Argthtjtr

The clerk misunderstood the order for rope. He reversed feet and inches, and the customer got only 30% of what he ordered.

How much rope had really been ordered?

Assume the order was for $x$ feet and $y$ inches, a total of $12x+y$ inches, and the customer got $y$ feet and $x$ inches, a total of $12y+x$ inches. We know that $12x+y=frac{10}{3}(12y+x)$, so that $36x+3y=120y+10x$, or $26x=117y$. As $gcd(26,117)=13$, we have $2x=9y$. So for any positive integer $k$, there is a solution $x=9k,y=2k$. As @Jaap points out, $x,ylt12$ can be assumed, therefore $k=1$, and so $x=9, y=2$.

With some trial and error:

Seems he ordered:

9 feet and 2 inches and received only 2 feet and 9 inches

First, we know that a foot is 12 inches.

Let the length of the rope be $x$ feet and $y$ inches.

In other words, the rope is $12x+y$ inches long.
Since the clerk misunderstood, the rope he got was $y$ feet and $x$ inches long, which can also be expressed as $12y+x$ inches.

With some trial and error, it is not hard to find that he ordered 9 feet and 2 inches of rope at first but received 2 feet and 9 inches of rope.

$0.3 times (12x + y)$ is what he received, with the ordered length being $xtext{'}ytext{"}$, $12y + x$ is what the clerk understood. $0 leq x < 12$ and $0 leq y < 12$

$0.3 times (12x + y) = 12y + x Rightarrow 3.6x + 0.3y = 12y + x Rightarrow 2.6x = 11.7y$
$y$ can further be limited because $frac{11.7}{2.6}y = 4.5y < 12$, leaving us with $0 leq y < 3$. $0$ and $1$ clearly don't work, which leaves us with $2$.

Set $y = 2 Rightarrow 2.6x = 23.4 Rightarrow x = 9$

$0.3 times (9text{'}2text{"}) = 33text{"} = 2text{'}9text{"}$

If you do not place a restriction on inches, there are an infinite number of answers. If you place the restriction that it must be less than 12, but can be any real number > 0, I believe there are still an infinite number of solutions. If it must be an integer, then I would think there is only a finite number of solution in this case.

Either way, below outlines my process for deriving this solution:

12i+f = 0.3(12f+i) -> Left side is swapped inches and feet, right side is 30% of expected amount of rope

12i+f = 0.3i+3.6f -> Reduction

11.7i = 2.6f -> Reduction

f=4.5i => i = 2/9f -> This is a relationship between the value for inches and feet that will satisfy the requirements of this problem.

Arbitrarily pick a value for f => 27

i = 2/9*27 = 6

Verify solution: 12*6+27 = 0.3(12*27+6) -> True


This will be valid for any pick for f or i.