An ore contains calcium hydroxide $\text{Ca}{{\left( \text{OH} \right)}_{\text{2}}}$ in association with calcium phosphate $\text{C}{{\text{a}}_{\text{3}}}{{\left( \text{P}{{\text{O}}_{\text{4}}} \right)}_{\text{2}}}$ . The analysis shows that calcium and phosphorus are present with a mole ratio of 5:3, which of the following best represents the composition of the ore?

A. $\text{Ca}{{\left( \text{OH} \right)}_{\text{2}}}\cdot \text{C}{{\text{a}}_{\text{3}}}{{\left( \text{P}{{\text{O}}_{\text{4}}} \right)}_{\text{2}}}$

B. $\text{Ca}{{\left( \text{OH} \right)}_{\text{2}}}\cdot \text{2}{{\left( \text{CO} \right)}_{\text{3}}}{{\left( \text{P}{{\text{O}}_{\text{4}}} \right)}_{\text{2}}}$

C. $\text{Ca}{{\left( \text{OH} \right)}_{\text{2}}}\cdot \text{3C}{{\text{a}}_{\text{3}}}{{\left( \text{P}{{\text{O}}_{\text{4}}} \right)}_{\text{2}}}$

D. $\text{Ca}{{\left( \text{OH} \right)}_{\text{3}}}\cdot \text{4C}{{\text{a}}_{\text{3}}}{{\left( \text{P}{{\text{O}}_{\text{4}}} \right)}_{\text{2}}}$

Hint: The process of representing an ore should be such that a portion is selected who composition will represent the average composition of the entire bulk. Keeping this in mind, we need to analyse the composition of the ore.

Complete step-by-step answer:
To know which of the following options show the correct representation of the ore, we need to analyse each of them.

Let us begin with $\text{Ca}{{\left( \text{OH} \right)}_{\text{2}}}\cdot \text{C}{{\text{a}}_{\text{3}}}{{\left( \text{P}{{\text{O}}_{\text{4}}} \right)}_{\text{2}}}$.
As we can see that in this case,
Number of moles of Ca = 4
Number of moles of P = 2

So the ratio of calcium: phosphorous is 4:2
This is not the correct answer.

Let us now check for $\text{Ca}{{\left( \text{OH} \right)}_{\text{2}}}\cdot \text{2}{{\left( \text{CO} \right)}_{\text{3}}}{{\left( \text{P}{{\text{O}}_{\text{4}}} \right)}_{\text{2}}}$ .
As we can see that in this case,
Number of moles of Ca = 7
Number of moles of P = 4
So the ratio of calcium: phosphorous is 7:4
This is not the correct answer.

Let us now check for $\text{Ca}{{\left( \text{OH} \right)}_{\text{2}}}\cdot \text{3C}{{\text{a}}_{\text{3}}}{{\left( \text{P}{{\text{O}}_{\text{4}}} \right)}_{\text{2}}}$

As we can see that in this case,
Number of moles of Ca = 5
Number of moles of P = 3
So the ratio of calcium: phosphorous is 5:3
This is the correct answer.

Let us check now for $\text{Ca}{{\left( \text{OH} \right)}_{\text{3}}}\cdot \text{4C}{{\text{a}}_{\text{3}}}{{\left( \text{P}{{\text{O}}_{\text{4}}} \right)}_{\text{2}}}$
As we can see that in this case,
Number of moles of Ca =13
Number of moles of P = 8
So the ratio of calcium: phosphorous is 13:8
This is not the correct answer.

So the correct answer is Option C.

Note: We need to calculate the number of molecules of each element present in a composition, to determine the mole ratio. Mole ratios are used as conversion factors between products and reactants in many chemistry problems.