document.write( "Question 498817: A normal duck has two legs.A lame duck has one leg.A sitting duck has no legs.There are 33 ducks with a total of 32 legs.The total number of normal ducks and lame ducks is twice the number of sitting ducks.What is the number of lame ducks? \n" ); document.write( "
Algebra.Com's Answer #337211 by Theo(13342)\"\" \"About 
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33 ducks in total with a total of 32 legs.
\n" ); document.write( "normal ducks have 2 legs.
\n" ); document.write( "lame ducks have 1 leg.
\n" ); document.write( "sitting ducks have 0 legs.
\n" ); document.write( "total number of normal ducks and lame ducks is twice the number of sitting ducks.
\n" ); document.write( "let x = number of normal ducks
\n" ); document.write( "let y = number of lame ducks
\n" ); document.write( "let z = number of sitting ducks
\n" ); document.write( "we know that x + y + z = 33
\n" ); document.write( "we also know that x + y = 2z
\n" ); document.write( "we can substitute for x + y in the equation to get:
\n" ); document.write( "2z + z = 33 which becomes:
\n" ); document.write( "3z = 33 which becomes:
\n" ); document.write( "z = 11
\n" ); document.write( "we know what the number of sitting ducks is equal to because z was equal to the number of sitting ducks.
\n" ); document.write( "since there were 33 ducks in total, subtract 11 from that to get 22 ducks that are either normal or lame.
\n" ); document.write( "the total number of legs is equal to 32 coming from 22 ducks.
\n" ); document.write( "22 ducks, if they were all normal, would have 44 legs.
\n" ); document.write( "this means that there are 12 ducks that have only 1 leg.
\n" ); document.write( "this means that there are 10 ducks that have 2 legs.
\n" ); document.write( "10 * 2 = 20 + 12 * 1 = 32 legs in total.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "a more formal way of solving this is as follos:\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "x = number of normal ducks with 2 legs.
\n" ); document.write( "y = number of lame ducks with 1 leg.
\n" ); document.write( "z = number of siting ducks with 0 legs.
\n" ); document.write( "total number of ducks = 33
\n" ); document.write( "this means that x + y + z = 33
\n" ); document.write( "since we know that x + y = 2z, then this equation becomes:
\n" ); document.write( "2z + z = 33 which becomes:
\n" ); document.write( "3z = 33 which becomes:
\n" ); document.write( "z = 11
\n" ); document.write( "this says that x + y = 33 - 11 = 22
\n" ); document.write( "total number of duck legs is equal to 32
\n" ); document.write( "this means that 2x + 1y + 0z = 32
\n" ); document.write( "0z drops out of the equation and we are left with:
\n" ); document.write( "2x + 1y = 32
\n" ); document.write( "we know that x + y = 22
\n" ); document.write( "we have 2 equations that need to be solved simultaneously.
\n" ); document.write( "they are:
\n" ); document.write( "2x + y = 32
\n" ); document.write( "x + y = 22
\n" ); document.write( "subtract the second equation from the first equation to get:
\n" ); document.write( "x = 10
\n" ); document.write( "that means that there are 10 normal ducks.
\n" ); document.write( "that also means that there are 22 - 10 = 12 lame ducks.
\n" ); document.write( "we already know that we have 11 sitting ducks.
\n" ); document.write( "10 normal ducks + 12 lame ducks + 11 sitting ducks = 33 ducks in total
\n" ); document.write( "20 normal duck legs + 12 lame duck legs + 0 sitting duck legs = 32 ducks legs in total.
\n" ); document.write( "your answer is:
\n" ); document.write( "the number of lame ducks is equal to 12.\r
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