# 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.

• The information your professor has given is incomplete. – user103816 Apr 5 '14 at 10:25
• This is apparently a riddle originated from Niccolò Fontana Tartaglia (1499-1557) and the answer is 4. Don't ask me why ;-p – achille hui Apr 5 '14 at 10:29
• You should highlight the phrase "Based on the same proportion", it is the key to properly answer this question. – achille hui Apr 5 '14 at 11:09
• But then, if in that country, $5/2=3$, then it seems that at least one of the symbols $5$, $2$ and $3$ has not the meaning we assign to it. We cannot know what meaning those people assign to $10$, therefore the question cannot be answered. Of course, it could also be that the numbers are the same, but they consistently round fractional results to the nearest odd integer. In that case, $10/3 = 3 + 1/3 = 3$. – celtschk Apr 5 '14 at 12:00
• Your professor should go back to that country and stay there. – Tyler Durden Apr 7 '14 at 0:04

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 $\pi^2$.

• That contradiction is just epic. Or just multiply both sides with $\pi^2-\frac{10}{3}$, $$0=1\implies 0=\pi^2 - \frac{10}{3} \implies \frac{10}{3} = \pi^2$$ – Guy Apr 5 '14 at 11:05
• It's not clear the OP's hypothesis is false; you've heard that "$\sqrt{3} > 2$ for large $3$"? – Andrew D. Hwang Apr 5 '14 at 11:05
• Physicists: $5=6$ for large values of $5$ and $6$. – Hakim Apr 5 '14 at 11:08
• But only for spherical '$=$'s in a vacuum. – Guy Apr 5 '14 at 11:08
• @user2345215: You completely missed the point of the answer above. – DumpsterDoofus Apr 5 '14 at 17:51

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.

• +1 The most mathematically consistent answer which achieves the desired result. – Mario Carneiro Apr 6 '14 at 10:17

As $5/2=3;$
it implies, $5/3=2$;
and $2*5/3=2*2=4;$
hence $10/3=4;$

• If division doesn't work as we expect, why would you think multiplication does? I.e., why is $2 \cdot 5/3 = 10/3$? – Pål GD Apr 5 '14 at 12:56

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.

I think your teacher wanted the following solution in which we only use the given relation that $\frac{5}{2}=3$: $$\frac{10}{3} = \frac{4}{3}\frac{5}{2} = 3\frac{4}{3} = 4.$$

• +1, but currently missing factor of $2$ starting after first equals sign. ;) – Andrew D. Hwang Apr 5 '14 at 11:04
• Actually you can only derive $10/3 = 10/(5/2) = 2\cdot 10/5$, since we neither know that in that foreign country $10/5=2$, nor that $2\cdot2=4$. – celtschk Apr 5 '14 at 12:04

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.

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.

• I think the question is "what's one-third of $10$?" not of $10_{a}$. By the way what does the subscript $a$ stand for? – user103816 Apr 5 '14 at 13:12
• There are two sets of numbers. The latter is base $10$. The first is in the mysterious unknown base, which we are asked to determine. Of course, the text of the problem was formulated five centuries ago, long before the current jargon was fully established, so there's no point in interpreting it anachronistically. Obviously, today we wouldn't phrase it in quite the same manner, but that's a different matter altogether. – Lucian Apr 5 '14 at 13:33

Although I agree with user2425, if there definitely is an answer then: $\dfrac{1}{2}5=3 \implies 5=6 \implies 10=12 \implies \dfrac{1}{3}10=4$

Given that $$$$\tag{1} \text{half of }5 = 3$$$$ this implies that $$$$\tag{2} \text{half of }10 = 6$$$$ $(2)$ then says that half of $1 = 0.6$ and therefore $$$$\tag{3} \frac{1}{3}\text{ of }1 = \frac{0.33 \times 0.6}{0.5} = 0.396$$$$ There for since $\frac{1}{3}$ of $1$ corresponds to $.396$ thus $\frac{1}{3}$ of $10 = 3.96$

Answer: $3.96$

• Welcome to math.SE: I have tried to improve the readability of your question by introducing Tex. For some basic information about writing math at this site see e.g. here, here, here and here. – Warren Moore Apr 6 '14 at 0:15
• I can't believe someone managed a rounding error here. Wow. – Guy Apr 6 '14 at 9:26
• Despite of the rounding errors, the way to the solution is straightly seen. – Michael Hoppe Dec 23 '15 at 21:04

Since 5/2 is 2.5, the convention clearly is to round up. Thus, 10/3, being 3.33..., rounds to 4.

• The fact 2.5 rounds to 3 doesn't mean that 3.33 rounds to 4, since most mathematicians will round up .5 and greater, but will round down anything less than .5 – Kenshin Apr 7 '14 at 3:02
• @Mew - the question isn't about mathematicians, but about a foreign country, where, obviously, different conventions apply. – Pete Becker Apr 7 '14 at 3:08
• you misunderstood my point. You said that they clearly round up because 5/2 is rounded up. In fact most countries will round up 5/2 but this doesn't mean they will clearly round up 10/3. Therefore if it is given that they round 5/2, it is no indication that they will round up 10/3. – Kenshin Apr 7 '14 at 3:37
• @Mew - you misunderstood the point of the question. It's a riddle, not a theorem. – Pete Becker Apr 7 '14 at 12:14

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: $$\frac{3}{2.5} = \frac{x}{3.\overline{3}} \implies x = 4$$

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.

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) $$\frac{5}{\square}=3\Rightarrow\square=\frac{5}{3}.$$