document.write( "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 \n" ); document.write( "
Algebra.Com's Answer #62223 by bucky(2189)\"\" \"About 
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A little tricky. First a long way of working it, followed by a much simpler method which
\n" ); document.write( "is explained at the end. The long way is just to get you thinking about how the prize money
\n" ); document.write( "is distributed, but the simpler method gets you to the answer really fast.
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\n" ); document.write( "Let P represent the total prize money.
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\n" ); document.write( "Then, the winner gets 1/2 of P or P/2
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\n" ); document.write( "The second runner up gets 1/4 of what the winner got. So the second runner up gets (1/4)*(P/2)
\n" ); document.write( "which multiplies out to be P/8.
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\n" ); document.write( "So if you add the winner's amount and the second runner-up's amount you get:
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\n" ); document.write( "P/2 + P/8
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\n" ); document.write( "Put these over a common denominator by multiplying P/2 times 4/4 (which is the same as
\n" ); document.write( "multiplying P/2 by 1) and this term becomes 4P/8. So the expression for the combination
\n" ); document.write( "of winnings for the winner and second place runner up is:
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\n" ); document.write( "4P/8 + P/8 = 5P/8
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\n" ); document.write( "which is 5/8 of P. Where's the missing 3/8 of P? Surprise ... it goes to the first place
\n" ); document.write( "runner up. So the total Prize money is split as follows which in order from left to right
\n" ); document.write( "is the winner's amount, the first place runner up's amount, and the second place runner up's amount:
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\n" ); document.write( "4P/8 + 3P/8 + P/8 = P
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\n" ); document.write( "If the winner gets $6000 then you can say that the second place runner up gets one fourth
\n" ); document.write( "of that or (1/4)* $6000 or $1500. Notice that the first place runner up gets 3P/8 and
\n" ); document.write( "the second place runner up gets P/8. Therefore, the first place runner up gets 3 times the
\n" ); document.write( "amount that the second place runner up gets. Since we know that the second place runner up
\n" ); document.write( "gets $1500, then the first place runner up gets 3 times that or $4500.
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\n" ); document.write( "So the total prize amount is the sum of the three prizes or:
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\n" ); document.write( "$6000 + $4500 + $1500 = $12000
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\n" ); document.write( "An easier way to look at this problem is to realize that the winner gets half of the total
\n" ); document.write( "prize money and the winner gets $6,000. So the total prize money must be twice the $6,000
\n" ); document.write( "which is $12,000. \n" ); document.write( "
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