Working 5 hours a day, A can complete a task in 8 days and working 6 hours a day, B can finish the same task in 10 days, working 8 hours a day, they can jointly complete the task in __________.
3 days
Understanding Work and Time Problems
This problem involves calculating the time taken by individuals and a group to complete a task, based on their working hours per day and total days. These are classic work and time problems, often encountered in quantitative aptitude sections of exams. The key is to determine the rate at which each person completes the work.
Calculating Individual Work Rates
First, let's find the total number of hours each person works to complete the task individually.
- A works 5 hours per day and finishes the task in 8 days.
- Total hours worked by A = Hours per day × Number of days
- Total hours worked by A = \(5 \text{ hours/day} \times 8 \text{ days} = 40 \text{ hours}\)
- B works 6 hours per day and finishes the same task in 10 days.
- Total hours worked by B = Hours per day × Number of days
- Total hours worked by B = \(6 \text{ hours/day} \times 10 \text{ days} = 60 \text{ hours}\)
Now, let's assume a 'Total Work Unit'. A common method is to take the Least Common Multiple (LCM) of the total hours worked by each person. The LCM of 40 and 60 is 120. Let's assume the total task is 120 units of work.
Now we can find the work rate of A and B per hour.
- A's hourly rate = Total Work Units / Total hours worked by A
- A's hourly rate = \(120 \text{ units} / 40 \text{ hours} = 3 \text{ units/hour}\)
- B's hourly rate = Total Work Units / Total hours worked by B
- B's hourly rate = \(120 \text{ units} / 60 \text{ hours} = 2 \text{ units/hour}\)
Calculating Combined Work Rate
When A and B work together, their work rates add up. They work 8 hours a day jointly.
- Combined hourly rate of A and B = A's hourly rate + B's hourly rate
- Combined hourly rate of A and B = \(3 \text{ units/hour} + 2 \text{ units/hour} = 5 \text{ units/hour}\)
They work for 8 hours each day. So, their combined work per day is:
- Combined daily rate = Combined hourly rate × Hours worked per day
- Combined daily rate = \(5 \text{ units/hour} \times 8 \text{ hours/day} = 40 \text{ units/day}\)
Calculating Days to Complete Task Jointly
To find the number of days they will take to complete the total task (120 units) while working together 8 hours a day, we use the formula:
Number of days = Total Work Units / Combined daily rate
Number of days = \(120 \text{ units} / 40 \text{ units/day}\)
Number of days = 3 days
So, working 8 hours a day, A and B can jointly complete the task in 3 days.
| Person | Hours/Day | Days to Complete | Total Hours to Complete | Hourly Rate (units/hour) |
|---|---|---|---|---|
| A | 5 | 8 | 40 | 3 |
| B | 6 | 10 | 60 | 2 |
| A & B (Jointly) | 8 | ? | - | 5 (combined) |
Step-by-Step Solution Recap
- Calculate total hours each person takes individually: A = \(5 \times 8 = 40\) hours, B = \(6 \times 10 = 60\) hours.
- Find the LCM of total hours to represent total work units: LCM(40, 60) = 120 units.
- Calculate individual hourly rates: A = \(120/40 = 3\) units/hour, B = \(120/60 = 2\) units/hour.
- Calculate their combined hourly rate: \(3 + 2 = 5\) units/hour.
- Calculate their combined daily rate when working 8 hours a day: \(5 \text{ units/hour} \times 8 \text{ hours/day} = 40\) units/day.
- Calculate the number of days to complete 120 units of work together: \(120 \text{ units} / 40 \text{ units/day} = 3\) days.
Revision Table: Key Concepts in Work and Time
| Concept | Explanation | Formula |
|---|---|---|
| Total Work | The total amount of work to be done (often assumed as 1 unit or an LCM value). | Rate × Time |
| Work Rate | The amount of work done per unit of time (e.g., per hour or per day). | Work / Time |
| Time Taken | The duration required to complete the work. | Work / Rate |
| Combined Rate | The sum of individual rates when people work together. | Rate\(_{A}\) + Rate\(_{B}\) + ... |
Additional Information: Variations in Work and Time Problems
Work and time problems can come in many forms. Some variations include:
- Problems involving efficiency ratios.
- Problems where some workers leave or join after a few days.
- Problems with alternating workdays.
- Problems comparing the work of different groups (e.g., men, women, children).
Understanding the basic principles of calculating individual and combined rates is essential for solving these variations. Always convert the given information into a standard rate (e.g., work per hour or work per day) relative to the total work.
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Important Questions from Work Efficiency
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