Final answer:
The mystery numbers Mrs. Martin is thinking of are 10 and -10, as they are opposites and their absolute values sum up to 20. For the scale factor problems, on the map, distances of 80, 40, double 80, and half of 80 meters are represented as 10, 5, 20, and 5 centimeters respectively.
Step-by-step explanation:
Mrs. Martin is thinking of two mystery numbers. The clues provided are that the two numbers are opposites and the distance between them on the number line is 20. This means that one number is the negative equivalent of the other, and when you add the absolute values of both (because distance on the number line is always positive), they must sum up to 20. So, the two mystery numbers must be 10 and -10 since |10| + |-10| equals 20.
Looking into the scale factor problem, Calvin has drawn a map with a scale factor of 1/800. To solve each problem, you must review and apply the scale factor accordingly:
- For the distance between Calvin's house and Frank's house which is 80 meters, the scaled distance on the map would be 80 meters × (1/800), which is 0.1 meters or 10 centimeters.
- If Calvin's house is 40 meters from the park, the scaled distance on the map would be 40 meters × (1/800), resulting in 0.05 meters or 5 centimeters.
- If the distance to the corner store is double that to Frank's, the distance on the map would be 160 meters × (1/800), giving us 20 centimeters.
- If the distance to his Grandmother's is halfway, meaning 40 meters in real life, the scaled distance on the map would be 40 meters × (1/800), which is again 5 centimeters.