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We are given that from a rope 11m long, two pieces of lengths \[2\dfrac{3}{5}\]m and \[3\dfrac{3}{{10}}\]m are cut off. We need a length of rope left.

First, we need to find the lengths of pieces of rope in proper fraction.

This is done by converting mixed fraction into proper fraction as follows –

$a\dfrac{b}{c} = a + \dfrac{b}{c}$

$a + \dfrac{b}{c} = \dfrac{{ac + b}}{c}$……. (1)

Length of piece 1 = \[2\dfrac{3}{5}\]m

Using the property in equation (1) we get,

\[2\dfrac{3}{5} = \dfrac{{2 \times 5 + 3}}{5} = \dfrac{{13}}{5}\]

Thus, Length of piece 1 = \[\dfrac{{13}}{5}\]m…………….. (2)

Similarly, we have to calculate the length of piece 2 in proper fraction.

Length of piece 2 = \[3\dfrac{3}{{10}}\]m

\[3\dfrac{3}{{10}} = \dfrac{{10 \times 3 + 3}}{{10}} = \dfrac{{33}}{{10}}\]

Thus, Length of piece 2 = \[\dfrac{{33}}{{10}}\]m……….. (3)

Now, let the length of third piece left be X(m)………… (4)

Total length of rope = Length of rope 1 + Length of rope 2 + Length of rope 3

From equation (2), (3) and (4) we get the lengths of pieces –

Total length of rope = \[\dfrac{{13}}{5}\]+\[\dfrac{{33}}{{10}}\]+$X$

Total length of rope is 11m.

Therefore, we get the equation –

11=\[\dfrac{{13}}{5}\]+\[\dfrac{{33}}{{10}}\]+$X$

Subtracting both sides by \[\dfrac{{13}}{5}\]and \[\dfrac{{33}}{{10}}\]we get,

\[X = 11 - \dfrac{{13}}{5} - \dfrac{{33}}{{10}}\]

Taking L.C.M. of 5 and 10 we get 10. Therefore,

\[X = \dfrac{{11(10) - 13(2) - 33}}{{10}}\]

\[X = \dfrac{{110 - 26 - 33}}{{10}}\]

Further simplifying we get,

\[X = \dfrac{{51}}{{10}}\]

Thus the length of the third piece will be \[\dfrac{{51}}{{10}}\]m.