How to calculate a price index

A price index measures how the price of a single item has changed relative to a chosen base year, expressed as a proportion of that base. You compute it with $I = \frac{P_{1}}{P_{0}} \times 100$, where $P_{0}$ is the price in the base year (always taken as index $100$) and $P_{1}$ is the price in the year of interest.

An index of $125$ means the price is $125\%$ of the base price, a rise of $25\%$; an index of $90$ means a fall of $10\%$. It answers questions such as 'find the price index in one year based on another', or, working backwards, 'given the index and the base price, find the later price'.

It is also the building block for the composite index that follows in this chapter.

Use a price index whenever a question gives the price of one item in two different years and asks how it has changed as an index or a percentage. Signals include the words 'price index', 'based on the year …', 'take … as the base year', or a table of prices for a base year and a later year.

If several items are involved and the question wants one overall figure, you will first find each item's price index this way, then combine them into a composite index. Also reach for the rearranged form $P_{1} = \frac{I}{100} \times P_{0}$ when the index and the base price are given and a missing price is wanted, a very common second part in exam questions.

The key is to keep clear which year is the base, because that price always goes underneath and always corresponds to an index of $100$.

1. 1

   Identify the base year

   Decide which year is the base; its price $P_{0}$ corresponds to index $100$.

2. 2

   Read off the two prices

   Note $P_{0}$, the base-year price, and $P_{1}$, the price in the year you are studying.

3. 3

   Apply the formula

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

4. 4

   Simplify to a number

   Do the division first, then multiply by $100$; keep exact values and round only at the end.

5. 5

   Interpret the index

   Compare with $100$: above means a rise, below means a fall; the difference from $100$ is the percentage change.

6. 6

   Rearrange if a price is missing

   If $I$ and $P_{0}$ are given, use $P_{1} = \frac{I}{100} \times P_{0}$.

• Dividing the wrong way round, the base-year price $P_{0}$ must be on the bottom: $\frac{P_{1}}{P_{0}}$, not $\frac{P_{0}}{P_{1}}$.
• Forgetting to multiply by $100$, leaving a ratio such as $1.25$ instead of the index $125$.
• Reading the index as the percentage change: an index of $125$ is a $25\%$ rise, not a $125\%$ rise.
• Mixing up which year is the base, the base year is always the one with index $100$.
• Rounding too early, so the final index is slightly off.

The single slip that costs marks is putting the base-year price in the wrong place, writing $\frac{P_{0}}{P_{1}}$ and getting $80$ instead of $125$. In one-to-one lessons our teachers fix the phrase 'later over base, times one hundred' and have you label $P_{0}$ and $P_{1}$ on the question before substituting, so the base year is always underneath.

We also separate the index from the percentage change out loud,'$125$ means up $25\%$', because that is where interpretation marks are won or lost. Since SPM Add Math is marked analytically, a correctly written $\frac{P_{1}}{P_{0}} \times 100$ already earns method marks even if the arithmetic slips.

Lessons are taught in English, while the SPM paper is set bilingually. Every teacher on spmaddmath.com.my is experienced, with lessons online at your own pace.