A fruit basket contains apples, oranges, and bananas. The ratio of the number of apples to the number of oranges is 3:4, and the ratio of the number of oranges to the number of bananas is 5:2. If there are a total of 180 fruits in the basket, what percentage of the fruits are oranges?
Let's use the ratio of apples to oranges to find the ratio of apples to bananas. Since the ratio of apples to oranges is 3:4, and the ratio of oranges to bananas is 5:2, we can multiply the two ratios to get the ratio of apples to bananas. This gives us 3 * 5 = 15, and 4 * 2 = 8, so the ratio of apples to bananas is 15:8.
If there are a total of 180 fruits in the basket, we can calculate the total number of oranges by multiplying the number of oranges per ratio unit by the number of ratio units. There are 4 units in the orange:apple ratio (3 + 4 = 7 total units), and 180/7 = 25.714 apples in one unit of the ratio. Multiplying this by the 4 units of oranges, we find there are 25.714 * 4 = <<25.714*4=102.856>>102.856 oranges in the basket.
Finally, to convert this to a percentage, we divide the number of oranges by the total number of fruits in the basket and multiply by 100. We find 102.856 / 180 * 100 = 57.14%, rounded to 2 decimal places.
Thus, the percentage of fruits that are oranges is 57.14%. Answer: \boxed{57.14}.