A can complete 50% of a work in 9 days and B can do 25% of the work in 9 days, if they work alone. If they work together then how much work (in percentage) can be completed in 6 days?
50
Understanding the Work and Time Problem
This question involves the concept of work and time. We are given the time taken by two individuals, A and B, to complete a certain percentage of a work when working alone. We need to find the percentage of the same work they can complete together in a specific number of days.
To solve this type of work and time problem, we first need to determine the individual work rates of A and B. The work rate is the amount of work done per unit of time, typically per day in these kinds of problems. Once we have their individual work rates, we can find their combined work rate when they work together. Finally, we can calculate the total work done by the combined entity in the given time.
Calculating Individual Work Rates of A and B
Let the total work be represented as 1 unit or 100%. The work rate is calculated as the fraction of work done per day.
- Work Rate of A:
- A completes 50% (or 0.5) of the work in 9 days.
- To find the time A takes to complete 100% (or 1 unit) of the work:
- Time taken by A for 100% work \( = \frac{\text{Days taken}}{\text{Fraction of work done}} = \frac{9 \text{ days}}{0.5} = 18 \text{ days} \).
- So, A's daily work rate \( = \frac{1}{\text{Time taken for 100% work}} = \frac{1}{18} \) of the work per day.
- Work Rate of B:
- B completes 25% (or 0.25) of the work in 9 days.
- To find the time B takes to complete 100% (or 1 unit) of the work:
- Time taken by B for 100% work \( = \frac{\text{Days taken}}{\text{Fraction of work done}} = \frac{9 \text{ days}}{0.25} = 36 \text{ days} \).
- So, B's daily work rate \( = \frac{1}{\text{Time taken for 100% work}} = \frac{1}{36} \) of the work per day.
Calculating Combined Work Rate
When A and B work together, their work rates add up to form the combined work rate.
- Combined daily work rate \( = \) A's daily work rate \( + \) B's daily work rate
- Combined daily work rate \( = \frac{1}{18} + \frac{1}{36} \)
- To add these fractions, we find a common denominator, which is 36.
- \( \frac{1}{18} = \frac{1 \times 2}{18 \times 2} = \frac{2}{36} \)
- Combined daily work rate \( = \frac{2}{36} + \frac{1}{36} = \frac{2+1}{36} = \frac{3}{36} \)
- Simplifying the fraction, the combined daily work rate \( = \frac{1}{12} \) of the work per day.
Calculating Work Done Together in 6 Days
If A and B work together for 6 days, the total work completed will be the combined daily work rate multiplied by the number of days.
- Work done together in 6 days \( = \) Combined daily work rate \( \times \) Number of days
- Work done together in 6 days \( = \frac{1}{12} \times 6 = \frac{6}{12} \)
- Simplifying the fraction, the work done together in 6 days \( = \frac{1}{2} \) of the work.
Converting Work Done to Percentage
To express the work done as a percentage, we multiply the fraction of work done by 100.
- Percentage of work completed \( = \) Fraction of work done \( \times 100\% \)
- Percentage of work completed \( = \frac{1}{2} \times 100\% = 50\% \)
Therefore, if A and B work together, they can complete 50% of the work in 6 days.
| Individual | Work Done | Time Taken (days) | Time for 100% Work (days) | Daily Work Rate (Fraction) |
|---|---|---|---|---|
| A | 50% (0.5) | 9 | \( \frac{9}{0.5} = 18 \) | \( \frac{1}{18} \) |
| B | 25% (0.25) | 9 | \( \frac{9}{0.25} = 36 \) | \( \frac{1}{36} \) |
| A + B (Combined) | - | - | \( \frac{1}{\frac{1}{18} + \frac{1}{36}} = \frac{1}{\frac{2+1}{36}} = \frac{36}{3} = 12 \) | \( \frac{1}{18} + \frac{1}{36} = \frac{3}{36} = \frac{1}{12} \) |
Work done by A and B together in 6 days \( = \) Combined daily rate \( \times \) 6 days \( = \frac{1}{12} \times 6 = \frac{1}{2} \).
Percentage of work done \( = \frac{1}{2} \times 100\% = 50\% \).
Revision Table: Key Concepts in Work and Time
| Concept | Explanation | Formula/Relation |
|---|---|---|
| Work Rate | Amount of work done per unit of time. | Work Rate \( = \frac{\text{Work Done}}{\text{Time Taken}} \) |
| Total Time for Work | Time taken to complete 100% or 1 unit of work. | Time \( = \frac{1}{\text{Work Rate}} \) |
| Combined Work Rate | Sum of individual work rates when multiple people work together. | \( R_{total} = R_1 + R_2 + \dots \) |
| Work Done Together | Combined rate multiplied by the time they work together. | Work Done \( = R_{total} \times \text{Time Taken} \) |
| Work in Percentage | Fraction of work done multiplied by 100. | Percentage Work \( = \text{Fraction Work} \times 100\% \) |
Additional Information: Understanding Efficiency
The concept of work rate is directly related to efficiency. A person with a higher work rate is more efficient because they can complete more work in the same amount of time, or complete the same amount of work in less time.
- If a person takes \(T\) days to complete the whole work, their efficiency (or daily work rate) is proportional to \( \frac{1}{T} \).
- When people work together, their efficiencies (work rates) combine additively if they are working independently on the same task.
- Work and time problems often assume constant work rates unless stated otherwise.
- Understanding the relationship between work, rate, and time (\( \text{Work} = \text{Rate} \times \text{Time} \)) is fundamental to solving these problems.
Important Questions from Work Efficiency
Five men and 2 boys can do in 30 days as much work as 7 men and 10 boys can do in 15 days. How many boys should join 40 men to do the same work in 4 days?
A man completes 7/8 of a job in 21 days. How many more days will it take him to finish the job if quantum of work further increased by 50%?
24 men and 12 women can do a piece of work in 30 days. In how many days can 12 men and 24 women do the same piece of work?
A and B together can do a piece of work in 4 days, B and C can do it in 6 days, A and C can do it in 8 days. Then A, B and C together can do the same work in :-
The efficiency of Genelia is 25% more than that of Jessica and Jessica can complete a work in 25 days. Genelia started the work alone and Jessica joined him just five days before the work was completed. For how many days did Genelia work alone?
