Q.For the same four-commodity data (Examples 5–8), verify the Factor Reversal Test for Fisher's Ideal Index — that is, check whether the Fisher price index multiplied by the corresponding Fisher quantity index equals the value index, V01 = (sum p1q1 / sum p0q0) x 100.

The Factor Reversal Test checks whether a price index times the matching quantity index reproduces the value index exactly — Fisher's Ideal Index is built to pass this check.

Step 1 — Fisher's Quantity Index (same formula as Fisher's price index, with p and q swapped).

Using the already-computed totals: $\sum q_0p_0=870$, $\sum q_1p_0=799$ (Example 6's denominator, re-read as a quantity aggregate), $\sum q_0p_1=1110$ (Example 5's numerator, re-read as a quantity aggregate), $\sum q_1p_1=1018$.

Step 2 — Value Index.

Step 3 — Check Factor Reversal: $F\times Q_{01}$ (as pure ratios, dividing by 100 twice, then re-expressing as a percentage). …

This test IS itself the cross-check. Arriving at the identical 117.01 both by the direct value-index formula (Step 2) and by multiplying the two independently-computed Fisher price and quantity indices (Step 3) confirms Fisher's Ideal I …

Multiplying the price and quantity indices as their PERCENTAGE values directly (127.50 × 91.77 = 11700.8) without dividing each by 100 first — the Factor Reversal Test is defined on the ratios (127. …