CAT 2024 Slot 3 · Quant · Question 8

Let the first increment be r as a decimal, so the second is 2r.

A rise of r multiplies the salary by \left( 1+r \right), and the second rise multiplies it by \left( 1+2r \right).

Ending at 187.5% of the start means \left( 1+r \right)\left( 1+2r \right)=1.875.

Expanding the left side, 1+3r+2{r}^{2}=1.875.

So 2{r}^{2}+3r-0.875=0.

Multiplying through by 8 to clear decimals, 16{r}^{2}+24r-7=0.

The discriminant is {24}^{2}+4\times 16\times 7=576+448=1024, whose square root is 32.

So r=\frac{-24\pm 32}{32}, giving r=\frac{8}{32}=0.25 or r=-\frac{56}{32}.

A negative increment makes no sense here, so r=0.25, that is 25% — the third option.

Shortcut. Note that 1.875=\frac{15}{8}, and \frac{15}{8}=\frac{5}{4}\times \frac{3}{2}. Those two factors are exactly a 25% rise followed by a 50% rise, and 50 is twice 25, so the condition is met — no quadratic at all.