Well, we know enough information to make some equations and solve them.
Things we know:
1) 52 legs total
2) Ducks have 2 legs
3) Cows have 4 legs
4) No other legs are being counted
5) 4 more cows than ducks
We have 2 unknowns: # of ducks, we'll call that D, and # of cows, we'll call that C.
So from facts 1 through 4 we can say that duck legs + cow legs = 52. Each duck has 2 legs and each cow has 4 legs. Therefore, we can write:
Equation 1: 2D + 4C = 52 (2 legs for each duck and 4 legs for each cow is 52 legs total)
From fact #5 we know that: C = D + 4. This is equation 2.
We have 2 equations and 2 unknowns so we can solve it.
From there you solve the equations using substitution. Take equation 2 and put "D+4" in place of "C" in equation 1.
Equation 1: 2D + 4C = 52
Substitute in "D + 4" for "C"
2D + 4(D + 4) = 52
Multiply the 4 through:
2D + 4D + 16 = 52
Add your D terms
6D + 16 = 52
Subtract 16 from both sides to get D on one side.
6D = 36
Divide by 6 to find D
D = 6
There are 6 ducks.
The problem doesn't ask for # of cows, but we know this too now. C = D + 4 so there are 10 cows.
You can easily do a quick check now. 10 cows with 4 legs each is 40 legs. 6 ducks with 2 legs each is 12 legs. 40 + 12 is 52 which is correct.
5/15/2013

Andrea T.