SOLUTION: 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 th

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?
Answer by Theo(13342) (Show Source): You can put this solution on YOUR website!
33 ducks in total with a total of 32 legs.
normal ducks have 2 legs.
lame ducks have 1 leg.
sitting ducks have 0 legs.
total number of normal ducks and lame ducks is twice the number of sitting ducks.
let x = number of normal ducks
let y = number of lame ducks
let z = number of sitting ducks
we know that x + y + z = 33
we also know that x + y = 2z
we can substitute for x + y in the equation to get:
2z + z = 33 which becomes:
3z = 33 which becomes:
z = 11
we know what the number of sitting ducks is equal to because z was equal to the number of sitting ducks.
since there were 33 ducks in total, subtract 11 from that to get 22 ducks that are either normal or lame.
the total number of legs is equal to 32 coming from 22 ducks.
22 ducks, if they were all normal, would have 44 legs.
this means that there are 12 ducks that have only 1 leg.
this means that there are 10 ducks that have 2 legs.
10 * 2 = 20 + 12 * 1 = 32 legs in total.

a more formal way of solving this is as follos:

x = number of normal ducks with 2 legs.
y = number of lame ducks with 1 leg.
z = number of siting ducks with 0 legs.
total number of ducks = 33
this means that x + y + z = 33
since we know that x + y = 2z, then this equation becomes:
2z + z = 33 which becomes:
3z = 33 which becomes:
z = 11
this says that x + y = 33 - 11 = 22
total number of duck legs is equal to 32
this means that 2x + 1y + 0z = 32
0z drops out of the equation and we are left with:
2x + 1y = 32
we know that x + y = 22
we have 2 equations that need to be solved simultaneously.
they are:
2x + y = 32
x + y = 22
subtract the second equation from the first equation to get:
x = 10
that means that there are 10 normal ducks.
that also means that there are 22 - 10 = 12 lame ducks.
we already know that we have 11 sitting ducks.
10 normal ducks + 12 lame ducks + 11 sitting ducks = 33 ducks in total
20 normal duck legs + 12 lame duck legs + 0 sitting duck legs = 32 ducks legs in total.
your answer is:
the number of lame ducks is equal to 12.

RELATED QUESTIONS
What is the fewest number of ducks if there are two ducks before a duck, two ducks behind (answered by richard1234)

A rubber duck race is an increasingly common and always popular fundraising idea for his... (answered by Alan3354)

A particular farm has cows and ducks. There are a total of 79 heads and 212 legs. How... (answered by edjones)

A game at a carnival has 75 rubber ducks floating in water. The ducks are numbered 1 to... (answered by stanbon)

A game at a carnival has 83 rubber ducks floating in water. The ducks are numbered 01 to... (answered by ikleyn)

Mrs. McDonald has a farm with ducks and pigs. There are 27 heads and 84 legs. How many... (answered by stanbon,ikleyn)

Julie sells 3 ducks and 4 chickens for $70.30. One chicken and one duck together are sold (answered by ankor@dixie-net.com)

A farmer has a combined total of 30 pigs and ducks and there is a combined total of 100... (answered by Edwin McCravy,lynnlo)

A duck and a chicken are walking to school. One way, the trip takes 2 hours for the duck... (answered by josmiceli,richwmiller)