2025 CSAT — Q75

The condition $X:Y = Y:Z$ means $Y^2 = XZ$ — the three scores form a geometric progression. Write $X = a^2 t,\ Y = abt,\ Z = b^2 t$ (so $Y^2 = XZ$). Then

Since $37$ is prime, $t = 1$ and $a^2 + ab + b^2 = 37$. The integer solution is $\{a,b\} = \{3,4\}$: $9 + 12 + 16 = 37$ ✓. This gives the GP scores $\{9, 12, 16\}$ — but in two valid orders, because $X:Y=Y:Z$ holds whether the ratio is $\tfrac34$ or $\tfrac43$:

• $(X,Y,Z) = (9, 12, 16)$: ratio $\tfrac{9}{12} = \tfrac{12}{16} = \tfrac34$ ✓ — gives I $<$ II $<$ III.
• $(X,Y,Z) = (16, 12, 9)$: ratio $\tfrac{16}{12} = \tfrac{12}{9} = \tfrac43$ ✓ — gives III $<$ II $<$ I.

Both satisfy every condition, so the ordering of $X, Y, Z$ is not fixed.

Answer: (d) Cannot be determined due to insufficient data.

The clean structural step is $X:Y=Y:Z \Rightarrow Y^2 = XZ$ (GP), reducing the sum to $a^2+ab+b^2 = 37$ with the unique unordered solution $\{9,12,16\}$. The trap — and the reason the answer is (d) — is that a GP condition is symmetric under reversal: increasing and decreasing GPs both satisfy it, so $X$ and $Z$ can swap. The exam reward is recognising that “ratio equals ratio” pins the set of scores but not their assignment to $X, Y, Z$, leaving two opposite orderings — hence insufficient data.