To find out how many cookies Homer had to begin with, we can work backwards step by step.
Step 1: Giving half of the cookies and half a cookie to the oldest brother
Let's say the number of cookies Homer had to begin with is C.
Homer gives half of the cookies and half a cookie to his oldest brother, which means he retains the other half and half a cookie.
So after giving, Homer has (C/2) + 0.5 cookies left.
Step 2: Giving half of what is now left and half a cookie to the second brother
Homer then gives half of what is now left, which is (C/2) + 0.5, along with half a cookie to his second brother.
After this, Homer retains the other half and half a cookie.
So after giving, Homer has [(C/2) + 0.5]/2 + 0.5 cookies left.
Step 3: Giving half of what is now left and half a cookie to the third brother
Finally, Homer gives half of what is now left, which is [(C/2) + 0.5]/2 + 0.5, along with half a cookie to his third brother.
After this, Homer retains the other half and half a cookie.
So after giving, Homer has [[(C/2) + 0.5]/2 + 0.5]/2 + 0.5 cookies left.
Since at the end Homer has zero cookies left, we can set up an equation to solve for C:
[[[(C/2) + 0.5]/2 + 0.5]/2 + 0.5] = 0
Now we can solve the equation step by step by simplifying and isolating C.
Step 1: Remove the brackets:
[(C/2) + 0.5]/2 + 0.5 = 0
Step 2: Multiply both sides by 2 to eliminate the denominators:
[(C/2) + 0.5] + 1 = 0
Step 3: Simplify:
C/2 + 0.5 + 1 = 0
Step 4: Combine like terms:
C/2 + 1.5 = 0
Step 5: Subtract 1.5 from both sides:
C/2 = -1.5
Step 6: Multiply both sides by 2 to isolate C:
C = -3
However, since the number of cookies cannot be negative, it appears there might be an error or inconsistency in the problem statement. Please double-check the information provided.