Question

The number of red balls in a bag is three more than the number of black balls. If the probability of drawing a red ball at random from the given bag is $\dfrac{12}{23}$, find the total number of balls in the given bag.

Hint:

Translate word problems into equations by defining variables clearly and use cross-multiplication in probability.

Answer 1

Problem:
A bag contains some black and red balls. The number of red balls is 3 more than the number of black balls. If the probability of drawing a red ball is \( \frac{12}{23} \), find the total number of balls in the bag.

Step 1: Let the number of black balls be \( x \)
Then, the number of red balls = \( x + 3 \)
Total number of balls = black balls + red balls = \( x + (x + 3) = 2x + 3 \)

Step 2: Use the probability information
The probability of drawing a red ball is given by: \[ \frac{\text{Number of red balls}}{\text{Total number of balls}} = \frac{12}{23} \Rightarrow \frac{x + 3}{2x + 3} = \frac{12}{23} \]
Step 3: Solve the equation
Cross-multiply: \[ 23(x + 3) = 12(2x + 3) \Rightarrow 23x + 69 = 24x + 36 \]
Now, bring like terms together: \[ 69 - 36 = 24x - 23x \Rightarrow 33 = x \]
Step 4: Calculate total number of balls
Total balls = \( 2x + 3 = 2(33) + 3 = 66 + 3 = 69 \)

Final Answer:
The total number of balls in the bag is: \( \boxed{69} \)