Answer:
Step-by-step explanation:
Because we have 3 unknowns, we need to come up with 3 equations. If the total number of animals is 440 and that number is made up of a combination of goats (g), ducks (d), and chickens (c) the first equation is
g + d + c = 440
The next equation is found in the fact that the number of ducks is one-quarter the number of chickens:
[tex]d=\frac{1}{4}c[/tex] and solving for c gives us that
c = 4d
The last equation says that of the total number of animals, 440, half the goats and half the ducks got away, leaving only 320 animals behind. The last equation, the tricky one, is:
[tex]440-\frac{1}{2}g-\frac{1}{2}d=320[/tex] and simplifying that:
[tex]-\frac{1}{2}g-\frac{1}{2}d=-120[/tex] and because nobody hates fractions more than I do, I'm going to get rid of them by multiplying everything by 2 to get:
-g - d = -240
We've got these equations now, but what I'm going to do is to sub in what c equals (c = 4d) for c in the first equation:
g + d + 4d = 440 and
g + 5d = 440 and pair that with the one right above:
-g - 1d = -240 and use elimination to solve. The g's cancel each other out, leaving us with 4d = 200 and d = 50. So there were 50 ducks originally. Now we will sub that in to solve for c:
c = 4d so
c = 4(50) and
c = 200. Now we will sub both those values into the very first equation we put together to solve for g:
g + 200 + 50 = 440 and
g + 250 = 440 so
g = 190.
Add them all together just to be sure we have the 440 that we were told we had in the beginning (and we do, so we're all done!)
