Question

The strength of an indigo solution in percentage is equal to the amount of indigo in grams per 100 cc of water. Two 800 cc bottles are filled with indigo solutions of strengths 33% and 17%, respectively. A part of the solution from the first bottle is thrown away and replaced by an equal volume of the solution from the second bottle. If the strength of the indigo solution in the first bottle has now changed to 21% then the volume, in cc, of the solution left in the second bottle is

Answer 1

Let's break down the problem step-by-step.

Step 1: Amount of Indigo in the First Bottle

Suppose \( x \) cc of the solution from the first bottle is thrown away. The amount of indigo in the solution that is thrown away is: \[ 0.33x \, \text{grams}. \] After this, the amount of solution left in the first bottle is \( 800 - x \) cc, and the amount of indigo left in the first bottle is: \[ 0.33(800) - 0.33x = 264 - 0.33x \, \text{grams}. \]

Step 2: Adding Solution from the Second Bottle

Now, \( x \) cc of the solution from the second bottle is added to the first bottle. The amount of indigo added from the second bottle is: \[ 0.17x \, \text{grams}. \] After this addition, the total volume of the solution in the first bottle remains 800 cc. The total amount of indigo in the first bottle becomes: \[ 264 - 0.33x + 0.17x = 264 - 0.16x \, \text{grams}. \]

Step 3: Setting up the Equation

It is given that after these operations, the strength of the solution in the first bottle changes to 21%. So, the amount of indigo in 800 cc of the solution is: \[ 0.21 \times 800 = 168 \, \text{grams}. \] Setting up the equation from the above information: \[ 264 - 0.16x = 168 \] Simplifying: \[ -0.16x = -96 \] Solving for \( x \): \[ x = \frac{-96}{-0.16} = 600 \]

Step 4: Finding the Volume Left in the Second Bottle

The volume of the solution left in the second bottle is: \[ \text{Original volume} - \text{Volume taken out} = 800 \, \text{cc} - 600 \, \text{cc} = 200 \, \text{cc}. \]

Final Answer:

The volume of the solution left in the second bottle is \( \boxed{200} \, \text{cc} \).

Answer 2

Let Bottle A have an indigo solution of strength 33% while Bottle B has an indigo solution of strength 17%. The task is to find the ratio in which these two solutions should be mixed to obtain a resultant solution of strength 21%.

Step 1: Set up the ratio using the alligation method

The ratio in which we mix the two solutions is given by: \[ \frac{A}{B} = \frac{21 - 17}{33 - 21} \] Simplifying: \[ \frac{A}{B} = \frac{4}{12} = \frac{1}{3} \] Hence, three parts of the solution from Bottle B are mixed with one part of the solution from Bottle A.

Step 2: Displacing the solution in Bottle A

For this process to happen, we need to displace \( 600 \, \text{cc} \) of solution from Bottle A and replace it with \( 600 \, \text{cc} \) of solution from Bottle B. Since both bottles contain \( 800 \, \text{cc} \), three parts of this volume equal \( 600 \, \text{cc} \), and the remaining \( 200 \, \text{cc} \) will stay in Bottle B.

Final Answer:

The volume of the solution remaining in Bottle B is \( \boxed{200} \, \text{cc} \).