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 $\frac{12}{23}$, find the total number of balls in the given bag.
Generated by claude-sonnet-4-6 · 2026-06-15 10:31 · grounding rag
Model Answer
Let the number of black balls = $x$
Then, number of red balls = $x + 3$
Total balls = $x + (x + 3) = 2x + 3$
Given: $P(\text{red ball}) = \dfrac{12}{23}$
$$\frac{x+3}{2x+3} = \frac{12}{23}$$
$$23(x+3) = 12(2x+3)$$
$$23x + 69 = 24x + 36$$
$$x = 33$$
Total number of balls $= 2(33) + 3 = \mathbf{69}$
Source: Chapter 14, Section 14.1
Explanation
- Set up a variable for black balls; red balls = black + 3; total = their sum.
- Use the theoretical probability formula: favourable outcomes ÷ total outcomes = 12/23.
- Cross-multiply and solve for $x$, then substitute back to find the total.
- Examiners award 1 mark for correct equation setup and 1 mark for the correct final answer (69).