A proportionality puzzle: If half of $5$ is $3$, then what's one-third of $10$? My professor gave us this problem.

In a foreign country, half of 5 is 3. Based on that same proportion, what's one-third of 10?

I removed my try because it's wrong.
 A: As $5/2=3;$
it implies, $5/3=2$;
and $2*5/3=2*2=4;$
hence $10/3=4;$
A: I imagine this to be a problem caused by this foreign country not having the concept of the number zero.
If you think about it as a number line, without  a 0:

If you were to divide this line into two equal parts, you would draw a line through the tick that corresponds to the number 3. Therefore, you could say that "half of 5 is 3"
The same goes for a number line that includes 1 through 10. If you wanted to divide that line into 3 equal parts, you would draw lines through the ticks that correspond to the number 4 and the number 7. Therefore, you could say that "One third of 10 is 4" and "two thirds of 10 is 7" which seems internally consistent because you could also claim that "one half of 7 is 4."
Of course, this makes no sense and only shows up because this country apparently doesn't consider any numbers less than 1.
A: I think your teacher wanted the following solution in which we only use the given relation that $\frac{5}{2}=3$:
$$
\frac{10}{3} = \frac{4}{3}\frac{5}{2} = 3\frac{4}{3} = 4.
$$
A: From a false assumption you can derive anything. Answer what you want: it will be correct.
For example: the answer is $\pi^2$, and I'll prove it. Suppose not. Then, by hypothesis, $5/2=3$, so $5=6$ and, substracting $5$ to each side of equation, $0=1$, a contradiction. So the answer is $\pi^2$.
A: I think this is more a question of language than of mathematics. (Indicated also by the fact that a "foreign country" is mentioned.)
A possible understanding of "half" in this case would be that "half" is an operation that assigns integers to integers by splitting them in to parts as evenly as possible and then taking the largest part. In other words, by "half" of $x$ could mean the smallest integer that is not less then half (with its usual meaning) of $x$, which we usually denote $\lceil\frac{x}{2}\rceil$.
Based on this same understanding, a "third" of $10$ would mean $\lceil\frac{10}{3}\rceil$, which is $4$.
But the result you will get in the end will ultimately depend on the way of thinking in that country.
A: If $\frac12\times 5=3$ tnen taking reciprocals gives $2\times\frac15=\frac13$. Then multiplying by $10$ gives
$$
4=2\times\frac15\times10=\frac13\times10
$$
Of course, assuming falsehood, one can prove anything.
A: The $5_a$ must be interpreted as being half of $10_a$. So $\dfrac{5_a}2=3$ is equivalent to saying $10_a=12$, a third of which is obviously $4$. Imagine for instance counting from $1$ to $5$ on fingers, and using a clenched fist to represent the $6$. One could then easily count in duodecimal on two hands.
A: Although I agree with user2425, if there definitely is an answer then:
$\dfrac{1}{2}5=3 \implies 5=6 \implies 10=12 \implies \dfrac{1}{3}10=4$
A: Given that
$$
\begin{equation}
\tag{1}
\text{half of }5 = 3
\end{equation}
$$
this implies that 
$$
\begin{equation}
\tag{2}
\text{half of }10 = 6
\end{equation}
$$
$(2)$ then says that half of $1 = 0.6$ and therefore 
$$
\begin{equation}
\tag{3}
\frac{1}{3}\text{ of }1 = \frac{0.33 \times 0.6}{0.5} = 0.396
\end{equation}
$$
There for since $\frac{1}{3}$ of $1$ corresponds to $.396$ thus $\frac{1}{3}$ of $10 = 3.96$
Answer: $3.96$
A: Since 5/2 is 2.5, the convention clearly is to round up. Thus, 10/3, being 3.33..., rounds to 4.
A: They say $\frac{5}{2}$ is $3$, when really it's $2.5$.
They say $\frac{10}{3}$ is $x$, when really it's $3.\overline{3}$.
Simple relational proportion:
$$ \frac{3}{2.5} = \frac{x}{3.\overline{3}} \implies x = 4 $$
A: The information your professor has given is incomplete. We cannot infer anything from the truth of the statement: "In a foreign country, half of 5 = 3."
In that country there would be different laws of nature. In current Math we have the fundamental principle of counting.
Acc. to fundamental principle of counting "The number of things in a group remains always same", that is if the number of balls in a bag are 5 then it will remain 5.
The commutative law, distributive law and associative laws in math are a consequence of fundamental principle of counting. If the fundamental principle of counting is inconsistent in that country then commutative law, distributive law and associative laws will also not hold in that country.
Truth of $\dfrac{5}{2}=3$  will cause a inconsistent universe. In that universe(country) a length of 2.5 units will be changable to a length of 3 units and vice-versa. We do not know whether the commutative,distributive and assoiative laws are true or not in that country.
Since your teacher does not mention what are the law's and axioms of Math in that country we cannot infer anything.
We cannot say that $\dfrac{5}{2}=3 \implies 5=6$ because $\dfrac{5}{2}$ is a fraction that may be used to represent a certain length, we cannot apply the rules of multiplication to $\dfrac{5}{2}$ in that country. 
A: Actually, the riddles is "3-quely" solvable if exactly one of the names "5", "2", or "halves" doesn't own their own meaning but the meaning of $\square$ instead.  So there are three cases.
1) $$\frac{\square}{2}=3\Rightarrow \frac{2\square}{3}=4.$$
2) $$\frac{5}{2}=\square \Rightarrow \frac{10}{\square}=4.$$
3) $$\frac{5}{\square}=3\Rightarrow\square=\frac{5}{3}.$$
A: Half of 5 is 3, ok so half of 10 would be 6?
Leaving 4 for the other half?
So 1/2 + 1/2 + 4 = 10?
1/3.   1/3.   1/3.
Answer is 4....
Let's say 10 = 10 pennies and to split them 3 ways
3 pennies for each with 1 left over, you cant split a penny unless you cut it. But you can split a pizza into 3 equal parts.
