A man ordered a length of rope from a hardware shop over the telephone. However, the order was miswritten by interchanging feet and inches. As a result, the length of rope received was only 30% of the length he had ordered. If the length of the rope ordered is between 6 feet and 12 feet, determine the actual length of the rope ordered.

Question

Accepted Answer

Let the ordered length be $$ L = 12x + y $$, where $$ x $$ represents feet and $$ y $$ represents inches (both less than 12). The miswritten length is $$ L' = 12y + x $$. According to the problem:  
$$
L' = 0.3L
$$

Substituting the expressions for $$ L $$ and $$ L' $$:  
$$
12y + x = 0.3(12x + y)
$$

Expanding and simplifying:  
$$
12y + x = 3.6x + 0.3y
$$  
$$
12y - 0.3y = 3.6x - x
$$  
$$
11.7y = 2.6x
$$

Dividing both sides by $$ 2.6 $$:  
$$
\frac{y}{x} = \frac{2.6}{11.7} = \frac{2}{9}
$$

From this ratio, express $$ x $$ in terms of $$ y $$:  
$$
x = \frac{9}{2}y
$$

Since both $$ x $$ and $$ y $$ must be integers, let $$ y = 2 $$:  
$$
x = \frac{9}{2} 	imes 2 = 9
$$

The ordered length is:  
$$
L = 12x + y = 12(9) + 2 = 108 + 2 = 110 	ext{ inches}
$$

Convert inches to feet:  
$$
110 	ext{ inches} = \frac{110}{12} = 9 	ext{ feet } + 2 	ext{ inches}
$$

Thus, the ordered length falls within the range **9 feet to 10½ feet**, corresponding to **Option C**.

Answer

Answer

Answer