SOLUTION: in a writing competition,the first place winner receives 1/2 of the prize money.the second runner up receives 1/4 of what the winner won. what was the total amount of prize money d

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Question 86108: in a writing competition,the first place winner receives 1/2 of the prize money.the second runner up receives 1/4 of what the winner won. what was the total amount of prize money distributed if the winner receives $6,000? please need help with this one
Answer by bucky(2189) About Me  (Show Source):
You can put this solution on YOUR website!
A little tricky. First a long way of working it, followed by a much simpler method which
is explained at the end. The long way is just to get you thinking about how the prize money
is distributed, but the simpler method gets you to the answer really fast.
.
Let P represent the total prize money.
.
Then, the winner gets 1/2 of P or P/2
.
The second runner up gets 1/4 of what the winner got. So the second runner up gets (1/4)*(P/2)
which multiplies out to be P/8.
.
So if you add the winner's amount and the second runner-up's amount you get:
.
P/2 + P/8
.
Put these over a common denominator by multiplying P/2 times 4/4 (which is the same as
multiplying P/2 by 1) and this term becomes 4P/8. So the expression for the combination
of winnings for the winner and second place runner up is:
.
4P/8 + P/8 = 5P/8
.
which is 5/8 of P. Where's the missing 3/8 of P? Surprise ... it goes to the first place
runner up. So the total Prize money is split as follows which in order from left to right
is the winner's amount, the first place runner up's amount, and the second place runner up's amount:
.
4P/8 + 3P/8 + P/8 = P
.
If the winner gets $6000 then you can say that the second place runner up gets one fourth
of that or (1/4)* $6000 or $1500. Notice that the first place runner up gets 3P/8 and
the second place runner up gets P/8. Therefore, the first place runner up gets 3 times the
amount that the second place runner up gets. Since we know that the second place runner up
gets $1500, then the first place runner up gets 3 times that or $4500.
.
So the total prize amount is the sum of the three prizes or:
.
$6000 + $4500 + $1500 = $12000
.
An easier way to look at this problem is to realize that the winner gets half of the total
prize money and the winner gets $6,000. So the total prize money must be twice the $6,000
which is $12,000.