Calculating Time To Complete Work Together Lilis And Faizhal Math Problem
Hey guys! Let's dive into a fun math problem today. We're going to figure out how long it takes Lilis and Faizhal to finish a job if they work together. This is a classic type of problem that involves understanding rates of work. So, buckle up, and let's get started!
Understanding the Problem
First, let's break down the information we have:
- Lilis can finish a job in 10 hours.
- Faizhal can finish the same job in 5 hours.
Our goal is to find out how long it takes if they work together. To do this, we need to think about how much of the job each person completes in one hour. This is where the concept of work rate comes in handy. Work rate is essentially the amount of work done per unit of time.
Lilis's Work Rate
Lilis's work rate is the amount of work she can complete in one hour. Since she finishes the entire job in 10 hours, she completes 1/10 of the job in one hour. Think of it like this: if the job is a pie, Lilis eats 1/10 of the pie every hour.
Faizhal's Work Rate
Similarly, Faizhal's work rate is the amount of work he can complete in one hour. Since he finishes the job in 5 hours, he completes 1/5 of the job in one hour. Faizhal is a faster worker, so he eats 1/5 of the pie every hour.
Combining Work Rates
When Lilis and Faizhal work together, their work rates combine. This means that the fraction of the job they complete together in one hour is the sum of their individual work rates. To find this combined work rate, we add Lilis's work rate (1/10) and Faizhal's work rate (1/5).
So, our equation looks like this:
Combined work rate = Lilis's work rate + Faizhal's work rate
Combined work rate = 1/10 + 1/5
Now, we need to add these fractions. To do this, we need a common denominator. The least common multiple of 10 and 5 is 10. So, we'll convert 1/5 to have a denominator of 10. 1/5 is equivalent to 2/10.
So, our equation becomes:
Combined work rate = 1/10 + 2/10
Combined work rate = 3/10
This means that together, Lilis and Faizhal complete 3/10 of the job in one hour.
Calculating the Time to Complete the Job Together
Now that we know they complete 3/10 of the job in one hour, we can find out how long it takes them to complete the entire job. If they do 3/10 of the job each hour, we need to find out how many "3/10s" fit into the whole job (which we can think of as 1). To do this, we take the reciprocal of the combined work rate.
The reciprocal of 3/10 is 10/3. This represents the number of hours it takes them to complete the job together.
So, the time it takes them to complete the job together is 10/3 hours. This is an improper fraction, so let's convert it to a mixed number to make it easier to understand. 10 divided by 3 is 3 with a remainder of 1. So, 10/3 is equal to 3 and 1/3 hours.
Converting to Hours and Minutes
We know it takes them 3 full hours, but what about the 1/3 of an hour? To convert 1/3 of an hour to minutes, we multiply it by 60 (since there are 60 minutes in an hour).
(1/3) * 60 minutes = 20 minutes
So, it takes them 3 hours and 20 minutes to complete the job together.
Step-by-Step Solution
Let's recap the steps we took to solve this problem:
- Identify individual work rates: Lilis's work rate is 1/10 (job per hour), and Faizhal's work rate is 1/5 (job per hour).
- Combine work rates: Add the individual work rates to find the combined work rate. 1/10 + 1/5 = 3/10 (job per hour).
- Find the reciprocal of the combined work rate: The reciprocal of 3/10 is 10/3, which represents the time it takes to complete the job together.
- Convert to mixed number (optional): 10/3 is equal to 3 and 1/3 hours.
- Convert fractional hours to minutes (optional): 1/3 of an hour is 20 minutes.
- State the final answer: It takes Lilis and Faizhal 3 hours and 20 minutes to complete the job together.
Why This Matters
This type of problem isn't just a math exercise; it has real-world applications. Think about project management, where you might have multiple team members working on different parts of a project. Understanding how to combine work rates can help you estimate how long it will take to complete the entire project. It also helps in understanding efficiency and how combining resources can speed up task completion.
Practice Problems
To really nail this concept, let's try a couple of practice problems.
Practice Problem 1
Sarah can paint a room in 8 hours, and John can paint the same room in 12 hours. How long will it take them to paint the room if they work together?
- Hint: Follow the same steps we used in the example problem. Find their individual work rates, combine them, and then find the reciprocal.
Practice Problem 2
Two pipes can fill a tank. Pipe A can fill the tank in 6 hours, and pipe B can fill the tank in 9 hours. If both pipes are opened at the same time, how long will it take to fill the tank?
- Hint: This is the same type of problem, just with pipes instead of people. The concept of work rate still applies.
Try these problems on your own, and let's see if you get the hang of it! The key is to break down the problem into smaller steps and understand what each step represents.
Common Mistakes to Avoid
When solving these types of problems, it's easy to make a few common mistakes. Let's go over some of these so you can avoid them.
Mistake 1: Adding the Times Directly
A common mistake is to simply add the times it takes each person to complete the job individually. For example, in our original problem, some might think that it takes 10 hours + 5 hours = 15 hours. This is incorrect because when people work together, they are completing the job at a faster rate than either of them could alone. Always remember to work with rates (fractions of the job completed per unit of time), not the times themselves.
Mistake 2: Forgetting to Find the Reciprocal
Another mistake is to calculate the combined work rate correctly (e.g., 3/10 in our example) but then forget to take the reciprocal to find the time it takes to complete the job together. Remember, the combined work rate represents the fraction of the job completed in one hour. To find the total time, you need to find the reciprocal.
Mistake 3: Incorrectly Adding Fractions
Adding fractions incorrectly can also lead to mistakes. Make sure you find a common denominator before adding the fractions. For example, when adding 1/10 and 1/5, you need to convert 1/5 to 2/10 before adding them together.
Mistake 4: Not Understanding the Concept of Work Rate
The fundamental concept of work rate is crucial for solving these problems. If you don't understand that work rate is the amount of work done per unit of time, you might struggle to set up the problem correctly. Make sure you understand this concept before attempting to solve these problems.
Real-World Examples
These types of problems aren't just theoretical; they show up in many real-world situations. Let's look at some examples.
Example 1: Construction
In construction, understanding work rates can help in project planning. If one team can build a wall in 8 hours and another team can build the same wall in 6 hours, you can calculate how long it will take if both teams work together. This helps in estimating project timelines and resource allocation.
Example 2: Manufacturing
In a manufacturing plant, different machines might have different production rates. If one machine can produce 100 units in an hour and another can produce 120 units in an hour, you can calculate the total production rate if both machines are running simultaneously. This helps in optimizing production schedules.
Example 3: Software Development
In software development, multiple developers might work on different parts of the same project. If one developer can complete a module in 20 hours and another can complete the same module in 15 hours, you can estimate how long it will take if they collaborate. This helps in project management and task assignment.
Example 4: Home Improvement
Even in home improvement projects, this concept is useful. If you can paint a room in 6 hours and your friend can paint the same room in 4 hours, you can figure out how long it will take if you both work together. This helps in planning your weekend projects!
Conclusion
So, there you have it! Solving problems involving combined work rates is all about understanding individual work rates, combining them, and then finding the total time. Remember to avoid common mistakes like adding times directly or forgetting to find the reciprocal. With a little practice, you'll be able to tackle these problems like a pro!
Remember our original problem? Lilis and Faizhal, working together, can complete the job in 3 hours and 20 minutes. Not bad, guys! Keep practicing, and you'll master these types of problems in no time.
