Quantitative Aptitude · Index Numbers

Tests of Adequacy: Time Reversal and Factor Reversal Tests

Updated 1 October 2026

Tests of adequacy check whether an index number formula behaves logically. The time reversal test needs P01 × P10 = 1. The factor reversal test needs P01 × Q01 = Σp1q1 ÷ Σp0q0. The circular test needs P01 × P12 × P20 = 1. Fisher's index passes the time reversal and factor reversal tests but not the circular test.

Understand Tests of Adequacy: Time and Factor Reversal Tests

An index number can be built in many ways. Tests of adequacy are logical checks that tell you whether a formula is consistent. No formula passes every test, so these tests are used to compare formulas.

The unit test says the index must not change if you change the units of measurement, such as kg to quintal. The simple aggregative index (Σp1 ÷ Σp0 × 100) fails this test because it adds prices quoted in different units. The simple average of price relatives and the weighted formulas such as Laspeyres, Paasche and Fisher pass it.

The time reversal test says that if you swap the base year and the current year, the new index should be the reciprocal of the old one. In symbols, P01 × P10 = 1. Think of it as going from 2020 to 2024 and back again. You should end where you started.

The factor reversal test says that if you swap the roles of price and quantity in the formula, the two indices multiplied together should give the true value ratio, Σp1q1 ÷ Σp0q0. So P01 × Q01 = V01.

The circular test extends time reversal to three or more periods. It needs P01 × P12 × P20 = 1. It is useful for shifting the base without recalculating. The simple aggregative index and the fixed-weight aggregative index pass it. In a fixed-weight index, the same weights are used for every year. This is not the same as Laspeyres, which uses base-year weights, and Paasche, which uses current-year weights. Laspeyres, Paasche and Fisher all fail the circular test.

Key formulas to remember

Unit test: Index is unchanged when units of price or quantity change; Passed by Laspeyres, Paasche, Fisher, Marshall-Edgeworth and the simple average of price relatives. Failed by the simple aggregative index.
Time reversal test: P01 × P10 = 1; P10 is found by swapping the subscripts 0 and 1 in the same formula. With indices on base 100, the product is 10,000.
Factor reversal test: P01 × Q01 = Σp1q1 ÷ Σp0q0; Q01 is found by swapping p and q in the price formula. Here P01 and Q01 are in ratio form, not multiplied by 100.
Circular test: P01 × P12 × P20 = 1; Extends time reversal to three periods. Passed by the simple aggregative index and the fixed-weight aggregative index (same weights for every year). Failed by Laspeyres, Paasche and Fisher.
Fisher price index: P01 = √[(Σp1q0 ÷ Σp0q0) × (Σp1q1 ÷ Σp0q1)] × 100; Geometric mean of Laspeyres and Paasche. Passes unit, time reversal and factor reversal tests. Fails the circular test.
Fisher quantity index: Q01 = √[(Σq1p0 ÷ Σq0p0) × (Σq1p1 ÷ Σq0p1)] × 100; Used in the factor reversal test.
Which formula passes which test: Laspeyres and Paasche: unit only. Marshall-Edgeworth: unit and time reversal. Fisher: unit, time reversal and factor reversal.; Fisher is called the ideal index for this reason. Learn this summary for direct MCQs.

How to solve Tests of Adequacy: Time and Factor Reversal Tests questions

Use this method when a question asks you to check a test or to use a test's result to find a missing value.

1Identify which test is asked: unit, time reversal, factor reversal or circular.
2Write the formula for the forward index P01 using the given formula (Laspeyres, Paasche, Fisher and so on).
3For time reversal, swap the subscripts 0 and 1 everywhere to get P10. For factor reversal, swap p and q to get Q01.
4Multiply the two indices. For time reversal this should equal 1. For factor reversal it should equal Σp1q1 ÷ Σp0q0.
5For the circular test, multiply P01, P12 and P20 and check that the result is 1.
6If the product matches, the formula satisfies the test. If it does not match in general, it fails. State this clearly.
7If the question gives numbers instead, use the rule directly. For example, P10 = 1 ÷ P01 or Q01 = V01 ÷ P01. Keep ratio form or base-100 form consistently.

Quickest way: Memorise the pass-fail table and use the reciprocal rule

When to use it: Use this in MCQs, where you rarely need to prove anything. Most questions are either a theory recall or a one-line calculation.

Remember: Fisher passes time reversal and factor reversal. Laspeyres and Paasche pass neither. Marshall-Edgeworth passes time reversal only.
Remember: Fisher, Laspeyres and Paasche fail the circular test. The simple aggregative index and the fixed-weight aggregative index (same weights every year) pass it.
If the question says a formula satisfies time reversal, then P10 = 1 ÷ P01. On base 100, P10 = 10,000 ÷ P01.
If the question says a formula satisfies factor reversal, then Q01 = (value ratio) ÷ P01. Divide the value ratio by the price ratio.
Check the options before calculating. If one option is a 'does not satisfy any test' type, it is rarely correct for Fisher.

