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6 buys drinks and says "I'm 1/2 Israeli and 1/2 Chinese."
15 orders appetizers for everyone and says "I'm 1/2 American and 1/2 Chinese."
12 orders the main course for everyone and says "I'm 2/3 Israeli and 1/3 Chinese."

What parts Israeli, Chinese, and American is 10?

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1  
Is this really a math puzzle, or is there some word-play or lateral thinking involved? – Gamow Mar 23 at 17:37
1  
Looks like math to me @Gamow – Z. Dailey Mar 23 at 17:37
16  
The nitpicker in me is just dying to point out that "2/3 Israeli and 1/3 Chinese" is impossible. Okay, I'm done, carry on enjoying the puzzle. – stacey Mar 23 at 19:08
11  
Because 7 ate 9 – Devsman Mar 23 at 20:44
11  
1/3 is possible if you're a time-traveler who's your own grandparent. – dan04 Mar 24 at 4:40

10 is

$\frac{1}{2}$ Israeli, $\frac{1}{2}$ American

Reasoning:

The prime $2$ is Israeli, the prime $3$ is Chinese, and the prime $5$ is American.
Then $6=2\cdot3$ is $\frac{1}{2}$ Israeli and $\frac{1}{2}$ Chinese.
Then $15=3\cdot5$ is $\frac{1}{2}$ American and $\frac{1}{2}$ Chinese.
Then $12=2\cdot2\cdot3$ is $\frac{2}{3}$ Israeli and $\frac{1}{3}$ Chinese.

Consequently $10=2\cdot5$ is $\frac{1}{2}$ Israeli and $\frac{1}{2}$ American.

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A different answer that's probably not what you intended, but:

If you take the numbers' statements to mean the system of equations:

$6 = (I + C) / 2$
$15 = (A + C) / 2$
$12 = (2I + C) / 3$

Then $A=42$, $C=-12$, and $I=24$.

There are an infinite number of ways to combine these to form the number 10, but if you continue the pattern of each number being a mixture of exactly two nationalities, then

$10$ is $7/18$ Chinese and $11/18$ Israeli.

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1  
In the two nationality case, this system also allows for 10 to be 11/27 American and 16/27 Chinese. (Or, as you pointed out, some mixture of the three, where the percentage of Chinese is between 7/18 and 16/27). – Matt Mar 24 at 13:48

10 is

1/2 American and 1/2 Israeli.

because

Israeli -> 2
Chinese -> 3
American -> 5
The fractions are the fraction of the numbers' prime factorization from the above mapping. So 10 = 2*5 has two prime factors. Half are 2 (Israeli) and half are 5 (American).

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10 is

1/2 Israeli and 1/2 American.

Explanation:

This problem refers to the prime factorization of the different numbers. "Israeli" corresponds to the power of $2$ in the factorization, "Chinese" corresponds to the power of $3$, and "American" corresponds to the power of $5$; each exponent is then divided by the sum of the exponents of all prime factors to obtain the fraction. $6 = 2^1 \cdot 3^1$, $15 = 3^1 \cdot 5^1$, $12 = 2^2 \cdot 3^1$, and $10 = 2^1 \cdot 5^1$, giving the fractions from the question and the answer.

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$6=2*3$, $15=3*5$ and $12=2^2*3$, so "Israeliness" is the ratio of the exponent of 2 to the sum of the exponents in the prime factor expansion of the given number. Same with Chineseness and 3, Americanness and 5. Therefore, 10 is 1/2 Israeli and 1/2 American.

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