Joy is planning a party. One game she decides to play is bobbing for apples. She divides her apples into three amounts, thinking that putting a different number in each same-sized bucket might make ti more fun. Three friends are helping her. Joy puts 1/2 of her apples plus 1/2 apple into Becky's bucket. She then puts 1/2 of the remaining apples plus 1/2 into Gayle's bucker. She then puts 1/2 of the remaining apples plus 1/2 apple into Bonnie's bucket. One apple is left over, which Joy eats. How many apples did she start with? How many apples went into each helper's bucket?
original amount: x apples
Becky gets (x+1)/2, leaving Joy (x-1)/2
Gayle gets (x+1)/4, leaving Joy (x-3)/4
Bonnie gets (x+1)/8, leaving 1 left over
x - (x+1)/2 - (x+1)/4 - (x+1)/8 = 1
x = 15
or, working backwards, there were
((1*2+1)*2+1)*2+1 = 15 apples to start
To solve this problem, let's break it down step by step.
Let's assume that the number of apples Joy started with is X.
1. Joy puts 1/2 (X) + 1/2 (apple) into Becky's bucket. So, Becky initially gets 1/2 (X) + 1/2 (apple) apples, which can be written as (1/2 X + 1/2).
After this step, the number of remaining apples is (X - 1/2 X - 1/2), which simplifies to (1/2 X - 1/2).
2. Joy puts 1/2 (1/2 X - 1/2) + 1/2 (apple) into Gayle's bucket. So, Gayle initially gets 1/2 (1/2 X - 1/2) + 1/2 (apple) apples, which can be written as (1/4 X - 1/4 + 1/2).
After this step, the number of remaining apples is (1/2 X - 1/2 - 1/4 X + 1/4 - 1/2), which simplifies to (1/4 X - 1/4).
3. Joy puts 1/2 (1/4 X - 1/4) + 1/2 (apple) into Bonnie's bucket. So, Bonnie initially gets 1/2 (1/4 X - 1/4) + 1/2 (apple) apples, which can be written as (1/8 X - 1/8 + 1/2).
After this step, the number of remaining apples is (1/4 X - 1/4 - 1/8 X + 1/8 - 1/2), which simplifies to (1/8 X - 1/8 - 1/2).
According to the given information, one apple is left over after all the buckets are filled, which means the number of remaining apples is 1.
So we can set up an equation to solve for X:
(1/8 X - 1/8 - 1/2) = 1
Simplifying this equation, we get:
1/8 X - 1/8 - 1/2 = 1
1/8 X - 4/8 = 1
1/8 X - 1/2 = 1
1/8 X = 1 + 1/2
1/8 X = 3/2
Multiplying both sides of the equation by 8, we get:
X = (3/2) * 8
X = 12
Therefore, Joy started with 12 apples.
To find out how many apples went into each helper's bucket, we can substitute X = 12 into the expressions we derived earlier:
Becky's bucket: 1/2 (X) + 1/2 (apple) = 1/2 (12) + 1/2 (1) = 6 + 1/2 = 6 1/2
Gayle's bucket: 1/2 (1/2 X - 1/2) + 1/2 (apple) = 1/2 (1/2 (12) - 1/2) + 1/2 (1) = 1/2 (6 - 1/2) + 1/2 = 1/2 (11/2) + 1/2 = 11/4 + 1/2 = 11/4 + 2/4 = 13/4 = 3 1/4
Bonnie's bucket: 1/2 (1/4 X - 1/4) + 1/2 (apple) = 1/2 (1/4 (12) - 1/4) + 1/2 (1) = 1/2 (3 - 1/4) + 1/2 = 1/2 (11/4) + 1/2 = 11/8 + 4/8 = 15/8 = 1 7/8
Therefore, the number of apples that went into each helper's bucket is as follows:
Becky: 6 1/2 apples
Gayle: 3 1/4 apples
Bonnie: 1 7/8 apples