Problem 30 - Intermediate Algebra

Question

End Zones. One groundskeeper can paint the end zone of a football field in 2 hours. Another can paint it in 1 hour 20 minutes. How many minutes will it take them working together to paint the end zone?

Step-by-Step Solution

Answer

It will take them 48 minutes to paint the end zone together.

Step 1: Convert Hours to Minutes

First, convert the time it takes each groundskeeper to paint the end zone into minutes. One groundskeeper takes 2 hours, which is equivalent to 120 minutes. The other takes 1 hour 20 minutes, which is 80 minutes.

Step 2: Calculate Rates of Work

Find the rate of work for each groundskeeper. The first one paints the field at a rate of \( \frac{1}{120} \) end zones per minute, and the second at a rate of \( \frac{1}{80} \) end zones per minute.

Step 3: Combine the Rates

Add the rates of both groundskeepers to find their combined rate of work: \( \frac{1}{120} + \frac{1}{80} \). To add these fractions, find a common denominator, which is 240. Thus, \( \frac{1}{120} = \frac{2}{240} \) and \( \frac{1}{80} = \frac{3}{240} \). So their combined rate is \( \frac{2}{240} + \frac{3}{240} = \frac{5}{240} \) end zones per minute.

Step 4: Calculate Time Working Together

The combined rate of \( \frac{5}{240} \) end zones per minute means they complete 1 end zone in \( \frac{240}{5} = 48 \) minutes when working together.

Key Concepts

Combined Work Rate

When solving rate of work problems, understanding the concept of a combined work rate is key. This refers to how quickly two or more workers can complete a task together. Each worker has their own rate, which signifies how much of the task they can accomplish in a unit of time, such as a minute. By summing these rates, you determine the collective speed of work. In the context of our exercise, the first groundskeeper's work rate is \(\frac{1}{120}\) end zones per minute, as they take 120 minutes to complete one zone. The second groundskeeper has a work rate of \(\frac{1}{80}\) end zones per minute. To find the combined work rate, simply add these individual rates. This gives the combined work rate, \(\frac{5}{240}\), which tells us that together, they can achieve \(\frac{5}{240}\) parts of the end zone every minute. Knowing the combined work rate is crucial because it simplifies the process of finding out how long a team can complete a task together. In practice, you'll often add fractions with differing denominators, which we'll cover next.

Fractions Addition

Adding fractions, especially with unlike denominators, is crucial for calculating combined work rates. In our example, the rates \(\frac{1}{120}\) and \(\frac{1}{80}\) need a common denominator to be added. This involves a few simple steps:

• Identify the denominators: 120 and 80.
• Find the least common multiple (LCM), which is the smallest number both denominators divide into evenly. Here, the LCM is 240.
• Convert each fraction to have the LCM as the new denominator: \(\frac{1}{120} = \frac{2}{240}\) and \(\frac{1}{80} = \frac{3}{240}\).

Now that the fractions have the same denominator, they can easily be added: \(\frac{2}{240} + \frac{3}{240} = \frac{5}{240}\). This sum represents the combined rate of work. Working with fractions can initially seem tricky, but with practice, this method becomes second nature, especially in word problems related to rates.

Time Conversion

Time conversion is an essential step in solving rate of work problems. It ensures that time is expressed in uniform units, making it easier to calculate rates. In this exercise, we must convert hours to minutes for both groundskeepers. This involves simple multiplication:

• For the groundskeeper taking 2 hours: Multiply by 60 to find \(2 \times 60 = 120\) minutes.
• The second groundskeeper takes 1 hour 20 minutes, which translates to \(60 + 20 = 80\) minutes.

By converting all time measurements into minutes, we can accurately assess the rates of work and combine them. Time conversion allows you to seamlessly integrate different time units into your calculations, thus avoiding errors in your final results. Remember, consistency in units is key to solving these types of mathematics problems efficiently.