At intermission, Xavier sold refreshments. Of the items he had, 1/5 were granola bars and 1/2 of the remaining items were containers of popcorn. The rest were divided equally among apples, rice cake, and containers of yogurt. Xavier sold the 8 apples he had. How many items did he originally have?
Let's break down the information step-by-step:
Step 1: Xavier had 1/5 of the items as granola bars.
Step 2: The remaining items after selling granola bars were 4/5 of the original items.
Step 3: 1/2 of the remaining items were containers of popcorn, so Xavier had 1/2 * 4/5 = 4/10 = 2/5 of the original items as containers of popcorn.
Step 4: The remaining items after selling granola bars and containers of popcorn were 3/5 of the original items.
Step 5: The remaining items were divided equally among apples, rice cake, and containers of yogurt. Since Xavier sold 8 apples, there were a total of 3 * 8 = 24 items in apples, rice cake, and containers of yogurt.
Step 6: The remaining items from Step 4 were 3/5 of the original items, and since there were 24 items in apples, rice cake, and containers of yogurt, the original number of items was 5/3 * 24 = 40.
Therefore, Xavier originally had 40 items.
To solve this problem, we need to work backward and calculate the number of items Xavier originally had.
We know that Xavier sold 8 apples. Since the remaining items were divided equally among apples, rice cakes, and containers of yogurt, each of these categories had the same number of items.
Let's assume the number of items in each category is represented by "x".
So, the total number of items in those three categories combined is 3 times "x", which is equal to 3x.
We also know that 1/2 of the remaining items were containers of popcorn. Therefore, the remaining number of items after selling the 8 apples and the containers of popcorn is equal to 2x.
We are also given that 1/5 of the original items were granola bars. Hence, the remaining number of items after selling the 8 apples, the containers of popcorn, and the granola bars is equal to 5x.
Since we want to find the original number of items, we need to find the value of "x".
We know that the remaining number of items is equal to 5x. Subtracting 5x from the original number of items gives us:
Original number of items - 5x = 0 (since all remaining items were sold)
Simplifying this equation gives:
Original number of items = 5x
So, we need to find the value of "x" for which the original number of items is 5 times "x".
Unfortunately, the given information does not provide us with enough details to determine the value of "x" or the original number of items. We would need additional information or data to solve this problem.
x = number of items
He have:
x / 5 of granola and x / 2 popcorn
x / 5 + x / 2 = 2 x / 10 + 5 x / 10 = 7 x / 10
He had 7 x / 10 granola and popcorn.
The rest = 10 x / 10 - 7 x / 10 = 3 x / 10
The rest were divided equally among apples, rice cake, and containers of yogurt mean:
The rest is divided into 3 equal parts each:
( 3 x / 10 ) / 3 = 3 x / 30 = x / 10
So 8 apples = x / 10
x / 10 = 8
Multiply both sides by 10
x = 80
He originally have 80 items.
Proof:
He had 7 x / 10 granola and popcorn.
He had 7 ∙ 80 / 10 = 560 / 10 = 56 granola and popcorn.
The rest = 80 - 56 = 24
The rest were divided equally among apples, rice cake, and containers of yogurt.
The rest / 3 = 24 / 3 = 8
He have 8 among apples, 8 rice cake, 8 containers of yogurt.