Final answer:
To find the number of winter sports participants at the resort, we use the ratios of skiers, snowboarders, and skaters and solve for the variable representing the ratio's common factor. After calculating the factor, we find the total participants to be 350.
Step-by-step explanation:
The question asks to find the number of winter sports participants currently at the resort given the ratio of downhill skiers to snowboarders to skaters and how this ratio changes when the number of skaters is increased by 75. We are given two ratios: the first being 5:7:2 and the second being 5:7:5 after adding 75 more skaters.
First, let's represent the number of skiers, snowboarders, and skaters as 5x, 7x, and 2x, respectively. The total number of participants is then 14x. When 75 skaters are added, the number of skaters becomes 2x + 75, and the new ratio is told to be 5:7:5. This means that the number of skiers and snowboarders remains the same, but the skaters become equal to the number of skiers and snowboarders individually, so 5x = 2x + 75.
Solving for x, we get 3x = 75, so x = 25. Now we can find the total number of participants by substituting x back into the total, 14x, which gives us 14(25) = 350 participants.