Question 498817
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.