Final answer:
Using the rates of each hose to fill the pool (1/5 of the pool per day and 1/2 per day for the garden hose and the larger hose, respectively), when combined, they fill the pool at a rate of 7/10 per day. This results in a total time of 10/7 days, or approximately 1.43 days, to fill the pool when both hoses are used together.
Step-by-step explanation:
To find out how long it will take to fill the pool when both hoses are used together, we first identify the individual rates at which each hose can fill the pool. The first garden hose can fill the pool in 5 days, which means its rate is 1/5 of the pool per day. The larger hose can fill the pool in 2 days, so its rate is 1/2 of the pool per day. To find the combined rate, we simply add these rates together:
Rate of garden hose + Rate of larger hose = 1/5 + 1/2
To combine these, we need a common denominator, which is 10:
(2/10) + (5/10) = 7/10
This means that together, the hoses can fill 7/10 of the pool in one day. To find out how many days it takes to fill the entire pool, we take the reciprocal of the combined rate:
1 / (7/10) = 10/7 days
Therefore, it will take 10/7 days, or approximately 1.43 days, to fill the pool when both hoses are used together.