Q.Which of the following index number formulae satisfies BOTH the Time Reversal Test and the Factor Reversal Test?

Only one of the four listed formulae is symmetrically built from both the base-year and current-year weights, which is exactly the property both tests of adequacy require. …

Why (a) fails: Laspeyres' Index, $\dfrac{\Sigma p_1q_0}{\Sigma p_0q_0}\times100$, uses only base-year quantities. Reversing the years does not give its reciprocal — it produces Paasche's formula for the reversed years instead, so $P_{01}^L\times P_{10}^L\neq1$ in general.

Why (b) fails: Paasche's Index has the mirror-image problem — it uses only current-year quantities, so it is equally asymmetric and also fails the Time Reversal Test.

Why (d) fails: the Simple Aggregative Index ignores quantities/weights altogether, so it cannot even be tested meaningfully against the Factor Reversal Test (which requires a companion quantity index built the same way), and it fails the Time Reversal Test too, since $\frac{\Sigma p_1}{\Sigma p_0}\times\frac{\Sigma p_0}{\Sigma p_1}=1$ only when the same commodities and prices are used consistently, which coincidentally does hold for this specific formula — but it is never used as an argument for calling it "ideal", because it still fails the Factor Reversal Test. …

Students often pick Laspeyres because it is the formula introduced first, or because WPI in practice actually uses a Laspeyres-type formula — but being the most commonly USED formula …