An empty bucket is put under two faucets. If one faucet is turned on alone the bucket fills in 6 minutes. If the other faucet is turned on alone the bucket fills in 4 minutes. If both are turned how many seconds will it take to fill the bucket?

rate of 1st bucket =1/6 buckets/minutes

rate of 2nd bucket = 1/4 buckets/minutes
combined rate= 1/6+1/4 or 5/12 buckets/minutes

time taken = 1/(5/12) = 12/5 or 2.4 minutes

To solve this problem, we need to calculate the rate at which each faucet can fill the bucket individually, and then add those rates together to find the combined rate when both faucets are turned on.

Let's denote the rate at which the first faucet can fill the bucket as 'F1' (in buckets per minute), and the rate at which the second faucet can fill the bucket as 'F2' (in buckets per minute).

We know that when the first faucet is turned on alone, the bucket fills in 6 minutes. So the rate of the first faucet, F1, can be calculated as:
F1 = 1 bucket / 6 minutes

Similarly, when the second faucet is turned on alone, the bucket fills in 4 minutes. So the rate of the second faucet, F2, can be calculated as:
F2 = 1 bucket / 4 minutes

Now, to find the combined rate when both faucets are turned on, we can simply add the rates:
Combined rate = F1 + F2

Let's calculate the combined rate:
Combined rate = (1/6) + (1/4)
= (2/12) + (3/12)
= 5/12

So, when both faucets are turned on, the bucket fills at a rate of 5/12 bucket per minute.

To find the time it takes to fill the bucket in seconds, we can calculate:
Time = 1 bucket / Combined rate

Let's calculate the time in seconds:
Time = 1 / (5/12) minutes
= 12/5 minutes
= (12/5) * 60 seconds
= 144 seconds

Therefore, it will take 144 seconds to fill the bucket when both faucets are turned on.

To find the time it takes to fill the bucket when both faucets are turned on, we need to determine the rate at which each faucet fills the bucket.

Let's assume that the amount of water filled in one minute by the first faucet is x gallons, and the amount of water filled in one minute by the second faucet is y gallons.

From the given information, we know that:
- The first faucet alone fills the bucket in 6 minutes, so in one minute it fills 1/6 of the bucket. Therefore, the rate of the first faucet is 1/6 of the bucket per minute.
- The second faucet alone fills the bucket in 4 minutes, so in one minute it fills 1/4 of the bucket. Therefore, the rate of the second faucet is 1/4 of the bucket per minute.

Now, let's consider what happens when both faucets are turned on together. Since they are working simultaneously, their rates of filling the bucket will add up.

So, the combined rate when both faucets are on will be:
Rate of the first faucet + Rate of the second faucet = 1/6 + 1/4

To add these fractions, we need a common denominator, which is 24 in this case:
(4/24) + (6/24) = 10/24

Hence, when both faucets are turned on, they fill 10/24 or 5/12 of the bucket per minute.

Now we can calculate the time it takes to fill the entire bucket:
1/(5/12) = 12/5

To convert this into seconds, we multiply by 60:
(12/5) x 60 = 144 seconds

Therefore, when both faucets are turned on, it will take 144 seconds to fill the bucket.