# Math Help - fraction problem

1. ## fraction problem

The king took a cup filled with juice and drank 1/5 of its contents. When the king looked away, the court jester refilled the cup by adding water to the remaining juice and then stirring. The king drank ¼ of this mixture. When the king looked away again, the jester refilled the cup with more water and stirred, and the king drank 1/3 of this mixture. When the king looked away a third time, the jester refilled the cup with more water. What percentage of this final mixture was water?

This problem has to be explained to elementary school age children they have no algebra concepts. It can be done or it would not have been asked of me.

2. Hello, Nancy!

The King took a cup filled with juice and drank $\tfrac{1}{5}$ of its contents.
When the king looked away, the Jester refilled the cup by adding water.

The King drank $\tfrac{1}{4}$ of this mixture.
Again the Jester refilled the cup with more water.

The King drank $\tfrac{1}{3}$ of this mixture.
The king looked away, and the Jester refilled the cup with water.

What percentage of this final mixture was water?

This problem has to be explained to elementary school age children.
They have no algebra concepts.
It can be done or it would not have been asked of me.
I may be able to explain it with a series of pictures.

The King began with a full cup of juice.
Code:
  -   *---------*
:   |/////////|
:   | - - - - |
:   |/////////|
:   | - - - - |
juice |/////////|
:   | - - - - |
:   |/////////|
:   | - - - - |
:   |/////////|
-   *---------*

The King drank $\tfrac{1}{5}$ of it.
The Jester refilled the cup with water.
Then the King drank $\tfrac{1}{4}$ of this mixture.
Code:
  -   *---------*
water |         |  He drank 1/4 of the water.
-   | - - - - |
:   |/////////|
:   | - - - - |
:   |/////////|
juice | - - - - |  He drank 1/4 of the juice.
:   |/////////|
:   | - - - - |
:   |/////////|
-   *---------*
We can see that " $\tfrac{1}{4}$ of the juice" is one of those shaded bars.

After the King's second sip (and the Jester's second refilling),
. . we have this diagram.
Then the King drank $\tfrac{1}{3}$ of this mixture.
Code:
  -   *---------*
:   |         |
water | - - - - |  He drank 1/3 of the water.
:   |         |
-   | - - - - |
:   |/////////|
:   | - - - - |
juice |/////////|  He drank 1/3 of the juice.
:   | - - - - |
:   |/////////|
-   *---------*
We see that " $\tfrac{1}{3}$ of the juice" is one of those shaded bars.

The final mixture looks like this:
Code:
  -   * - - - - *
:   |         |
:   | - - - - |
water |         |
:   | - - - - |
:   |         |
-   | - - - - |
:   |/////////|
juice | - - - - |
:   |/////////|
-   * - - - - *

And we see that the water is: $\tfrac{3}{5} = 60$% of the final mixture.