Problem 14 - Intermediate Algebra

Question

Groundskeeping. It takes a groundskeeper 45 minutes to prepare a Little League baseball field for a game. It takes his assistant 55 minutes to prepare the same field. How long will it take if they work together to prepare the field?

Step-by-Step Solution

Answer

They will take 24.75 minutes to prepare the field together.

Step 1: Determine each person's work rate

The work rate of the groundskeeper is calculated as the reciprocal of the time it takes for him to complete the job alone. Therefore, the groundskeeper's work rate is \( \frac{1}{45} \) fields per minute. Similarly, the assistant's work rate is \( \frac{1}{55} \) fields per minute.

Step 2: Combine their work rates

When two people work together, their combined work rate is the sum of their individual work rates. Hence, the combined work rate is \( \frac{1}{45} + \frac{1}{55} \) fields per minute.

Step 3: Calculate the combined work rate

To add the work rates, find a common denominator. The Least Common Multiple (LCM) of 45 and 55 is 495. Convert each work rate to have this common denominator: \( \frac{1}{45} = \frac{11}{495} \) and \( \frac{1}{55} = \frac{9}{495} \). Their combined work rate is \( \frac{11}{495} + \frac{9}{495} = \frac{20}{495} \).

Step 4: Determine time taken to complete the job

The combined work rate, \( \frac{20}{495} \) fields per minute, implies that together they can prepare 1 field in \( \frac{495}{20} \) minutes. Simplifying this fraction gives \( 24.75 \) minutes.

Key Concepts

Collaborative Work

When two or more people work together on a task, we refer to this as collaborative work. The goal is to harness the collective strengths of the individuals involved. In our groundskeeper example, the groundskeeper and his assistant are teaming up to prepare a baseball field. Each has their own pace and efficiency, yet by combining their efforts, they can complete the task quicker than either could on their own.
However, simply working together doesn't automatically lead to efficiency. Calculating their combined work rate is crucial to understanding how much time they will save. By determining the rates at which each person works and summing them up, we can see how their collaboration affects the total time needed to finish the job. This principle of combined work exemplifies how pooling resources and skills can lead to increased productivity.

Fraction Addition

In order to determine the collaborative work rate, we first need to add the individual work rates, expressed as fractions, of both workers. Fraction addition is a common math process that requires careful attention to detail, especially regarding denominators.
To add fractions accurately, the fractions must have the same denominator. If they don't, we must convert them to equivalent fractions that do. This step is essential in ensuring that the sum is calculated correctly. For the groundskeeper and his assistant, their work rates were initially fractions with 45 and 55 as denominators, respectively. The next step is finding a common denominator which in this case will directly help in summing up their efforts.

Least Common Multiple

The least common multiple (LCM) is a fundamental concept in math used when dealing with fractions that have different denominators. The LCM of two numbers is the smallest number that both numbers divide into evenly. In our scenario, we need the LCM of 45 and 55 to add their work rates.

• First, list the prime factors of each number: 45 is composed of 3x3x5, and 55 is 5x11.
• The LCM is then found by taking the highest power of each prime number present. For 45 and 55, this would be 3² (from 45), 5 (from both), and 11 (from 55), giving an LCM of 495.

Once we determine that 495 is the LCM, we convert each worker's rate to have this common denominator. This allows us to compare and sum up their joint efficiencies properly, culminating in the calculation of the total time required to complete the job together.