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CA Foundation · Quantitative Aptitude

Index Numbers for CA Foundation Quantitative Aptitude

An index number measures the relative change in a variable, such as price or quantity, between a base period and a current period, usually as a percentage. To solve questions, identify the type asked, pick the right formula, substitute carefully, and use options to eliminate wrong answers.

What this chapter covers

Index Numbers is a formula-driven chapter in the Statistics part of Paper 3. It shows how to compress many price or quantity changes into one number. You compare a current period with a base period, where the base index is 100.

The chapter has a clear core. You learn simple and aggregative methods, then weighted indices (Laspeyres, Paasche, Fisher), then tests that check whether a formula is consistent. After that come practical tools: base shifting, splicing, deflating, and the cost of living index.

It links to other parts of the paper. You use averages (arithmetic and geometric mean) and ratios from earlier chapters. Calculation speed from Business Mathematics helps too. Since Paper 3 is an MCQ paper with 0.25 negative marking, the chapter rewards fast, accurate substitution and smart option checking.

Index Numbers is one of the more predictable chapters in Statistics. Questions are mostly direct: a formula, a table of prices and quantities, and four options. Once you know the formulas and the table layout, each question can be done in a minute or less. Few other chapters give such reliable returns for the effort. Since wrong answers cost 0.25 marks each, this chapter is a good place to gain safe marks while you stay selective elsewhere in the paper.

Index Numbers: topics in the order to study them

  1. 1Meaning and Uses of Index NumbersIt gives you the base-period idea and the vocabulary that every later formula depends on.
  2. 2Methods of Constructing Index NumbersSimple aggregative and price relative methods are the building blocks before weights are added.
  3. 3Weighted Index Numbers: Laspeyres, Paasche, FisherThis is the core, most-tested part, so study it once the unweighted methods feel easy.
  4. 4Tests of Adequacy: Time and Factor Reversal TestsThese tests only make sense after you know the weighted formulas they are applied to.
  5. 5Base Shifting, Splicing and DeflatingThese are short, mechanical conversions that rely on understanding what an index of 100 means.
  6. 6Cost of Living Index and Consumer Price IndexIt applies weighted indices to a real use, so learn it after the formulas are secure.
  7. 7Limitations and Problems in Index NumbersIt is theory-based and quick to learn, so finish with it and use it for revision.

How to prepare Index Numbers

Treat this chapter as a formula set plus a table-reading skill. Aim for accuracy first, then speed.

  1. Write each formula on one page in plain symbols and say what each letter means: p0, p1 for base and current prices, q0, q1 for quantities.
  2. Solve two or three simple examples by hand for each method until you can set up the table of p0q0, p1q0, p0q1, p1q1 without hesitation.
  3. Practise Laspeyres, Paasche and Fisher on the same data. Fisher is the geometric mean of the other two, so you reuse your sums and save time.
  4. Learn the two reversal tests as checks: time reversal needs P01 × P10 = 1, and factor reversal needs P01 × Q01 = V01, the ratio of total values. Here Q01 is found by interchanging p and q in the same price formula. Know which formula satisfies which test.
  5. Drill base shifting, splicing and deflating with a few short problems each. Remember that deflating divides by the price index and multiplies by 100.
  6. Do timed MCQ sets. Estimate first, then eliminate options that are clearly too high or too low. For example, Fisher always lies between Laspeyres and Paasche, so if you have both, discard any Fisher option outside that range. Another check: Laspeyres equals the weighted arithmetic mean of the price relatives (p1 ÷ p0) × 100 with weights p0q0. So a Laspeyres result far outside the range of the individual price relatives is wrong.
  7. In the last week, read the limitations and the cost of living index points once a day, and redo the questions you got wrong.

Common mistakes in Index Numbers

  • Mixing up Laspeyres and Paasche weights

    Fix: Remember that L is for base quantities (q0) and P is for current quantities (q1). Check the subscripts before you compute.

  • Forgetting to multiply by 100

    Fix: Read the options. If they look like 1.25 and not 125, you can spot the issue. Always finish with the ×100 step unless a test or product is asked.

  • Taking Fisher as the arithmetic mean of Laspeyres and Paasche

    Fix: Write Fisher = √(L × P) every time. It is the geometric mean, and it satisfies both the time reversal and factor reversal tests (but not the circular test).

  • Applying the wrong direction in base shifting or deflating

    Fix: Divide by the index of the new base year in base shifting. Divide by the price index in deflating. Then multiply by 100.

  • Errors in the table of products

    Fix: Compute only the columns your formula needs, add row by row, and re-add totals once. Use the options to catch large errors.

  • Treating limitations as unimportant theory

    Fix: Learn five or six limitations in your own words. Theory questions are fast marks that need no calculation.

Last-day revision: Index Numbers

  • An index number is a relative measure with the base period set at 100.
  • Simple aggregative price index = (Σp1 ÷ Σp0) × 100.
  • Price relative method: average the relatives (p1 ÷ p0) × 100 using AM or GM as asked.
  • Laspeyres price index uses base-year quantities: (Σp1q0 ÷ Σp0q0) × 100.
  • Paasche price index uses current-year quantities: (Σp1q1 ÷ Σp0q1) × 100.
  • Fisher index = √(Laspeyres × Paasche), the geometric mean of the two.
  • Time reversal test: P01 × P10 = 1. Fisher satisfies it. Laspeyres and Paasche do not.
  • Factor reversal test: P01 × Q01 = Σp1q1 ÷ Σp0q0, where Q01 is obtained by interchanging p and q in the same formula. Fisher satisfies it.
  • Fisher does not satisfy the circular test.
  • Base shifting: new index = (old index ÷ index of the new base year) × 100.
  • Real value = (money value ÷ price index) × 100. This is deflating.
  • Cost of living index weights prices by the share of each item in household spending.
  • Limitations include choice of base, items and weights, and the index becoming outdated over time.

Index Numbers practice questions

Index Numbers: frequently asked questions

Which formulas matter most in Index Numbers for CA Foundation?

Laspeyres, Paasche and Fisher price indices matter most, along with the two reversal tests. Base shifting and deflating are short and easy to score on. Learn the formulas with the meaning of each symbol.

Why is Fisher's index called the ideal index?

It is the geometric mean of Laspeyres and Paasche, and it satisfies both the time reversal and factor reversal tests. It also uses both base and current quantities, so it reduces the bias of either index alone.

How do I save time on Index Numbers MCQs?

Compute only the sums your formula needs and reuse them for Fisher. Estimate the answer first and eliminate options that are far off. If a question needs a long table and you are short on time, skip it and return later.

What is the difference between base shifting and splicing?

Base shifting changes the base period of one index series to a new year. Splicing joins two overlapping index series with different bases into one continuous series. Both use a ratio with the common year's index.