KBAT: Deciding with a Composite Index

A routine index-numbers question says 'find the composite index' and stops. A decision KBAT question keeps going: it hands you a rule, raise the retail price only if costs climb past a threshold, or a target composite index, and asks you to reason your way there.

That is Kemahiran Berfikir Aras Tinggi, higher-order thinking: computing $\bar{I}=\frac{\sum I_i w_i}{\sum w_i}$ is ordinary Form 4 work, but reading what the single number means for a real budget, and especially working backwards from a target index to the price change one ingredient is allowed, is not spelt out. In Add Math this rewards students who treat the composite index as a weighted story about many prices, not just a figure to hand up, students who can turn '$133.5$' into 'costs rose about a third' and then decide what to do about it.

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

A small bakery makes one butter cake. The table shows the price index of each ingredient for the year 2024 based on 2020, and the weightage of each ingredient in the recipe.

The dependable first move on any composite-index question is to read the table carefully and write the formula $\bar{I}=\frac{\sum I_i w_i}{\sum w_i}$ before touching the calculator. Check which year is the base, here 2020, so every index is measured against $100$.

Then form the product $I_i w_i$ for each ingredient in its own line, rather than trying to do all four at once; that is where slips hide. Add the products for the numerator and add the weightages for the denominator, never divide by the number of ingredients, because the whole point of a weightage is that ingredients count unequally.

Getting these two sums right, and knowing the denominator is $\sum w = 20$, makes every later part, the cost, the decision, the reverse calculation, a short and safe step.

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

• The correct formula $\bar{I}=\frac{\sum I_i w_i}{\sum w_i}$ written down.
• Each product formed correctly, $600,\ 220,\ 1200,\ 650$, and summed to $2670$.
• Division by the total weightage $\sum w = 20$, not by the count of ingredients, giving $\bar{I}=133.5$.
• The 2024 cost as $8.00\times\frac{133.5}{100}=\text{RM}10.68$.
• A clear decision tied to the rule, a $33.5\%$ rise exceeds $30\%$, so the price rise is triggered.
• The reverse equation set up and solved: $1470+8I_b=2500\Rightarrow I_b=128.75$.
• A sentence interpreting the result, butter no more than about $28.75\%$ above the 2020 price.

Decision questions reward students who compute cleanly and then explain in words, and both habits sharpen fastest with feedback. In a one-to-one lesson our teachers ask you to lay the products out line by line, to say what the composite index means before using it, and to rehearse the reverse step, setting the formula equal to a target and solving for the one unknown.

We treat the final interpretation as part of the answer, because that sentence earns the reasoning mark. 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.