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
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
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.)
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.
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.
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.