Problem 21 - Elementary Algebra

Question

An inlet pipe can fill a tank in 5 hours. An outlet pipe can empty the tank in 4 hours. If both pipes are open, can the tank be filled? Explain.

Step-by-Step Solution

Answer

Answer: The tank is being emptied.

Step 1: Find the rate per hour for the inlet pipe.

As the inlet pipe fills the tank in 5 hours, the rate at which it fills the tank per hour is 1/5 of the total tank.

Step 2: Find the rate per hour for the outlet pipe.

As the outlet pipe empties the tank in 4 hours, the rate at which it empties the tank per hour is 1/4 of the total tank.

Step 3: Calculate the rate per hour when both pipes are open

To find the rate at which the tank is filled when both pipes are open, we will subtract the rate at which the outlet pipe empties the tank from the rate at which the inlet pipe fills the tank. So, the equation will be: Rate per hour = (inlet pipe rate per hour) - (outlet pipe rate per hour) Rate per hour = (1/5) - (1/4)

Step 4: Simplify the expression and determine the rate per hour

To subtract the two fractions, they first need to have the same denominator, which in this case would be 20. The equation becomes: Rate per hour = (4/20) - (5/20) Now, subtract the numerators: Rate per hour = -1/20

Step 5: Interpret the result

Since the rate per hour is negative (-1/20), it means that the tank is actually being emptied and not filled when both pipes are open. Therefore, the tank cannot be filled when both pipes are open.

Key Concepts

Rate Problems

Rate problems are about determining how fast something occurs over a given time. Imagine you're filling a pool using a hose. The rate tells us how quickly the water enters the pool. In this exercise, we're dealing with two rates: one for filling and one for emptying a tank. When dealing with rate problems, it's essential to break down the task into smaller parts:

• Determine the rate for each component separately.
• Combine the rates to see the overall effect.
  

For example, if a pipe can fill a tank in 5 hours, its rate of filling is the reciprocal of those hours, or \(\frac{1}{5}\) of the tank per hour. This exercise also deals with an outlet pipe that empties the tank, creating an opposite rate that needs to be considered.

Fractions Subtraction

Subtracting fractions is an essential skill in algebraic problem-solving, especially for rate problems. When subtracting fractions, the first step is to ensure the fractions share a common denominator. This means finding a number into which both denominators can evenly divide. In this case, we're subtracting \(\frac{1}{4}\) from \(\frac{1}{5}\).

Here's how:

• Determine the least common multiple of the two denominators (4 and 5), which is 20.
• Convert each fraction to have this common denominator: \(\frac{1}{4}\) becomes \(\frac{5}{20}\) and \(\frac{1}{5}\) becomes \(\frac{4}{20}\).
• Subtract the numerators, keeping the denominator: \(\frac{4}{20} - \frac{5}{20} = \frac{-1}{20}\).

This process allows us to directly compare and manipulate the rates in the exercise.

Negative Numbers

Working with negative numbers can be challenging but is crucial in many algebraic and real-world problems. A negative result indicates something being removed or reduced. In the context of this problem, when both the inlet and outlet pipes are open, the resulting rate is \(-\frac{1}{20}\), indicating a decrease in the tank's water level.

Negative numbers remind us of:

• A subtraction or removal process.
• Opposite or reverse action, such as draining instead of filling.

Being comfortable with negative numbers means understanding that they reveal the direction of change, which is vital in analyzing scenarios like this.

Tank Problems

Tank problems often involve understanding flow rates and the impact of those rates when combined. They simulate real-world situations where addition and subtraction of rates occur, such as with filling and draining mechanisms. In this exercise, we have two opposing actions with the tank: filling and emptying.

Key steps in solving tank problems include:

• Calculating individual rates, as in how much a pipe fills or empties per hour.
• Determining the net effect by adding or subtracting these rates.
• Interpreting the result to understand how the tank's level changes over time.

Through careful calculation and understanding of the problem, we discover that the tank's water level decreases when both pipes are in action, exemplifying the complexity and importance of these problems.