Index Numbers Worked Examples (KBAT)

Q: How do I combine indices that use different base years?

Anchor to one price. If the 2020 index based on 2015 is 120120120 and the 2025 index based on 2020 is 115115115, then I2025/2015=115×120100=138I_{2025/2015}=\frac{115\times 120}{100}=138I2025/2015=100115×120=138. The shortcut is to multiply the indices and divide by 100100100.

Q: How is a composite index linked to total spending?

A composite index of 125125125 means the whole basket costs 125%125\%125% of its base-year cost, a 25%25\%25% rise. So later spending is E1=Iˉ100×E0E_1=\frac{\bar{I}}{100}\times E_0E1=100Iˉ×E0, the same scaling used for a single price.

What these examples cover

These hard Index Numbers examples put two ideas together in each question, the way the longer SPM parts do. You will chain a price index across a change of base year, read the composite formula $\bar{I}=\frac{\sum I w}{\sum w}$ backwards to recover a weight ratio, and combine a missing index with a household's expenditure.

The numbers stay small and clean so the difficulty is in the reasoning, not the arithmetic. Cover the solution, attempt each question in full on paper, and only then check line by line.

When your route differs from ours, that is often fine in Index Numbers, but confirm both routes reach the same value, because a second independent method is the strongest possible check under analytic marking.

Worked examples

Work through all three. Each one asks you to hold two ideas at once, so plan the steps before you start writing, and keep every substitution explicit for the method marks.

Q1[5 marks]

The price index of an item for 2020 based on 2015 is $120$ . The price index for 2025 based on 2020 is $115$ .

(a) Find the price index for 2025 based on 2015. (b) If the price in 2015 was RM250, find the price in 2025.

Show worked solution

(a) The two indices use different base years, so anchor everything to one price. Let the 2015 price be $P$ .

Since the 2020 index based on 2015 is $120$ , the 2020 price is:

P_{2020}=\frac{120}{100}\times P=1.2P

The 2025 index based on 2020 is $115$ , so the 2025 price is $1.15$ times the 2020 price:

P_{2025}=\frac{115}{100}\times 1.2P=1.38P

Now the 2025 index based on 2015 compares $P_{2025}$ with the 2015 price $P$ :

I_{2025/2015}=\frac{1.38P}{P}\times 100=138

(b) Use this combined index to scale the 2015 price directly to 2025:

P_{2025}=\frac{138}{100}\times 250=345

Answer

The 2025 index based on 2015 is $138$ , and the 2025 price is RM345. This matches the chaining shortcut $I_{2025/2015}=\frac{I_{2025/2020}\times I_{2020/2015}}{100}=\frac{115\times 120}{100}=138$ .

Check part (b) stepwise: the 2020 price is $1.2\times 250=300$ , and $1.15\times 300=345$ .

Q2[5 marks]

A basket contains two goods A and B. Their price indices for 2024 based on 2021 are $140$ and $110$ , with weights $m$ and $n$ .

The composite index is $122$ . Find the ratio $m:n$ .

Show worked solution

Start from the composite formula and substitute the two indices, keeping the weights as unknowns:

\bar{I}=\frac{140m+110n}{m+n}=122

Multiply both sides by $m+n$ to clear the fraction:

140m+110n=122(m+n)=122m+122n

Gather the $m$ terms on one side and the $n$ terms on the other:

140m-122m=122n-110n \;\Rightarrow\; 18m=12n

Divide both sides by the common factor $6$ and write as a ratio:

3m=2n \;\Rightarrow\; \frac{m}{n}=\frac{2}{3}

Answer

The ratio is $m:n=2:3$ . Check with the simplest whole numbers $m=2,\,n=3$ : $\frac{140(2)+110(3)}{2+3}=\frac{280+330}{5}=\frac{610}{5}=122$ , the given composite.

Q3[5 marks]

A household's spending falls into three categories P, Q and R with weights in the ratio $3:2:1$ . The price indices for 2024 based on 2020 are $135$ for P, $120$ for Q and an unknown value for R.

The composite index is $125$ . (a) Find the price index of category R.

(b) The household spent RM600 on this basket in 2020; estimate the amount needed for the same basket in 2024, and state the percentage increase.

