Problem 16 - Prealgebra and Introductory Algebra

Question

One grocery clerk can stock a shelf in 20 min. A second clerk requires 30 min to stock the same shelf. How long would it take to stock the shelf if the two clerks worked together?

Step-by-Step Solution

Answer

If both clerks work together, it will take them 12 minutes to stock the shelf.

Step 1: Determine the Rates

To start with, determine the rates at which each clerk can stock the shelf. The first clerk can stock a shelf in 20 minutes, so their rate of work is \(\frac{1}{20}\) shelves per minute. The second clerk can stock the shelf in 30 minutes, hence their work rate is \(\frac{1}{30}\) shelves per minute.

Step 2: Combine the Rates

When the two clerks work together, their individual rates of work are combined. This means that each minute, they together stock \(\frac{1}{20} + \frac{1}{30}\) of a shelf. To find a numeric value, compute this sum which gives \(\frac{1}{12}\) shelves per minute.

Step 3: Calculate the Time

To find out how long it would take for the clerks to stock the shelf together, the reciprocal of their combined rate needs to be taken, as that represents the time taken to complete 1 unit of work (i.e., stocking one shelf). Therefore, the time taken is \(\frac{1}{(\frac{1}{12})} = 12\) minutes.

Key Concepts

Collaborative Work

In collaborative work scenarios, the main idea is how two or more people working together can accomplish a task faster than one person alone. This concept is especially useful in scenarios like the grocery store problem, where multiple clerks stock shelves. Each person contributes to the task based on their individual ability.
Understanding collaborative work helps one to efficiently allocate resources and manage team productivity. The key takeaway is that combining efforts can dramatically reduce total completion time.
When dealing with collaborative work problems, remember that:

• Each worker has a specific rate at which they can complete a job.
• The total work done together is the sum of each individual's rate.
• Collaboration can achieve faster results compared to working alone.

Effective teamwork means blending these individual contributions to work smarter and finish projects quicker.

Rates of Work

The concept of rates of work in mathematics involves understanding how much work can be completed in a specific amount of time. This is typically expressed as the "rate" of completing a task, which in the grocery clerk problem is how much of a shelf is stocked per minute.
Each clerk has their own rate of work. For the clerks:

• The first clerk stocks at \(\frac{1}{20}\) shelves per minute.
• The second clerk stocks at \(\frac{1}{30}\) shelves per minute.

To find out how fast they work together, you simply add these rates. Summing up the individual rates allows us to understand how both clerks, as a team, can complete the task together more efficiently.
Understanding rates of work helps in predicting total time and managing workflow effectively, benefiting both individual and collective tasks.

Algebraic Fractions

Algebraic fractions come into play in work rate problems when dealing with collaboration. These are fractions that represent parts of a whole task being done over time. For the clerk problem, we see algebraic fractions like \(\frac{1}{20}\) and \(\frac{1}{30}\), which denote partial shelf stocking per minute.
Combining these fractions follows the same rules as regular fraction addition, requiring a common denominator:

• Convert rates of work to common denominators: \(\frac{1}{20}\) to \(\frac{3}{60}\) and \(\frac{1}{30}\) to \(\frac{2}{60}\).
• Add them: \(\frac{3}{60} + \frac{2}{60} = \frac{1}{12}\).

These calculations show how algebraic fractions make it possible to determine a combined rate of work. This concept not only applies to collaborative work but can also be extended to any work rate-based problems where collaboration or division of tasks is involved.