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Hint: We will first consider the given values and consider the first set of values as the weight of iron rod is \[23.4\]kg and length as 13m and the second set of values as the weight be \[x\] kg and length as 6m. After this we will use the ratio and proportion method to find the value of \[x\] by writing them in the form of \[\dfrac{{{\text{weight}}}}{{{\text{length}}}}\] and put the values of both sets equal and simplifying the expression gives the desired result.

Consider the given values in the question,

The weight of the 13 m long rod is 23.4 Kg.

We need to find the value of the weight of 6 m long rod.

Let the weight of 6 m long iron rod will be $x$

Next, we will write the proportion by equating two ratios $23.4{\text{ kg}}$ to 13 m and $x$ to 6 m.

Thus, we get,

$ \Rightarrow \dfrac{{23.4{\text{ }}}}{{13{\text{ }}}} = \dfrac{x}{{6{\text{ }}}}$

Here, we will apply cross multiplication on the expression above to obtain the value of $x$.

We get,

$ \Rightarrow 23.4{\text{ }} \times 6{\text{ }} = x\left( {13{\text{ }}} \right)$

Next, we will divide both sides of the above equation by 13.

Thus, we get,

$

\Rightarrow \dfrac{{23.4{\text{ }} \times 6{\text{ }}}}{{13{\text{ }}}} = \dfrac{{x\left( {13{\text{ }}} \right)}}{{13{\text{ }}}} \\

\Rightarrow \dfrac{{23.4{\text{ }} \times 6}}{{13}} = x \\

$

Further, we will perform multiplication and division to solve for $x$.

$

\Rightarrow \dfrac{{{\text{140.4 kg}}}}{{13}} = x \\

\Rightarrow 10.8{\text{ kg}} = x \\

$

Hence, the weight of the rod having length 6 m is $10.8{\text{ kg}}$.

Therefore, option (C) is correct.

Note: Ratios of the proportion should be from weight to length, because we have to find the weight of the other rod. As we are given two values of one set and one value is given for the second set and need to find the other value, we must use the ratio and proportion method to determine the other value. Also the units of the data should be checked while solving the equation.

__Complete step by step solution:__Consider the given values in the question,

The weight of the 13 m long rod is 23.4 Kg.

We need to find the value of the weight of 6 m long rod.

Let the weight of 6 m long iron rod will be $x$

Next, we will write the proportion by equating two ratios $23.4{\text{ kg}}$ to 13 m and $x$ to 6 m.

Thus, we get,

$ \Rightarrow \dfrac{{23.4{\text{ }}}}{{13{\text{ }}}} = \dfrac{x}{{6{\text{ }}}}$

Here, we will apply cross multiplication on the expression above to obtain the value of $x$.

We get,

$ \Rightarrow 23.4{\text{ }} \times 6{\text{ }} = x\left( {13{\text{ }}} \right)$

Next, we will divide both sides of the above equation by 13.

Thus, we get,

$

\Rightarrow \dfrac{{23.4{\text{ }} \times 6{\text{ }}}}{{13{\text{ }}}} = \dfrac{{x\left( {13{\text{ }}} \right)}}{{13{\text{ }}}} \\

\Rightarrow \dfrac{{23.4{\text{ }} \times 6}}{{13}} = x \\

$

Further, we will perform multiplication and division to solve for $x$.

$

\Rightarrow \dfrac{{{\text{140.4 kg}}}}{{13}} = x \\

\Rightarrow 10.8{\text{ kg}} = x \\

$

Hence, the weight of the rod having length 6 m is $10.8{\text{ kg}}$.

Therefore, option (C) is correct.

Note: Ratios of the proportion should be from weight to length, because we have to find the weight of the other rod. As we are given two values of one set and one value is given for the second set and need to find the other value, we must use the ratio and proportion method to determine the other value. Also the units of the data should be checked while solving the equation.

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