Problem 22 - Elementary Algebra

Question

An inlet pipe can fill a tank in \(a\) units of time. An outlet pipe can empty the tank in \(b\) units of time. If both pipes are open, how many units of time are required to fill the tank? Are there any restrictions on \(a\) and \(b\) ?

Step-by-Step Solution

Answer

Answer: The formula for calculating the time required to fill the tank with both pipes open is given by: \(\text{Time required} = \frac{1}{\frac{1}{a} - \frac{1}{b}}\). The restriction on the values of \(a\) and \(b\) is that \(a > b\), which means the inlet pipe must fill the tank faster than the outlet pipe empties it.

Step 1: Determine the rate of filling and emptying the tank

We are given that the inlet pipe fills the tank in \(a\) units of time, and the outlet pipe empties the tank in \(b\) units of time. We can express the rate of filling/emptying the tank using the unit of "tank per time unit," with 1 being 100% of the tank. Rate of filling = \(\frac{1}{a}\) (tank per time unit) Rate of emptying = \(\frac{1}{b}\) (tank per time unit)

Step 2: Determine the net rate of filling

Now that we have the rate of filling and emptying the tank, we can find the net rate of filling by subtracting the rate of emptying from the rate of filling. Net rate of filling = Rate of filling - Rate of emptying Let's denote the net rate of filling by N: $$N = \frac{1}{a} - \frac{1}{b}$$

Step 3: Find the time required to fill the tank

Assuming the net rate of filling N is positive, we can find the time required to fill the tank by dividing the size of the tank (1) by the net rate of filling. Time required = \(\frac{1}{N}\) By substituting the value of N from step 2: $$\text{Time required} = \frac{1}{\frac{1}{a} - \frac{1}{b}}$$

Step 4: Consider restrictions on \(a\) and \(b\)

In order for the tank to fill up, the net rate of filling (N) must be positive. That is, $$\frac{1}{a} - \frac{1}{b} > 0$$ The above inequality holds true if \(a > b\). So, the time required to fill the tank with both pipes open is given by: $$\text{Time required} = \frac{1}{\frac{1}{a} - \frac{1}{b}}$$ And the restriction on \(a\) and \(b\) is that \(a > b\).

Key Concepts

Algebraic Expressions

Algebraic expressions are foundational in understanding and solving various algebra problems. They are comprised of variables, coefficients, and arithmetic operations (like addition, subtraction, multiplication, and division). In the context of our exercise regarding the filling and emptying of a tank, algebraic expressions are used to represent the rates of work.

For instance, the rate at which the inlet pipe fills the tank is expressed as \( \frac{1}{a} \) tanks per time unit. Similarly, the outlet pipe's emptying rate is \( \frac{1}{b} \) tanks per time unit. It's essential to understand that these expressions describe relationships between quantities. They simplify complex scenarios, allowing for effective problem-solving. The ability to manipulate these algebraic expressions by combining them to find the net rate of filling, denoted as \( N = \frac{1}{a} - \frac{1}{b} \) in this case, is an essential skill in algebra.

To help students grasp these concepts:

• Use specific examples with numbers to illustrate the abstract expressions.
• Break down the steps for adding, subtracting, and inverting fractions as this is a common hurdle.
• Encourage the practice of simplifying complex expressions to improve comprehension and solve problems efficiently.

Solving Inequalities

Solving inequalities is another critical skill within algebra that is closely related to solving equations. The main difference is that, while an equation will have a specific solution, an inequality will have a range of solutions. Inequalities use symbols like \( > \) (greater than), \( < \) (less than), \( \geq \) (greater than or equal to), and \( \leq \) (less than or equal to) to describe the relative sizes of two expressions.

In our tank example, we encounter an inequality to express the condition for the tank to fill: \( \frac{1}{a} - \frac{1}{b} > 0 \) implies that \( a > b \) must be true. This showcases that for the tank to fill, the rate at which it fills must exceed the rate it empties. The analysis of this inequality leads us to understand the restriction on the variables \( a \) and \( b \).

To enhance student understanding of inequalities:

• Walk through different scenarios detailing when the inequality holds versus when it doesn't.
• Use number line visuals to conceptualize the range of solutions for inequalities.
• Practice reversing the inequality sign when multiplying or dividing by negative numbers, as this is a pivotal concept often misunderstood.

Unit Rates

Unit rates provide a way to compare different quantities and are a staple in rate of work problems. A 'unit rate' describes how many units of one quantity correspond to one unit of another quantity. It is used extensively to express efficiency or speed, like miles per hour or, in our tank example, tanks per time unit.

The inlet and outlet pipes have their efficiencies represented as unit rates. Understanding unit rates allows us to figure out how the combined action of two rates affects the filling of the tank. When solving for the time required to fill the tank using both pipes, we essentially determine the combined unit rate and then calculate the inverse to find the time.

For further student comprehension of unit rates:

• Relate unit rates to real-world situations, such as speed of vehicles, to make them tangible.
• Explain that the inverse of a unit rate gives us a different perspective, often necessary to solve a problem—like finding the time to complete a task given the rate.
• Highlight that comparing unit rates directly is crucial, especially when dealing with efficiency and work rate problems.