Calculating Time to Fill a Bathtub Using Both Cold and Hot Water Faucets

Are you wondering how long it will take to fill a bathtub if you use both the cold and hot water faucets together? This article will guide you through the process of determining the total time needed based on the individual filling rates of each faucet.

Determining the Rates of Each Faucet

Let's start by calculating the rates of the cold and hot water faucets individually. We know that:

The cold water faucet takes 7 minutes to fill the bathtub. The hot water faucet takes 12 minutes to fill the bathtub.

Cold Water Faucet Rate

The rate at which the cold water faucet fills the bathtub is:

Ratecold (frac{1 , text{tub}}{7 , text{minutes}} frac{1}{7} , text{tubs per minute})

Hot Water Faucet Rate

The rate at which the hot water faucet fills the bathtub is:

Ratehot (frac{1 , text{tub}}{12 , text{minutes}} frac{1}{12} , text{tubs per minute})

Combined Rate

To find the combined rate when both faucets are used simultaneously, we need to add their individual rates:

Combined Rate Ratecold Ratehot (frac{1}{7} frac{1}{12})

To add these fractions, we need a common denominator. The least common multiple (LCM) of 7 and 12 is 84. Converting each rate to have this common denominator:

(frac{1}{7} frac{12}{84})

(frac{1}{12} frac{7}{84})

Now, we can add them:

Combined Rate (frac{12}{84} frac{7}{84} frac{19}{84} , text{tubs per minute})

Time to Fill the Tub

To find the time needed to fill one tub with both faucets, we take the reciprocal of the combined rate:

t (frac{1 , text{tub}}{frac{19}{84} , text{tubs per minute}} frac{84}{19} , text{minutes} approx 4.42 , text{minutes})

Therefore, it will take approximately 4.42 minutes to fill the bathtub using both the cold and hot water faucets.

Solving for x

If you are given the following equation to solve for (x):

(frac{1}{x} frac{1}{7} frac{1}{12})

Let's solve for (x).

First, find the common denominator of 7 and 12, which is 84:

(frac{1}{7} frac{12}{84})

(frac{1}{12} frac{7}{84})

Then add the fractions:

(frac{1}{x} frac{12}{84} frac{7}{84} frac{19}{84})

Now, take the reciprocal of (frac{19}{84}) to solve for (x):

(frac{1}{x} frac{19}{84})

(frac{84}{19} x)

Thus, (x approx 4.42 , text{minutes}).

Conclusion

Understanding the rates and combined rates of water flow from both faucets not only helps in practical scenarios but also reinforces the concept of rates in mathematics. By breaking down the problem into simple steps, we can derive useful solutions in a variety of real-world situations.