Final answer:
To solve the problem, use the concept of rates and set up an equation with the combined rates of the hoses. The equation is 1/30 + 1/20 = 1/x, where x represents the time it takes for both hoses to fill the pool. Solve for x to find that it will take 12 minutes for both hoses to fill the pool.
Step-by-step explanation:
To solve this problem, we can use the concept of rates. Let x represent the number of minutes it takes for both hoses to fill the pool. Based on the given information, the rate at which the medium water hose fills the pool is 1 pool per 30 minutes, which can be written as 1/30 pools per minute. Similarly, the rate at which the larger water hose fills the pool is 1 pool per 20 minutes, or 1/20 pools per minute. When both hoses are turned on at the same time, their rates of filling the pool are combined. So, we can set up the equation:
1/30 + 1/20 = 1/x
To solve for x, we can find a common denominator of 60x and multiply both sides of the equation by it:
2x + 3x = 60
5x = 60
x = 12
Therefore, it will take both hoses 12 minutes to fill the pool.