Problem 32 - Elementary and Intermediate Algebra

Question

Holiday Decorating. One crew can put up holiday decorations in the mall in 8 hours. A second crew can put up the decorations in 10 hours. How long will it take if both crews work together to decorate the mall?

Step-by-Step Solution

Answer

It takes approximately 4.44 hours for both crews to decorate the mall together.

Step 1: Determine Individual Rates of Work

For the first crew, they can complete the decoration job in 8 hours, so their rate of work is \( \frac{1}{8} \) of the job per hour. For the second crew, they can complete the job in 10 hours, leading to a rate of \( \frac{1}{10} \) of the job per hour.

Step 2: Add Rates of Work

Since both crews are working together, their combined rate of work is the sum of their individual rates. Add the two rates: \( \frac{1}{8} + \frac{1}{10} \). To add these fractions, find a common denominator, which is 40. Thus, \( \frac{1}{8} = \frac{5}{40} \) and \( \frac{1}{10} = \frac{4}{40} \). Adding these gives \( \frac{5}{40} + \frac{4}{40} = \frac{9}{40} \).

Step 3: Calculate Time To Complete The Job Together

With a combined rate of \( \frac{9}{40} \) of the job per hour, we need to find how much time, \( t \), it will take for them to complete the job together, so we set up the equation \( \frac{9}{40} \times t = 1 \). Solving for \( t \), we have \( t = \frac{40}{9} \).

Step 4: Simplify and Conclude

Calculate \( \frac{40}{9} \) which is approximately 4.44 hours. Therefore, it will take them about 4 hours and 26.4 minutes to decorate the mall together.

Key Concepts

Algebra

Algebra is a powerful tool used to solve equations and understand relationships between different quantities. In work rate problems like decorating the mall, algebra helps us calculate how long it takes to complete a task by combining rates of work.
The exercise starts by expressing each crew's rate of work as a fraction of the job completed per hour. For example, the first crew completes the job in 8 hours, so their rate is \( \frac{1}{8} \) of the job per hour. Similarly, the second crew's rate is \( \frac{1}{10} \).
To find out how long both crews will take working together, we add their rates to find a combined rate \( \left( \frac{1}{8} + \frac{1}{10} \right) \). Once we calculate the combined rate, we use it in the equation \( \frac{9}{40} \times t = 1 \) to find the time \( t \) needed to complete the task together. Algebraic manipulation lets us solve for \( t \), revealing that the job can be finished in approximately 4.44 hours.

Collaborative Work

Collaborative work involves multiple parties working together to complete a task more efficiently than they could individually. The mall decoration task exemplifies this as both crews combine efforts. Each crew's work rate contributes to the overall rate, allowing the job to be completed quicker than if each worked alone.
Here's how it works: When two or more entities collaborate, their individual capabilities blend into a more efficient process. Adding up their work rates, as we did with \( \frac{1}{8} \) and \( \frac{1}{10} \), shows how collaboration brings about synergy. The total effort exerted by combining capabilities effectively reduces the total time needed.
This principle is commonly used in workplaces to maximize productivity and reduce task timelines. It highlights how teamwork and combined efforts can lead to practical solutions that leverage individual strengths.

Fraction Addition

Fraction addition is an essential mathematical process, especially when determining combined work rates, as seen in this problem-solving exercise. When combining fractions like the work rates \( \frac{1}{8} \) and \( \frac{1}{10} \), the first step is to find a common denominator. This enables us to add them accurately.
In this instance, the common denominator for 8 and 10 is 40. We convert each fraction accordingly: \( \frac{1}{8} = \frac{5}{40} \) and \( \frac{1}{10} = \frac{4}{40} \). Adding these results in \( \frac{9}{40} \).
Fraction addition allows for an accurate representation of the combined work rate, crucial for calculating the total time needed to complete joint tasks. This skill is invaluable in solving various math problems where combining parts into a whole is necessary.