Problem 79 - Algebra A Combined Function

Question

One pump fills a tank 3 times as fast as another pump. If the pumps work together, they fill the tank in 21 minutes. How long does it take each pump to fill the tank?

Step-by-Step Solution

Answer

The slower pump takes 84 minutes, and the faster pump takes 28 minutes to fill the tank individually.

Step 1: Define Variables

Let the slower pump take \( x \) minutes to fill the tank by itself. Thus, the faster pump fills the tank in \( \frac{x}{3} \) minutes. We need to find the values of \( x \) and \( \frac{x}{3} \).

Step 2: Use the Combined Work Formula

When both pumps are working together, their combined rate is the sum of their individual rates. The formula is given by \( \frac{1}{x} + \frac{1}{\frac{x}{3}} = \frac{1}{21} \).

Step 3: Simplify the Equation

The equation from the previous step can be simplified as \( \frac{1}{x} + \frac{3}{x} = \frac{1}{21} \), leading to \( \frac{4}{x} = \frac{1}{21} \).

Step 4: Solve for x

To find \( x \), solve \( \frac{4}{x} = \frac{1}{21} \). This leads to \( x = 84 \).

Step 5: Determine the Faster Pump's Time

Since the slower pump takes \( x = 84 \) minutes, and the faster pump fills the tank in \( \frac{x}{3} \), the faster pump takes \( \frac{84}{3} = 28 \) minutes.

Key Concepts

Rate Problems

Rate problems involve finding out how fast something is done. In everyday life, we encounter rate problems like driving speed, filling pools, and pumping water, just like in the example above. To solve such problems, we look at how each part contributes to the overall task.
Understanding rates means knowing the amount of work done per unit of time. For example:

• If a pump fills a tank in an hour, its rate is "one tank per hour".
• Similarly, if a pump takes "x minutes" to fill a tank, its rate becomes \( \frac{1}{x} \) of the tank per minute.

Rate problems often include combining rates, which is done by adding them together if entities work simultaneously. Breaking problems down this way makes it easier to see how parts work together to achieve a goal.

Work Formula

The work formula is a powerful tool in solving problems involving multiple entities working together. When two pumps are involved, their work rates add up. This is represented by the formula:
\[ \frac{1}{x} + \frac{1}{y} = \frac{1}{T} \]
Where:

• \( \frac{1}{x} \) is the rate of the first pump.
• \( \frac{1}{y} \) is the rate of the second pump.
• \( T \) is the total time taken when working together.

In our problem, we simplify the pump's rates to find a common total rate in filling the tank. This approach allows us to work backward to find how long it would take individually for each pump to complete the task. Using the work formula effectively turns a complicated question into a straightforward calculation.

Ratio and Proportion

Ratios and proportions help compare different quantities. In our example, the two pumps fill the tank at different speeds—one three times faster than the other. This is a ratio of 1:3.
Understanding ratios involves grasping relative speeds or capabilities. Here’s how you might think about it:

• For every unit of time the slower pump works, the faster pump completes three units of the same task.

To compute how long each pump takes, it’s necessary to establish the proportion between their rates of work. This concept is critical to deducing how individual performances add up when working together. Aligning ratios and proportions with real-life problems helps hone problem-solving skills.

Problem-Solving Strategies

Problem-solving strategies are essential in tackling algebra word problems effectively. One strategy is to start by defining variables clearly, as seen in the solution above with \( x \) representing the time taken by the slower pump.
Here are some key strategies:

• Define Variables: Choose variables touching all aspects of the problem, helping translate word problems into equations easily.
• Use Equations: Work out a step-by-step approach to simplify complex relationships and calculate unknowns.
• Check the Solution: Verify results by substituting values back into the equation to ensure logical consistency.

Practicing these strategies strengthens your ability to break down challenging algebra problems, making them manageable and, ultimately, solvable.