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Father and son teamwork speeds up a job, but how fast is the son alone? Put your math skills to the test with this rate and time problem!
A man can do a piece of work in 60 hours. If he takes his son with him and both work together Then the work is finished in 40 hours. How many hours will the son take to do the same job, If he worked alone on the job?
The son will take 240 hours to complete the job if he works alone.
Explanation
To find how many hours it would take the son to do the job alone, we can use the concept of combined rate. This principle states that when multiple people work together on a task, their individual rates add up to determine the combined rate at which the task is completed.
In this case, let:
- x be the time it takes the son to complete the job alone.
- y be the combined rate of the father and son working together.
We know that:
- The father alone can complete the job in 60 hours, so his individual rate is 1/60 of the job per hour.
- Together, they can finish the job in 40 hours, so their combined rate is 1/40 of the job per hour.
Using the combined rate formula, we can write the equation:
1/60 + 1/x = 1/40
Now, we can solve for x, the time it takes the son to complete the job alone:
1/x = 1/40 – 1/60 1/x = (3 – 2) / 240 1/x = 1/240 x = 240 hours
Therefore, the son would take 240 hours to complete the job if he worked alone.
Applications of Rates and Work Problems in Mathematics
Rates and work problems are fundamental concepts in mathematics with a wide range of real-world applications. They involve understanding relationships between speed, distance, time, and work, and using this knowledge to solve problems in various fields. Here are some examples of how rates and work problems are applied in different contexts:
Daily Life:
- Calculating travel times: Determining how long it will take to travel a certain distance by car, bike, or public transportation based on the speed and possible traffic conditions.
- Planning work schedules: Estimating how long it will take to complete tasks like cleaning, cooking, or yard work based on your own efficiency and potential help from others.
- Managing budgets: Calculating the cost of materials or services based on hourly rates or unit prices.
Business and Industry:
- Production planning: Optimizing production schedules and resource allocation by considering the speed of machines, labor requirements, and deadlines.
- Inventory management: Determining how much stock to keep on hand based on sales rates and lead times for new orders.
- Cost-benefit analysis: Evaluating the cost-effectiveness of different projects by considering the rate of return on investment.
Science and Engineering:
- Calculating chemical reaction rates: Understanding how quickly a chemical reaction occurs based on the concentration of reactants and temperature.
- Monitoring environmental changes: Analyzing data on pollution levels, resource depletion, or population growth to understand the rate of change and predict future trends.
- Designing efficient systems: Optimizing the flow of materials or energy in systems like pipelines, electrical grids, or transportation networks.
These are just a few examples, and the applications of rates and work problems extend far beyond these. The ability to solve these problems is a valuable skill in many fields, and understanding the underlying principles can help you make informed decisions in various aspects of your life.
Here are some additional points to consider:
- Rate can be expressed in various units: miles per hour, gallons per minute, pages printed per second, etc. It’s crucial to ensure consistent units when solving problems.
- Work is often measured as the total amount of something completed: distance traveled, gallons pumped, pages printed, etc.
- The fundamental equation for work problems is work = rate × time. This equation can be rearranged to solve for any of the three variables.
- Real-world problems may involve additional complexities: multiple rates, varying work requirements, unforeseen obstacles, etc. Adapt your approach to consider these factors.
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Source: Math Hello Kitty
Categories: Math
