Question

A painter mixes white and black paints to create different shades in two different buckets. He prepares 10 kg in one bucket with 20% black paint in it. He then prepares 20 kg of another shade in the second bucket with 20% black paint in it. He pours the second bucket into the first one. What is the percentage of black paint in the mixture?

Hint:

For mixture problems, if the percentage concentration of a component is the same in all the solutions being mixed, the final mixture's concentration will also be the same. This can save you from performing any calculations.

Answer 1

Correct Answer: 20

Step 1: Understanding the Question:
The problem asks for the final percentage of black paint after mixing two different quantities of paint that have the same initial percentage of black paint.
Step 2: Key Formula or Approach:
The percentage of a component in a mixture is calculated by the formula:
\[ \text{Percentage} = \left( \frac{\text{Total amount of the component}}{\text{Total amount of the mixture}} \right) \times 100% \] Step 3: Detailed Explanation:
Method 1: Direct Calculation
First, calculate the amount of black paint in each bucket.
Amount of black paint in the first bucket = 20% of 10 kg
\[ = \frac{20}{100} \times 10 \text{ kg} = 2 \text{ kg} \] Amount of black paint in the second bucket = 20% of 20 kg
\[ = \frac{20}{100} \times 20 \text{ kg} = 4 \text{ kg} \] Now, calculate the total amount of black paint and the total amount of the mixture.
Total black paint = 2 kg + 4 kg = 6 kg
Total mixture = 10 kg + 20 kg = 30 kg
Finally, calculate the percentage of black paint in the final mixture.
\[ \text{Percentage of black paint} = \left( \frac{6 \text{ kg}}{30 \text{ kg}} \right) \times 100% = \frac{1}{5} \times 100% = 20% \] Method 2: Logical Approach
Both paint mixtures have the same concentration of black paint (20%). When you mix two or more solutions that have the same concentration, the final mixture will also have that same concentration, regardless of the volumes or weights being mixed.
Step 4: Final Answer:
The percentage of black paint in the final mixture is 20%.