A prize of $27,000 is to be divided among three people in the ratio 3:5:7. What is the largest share? 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
 A: 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.
A: 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.)
A: 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.
A: 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.
