When only the cold faucet is fully turned on, a bathtub fills in 8 minutes. When only the hot faucet is turned on, the same bathtub fiills in 12 minutes. If both the cold and hot faucet are fully turned on, how long will it take to fill the bathtub?

hot ---> 1/12 tub/min

cold --> 1/8 tub/min

hot + cold ---> (1/12 + 1/8 )tub/min

= (2/24 + 3/24) = 5/24 tub/min

so 24/5 min = 4.8 minutes

To solve this problem, we'll use the concept of rates.

Let's say the cold faucet has a rate of C bathtubs per minute and the hot faucet has a rate of H bathtubs per minute.

From the information given, we know that when only the cold faucet is fully turned on, the bathtub fills in 8 minutes. So the rate of the cold faucet is 1 bathtub per 8 minutes, which can be written as:
C = 1/8 bathtubs per minute.

Similarly, when only the hot faucet is turned on, the same bathtub fills in 12 minutes. So the rate of the hot faucet is 1 bathtub per 12 minutes, which can be written as:
H = 1/12 bathtubs per minute.

Now, to find the combined rate of both faucets, we can simply add the rates together:
C + H = 1/8 + 1/12 bathtubs per minute.

To add these fractions, we need to find a common denominator, which in this case is 24. So we rewrite the equation as:
C + H = 3/24 + 2/24 bathtubs per minute.

Now we can add the fractions:
C + H = 5/24 bathtubs per minute.

Therefore, the combined rate of both faucets is 5/24 bathtubs per minute.

To find the time it takes to fill the bathtub when both faucets are fully turned on, we can use the formula:
Time = Volume / Rate.

However, since the volume of the bathtub is not given, we can assume it to be 1 (since we are dealing with rates of 1 bathtub per time).

So the time it takes to fill the bathtub when both faucets are fully turned on is:
Time = 1 / (5/24) minutes.

To divide by a fraction, we can multiply by its reciprocal:
Time = 1 * (24/5) minutes.

Simplifying, we get:
Time = 24/5 minutes.

Therefore, it will take 24/5 minutes or 4.8 minutes to fill the bathtub when both the cold and hot faucet are fully turned on.