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We are given the total length of rope which is $7\dfrac{2}{3}$ metre.

The cost of $7\dfrac{2}{3}$m rope is also provided in the given question which is Rs. $12\dfrac{3}{4}$.

We need to find the cost of rope per metre.

Now, we will convert the mixed fractions into a simpler fraction by the formula $a\dfrac{b}{c} = \dfrac{{(a \times c) + b}}{c}$.

The total length of the rope is $7\dfrac{2}{3}$m which can be written as $7\dfrac{2}{3}m = \dfrac{{(7 \times 3) + 2}}{3}m = \dfrac{{23}}{3}m$.

The total cost of the rope is Rs. $12\dfrac{3}{4}$ which can be written as $Rs.12\dfrac{3}{4} = Rs.\dfrac{{\left( {12 \times 4} \right) + 3}}{4} = Rs.\dfrac{{51}}{4}$.

We know that the total cost of a rope is given by the product of the cost of 1 m of rope and the total length of the rope i. e., the total cost of the rope = (cost per metre $ \times $total length) of the rope.

Now, we will put the values in the formula mentioned above. On substituting the values, we get

$\therefore $$\dfrac{{51}}{4}$= cost per metre of the rope $ \times $ $\dfrac{{23}}{3}$

$ \Rightarrow $cost per metre = $\dfrac{{51}}{4} \times \dfrac{3}{{23}} = 1\dfrac{{61}}{{92}} = 1.66$