It Takes 5 Minutes To Fill The Sink Using Only The Cold Water. If Both The Hot Water And The Cold Water Are Turned On, It Takes 2 Minutes To Fill The Sink. Which Is The Correct Table That Can Be Used To Determine $x$, The Time In Minutes It

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Introduction

In this article, we will explore a classic problem in mathematics that involves the concept of rates and ratios. The problem states that it takes 5 minutes to fill a sink using only the cold water, but when both the hot water and the cold water are turned on, it takes only 2 minutes to fill the sink. We will use a table to determine the time in minutes, denoted as xx, it takes to fill the sink when both the hot and cold water are turned on.

Understanding the Problem

Let's break down the problem and understand what is being asked. We have two scenarios:

  1. Cold Water Only: It takes 5 minutes to fill the sink using only the cold water.
  2. Both Hot and Cold Water: It takes 2 minutes to fill the sink when both the hot and cold water are turned on.

We need to find the time in minutes, denoted as xx, it takes to fill the sink when both the hot and cold water are turned on.

Creating a Table to Determine xx

To solve this problem, we can create a table that shows the rate at which the sink is being filled in both scenarios. Let's assume that the rate at which the cold water fills the sink is rcr_c and the rate at which the hot water fills the sink is rhr_h.

Scenario Time (minutes) Rate (sink/minute)
Cold Water Only 5 rcr_c
Both Hot and Cold Water 2 rc+rhr_c + r_h

We can now use this table to determine the value of xx, the time in minutes it takes to fill the sink when both the hot and cold water are turned on.

Analyzing the Table

From the table, we can see that the rate at which the sink is being filled when both the hot and cold water are turned on is rc+rhr_c + r_h. We also know that it takes 2 minutes to fill the sink in this scenario.

We can set up an equation using the table:

1rc+rh=2\frac{1}{r_c + r_h} = 2

We can now solve for rc+rhr_c + r_h:

rc+rh=12r_c + r_h = \frac{1}{2}

We can now substitute this value back into the table:

Scenario Time (minutes) Rate (sink/minute)
Cold Water Only 5 rcr_c
Both Hot and Cold Water 2 12\frac{1}{2}

Solving for xx

We can now use the table to determine the value of xx, the time in minutes it takes to fill the sink when both the hot and cold water are turned on.

We can set up an equation using the table:

1rc+rh=x\frac{1}{r_c + r_h} = x

We can now substitute the value of rc+rhr_c + r_h that we found earlier:

112=x\frac{1}{\frac{1}{2}} = x

Simplifying the equation, we get:

2=x2 = x

Therefore, the time in minutes it takes to fill the sink when both the hot and cold water are turned on is x=2x = 2 minutes.

Conclusion

In this article, we used a table to determine the time in minutes it takes to fill a sink when both the hot and cold water are turned on. We found that the time in minutes is x=2x = 2 minutes. This problem is a classic example of how rates and ratios can be used to solve problems in mathematics.

References

Introduction

In our previous article, we explored a classic problem in mathematics that involves the concept of rates and ratios. The problem states that it takes 5 minutes to fill a sink using only the cold water, but when both the hot water and the cold water are turned on, it takes only 2 minutes to fill the sink. We used a table to determine the time in minutes, denoted as xx, it takes to fill the sink when both the hot and cold water are turned on.

In this article, we will answer some of the most frequently asked questions related to this problem.

Q: What is the rate at which the cold water fills the sink?

A: The rate at which the cold water fills the sink is rcr_c. We can find this rate by dividing the volume of the sink by the time it takes to fill the sink using only the cold water. In this case, the rate is 15\frac{1}{5} sink/minute.

Q: What is the rate at which the hot water fills the sink?

A: The rate at which the hot water fills the sink is rhr_h. We can find this rate by dividing the volume of the sink by the time it takes to fill the sink when both the hot and cold water are turned on. In this case, the rate is 12\frac{1}{2} sink/minute.

Q: How do we find the value of xx, the time in minutes it takes to fill the sink when both the hot and cold water are turned on?

A: To find the value of xx, we can use the table we created earlier. We can set up an equation using the table:

1rc+rh=x\frac{1}{r_c + r_h} = x

We can now substitute the values of rcr_c and rhr_h that we found earlier:

115+12=x\frac{1}{\frac{1}{5} + \frac{1}{2}} = x

Simplifying the equation, we get:

2=x2 = x

Therefore, the time in minutes it takes to fill the sink when both the hot and cold water are turned on is x=2x = 2 minutes.

Q: What if the rate at which the hot water fills the sink is different from the rate at which the cold water fills the sink?

A: If the rate at which the hot water fills the sink is different from the rate at which the cold water fills the sink, we can still use the table to determine the value of xx. We can set up an equation using the table:

1rc+rh=x\frac{1}{r_c + r_h} = x

We can now substitute the values of rcr_c and rhr_h that we found earlier:

1rc+rh=x\frac{1}{r_c + r_h} = x

Simplifying the equation, we get:

115+rh=x\frac{1}{\frac{1}{5} + r_h} = x

We can now solve for xx:

x=115+rhx = \frac{1}{\frac{1}{5} + r_h}

Therefore, the time in minutes it takes to fill the sink when both the hot and cold water are turned on is x=115+rhx = \frac{1}{\frac{1}{5} + r_h} minutes.

Q: Can we use this problem to solve other problems involving rates and ratios?

A: Yes, we can use this problem to solve other problems involving rates and ratios. The concept of rates and ratios is a fundamental concept in mathematics, and it can be applied to a wide range of problems.

For example, we can use this problem to solve problems involving the rate at which a liquid flows through a pipe, or the rate at which a gas expands in a container. We can also use this problem to solve problems involving the rate at which a population grows or declines.

Conclusion

In this article, we answered some of the most frequently asked questions related to the problem of filling a sink using only the cold water and both the hot and cold water. We used a table to determine the time in minutes, denoted as xx, it takes to fill the sink when both the hot and cold water are turned on. We also discussed how to use this problem to solve other problems involving rates and ratios.

References