Problem 36 - Elementary Algebra

Question

Jose can build a small shed by himself in 26 hours. Alex builds the same small shed in 2 days. How long would it take them to build the shed working together?

Step-by-Step Solution

Answer

Jose and Alex together can build the shed in approximately 16.86 hours.

Step 1: Find Alex's Rate

First, we need to convert the time Alex takes from days to hours since Jose's rate is already in hours. We know that 1 day = 24 hours. Therefore, in 2 days, Alex takes \(2 \times 24 = 48\) hours to build the shed on his own. So, Alex's rate of work is \(\frac{1}{48}\) of the shed per hour.

Step 2: Find Jose's Rate

Jose can build the shed by himself in 26 hours, so his rate of working is \(\frac{1}{26}\) of the shed per hour.

Step 3: Find Combined Rate

To find their combined rate of working together, add their individual rates: \(\frac{1}{26} + \frac{1}{48} \). We need a common denominator to add these fractions, which is 624 (the least common multiple of 26 and 48).

Step 4: Calculate Combined Rate

Convert \(\frac{1}{26}\) and \(\frac{1}{48}\) with a denominator of 624. \(\frac{1}{26} = \frac{24}{624}\) and \(\frac{1}{48} = \frac{13}{624}\). Add the fractions: \(\frac{24}{624} + \frac{13}{624} = \frac{37}{624}\). Therefore, together they complete \(\frac{37}{624}\) of the shed per hour.

Step 5: Calculate Time to Complete Task

Now, we need to find the number of hours it takes to complete the entire shed together. We do this by taking the reciprocal of their combined rate: \(\frac{624}{37}\). Calculate \(\frac{624}{37} \approx 16.86\) hours.

Key Concepts

Understanding Rates of Work

When tackling work problems like the one with Jose and Alex building a shed, understanding rates of work is essential. A rate of work tells us how much of a task can be completed in a certain amount of time. For example:

• Jose can finish the entire shed in 26 hours. Therefore, in one hour, he completes \(\frac{1}{26}\) of the shed.
• Alex, on the other hand, takes 48 hours to build the same shed. Hence, in one hour, his rate is \(\frac{1}{48}\).

Recognizing individual rates is the foundation of solving combined work problems and finding out how two people can finish a task together more efficiently. It simplifies our calculations by breaking down the total work into manageable pieces.

The Combined Work Formula

The combined work formula helps us determine how long it takes for two individuals working together to complete a task. The formula is an extension of individual rates of work:

• The combined rate is simply the sum of individual rates. For Jose and Alex, this is: \(\frac{1}{26} + \frac{1}{48}\).
• Once you find the sum, it becomes the rate per hour they work together on the task.

This combined rate tells us how much of the task they can complete together in one hour. Calculating their combined rate means we can quickly figure out the total time needed to finish the job together.

Using Least Common Multiple (LCM)

Working with fractions, as seen in the combined work formula, often requires finding a least common multiple (LCM). This step is necessary to add fractions with different denominators.

• LCM is the smallest number that each of the denominators can divide into without leaving a remainder.
• In Jose and Alex's case, the denominators 26 and 48 have an LCM of 624. Using this LCM, the fractions \(\frac{1}{26}\) and \(\frac{1}{48}\) are converted to \(\frac{24}{624}\) and \(\frac{13}{624}\) respectively.

Having a common denominator allows us to directly add the fractions, which is a crucial step in finding the combined work rate.

Understanding Reciprocals

In math, a reciprocal is what you multiply a number by to get one. For a fraction \(\frac{a}{b}\), its reciprocal is \(\frac{b}{a}\). Reciprocals are crucial when dealing with rate problems:

• To calculate how long it takes for Jose and Alex to finish the shed together, we first find their combined rate as fractions added together, \(\frac{37}{624}\).
• Next, the reciprocal of this rate tells us how much time the whole task will take: \(\frac{624}{37}\).
• The result gives us the total hours they need when working together, simplifying the whole process into a clear time estimate.

Understanding reciprocals helps convert from portions of tasks per hour to total time, making it very useful in problem-solving for work rates.