Five men can complete a work in 20 days. Ten women can complete the same work in 15 days. Two men and six women started working together. After 5 days, three women left the work a new man joined the work. The group continued working together till the end of the work. In how many days will they be able to do the remaining work?
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Understanding the Work and Time Problem
This problem involves understanding the concept of work rate and how it changes when the number of workers or the type of workers changes. We are given information about the time taken by a certain number of men and women to complete a task, and then we analyze the progress of the work under different group compositions.
Calculating Individual Efficiencies
First, let's determine the individual work rate or efficiency of one man and one woman. We can assume a total amount of work that is easily divisible by the time taken by both groups. The least common multiple (LCM) of 20 days (for men) and 15 days (for women) is a good choice for the total work units.
LCM(20, 15) = 60 units.
Now, we can find the efficiency:
- Efficiency of 5 men: If 5 men complete 60 units in 20 days, their combined efficiency is $\frac{60 \text{ units}}{20 \text{ days}} = 3$ units per day.
- Efficiency of 1 man: Since the 5 men have equal efficiency, the efficiency of 1 man is $\frac{3 \text{ units/day}}{5 \text{ men}} = 0.6$ units per day.
- Efficiency of 10 women: If 10 women complete 60 units in 15 days, their combined efficiency is $\frac{60 \text{ units}}{15 \text{ days}} = 4$ units per day.
- Efficiency of 1 woman: Since the 10 women have equal efficiency, the efficiency of 1 woman is $\frac{4 \text{ units/day}}{10 \text{ women}} = 0.4$ units per day.
We can summarise the individual efficiencies:
| Worker Type | Efficiency (units/day) |
|---|---|
| Man | 0.6 |
| Woman | 0.4 |
Work Done in the First 5 Days
Initially, a group of 2 men and 6 women started working together. Let's calculate their combined efficiency and the amount of work they completed in the first 5 days.
- Efficiency of 2 men = $2 \times 0.6$ units/day = 1.2 units/day.
- Efficiency of 6 women = $6 \times 0.4$ units/day = 2.4 units/day.
- Combined efficiency of the initial group = 1.2 + 2.4 = 3.6 units/day.
- Work done in 5 days = Combined efficiency $\times$ Time = $3.6 \text{ units/day} \times 5 \text{ days} = 18$ units.
Remaining Work Calculation
The total work is 60 units. After 5 days, 18 units of work are completed. The remaining work is:
Remaining Work = Total Work - Work Done = 60 units - 18 units = 42 units.
New Group Composition and Efficiency
After 5 days, the group changes: three women left, and a new man joined.
- Number of men in the new group = Initial men + New man = 2 + 1 = 3 men.
- Number of women in the new group = Initial women - Women who left = 6 - 3 = 3 women.
The new group consists of 3 men and 3 women. Let's calculate their combined efficiency.
- Efficiency of 3 men = $3 \times 0.6$ units/day = 1.8 units/day.
- Efficiency of 3 women = $3 \times 0.4$ units/day = 1.2 units/day.
- Combined efficiency of the new group = 1.8 + 1.2 = 3 units/day.
Time to Complete the Remaining Work
The remaining work is 42 units, and the new group works at a combined efficiency of 3 units/day. The time taken to complete the remaining work is:
Time = $\frac{\text{Remaining Work}}{\text{New Combined Efficiency}} = \frac{42 \text{ units}}{3 \text{ units/day}} = 14$ days.
Therefore, they will be able to do the remaining work in 14 days.
Summary of Steps for Work Problems
To solve problems like this, follow these steps:
- Determine a common amount for the total work (often the LCM of the given time periods).
- Calculate the efficiency (work rate) of individuals or groups based on the total work and time.
- Calculate the work done during the initial period by the initial group.
- Find the remaining work.
- Determine the composition of the new group.
- Calculate the combined efficiency of the new group.
- Calculate the time required by the new group to complete the remaining work.
Revision Table for Work and Time Concepts
| Concept | Explanation | Formula |
|---|---|---|
| Total Work | The entire task to be completed. Often assumed as a unit (e.g., 1) or LCM of days. | Assumed Value (LCM) |
| Efficiency (Work Rate) | Amount of work done by a person or group per unit of time. | $\frac{\text{Total Work}}{\text{Time Taken}}$ |
| Time Taken | Duration required to complete the work. | $\frac{\text{Total Work}}{\text{Efficiency}}$ |
| Work Done | Amount of work completed in a specific time period. | Efficiency $\times$ Time |
| Remaining Work | Work left after some work is completed. | Total Work - Work Done |
Additional Information on Work and Time Problems
Work and time problems often involve different types of workers (like men, women, children), or different machines, or pipes filling/emptying tanks. The core idea remains the same: standardizing the work and calculating individual or combined efficiencies. Efficiency is inversely proportional to the time taken; if someone is more efficient, they take less time to do the same amount of work.
When groups of different workers work together, their efficiencies are added up to find the combined efficiency. Changes in the group composition lead to changes in the combined efficiency, which in turn affects the time taken for the remaining work. It's crucial to carefully track the work completed, the remaining work, and the efficiency of the group working on the remaining portion.
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Important Questions from Work Efficiency
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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 :-
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?
