KBAT: Interpreting Index Numbers to Decide

A routine index-number question asks for one value: 'find the composite index'. A decision KBAT question buries several indices in a table, brings in more than one base year, and finishes with a money question, what price is fair, does this profit target hold?

That is higher-order thinking, Kemahiran Berfikir Aras Tinggi: the composite-index formula and the chaining rule are familiar, but you must decide which formula applies, keep track of which year is the base, and then turn the index back into ringgit to answer the real question. Nothing tells you which step comes first.

In Add Math this rewards students who can carry a number through several stages and read it as a decision, not only students who can compute a single index in isolation.

This is an original question written in the style of SPM Paper 2. Try it before reading the solution.

Notice what the index did for the bakery. It turned a memory of old costs into a defensible 2024 figure, RM34.10, and from there the pricing question answered itself.

That is the whole point of an index number: it carries a change faithfully through time so that a decision can rest on it, rather than on a guess about how much prices 'feel' like they have risen. A KBAT question tests exactly that final move, from the tidy index back to a number the bakery can act on.

When an index-number question is a table plus a paragraph, the first step is to name which index is wanted and which year is the base. A single price index compares one item across years, $I=\dfrac{P_{1}}{P_{0}}\times 100$.

A composite index combines several items by weight, $\bar{I}=\dfrac{\sum I\,w}{\sum w}$. If two base years appear, 2020, 2022, 2024, you will chain: $\bar{I}_{A/C}=\dfrac{\bar{I}_{A/B}\times \bar{I}_{B/C}}{100}$.

Read the weights as shares by cost, not as prices. Once you know which index and which base year, drop the numbers into the matching formula.

For a decision part, convert the index back into money: a base-year amount times the index over 100 gives the later amount, and from there the profit or price follows.

Marking is analytic, so method marks are awarded line by line. On an index-number decision question a marker looks for:

• The correct formula chosen, $\sum I\,w \div \sum w$ for a composite index, product-over-100 for chaining.
• The substitution shown, such as $(120\times5+110\times2+140\times3)\div 10$.
• The weights summed correctly in the denominator.
• Chaining done as index $\times$ index $\div 100$, not by adding the two indices.
• The index converted back into money for the decision, cost $\times$ index $\div 100$.
• A final sentence answering the decision, such as 'RM45 meets the target', with the profit shown.

The slips in these questions come from mixing up base years and from stopping at the index instead of answering the money question, so that is what our teachers work on. In a one-to-one lesson we label every index with its two years before doing anything, rehearse the chaining rule until the divide-by-100 is automatic, and practise the last move, turning an index back into ringgit and stating the decision.

Teachers at spmaddmath.com.my are experienced, and lessons are online and taught in English. Message us on WhatsApp to arrange a one-hour paid class from RM50/hr at the teacher's rate.