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.
Similar Questions
X and Y can complete a work in 9 days and 36 days, respectively. X begins to do the work and they work alternately one at a time for one day each. The whole work will be complete in:
P is two times as efficient as Q. P is able to complete a piece of work in 40 days less than Q. Working together, the whole number of days taken by them to complete the work is:
(Round off to the nearest integer)
4 women or 6 boys can finish a work in the same number of days. A boy can finish it in 60 days. In how many days can 5 women finish the work, working together every day?
To do a certain work, Ajay and Bharat work on alternate days, with Bharat starting the work on the first day. Ajay can finish the work alone in 32 days. If the work gets completed in exactly 8 days, then Bharat alone can finish 7 times the same work in ____________ days.
A, B and C can separately complete a work in 12, 15 and 20 days, respectively. They worked together 4 days. What will be the remaining work?
A, B and C, working alone, can complete a job in 16, 24 and 36 days, respectively. In how many days can they complete the job if they work together?
Rakshit, Ajay, and Satish are sanitation workers in a Municipal Corporation. Rakshit alone takes 20 hours to clean a drain while Ajay takes 12 hours when working alone to do the same. All three together take only 5 hours to clean the drain. In how many hours, can Satish complete the work alone?
15 men and 25 women can complete a piece of work in 9.6 days. If 16 women can complete the same work in 27 days, find the number of days in which 16 men can complete the same work.
A,B and C can do a piece of work in 30 days, 40 days and 50 days, respectively. Beginning with A, if A, B and C do the work alternatively then in how many days will the work be finished?
Ravi can do a piece of work in 40 days and Sudha can do the same piece of work in 60 days. If they work on alternative days starting with Sudha on the first day, then in how many days will the work be completed?
Important Questions from Work Efficiency
A and B working together can complete a job in 30 days. The ratio of their efficiencies is 3 : 2. In how many days can the faster person complete the job?
A takes 15 days to complete \(\frac{5}{7} \) of a work. With the help of B, they finish the whole work in 12 days. In how many days, B alone will complete the same work
A alone can complete a work in 14 days and B alone can complete the same work in 21 days. A and B start the work together but A leaves the work after 4 days of the starting of work. In how many days B will complete the remaining work?
For completing a certain work, A is 50% less efficient than B and B is 50% more efficient than C. Working together A, B and C can complete the work in 48 days. A alone can complete the same work in:
30 persons can do a piece of work in 24 days. How many more persons are required to complete the work in 20 days?
