A man ordered a length of rope by telephone from his nearest hardware shop. But when a worker in the shop brought the rope, he found that the man on the telephone had miswritten the order by interchanging feet and inches. As a result of this, the length of rope received was only $30\%$ of the length he had ordered. The length of the rope which the man ordered was between

### Concept & Logic The problem relies on forming a linear equation by establishing a relationship between standard units of measurement. The key is converting everything into the smaller unit (inches) to maintain consistency. $$1 \text{ foot} = 12 \text{ inches}$$ ### Step-by-Step Solution * **Given:** * Let the ordered length be $x$ feet and $y$ inches. * Because it's a standard measurement, $y$ must be less than $12$. * Total ordered length in inches = $12x + y$. * The received length was interchanged: $y$ feet and $x$ inches. * Total received length in inches = $12y + x$. * Received length is $30\%$ of the ordered length. * **Calculation / Deduction:** * Set up the equation based on the $30\%$ condition: * $12y + x = \frac{30}{100}(12x + y)$ * $12y + x = \frac{3}{10}(12x + y)$ * Multiply both sides by $10$ to clear the fraction: * $10(12y + x) = 3(12x + y)$ * $120y + 10x = 36x + 3y$ * Rearrange terms to group variables: * $120y - 3y = 36x - 10x$ * $117y = 26x$ * Simplify the ratio by dividing by $13$: * $9y = 2x$ * $x = 4.5y$ * Since $x$ and $y$ must be whole numbers (integers) in standard notation, and $y$ must be an inch value less than $12$, let's test possible values for $y$. * If $y = 2$, then $x = 4.5(2) = 9$. * This gives an ordered length of $9$ feet and $2$ inches. * Let's verify: $9$ ft $2$ in is $110$ inches. Swapped, it is $2$ ft $9$ in, which is $33$ inches. $33$ is exactly $30\%$ of $110$. * The ordered length is $9$ feet $2$ inches, which falls between $9$ ft and $10 \frac{1}{2}$ ft. ### Exam Strategy & Shortcut Instead of heavy algebra, test the constraints. The ratio reduces to $x = 4.5y$. For $x$ to be an integer, $y$ must be an even number ($2, 4, 6, 8, 10$). If $y=4$, $x=18$, meaning $18$ feet $4$ inches. This is not in the options. Thus, $y$ must be $2$, making $x=9$. ### Common Pitfall A common mistake is failing to apply the real-world constraint that the "inches" portion ($y$) of a standard measurement must be less than $12$. Without this logical boundary, the equation has infinite solutions and students get stuck. ### Final Answer **Therefore, the correct answer is 9 ft and 10 1/2 ft.**