Arnav ordered $4$ pairs of black socks and some additional pairs of blue socks. The price of the black socks per pair was twice that of the blue socks. When the order was filled, it was found that the number of pairs of the two colours had been interchanged. This increased the bill by $50\%$. The ratio of the number of pairs of black socks to the number of pairs of blue socks in the original order was

### Concept & Strategy This problem is based on setting up a linear equation from word problems involving price and quantity. When quantities are interchanged, you formulate a new total cost and compare it to the original cost using the given percentage increase. ### Step-by-Step Solution * **Given:** * Original order: $4$ pairs of black socks, $x$ pairs of blue socks. * Price of black socks = $2P$, Price of blue socks = $P$. * New order (interchanged): $x$ pairs of black socks, $4$ pairs of blue socks. * New bill is $50\%$ more than the original bill. * **Calculation / Deduction:** * Calculate the original bill: $4 \times (2P) + x \times (P) = 8P + xP = P(8 + x)$. * Calculate the new bill: $x \times (2P) + 4 \times (P) = 2xP + 4P = P(2x + 4)$. * The new bill is $50\%$ higher, which means New Bill = $1.5 \times$ Original Bill. * $P(2x + 4) = 1.5 \times P(8 + x)$ * Divide both sides by $P$ to simplify: $2x + 4 = 1.5(8 + x)$ * Expand the right side: $2x + 4 = 12 + 1.5x$ * Subtract $1.5x$ from both sides: $0.5x = 8$ * $x = 16$. So, Arnav originally ordered $16$ pairs of blue socks. * We need the original ratio of black to blue socks: $4 : 16$. * Simplify the ratio: $1 : 4$. ### Exam Strategy & Shortcut Since options are provided, you can use **Option Elimination** to bypass the algebra entirely. * Try Option (a) $1 : 2$. Black=$4$, so Blue=$8$. Original cost = $4(2) + 8(1) = 16$. Swapped cost = $8(2) + 4(1) = 20$. Increase is $4/16 = 25\%$. Incorrect. * Try Option (b) $1 : 4$. Black=$4$, so Blue=$16$. Original cost = $4(2) + 16(1) = 24$. Swapped cost = $16(2) + 4(1) = 36$. Increase is $12/24 = 50\%$. Correct! ### Common Pitfall A frequent error is misreading which variable represents which item. Finding $x = 16$ and immediately picking an option like $4 : 1$ (if mistakenly reversing the final ratio requirement of black to blue) is a common trap. Always double-check the required order of the ratio in the last sentence. ### Final Answer **Therefore, the correct answer is 1 : 4.**