Question 1198607: Four families went to a baseball game. A vendor selling bags of popcorn came by. The Wilson family bought half of the bags of popcorn plus one. The Matinez family bought half of the remaining bags of popcorn plus one. The Brightfeather family bought half of the remaining bags of popcorn plus one. The Wimberly family bought half of the remaining bags of popcorn plus one, leaving the vendor with no bags of popcorn. If the Wimberlys bought 2 bags of popcorn, how many bags did each of the four families buy?
Found 2 solutions by greenestamps, mccravyedwin:
Answer by greenestamps(13209)   (Show Source):
You can put this solution on YOUR website!
If we treat it with formal mathematics, we have, for each family, the number of bags remaining is the current number of bags, minus one-half the current number of bags, minus one more. As a function,
f(x)=x-(1/2)x-1
f(x)=(1/2)x-1
The input to this function is the number of bags remaining before a family buys theirs; the output is the number of bags remaining after that family buys theirs.
Then, if n is the original number of bags,
f(n) = bags remaining after the first family buys theirs;
f(f(n)) = bags remaining after the second family buys theirs;
...
Then, since the number of bags remaining after the fourth family buys theirs is 0,
f(f(f(f(n))))=0
The problem can be solved from there; but it is awkward.
The problem is solved far more easily by working "backwards", starting with the 0 bags left at the end and using the inverse function.
Given the function f(x)=(1/2)x-1, the inverse function (call it g(x)) is g(x)=2(x+1).
The input to this inverse function is the number of bags remaining AFTER a family buys theirs; the output is the number remaining before that family buys theirs.
Working backwards starting with the 0 bags left at the end and using this inverse function...
bags remaining before the fourth family bought theirs: g(0) = 2(0+1) = 2
bags remaining before the third family bought theirs: g(2) = 2(2+1) = 6
bags remaining before the second family bought theirs: g(6) = 2(6+1) = 14
bags remaining before the first family bought theirs: g(14) = 2(14+1) = 30
ANSWERS:
The first family bought 30-14 = 16 bags
The second family bought 14-6 = 8 bags
The third family bought 6-2 = 4 bags
The fourth family bought 2-0 = 2 bags
CHECK:
30 bags to start
First family buys (30/2)+1 = 16, leaving 14
Second family buys (14/2)+1 = 8, leaving 6
Third family buys (6/2)+1 = 4, leaving 2
Fourth family buys (2/2)+1 = 2, leaving 0
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NOTE: The "working backwards" solution can be performed informally, without using the mathematical concept of inverse functions; however, seeing the formal presentation of the solution should be useful to the student.
Answer by mccravyedwin(409)  (Show Source):
You can put this solution on YOUR website!
The reverse of the customer buying half of what the vendor has, then the customer buying 1 more --
--is--
The customer giving back 1 to the vendor, and then in addition, giving him back
enough bags to double what the vendor then has.
So let's "show the movie backwards". There are 4 families.
The vendor has 0 bags.
The 4th family gives back 1 bag to the vendor, so the vendor now has 1 bag.
Then they also give back enough bags to double what the vendor has.
So they give the vendor back 1 bag
So now the vendor then has 2 bags.
The 4th family therefore bought 1+1=2 bags.
Now the vendor has 2 bags.
The 3rd family gives back 1 bag to the vendor, so the vendor now
has 2+1=3 bags.
Then they also give back enough bags to double what the vendor has.
So they give the vendor back 3 bags
So now the vendor has 3+3=6 bags.
The 3rd family therefore bought 1+3=4 bags.
Now the vendor has 6 bags.
The 2nd family gives back 1 bag to the vendor, so the vendor now
has 6+1=7 bags.
Then they also give back enough bags to double what the vendor has.
So they give the vendor back 7 bags
So now the vendor has 7+7=14 bags.
The 2nd family therefore bought 1+7=8 bags.
Now the vendor has 14 bags.
The 1st family gives back 1 bag to the vendor, so the vendor now
has 14+1=15 bags.
Then they also give back enough bags to double what the vendor has.
So they give the vendor back 15 bags
So now the vendor has 15+15=30 bags.
The 1st family therefore bought 1+15=16 bags.
Edwin
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