Show worked solution

(a) Weights in the ratio $3:2:1$ can be used as $3$ , $2$ and $1$ , so the total weight is $3+2+1=6$ . Substitute into the composite formula, with $I_R$ unknown:

125=\frac{135(3)+120(2)+I_R(1)}{6}

Work out the known products, $135(3)=405$ and $120(2)=240$ , then multiply both sides by $6$ :

125\times 6=405+240+I_R \;\Rightarrow\; 750=645+I_R

I_R=750-645=105

(b) Expenditure on the same basket scales with the composite index, so the 2024 amount is $\frac{125}{100}$ of the 2020 amount:

E_{2024}=\frac{125}{100}\times 600=750

Answer

The index of category R is $105$ , the 2024 amount is RM750, and the percentage increase is $\frac{750-600}{600}\times 100=25\%$ . Check part (a): $\frac{405+240+105}{6}=\frac{750}{6}=125$ ; and a composite of $125$ is a $25\%$ rise, matching part (b).

Q4[4 marks]

The price index of a sack of rice in 2024 based on 2019 is $125$ . (a) Find the price index of the rice in 2019 based on 2024.

(b) The price of the rice in 2024 is RM7.50 per sack. Find the price in 2019, and verify your answer to (a) directly from the two prices.

Show worked solution

(a) Let the 2019 price be $P$ . Since the 2024 index based on 2019 is $125$ , the 2024 price is:

P_{2024}=\frac{125}{100}\times P=1.25P

The 2019 index based on 2024 compares the earlier price $P$ with the later price $1.25P$ , so the two years swap which one sits on top:

I_{2019/2024}=\frac{P}{1.25P}\times 100=80

(b) Apply this reversed index to the 2024 price to get the 2019 price:

P_{2019}=\frac{80}{100}\times 7.50=6.00

Answer

The 2019-based-on-2024 index is $80$ , and the 2019 price is RM6.00. Check directly: $\frac{6.00}{7.50}\times100=80$ , and reversing again, $\frac{7.50}{6.00}\times100=125$ , the index given in the question, both routes agree.

Q5[5 marks]

A canteen's weekly basket in 2020 contains $20$ kg of rice at RM $2.00$ per kg and $12$ litres of cooking oil at RM $5.00$ per litre. The price indices for 2024 based on 2020 are $110$ for rice and $130$ for oil.

(a) Find the composite price index for this basket based on 2020. (b) Estimate the total cost of the same basket in 2024, and state the percentage increase in cost.

Show worked solution

(a) The weight of each item in the composite formula is its own base-year expenditure, not just the quantity bought. Find each expenditure in 2020:

w_{rice}=20\times 2.00=40 \qquad w_{oil}=12\times 5.00=60

Substitute these weights and the two price indices into the composite formula:

\bar{I}=\frac{110(40)+130(60)}{40+60}=\frac{4400+7800}{100}=122

(b) The total 2020 cost of the basket is the sum of the two expenditures, RM $40+60=100$ . Scale this by the composite index to estimate the 2024 cost:

E_{2024}=\frac{122}{100}\times 100=122

Answer

The composite index is $122$ , the basket costs about RM122 in 2024, and the percentage increase is $22\%$ . Check: $\frac{122-100}{100}\times100=22\%$ , matching the composite index directly.

Q6[6 marks]

A family's composite "cost of travel" index combines petrol (P) and public transport (Q), weighted $3$ and $7$ based on their 2019 spending share. In 2023 based on 2019, petrol's price index is already $150$ .

For the family's budget to hold, the composite index must not exceed $122$ . Find the maximum possible price index of Q this year, and state this as a maximum percentage increase.

Show worked solution

Set the composite formula with the budget cap as an inequality, substituting the known weights and petrol's index, and let $I_Q$ be the unknown transport index:

\frac{150(3)+I_Q(7)}{3+7}\leq 122

Multiply both sides by $10$ and simplify the known term:

450+7I_Q\leq 1220

Isolate $I_Q$ :

7I_Q\leq 770 \;\Rightarrow\; I_Q\leq 110

Answer

The price index of public transport must not exceed $110$ , a maximum increase of $10\%$ . Check at the boundary: $\frac{150(3)+110(7)}{10}=\frac{450+770}{10}=\frac{1220}{10}=122$ , exactly the cap, confirming $I_Q=110$ is the largest value allowed.

Q7[5 marks]

A stall's two main ingredients, chicken and onions, are weighted $5$ and $3$ based on their 2021 spending share. From 2021 to 2024, chicken's price rose by $16\%$ while onions' price fell by $8\%$ .

(a) Write down the 2024-based-on-2021 price index for each ingredient. (b) Find the composite price index for the pair.

Show worked solution

(a) Convert each percentage change to an index directly: a rise adds to $100$ , a fall subtracts from it.

