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This is not homework; I was just reviewing some old math flash cards and I came across this one I couldn't solve. I'm not interested in the solution so much as the reasoning.

Thanks

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Hint: $3 + 5 + 7 = 15$. So separate the money into $15$ distinct piles of equal amounts (why can we do that?). Give $3$ piles to the first person, $5$ piles to the second, and $7$ piles to the third. This now amounts to finding how much money was given to the third person. Hope that helps. –  JavaMan Jul 20 '11 at 2:38
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@DJC: I think that's great - why don't you make that a full answer instead of just a comment? –  mixedmath Jul 20 '11 at 2:41

4 Answers 4

up vote 8 down vote accepted

You can think of splitting the money in the ratio $3:5:7$ as dividing it into $3+5+7=15$ equal parts and giving $3$ of these parts to one person, $5$ to another, and $7$ to the third. One part, then, must amount to $\frac{27000}{15}=1800$ dollars, and the shares must then be $3 \cdot 1800 = 5400$, $5 \cdot 1800 = 9000$, and $7 \cdot 1800 = 12600$ dollars, respectively. (As a quick check, $5400+9000+12600=27000$, as required.)

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haha it's so easy when you think about it this way. Ahh, good ol' math. thanks! –  oldmanharry Jul 20 '11 at 2:46

Hint: 3+5+7=15. So separate the money into 15 distinct piles of equal amounts (why can we do that?). Give 3 piles to the first person, 5 piles to the second, and 7 piles to the third. This now amounts to finding how much money was given to the third person. Hope that helps.

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You can consider the numbers in the ratio as shares in the prize -- that is, the prize is to be divided into $3+5+7=15$ shares, and the three people get $3/15$, $5/15$ and $7/15$ of the prize, respectively. The idea is that the fractions of the prize have to add up to $1$, and you can make sure that they do by putting their sum in the denominator. $27000/15=1800$, so the three shares are $3\cdot1800=5400$, $5\cdot1800=9000$ and $7\cdot 1800=12600$, respectively.

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is there no way to do multiple accepted answers? :\ –  oldmanharry Jul 20 '11 at 2:48
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@oldmanharry: I'm afraid not. The idea is to not give too much incentive for people to only look at old questions rather than looking at new questions. At least, that's what I would think. –  mixedmath Jul 20 '11 at 3:08
    
@oldmanharry: You could just visit some of joriki's answers and upvote any that melt your face. (I think I'm responsible for about 5k of his reputation, for face-melting votes alone!) –  The Chaz 2.0 Jul 20 '11 at 4:15

Lets $a$ amount that get first person $b$ amount of second and $c$ amount of third person, from conditions in question follow system $$a+b+c=27000$$ $$a:b:c=3:5:7$$ from second equation follow $a:3=b:5=c:7=k$ and $a=3k,b=5k,c=7k$ if these values put in first equation of system above we get $$3k+5k+7k=27000, 15k=27000, k=27000/15=1800$$ $k$ is coefficient of proportionality, clearly $$a=3k=5400$$ $$b=5k=9000$$ $$c=7k=12600$$ This method can be generalized.

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