Jane had a bottle filled with juice. At first she drank 1/5 of the juice in the bottle. After 1 hours she drank 1/4 of the juice remaining in the bottle. After another 2 hours she drank 1/3 of the remaining juice in the bottle. At that point Jane checked how much juice was left in the bottle: There was 2/3 of a cup left. No other juice was added to or removed to the bottle. How much juice was in the bottle originally?
What is the equation? By using a strip diagram, 1/5 will be 4/5, 1/4 will be 3/4, and 1/3 will be 2/3. So, will the equation be 2/3x times 3/4x times 4/5x = 2/3. Then, 2/5x = 2/3. 5/2 times 2/5 = 2/3 times 5/2. X is 5/3 of the juice in the bottle
yes
1 - 1/5 = 5/5 - 1/5 = 4/5 of a bottle remaining.
4/5 - 1/4 = 16/20 - 5/20 = 11/20 of a bottle remaining.
11/20 - 1/3 = 33/60 - 20/60 = 13/60 of a bottle remaining.
Originally, she had X cups of juice in the bottle.
13x/60 = 2/3 cups.
13x = 40,
X = 40/13 = 3 1/13 cups, originally.
Check:
4/5 * 40/13 = 32/13 = 2 6/13 cups remaining.
11/20 * 40/13 = 22/13 = 1 9/13 cups remaining.
13/60 * 40/13 = 40/60 = 2/3 cup remaining.
Let's break down the problem step by step:
Step 1: Jane initially drank 1/5 of the juice in the bottle.
Step 2: After 1 hour, she drank 1/4 of the remaining juice.
Step 3: After another 2 hours, she drank 1/3 of the remaining juice.
Step 4: At this point, only 2/3 of a cup of juice was left.
To find out how much juice was in the bottle originally, we can use the strip diagram approach:
Let's assume the original amount of juice in the bottle is represented by x.
After the first step, 1/5 of x was consumed, which leaves us with 4/5 of x.
After the second step, 1/4 of the remaining juice (4/5 of x) was consumed, leaving us with 3/4 of 4/5x, or (3/4) * (4/5) * x, which simplifies to 3/5x.
After the third step, 1/3 of the remaining juice (3/5x) was consumed, leaving us with 2/3 of 3/5x, or (2/3) * (3/5) * x, which simplifies to 2/5x.
At this point, 2/3 of a cup of juice is left, so we have the equation:
2/5x = 2/3
To solve this equation, we can cross-multiply:
(2/5)x = (2/3)
3 * (2/5)x = 2 * (2/3)
6/5x = 4/3
To isolate x, we need to divide both sides of the equation by (6/5):
(6/5x)/(6/5) = (4/3)/(6/5)
x = (4/3) * (5/6)
Simplifying this expression gives us:
x = (20/18)
x = (10/9)
Therefore, the original amount of juice in the bottle was (10/9) of a cup.
The equation to represent the problem is: (1 - 1/5) * (1 - 1/4) * (1 - 1/3) * x = 2/3.
Here's an explanation of each step:
1. The first step is to represent the amount of juice remaining after each time Jane drinks a portion. We start with the original amount of juice in the bottle, x, and subtract the fraction Jane drinks each time.
2. After the first time Jane drinks, there is 1 - 1/5 = 4/5 of the juice remaining in the bottle.
3. After the second time Jane drinks, there is (1 - 1/4)(4/5) = 3/4 * 4/5 = 12/20 = 3/5 of the juice remaining.
4. After the third time Jane drinks, there is (1 - 1/3)(3/5) = 2/3 * 3/5 = 6/15 = 2/5 of the juice remaining.
5. The final step is to set up the equation by multiplying all the remaining fractions together and setting it equal to the given amount of juice left, which is 2/3.
So, the equation becomes (2/3)*x = 2/3.
To solve for x, we can multiply both sides of the equation by the reciprocal of 2/3, which is 3/2:
(2/3)*(3/2)*x = (2/3)*(3/2)
x = 1 * (3/2)
x = 3/2
Therefore, the original amount of juice in the bottle was 3/2, or 1.5 cups.