A 27 quartz capacity car radiator is filled with 18% alcohol solution. How many quartz be drained and then be replaced by a 90% alcohol solution for resulting solution to contain 42% alcohol?

### Concept & Strategy This is a **Replacement Problem** that can be brilliantly solved using Alligation. When a volume is drained, the remaining liquid maintains the original $18\%$ concentration. We are essentially mixing this remaining $18\%$ base solution with a purely added $90\%$ solution to reach a final $42\%$ mixture. ### Step-by-Step Solution * **Given:** * Total capacity = $27$ quartz * Concentration of leftover base solution = $18\%$ * Concentration of added replacement solution = $90\%$ * Target final concentration = $42\%$ * **Calculation / Deduction:** * Apply the Alligation rule to the two components forming the final mixture: the leftover $18\%$ solution and the added $90\%$ solution. * Place $18\%$ on the left and $90\%$ on the right. Target $42\%$ in the middle. * Diagonal difference from $90\%$ to $42\%$ is $48$. (This corresponds to the parts of the $18\%$ leftover solution). * Diagonal difference from $18\%$ to $42\%$ is $24$. (This corresponds to the parts of the added $90\%$ solution). * The ratio of (Leftover Base) : (Added Solution) is $48 : 24$, which simplifies to $2 : 1$. * This means the final radiator mixture is made of $2$ parts leftover solution and $1$ part added solution. * Total parts $= 2 + 1 = 3$ parts. * These $3$ parts constitute the full $27$ quartz capacity. * $3$ parts $= 27$ quartz $\implies 1$ part $= 9$ quartz. * The amount replaced is the added solution, which is $1$ part. * Therefore, $9$ quartz were drained and replaced. ### Exam Strategy & Shortcut **Equation Method (Alternative execution if you forget Alligation):** Let $x$ be the replaced volume. Total pure alcohol = Alcohol in remaining part + Alcohol in new part. $0.18(27 - x) + 0.90(x) = 0.42(27)$ Instead of fully multiplying, divide the whole equation by the common factor $0.06$: $3(27 - x) + 15x = 7(27)$ $81 - 3x + 15x = 189$ $12x = 108 \implies x = 9$. This is efficient, but the Alligation ratio ($2:1$) allows you to solve it visually without paper. ### Common Pitfall The most dangerous pitfall in replacement alligation is misunderstanding what the final parts represent. The ratio ($2:1$) gives the fractions of the *final total mixture*. The amount replaced is *only* the added portion ($1$ part), while students often mistakenly identify the leftover portion ($2$ parts = $18$ quartz) as the drained amount. Always double-check which side of the ratio you are using. ### Final Answer **Therefore, the correct answer is 9 quartz.**