1. A recipe for a cake requires 3 cups of flour, 2 cups of sugar, and 4 eggs. How much flour and sugar are needed to make a cake with 6 eggs? Answer: 4.5 cups of flour and 3 cups of sugar. Rationale: To find the amount of flour and sugar needed, we can use the ratio of eggs to flour and sugar, which is 4:3:2. If we multiply both sides of this ratio by 1.5, we get 6:4.5:3, which means that for every 6 eggs, we need 4.5 cups of flour and 3 cups of sugar. 2. A map has a scale of 1 cm : 5 km. What is the actual distance between two towns that are 8 cm apart on the map? Answer: 40 km. Rationale: To find the actual distance, we can use the proportion 1 cm / 5 km = 8 cm / x km, where x is the unknown distance. By cross-multiplying, we get x = (8 * 5) / 1 = 40 km. 3. A bag contains 12 red balls, 8 blue balls, and 10 green balls. What is the probability of drawing a red ball or a blue ball from the bag? Answer: 0.67 or 67%. Rationale: To find the probability of drawing a red ball or a blue ball, we can use the formula P(A or B) = P(A) + P(B) - P(A and B), where A and B are the events of drawing a red ball and a blue ball, respectively. The probability of drawing a red ball is P(A) = 12 / (12 + 8 + 10) = 0.4. The probability of drawing a blue ball is P(B) = 8 / (12 + 8 + 10) = 0.267. The probability of drawing both a red ball and a blue ball is P(A and B) = 0, since it is impossible to draw two balls at once from the bag. Therefore, P(A or B) = 0.4 + 0.267 - 0 = 0.667, which can be rounded to 0.67 or expressed as a percentage as 67%. 4. A car travels at an average speed of 60 km/h for the first two hours of a trip, then at an average speed of 80 km/h for the next hour, and then at an average speed of 40 km/h for the last hour. What is the average speed of the car for the whole trip? Answer: 55 km/h. Rationale: To find the average speed of the car for the whole trip, we can use the formula average speed = total distance / total time. The total distance traveled by the car is (60 * 2) + (80 * 1) + (40 * 1) = 240 km. The total time taken by the car is 2 + 1 + 1 = 4 hours. Therefore, average speed = 240 / 4 = 60 km/h.