1. **State the problem:** We have a 30 cm by 30 cm square board with two rectangular cards attached: one 8 cm by 12 cm (Card A) and one 15 cm by 20 cm (Card B). The larger card covers one-quarter of the smaller card where they overlap. We want to find the probabilities that a dart thrown randomly at the board: a) hits both cards b) hits at least one card c) hits exactly one card 2. **Calculate areas:** - Area of board = $30 \times 30 = 900$ cm$^2$ - Area of Card A = $8 \times 12 = 96$ cm$^2$ - Area of Card B = $15 \times 20 = 300$ cm$^2$ - Overlap area = one-quarter of Card A = $\frac{1}{4} \times 96 = 24$ cm$^2$ 3. **Calculate probabilities:** - Probability dart hits both cards (overlap area): $$P(\text{both}) = \frac{24}{900} = \frac{2}{75} \approx 0.0267$$ - Probability dart hits at least one card (union of areas): $$P(\text{A} \cup \text{B}) = \frac{\text{Area A} + \text{Area B} - \text{Overlap}}{\text{Area board}} = \frac{96 + 300 - 24}{900} = \frac{372}{900} = \frac{62}{150} \approx 0.4133$$ - Probability dart hits exactly one card (either A or B but not both): $$P(\text{exactly one}) = P(\text{A} \cup \text{B}) - P(\text{both}) = \frac{372}{900} - \frac{24}{900} = \frac{348}{900} = \frac{58}{150} \approx 0.3867$$ 4. **Answer:** - a) Probability hits both cards: $\boxed{\frac{2}{75} \approx 0.0267}$ - b) Probability hits at least one card: $\boxed{\frac{62}{150} \approx 0.4133}$ - c) Probability hits exactly one card: $\boxed{\frac{58}{150} \approx 0.3867}$ 5. **Additional:** - Probability that neither card is hit is $1 - P(\text{A} \cup \text{B}) = 1 - \frac{62}{150} = \frac{88}{150} \approx 0.5867$