachers are experienced, and lessons run online in English, which also builds comfort with the English terms used in this chapter, while SPM papers themselves are set bilingually in Bahasa Melayu and English.

If you'd like to try a session, the first lesson is a one-hour paid class at the teacher's rate, from RM50 an hour, depending on the teacher's experience, quoted on WhatsApp. Whether you're still getting comfortable with the basic price index or working through weighted composite-index problems, we start from wherever you are.