Sink Filling Puzzle Determining Hot Water Time With Math

Aug 1, 2025 by Felix Dubois 57 views

Unlocking the Sink-Filling Puzzle: A Mathematical Exploration

Hey guys! Today, we're diving into a classic math problem that involves filling a sink with hot and cold water. It's a fun little puzzle that combines rates and time, and we're going to break it down step-by-step. So, grab your thinking caps, and let's get started!

The Sink-Filling Scenario

Here's the scenario we're working with: Imagine you're filling your sink. You notice that it takes 5 minutes to fill the sink using only the cold water tap. Now, if you turn on both the hot and cold water taps together, it only takes 2 minutes to fill the same sink. The challenge is to figure out how long it would take to fill the sink using only the hot water tap. To do this, we need to find the correct table that can be used to determine $x$, which represents the time in minutes it takes to fill the sink with just the hot water.

Breaking Down the Problem

To solve this, we need to think about the rates at which the sink is being filled. Let's define some variables to make things clearer:

Let $C$ be the rate at which the cold water tap fills the sink (in sinks per minute).
Let $H$ be the rate at which the hot water tap fills the sink (in sinks per minute).
Let $x$ be the time it takes to fill the sink using only the hot water tap (in minutes).

Understanding Rates: The rate at which something happens is simply the amount of work done per unit of time. In this case, the "work" is filling the sink, and the "time" is measured in minutes. So, if the cold water tap fills the sink in 5 minutes, its rate is 1/5 sinks per minute. Similarly, if both taps together fill the sink in 2 minutes, their combined rate is 1/2 sinks per minute.

Setting Up the Equations

From the given information, we can set up the following equations:

The cold water tap fills the sink in 5 minutes, so its rate is:

$C = \frac{1}{5}$
When both taps are on, they fill the sink in 2 minutes, so their combined rate is:

$C + H = \frac{1}{2}$
We want to find the time it takes for the hot water tap to fill the sink alone, which we've called $x$ . So, the rate of the hot water tap is:

$H = \frac{1}{x}$

Now, we have a system of equations that we can use to solve for $x$ . The key is to substitute the known values and solve for the unknown.

Solving for the Hot Water Time

Let's substitute the value of $C$ from equation (1) into equation (2):

$\frac{1}{5} + H = \frac{1}{2}$

Now, we can solve for $H$ :

$H = \frac{1}{2} - \frac{1}{5}$

To subtract these fractions, we need a common denominator, which is 10:

$H = \frac{5}{10} - \frac{2}{10}$

$H = \frac{3}{10}$

So, the rate at which the hot water tap fills the sink is 3/10 sinks per minute. Now, we can use equation (3) to find $x$ :

$\frac{1}{x} = \frac{3}{10}$

To solve for $x$ , we can take the reciprocal of both sides:

$x = \frac{10}{3}$

Therefore, it takes 10/3 minutes, or 3 minutes and 20 seconds, to fill the sink using only the hot water tap.

Identifying the Correct Table

Now that we've solved the problem, let's think about how we could represent this information in a table. The table should help us visualize the rates and times involved. We need to focus on a table that correctly sets up the relationship between the rates of the cold water, the hot water, and their combined rate.

What Makes a Table Correct?

A correct table for this problem will typically show the individual rates of the cold and hot water taps, as well as the combined rate when both taps are on. It should also help in setting up an equation to solve for the unknown time, $x$ . The key is to represent the rates as fractions of the sink filled per minute.

Key Elements of a Correct Table:

Individual Rates: The table should clearly show the rate of the cold water tap (1/5) and the rate of the hot water tap (1/x). These rates represent the fraction of the sink each tap fills in one minute.
Combined Rate: The table should also show the combined rate when both taps are on (1/2). This represents the fraction of the sink filled in one minute when both taps are running.
Equation Setup: The table should lead to the equation that relates these rates: (Rate of Cold Water) + (Rate of Hot Water) = (Combined Rate), which is $\frac{1}{5} + \frac{1}{x} = \frac{1}{2}$ .

Common Mistakes in Tables

When looking at different tables, watch out for common mistakes:

Incorrect Rates: Some tables might incorrectly represent the rates as the time it takes to fill the sink, rather than the fraction of the sink filled per minute. For example, they might use 5 instead of 1/5 for the cold water tap.
Wrong Equation: A table might lead to an incorrect equation that doesn't properly relate the individual rates to the combined rate. For example, it might add times instead of rates.
Missing Information: Some tables might not include all the necessary information, such as the individual rates or the combined rate.

Constructing the Ideal Table

To make sure we understand what the correct table should look like, let's construct one ourselves:

Water Source	Time to Fill Sink (minutes)	Rate (sinks per minute)
Cold Water	5	1/5
Hot Water	x	1/x
Combined (Hot + Cold)	2	1/2

This table clearly shows the time it takes each water source to fill the sink and the corresponding rate. It also sets up the equation:

$\frac{1}{5} + \frac{1}{x} = \frac{1}{2}$

This is the equation we used to solve for $x$ . So, any correct table should lead to this equation.

Real-World Applications

This type of problem isn't just a math exercise; it has real-world applications. Understanding rates and how they combine is crucial in various fields:

Engineering: Engineers use rate calculations to design systems involving fluid flow, such as pipelines or drainage systems. They need to know how different flow rates combine to ensure efficient operation.
Construction: In construction, rate calculations are used to estimate how long it will take to complete a project. For example, if one team can lay 100 bricks per hour and another team can lay 120 bricks per hour, their combined rate is essential for project planning.
Manufacturing: Manufacturers use rate calculations to optimize production processes. They need to know how quickly machines can produce goods and how different machines working together will affect overall output.
Everyday Life: We use rate calculations in everyday life, often without even realizing it. For example, when filling a pool with multiple hoses, we're essentially combining rates to determine how long it will take to fill the pool.

By understanding these concepts, we can solve similar problems in various contexts, making this a valuable skill beyond the classroom.

Conclusion: Mastering Rate Problems

So, there you have it! We've successfully navigated the sink-filling problem by breaking it down into manageable steps. We defined variables, set up equations, solved for the unknown, and even identified what the correct table should look like. Remember, the key to solving these types of problems is to understand the concept of rates and how they combine.

Final Thoughts

Key Takeaways:

Rates are crucial: Always think in terms of rates (amount per unit of time) when dealing with these problems.
Set up equations: Translate the word problem into mathematical equations.
Check your work: Make sure your solution makes sense in the context of the problem.

By mastering these skills, you'll be well-equipped to tackle similar problems and apply these concepts in real-world situations. Keep practicing, and you'll become a pro at solving rate problems in no time! And always remember, math can be fun – especially when it involves filling sinks!