Answer:
4.847 cups
Step-by-step explanation:
Let's say x is number of cups of orange juice originally in the container.
The boy takes 1 cup of orange juice out, so there is x−1 cups left out of a total volume of x cups. So the new concentration in the container is:
(x − 1) / x
Next, he takes another cup out, but this time, it isn't 100% orange juice any more. So the number of cups of orange juice left in the container is x − 1 − (x − 1) / x. The total volume is still x cups, so the new concentration is:
[x − 1 − (x − 1) / x] / x
Repeating this logic, after he replaces the third cup with water, the final concentration is:
{x − 1 − (x − 1) / x − [x − 1 − (x − 1) / x] / x} / x
This final concentration is equal to 1/2.
1/2 = {x − 1 − (x − 1) / x − [x − 1 − (x − 1) / x] / x} / x
1/2 x = x − 1 − (x − 1) / x − [x − 1 − (x − 1) / x] / x
1/2 x² = x (x − 1) − (x − 1) − [x − 1 − (x − 1) / x]
1/2 x² = x (x − 1) − (x − 1) − (x − 1) + (x − 1) / x
1/2 x² = x (x − 1) − 2 (x − 1) + (x − 1) / x
1/2 x³ = x² (x − 1) − 2x (x − 1) + (x − 1)
1/2 x³ = x³ − x² − 2x² + 2x + x − 1
0 = 1/2 x³ − 3x² + 3x − 1
0 = x³ − 6x² + 6x − 2
Using a calculator to solve this:
x = 4.847
There are originally 4.847 cups in the container.
