Aboy thinks he has discovered a way to drink extra orange juice without alerting his parents. for every cup of orange juice he takes from a container of orange juice, he pours one cup of water back into the container. if he completes this process three times on the same container of juice, the resulting mixture will be exactly 50% water and 50% juice. how many cups of orange juice were originally in the container. (p. s its not 6)

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.

7 weeks

step-by-step explanation:

convert each fraction to the least common multiple of 20, so 3/4 = 15/20 & 1/10 = 2/20, then divide 15 by 2 and you get a little over 7, so the food will last 7 weeks.