four families bought popcorn at a game.The Wilson's bought half the bags of popcorn plus one. the Martinez's bought half of the remaining bags plus one. the Brightfeather's bought half of the remaining bags plus one. The Wilberly's bought half the remaining bags plus one. If the Wilbery's bought 2 bags of popcorn, how many bags did each of the four families buy?
Let's assign variables for each family:
Let's call the number of bags the Wilson's bought as W.
Let's call the number of bags the Martinez's bought as M.
Let's call the number of bags the Brightfeather's bought as B.
Let's call the number of bags the Wilberly's bought as X.
Given information:
The Wilberly's bought 2 bags of popcorn (X = 2).
Now, let's calculate the number of bags purchased by each family step-by-step:
1. The Wilson's bought half the bags of popcorn plus one:
W = (1/2)X + 1
W = (1/2)2 + 1
W = 1 + 1
W = 2
2. The Martinez's bought half of the remaining bags plus one:
M = (1/2)(X - W) + 1
M = (1/2)(2 - 2) + 1
M = 0 + 1
M = 1
3. The Brightfeather's bought half of the remaining bags plus one:
B = (1/2)(X - W - M) + 1
B = (1/2)(2 - 2 - 1) + 1
B = (1/2)(-1) + 1
B = -1/2 + 1
B = 1/2
So far, The Wilson's bought 2 bags (W = 2), the Martinez's bought 1 bag (M = 1), and the Brightfeather's bought 1/2 bag (B = 1/2).
4. The Wilberly's bought half the remaining bags plus one:
X = (1/2)(X - W - M - B) + 1
2 = (1/2)(2 - 2 - 1 - 1/2) + 1
2 = (1/2)(2 - 4/2 - 1/2) + 1
2 = (1/2)(2 - 5/2) + 1
2 = (1/2)(4/2 - 5/2) + 1
2 = (1/2)(-1/2) + 1
2 = -1/4 + 1
2 = 3/4
Therefore, the Wilberly's bought 2 bags of popcorn (X = 2), the Wilson's bought 2 bags, the Martinez's bought 1 bag, the Brightfeather's bought 1/2 bag, and the remaining bags were 3/4.
In conclusion, each of the four families bought the following number of bags of popcorn:
- The Wilson's bought 2 bags.
- The Martinez's bought 1 bag.
- The Brightfeather's bought 1/2 bag.
- The Wilberly's bought 2 bags.
To solve this problem, we can work backwards by reversing the steps each family took to buy popcorn.
Step 1: The Wilberlys bought 2 bags of popcorn.
Step 2: The Brightfeathers bought half of the remaining bags plus one.
Step 3: The Martinezes bought half of the remaining bags plus one.
Step 4: The Wilsons bought half the bags of popcorn plus one.
Let's solve it step by step:
Step 4:
If the Wilberlys bought 2 bags of popcorn, that means there were (2 * 2) - 1 = 3 bags of popcorn remaining before they made their purchase.
Step 3:
The Martinezes bought half of the remaining bags plus one. Since there were 3 bags remaining, the Martinezes bought (3 / 2) + 1 = 2 bags of popcorn.
Step 2:
The Brightfeathers bought half of the remaining bags plus one. Since there were 1 bag remaining, the Brightfeathers bought (1 / 2) + 1 = 1.5 + 1 = 2.5 bags of popcorn.
Now we encounter a problem because we can't have half a bag of popcorn. We need to reconsider our previous steps and adjust the numbers accordingly.
Let's go back to Step 4 and make changes:
Step 4:
Since the Brightfeathers couldn't have bought 2.5 bags, it means there were (2 * 3) - 1 = 5 bags of popcorn remaining before Step 3.
Step 3:
The Martinezes bought half of the remaining bags plus one. Since there were 5 bags remaining, the Martinezes bought (5 / 2) + 1 = 2.5 + 1 = 3.5 bags of popcorn.
This doesn't work either, so let's go back once again.
Step 4:
Since the Martinezes couldn't have bought 3.5 bags, it means there were (2 * 5) - 1 = 9 bags of popcorn remaining before Step 3.
Step 3:
The Martinezes bought half of the remaining bags plus one. Since there were 9 bags remaining, the Martinezes bought (9 / 2) + 1 = 4.5 + 1 = 5.5 bags of popcorn.
Again, this doesn't work, so let's go back one more time.
Step 4:
Since the Martinezes couldn't have bought 5.5 bags, it means there were (2 * 9) - 1 = 17 bags of popcorn remaining before Step 3.
Step 3:
The Martinezes bought half of the remaining bags plus one. Since there were 17 bags remaining, the Martinezes bought (17 / 2) + 1 = 8.5 + 1 = 9.5 bags of popcorn.
Once again, this doesn't work. Let's make one more adjustment.
Step 4:
Since the Martinezes couldn't have bought 9.5 bags, it means there were (2 * 17) - 1 = 33 bags of popcorn remaining before Step 3.
Step 3:
The Martinezes bought half of the remaining bags plus one. Since there were 33 bags remaining, the Martinezes bought (33 / 2) + 1 = 16.5 + 1 = 17.5 bags of popcorn.
Unfortunately, this doesn't work out either. It seems we are getting fractional values for the number of bags bought by each family, which is not possible.
Therefore, it seems like there may be an error in the information or calculations provided. Please double-check the problem statement or provide additional information to proceed further.