I_{chicken}=100+16=116 \qquad I_{onion}=100-8=92

(b) Substitute both indices and their weights into the composite formula:

\bar{I}=\frac{116(5)+92(3)}{5+3}=\frac{580+276}{8}=107

Answer

The composite index is $107$ , so the stall's combined ingredient cost rose overall by $7\%$ , even though onions became cheaper, chicken carries the heavier weight ( $5$ against $3$ ), so its rise dominates the average. Check with a second route: weight each deviation from $100$ directly, $5(+16)+3(-8)=80-24=56$ , then divide by the total weight, $\frac{56}{8}=7$ , the same net rise of $7\%$ .

Q8[6 marks]

Two items C and D have weights $8$ and $12$ based on 2020 spending. The composite price index for 2023 based on 2020 is $116$ .

The price index of D exceeds that of C by $10$ , that is $I_D=I_C+10$ . Find the price index of each item.

Show worked solution

Substitute the weights, the composite value, and the relationship $I_D=I_C+10$ into the composite formula, leaving $I_C$ as the only unknown:

\frac{I_C(8)+(I_C+10)(12)}{20}=116

Multiply both sides by $20$ and expand:

8I_C+12I_C+120=2320 \;\Rightarrow\; 20I_C=2200

Solve for $I_C$ , then find $I_D$ :

I_C=110 \;\Rightarrow\; I_D=110+10=120

Answer

The price index of C is $110$ and of D is $120$ . Check by substituting both back into the composite formula: $\frac{110(8)+120(12)}{20}=\frac{880+1440}{20}=\frac{2320}{20}=116$ , the given composite.

These three show what "hard" really means in Index Numbers: not heavier arithmetic, but two linked ideas in one question. Anchor a chain of indices to a single price, read the composite formula in whichever direction the unknown sits, and remember that a composite index doubles as a percentage change in total spending.

Key method points

These harder examples reward a clear plan and a second check. Keep the following in mind as you tackle the longer parts.

To chain a base-year change, let the earliest price be $P$ and multiply through: $I_{C/A}=\frac{I_{C/B}\times I_{B/A}}{100}$ .
Solve a composite backwards by clearing the denominator first: $\bar{I}(\sum w)=\sum I w$ , then collect the unknown.
A weight ratio comes out of $\frac{\sum I w}{\sum w}=\bar{I}$ by gathering like terms and simplifying to lowest form.
A composite index is also a percentage: an index of $125$ means total spending on the same basket rises by $25\%$ .
Estimate later expenditure with $E_1=\frac{\bar{I}}{100}\times E_0$ , the same scaling as a single price.
Always confirm with a second route, stepwise prices, or substituting the simplest whole-number weights, since analytic marking rewards a secured answer.

How a teacher helps

In the harder parts, students usually know each formula but stall on how the two ideas connect, whether to chain the indices or average them, or how a composite becomes a percentage rise in spending. In a one-to-one lesson our teacher helps you plan the steps before writing, so the connection is clear from the first line.

Because our teachers are experienced, you work with someone who models the second-route check that makes hard answers secure. Lessons are taught in English, while SPM papers are set in both Malay and English, so the notation reads the same to you in either version.

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Frequently asked questions

How do I combine indices that use different base years?

Anchor to one price. If the 2020 index based on 2015 is $120$ and the 2025 index based on 2020 is $115$ , then $I_{2025/2015}=\frac{115\times 120}{100}=138$ .

The shortcut is to multiply the indices and divide by $100$ .

Why can I use the ratio 3:2:1 directly as weights 3, 2 and 1?

Because the composite index is a ratio itself, scaling every weight by the same factor cancels top and bottom. Using the simplest whole numbers in the ratio gives the same composite as any equivalent set of weights.

How is a composite index linked to total spending?

A composite index of $125$ means the whole basket costs $125\%$ of its base-year cost, a $25\%$ rise. So later spending is $E_1=\frac{\bar{I}}{100}\times E_0$ , the same scaling used for a single price.

What is the best way to check a hard Index Numbers answer?

Use a different route. Recompute a chained index through stepwise prices, or substitute the simplest whole-number weights back into $\frac{\sum I w}{\sum w}$ .

Two independent methods agreeing is the strongest check.

Source:SRC-DSKP-EN

Written by the spmaddmath.com.my editorial team.· Last updated 5 September 2026

Index Numbers, Worked Examples (KBAT)

What these examples cover

Worked examples

Key method points

How a teacher helps

Frequently asked questions

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