Problem 40 - Elementary and Intermediate Algebra

Question

Office Work. In 5 hours, a secretary can address 100 envelopes. Another secretary can address 100 envelopes in 6 hours. How long would it take the secretaries, working together, to address 300 envelopes? (Hint: Think of addressing 300 envelopes as three 100 -envelope jobs.

Step-by-Step Solution

Answer

Together, the secretaries can address 300 envelopes in approximately 8.18 hours.

Step 1: Determine the Rate of Each Secretary

The rate of work can be expressed as envelopes per hour. The first secretary's rate is \( \frac{100 \text{ envelopes}}{5 \text{ hours}} = 20 \text{ envelopes per hour} \). Similarly, the second secretary's rate is \( \frac{100 \text{ envelopes}}{6 \text{ hours}} \approx 16.67 \text{ envelopes per hour} \).

Step 2: Calculate the Combined Rate

To find the rate at which both secretaries work together, add their individual rates: \( 20 + 16.67 \approx 36.67 \text{ envelopes per hour} \).

Step 3: Find the Time to Address 300 Envelopes

Use the formula \( \text{Time} = \frac{\text{Number of Envelopes}}{\text{Combined Rate}} \). For 300 envelopes: \( \text{Time} = \frac{300}{36.67} \approx 8.18 \text{ hours} \).

Step 4: Final Answer Rounded

Therefore, working together, the secretaries can address 300 envelopes in about 8.18 hours, or approximately 8 hours and 11 minutes.

Key Concepts

Understanding Rate of Work

The concept of the rate of work is much like speed but applied to tasks. It tells us how much of a task is completed in a unit of time, such as an hour. In our example with the secretaries, each has a specific rate indicating how many envelopes they can address per hour. This is determined by dividing the total task accomplished by the time taken to complete it. For the first secretary, who addresses 100 envelopes in 5 hours, the rate is calculated as follows:

• Rate of Secretary 1 = \( \frac{100 \text{ envelopes}}{5 \text{ hours}} = 20 \text{ envelopes per hour} \)

Similarly, the second secretary, who takes 6 hours to address the same number of envelopes, works at a slightly slower rate:

• Rate of Secretary 2 = \( \frac{100 \text{ envelopes}}{6 \text{ hours}} \approx 16.67 \text{ envelopes per hour} \)

This concept is fundamental as it allows us to predict how long someone might take to complete any portion of a task based on their efficiency.

Using Combined Work Rates

When two or more entities work together, we can combine their work rates to determine their overall effectiveness as a team. This is done by simply adding their individual rates of work. For the secretaries, adding their rates yields the combined work rate:

• Combined Rate = \( 20 + 16.67 \approx 36.67 \text{ envelopes per hour} \)

This combined rate then describes how efficiently the secretaries can address envelopes as a team. It indicates that if both work together, they can address around 36.67 envelopes in an hour. Understanding combined work rates allows us to tackle more complex tasks by leveraging the strengths of different workers or machines together, optimizing productivity and efficiency.

Calculating Time with Combined Work

Once we have a combined work rate, calculating the time required to complete a certain amount of work becomes straightforward. We use the formula:

• \( \text{Time} = \frac{\text{Number of Envelopes}}{\text{Combined Rate}} \)

For the secretaries tasked with addressing 300 envelopes, this translates to:

• \( \text{Time} = \frac{300}{36.67} \approx 8.18 \text{ hours} \)

This indicates that together, they can complete the task in a little over 8 hours, specifically about 8 hours and 11 minutes.

Time calculations with work rates are practical tools not only in office settings but in various fields requiring project planning and time management, helping to establish realistic timelines and expectations.