Problem 18 - Intermediate Algebra

Question

Smoke Damage. Three ventilation fans, each working alone, can clear the smoke out of a room in 12 hours, 16 hours, and 24 hours, respectively. How long would it take to clear out the smoke in the room if all three fans are used?

Step-by-Step Solution

Answer

When all three fans are used, it takes approximately 5.33 hours to clear the smoke.

Step 1: Determine Individual Rates

First, calculate how much of the room each fan clears in one hour. Fan 1 clears the room in 12 hours, so its rate is \( \frac{1}{12} \) rooms per hour. Fan 2 clears the room in 16 hours, so its rate is \( \frac{1}{16} \) rooms per hour. Fan 3 clears the room in 24 hours, so its rate is \( \frac{1}{24} \) rooms per hour.

Step 2: Calculate Total Rate

Add up the rates of all three fans to find the combined rate. This is done by summing the individual rates:\[ \text{Total Rate} = \frac{1}{12} + \frac{1}{16} + \frac{1}{24} \] To add these fractions, find a common denominator. The least common multiple of 12, 16, and 24 is 48. Hence:\[ \frac{1}{12} = \frac{4}{48}, \quad \frac{1}{16} = \frac{3}{48}, \quad \frac{1}{24} = \frac{2}{48} \] So, \( \text{Total Rate} = \frac{4}{48} + \frac{3}{48} + \frac{2}{48} = \frac{9}{48} \).

Step 3: Solve for Time

The total rate \( \frac{9}{48} \) rooms per hour tells us how much of the room is cleared per hour. To clear 1 room: Use the formula: \[ \text{Time} = \frac{1}{\text{Total Rate}} = \frac{1}{\frac{9}{48}} \]Simplify this by multiplying:\[ \text{Time} = \frac{48}{9} \] Divide 48 by 9 to get \( 5.3333 \ldots \). Therefore, it takes approximately 5.33 hours when rounded to two decimal places.

Key Concepts

Rates and Work Problems

In algebra, rates and work problems involve calculating how quickly different resources can complete a task.
These problems can be tricky but become manageable if broken down into smaller steps. In this exercise, we have three different ventilation fans, each with a unique rate of working.

• Fan 1 clears the room in 12 hours, meaning its work rate is \( \frac{1}{12} \) of the room per hour.
• Fan 2 takes 16 hours, allowing it to clear \( \frac{1}{16} \) of the room per hour.
• Fan 3 completes the task in 24 hours, equaling a rate of \( \frac{1}{24} \) of the room per hour.

Finding the combined rate involves adding these individual rates, which requires a common denominator to make addition possible.

Least Common Multiple

When adding fractions, especially those with different denominators, finding the least common multiple (LCM) is crucial.
The LCM is the smallest number that all denominators can divide into without a remainder.
In our smoke clearing problem, the denominators are 12, 16, and 24.

• To find the LCM, we list the multiples of each number.
• The smallest multiple all three share is 48.

This means all fractions need to be expressed with 48 as their denominator before being added, which simplifies the math greatly.

Fraction Addition

Once the common denominator is determined, adding fractions becomes straightforward.
Each fraction from the problem is converted like this:

• \( \frac{1}{12} \) becomes \( \frac{4}{48} \)
• \( \frac{1}{16} \) becomes \( \frac{3}{48} \)
• \( \frac{1}{24} \) becomes \( \frac{2}{48} \)
• The fractions are then added as \( \frac{4}{48} + \frac{3}{48} + \frac{2}{48} = \frac{9}{48} \)

This final fraction \( \frac{9}{48} \) represents the room-clearing rate per hour when all fans are working together.

Problem Solving Steps

To solve work and rate problems, follow a structured approach.
Begin by breaking down each task into smaller, manageable rates as we did with individual fan rates.

• Identify the rate for each task component.
• Determine a common denominator to add rates accurately.
• Add up all rates to find the overall or total rate.
• Calculate the time taken to complete the task by inverting the total rate, which gives the overall time needed.

By following these steps, such complex problems become easy to solve, demystifying the process one stage at a time.