Q.Verify that Fisher's Ideal Index Number satisfies the Time Reversal Test, using the data of Commodities A, B, C, D from the worked examples above ($p_0, p_1, q_0, q_1$ as given there).

The Time Reversal Test requires that swapping the base and current years and multiplying the two resulting index ratios together gives exactly 1.

Step 1 — Recall $P_{01}^{F}$ from the worked example above.

Step 2 — Compute $P_{10}^{F}$ by swapping every $p_0\leftrightarrow p_1$ and $q_0\leftrightarrow q_1$ together.

Step 3 — Multiply the two ratios (dual-check against the algebraic identity).

This also follows algebraically: $P_{01}^F\times P_{10}^F=\sqrt{\dfrac{\Sigma p_1q_0}{\Sigma p_0q_0}\cdot\dfrac{\Sigma p_1q_1}{\Sigma p_0q_1}\cdot\dfrac{\Sigma p_0q_1}{\Sigma p_1q_1}\cdot\dfrac{\Sigma p_0q_0}{\Sigma p_1q_0}}=\sqrt{1}=1$ exactly — the small 0.0000-level gap above is only rounding.

When reversing the years, EVERY $p_0$ must swap with $p_1$ AND every $q_0$ must swap with $q_1$ together — swapping only the prices while leaving the quantities unchanged is a common slip that makes the test appear to fail when the formula is actually fine.