Common mistakes in Tests of Adequacy: Time and Factor Reversal Tests

Saying Fisher's index satisfies the circular test.
Students know Fisher is 'ideal' and assume it passes everything.
Fix: Fisher passes unit, time reversal and factor reversal only. It fails the circular test.
Writing the time reversal condition as P01 + P10 = 1 or P01 = P10.
The product and sum are confused, or the idea of reversing is misread.
Fix: It is a product: P01 × P10 = 1. The two indices are reciprocals.
Forgetting to divide by 100 when indices are on base 100.
The tests are written in ratio form, but the data are given as 125 or 80.
Fix: Convert to ratios first (125 becomes 1.25), or use 10,000 as the product for time reversal.
Swapping only one subscript when forming P10.
Students rush and swap p only or q only.
Fix: Swap 0 and 1 in every term, in both numerator and denominator. Laspeyres P01 = Σp1q0 ÷ Σp0q0 becomes P10 = Σp0q1 ÷ Σp1q1.
Using the factor reversal test with Σp1q1 ÷ Σp0q0 multiplied by 100 on one side only.
The price and quantity indices are written with ×100 but the value ratio is not.
Fix: Keep everything in ratio form, or compare P × Q ÷ 100 with the value ratio × 100.
Believing the simple aggregative index passes the unit test.
It looks simple and so seems safe.
Fix: It adds prices in different units, so changing units changes the index. It fails the unit test.

Worked examples

Example 1

Which of the following index numbers satisfies both the time reversal test and the factor reversal test? (A) Laspeyres' index (B) Paasche's index (C) Fisher's ideal index (D) Marshall-Edgeworth index

Show the solution

Laspeyres and Paasche satisfy neither test in general, so A and B are out.
Marshall-Edgeworth satisfies the time reversal test but not the factor reversal test, so D is out.
Fisher's index satisfies both. Its P01 × P10 equals 1, and its P01 × Q01 equals Σp1q1 ÷ Σp0q0.

Answer: (C) Fisher's ideal index

Example 2

A price index number on base 100 for 2024 with respect to 2019 is 125 using a formula that satisfies the time reversal test. What is the index number for 2019 with respect to 2024? (A) 75 (B) 80 (C) 100 (D) 125

Show the solution

Time reversal gives P01 × P10 = 1 in ratio form.
P01 = 125 ÷ 100 = 1.25.
P10 = 1 ÷ 1.25 = 0.80.
On base 100, this is 80.

Answer: (B) 80

Example 3

The total value of goods bought rose from ₹40,000 in the base year to ₹60,000 in the current year. Fisher's price index is 120. What is Fisher's quantity index? (A) 90 (B) 125 (C) 130 (D) 150

Show the solution

Value ratio V01 = 60,000 ÷ 40,000 = 1.5.
Fisher's index satisfies the factor reversal test, so P01 × Q01 = V01.
P01 = 120 ÷ 100 = 1.2.
Q01 = 1.5 ÷ 1.2 = 1.25.
On base 100, Q01 = 125.

Answer: (B) 125

Exam tips

Most questions are direct recall. Learn the table of which formula passes which test, especially Fisher's.
For numerical questions, convert indices to ratios first. Convert back at the end.
Watch for the word 'not'. Questions often ask which formula does not satisfy a test.
Do not spend time proving a test algebraically in an MCQ. Use the known result and move on.
The circular test is often a trap option. Fisher fails it, so do not tick it for Fisher.

Practice questions from Index Numbers

Tests of Adequacy: Time and Factor Reversal Tests: frequently asked questions

What is the time reversal test in index numbers?

It checks that swapping the base year and the current year gives the reciprocal index. The condition is P01 × P10 = 1. Fisher's and Marshall-Edgeworth indices satisfy it.

Does Fisher's index satisfy the circular test?

No. Fisher's index satisfies the unit, time reversal and factor reversal tests, but it fails the circular test. This is a frequent exam question.

How do I check the factor reversal test?

Find the price index P01, then find the quantity index Q01 by swapping p and q in the formula. Multiply them and compare with Σp1q1 ÷ Σp0q0. If they are equal, the test is satisfied.

Which index numbers satisfy the circular test?

The simple aggregative index, the fixed-weight aggregative index and the simple geometric mean of price relatives satisfy it. Fixed-weight means the same weights are used for every year. This is different from Laspeyres, which uses base-year weights. Laspeyres, Paasche and Fisher do not satisfy it.