The contrapositive of the statement 'If two numbers are not equal, then their squares are not equal', is

Let the two numbers be represented by the real variables $$a$$ and $$b$$.

Define the following two simple statements:

$$P : a \ne b \quad\text{(the numbers are not equal)}$$

$$Q : a^{2} \ne b^{2} \quad\text{(the squares are not equal)}$$

The sentence given in the question, “If two numbers are not equal, then their squares are not equal”, is therefore written symbolically as

$$P \rightarrow Q.$$

To form the contrapositive, recall the logical rule that the contrapositive of any implication $$P \rightarrow Q$$ is $$\lnot Q \rightarrow \lnot P.$$ Here $$\lnot$$ means “not”. We now compute each negation explicitly.

Starting with $$Q$$:

$$Q : a^{2} \ne b^{2}$$

$$\therefore \;\lnot Q : a^{2} = b^{2}$$

Next, deal with $$P$$:

$$P : a \ne b$$

$$\therefore \;\lnot P : a = b$$

Substituting these negations into $$\lnot Q \rightarrow \lnot P$$ gives

$$a^{2} = b^{2} \rightarrow a = b.$$

Expressed verbally, this reads:

“If the squares of two numbers are equal, then the numbers are equal.”

Comparing with the options provided, this statement matches Option D.

Hence, the correct answer is Option D.