There is a basket containing red and blue balls. 40% of the balls are red and 60% of the balls are blue. Two red balls are added into the basket to make them 50% red and 50% blue balls in the basket. How many balls are in the basket now?
A. 4 B. 10 C. 12 D. 6
Correct Answer: C
Explanation
Let's solve the problem step by step:
1. Define variables: - Let \( R \) be the number of red balls originally in the basket. - Let \( B \) be the number of blue balls originally in the basket.
2. Initial proportions: - 40% of the balls are red, so \( R = 0.4(R + B) \). - 60% of the balls are blue, so \( B = 0.6(R + B) \).
3. Simplify: - Substitute \( R \) into the equation: \( R = 0.4(R + B) \) implies \( R = 0.4R + 0.4B \). - Rearrange to find \( R \): \( 0.6R = 0.4B \) or \( R = \frac{2}{3}B \).
4. Add two red balls: - New number of red balls = \( R + 2 \). - New total number of balls = \( R + B + 2 \).
5. New proportions: - The new proportion of red balls is 50%. Thus: \[ \frac{R + 2}{R + B + 2} = 0.5 \]
6. Solve for \( R \) and \( B \): \[ R + 2 = 0.5(R + B + 2) \] \[ R + 2 = 0.5R + 0.5B + 1 \] \[ R + 1 = 0.5R + 0.5B \] \[ 0.5R = 0.5B - 1 \] \[ R = B - 2 \]
7. Substitute \( R = B - 2 \) into \( R = \frac{2}{3}B \): \[ B - 2 = \frac{2}{3}B \] \[ B - \frac{2}{3}B = 2 \] \[ \frac{1}{3}B = 2 \] \[ B = 6 \]
8. Find \( R \): \[ R = B - 2 = 6 - 2 = 4 \]
9. Total number of balls now: \[ \text{New total number of balls} = R + B + 2 = 4 + 6 + 2 = 12 \]