Nine boys can complete a work in 360 days, 18 men can complete the same work in 72 days, and 12 women can complete it in 162 days. In how many days can 4 men, 12 women, and 10 boys together complete the work?
81
Solving Work and Time Problems with Different Groups
This question involves a classic work and time scenario where different groups of people (boys, men, and women) have different efficiencies in completing the same work. We need to find the time it takes for a specific combination of these groups to complete the task.
Understanding Work Rate and Efficiency
The core concept here is that the total work is constant, regardless of who does it. The rate at which work is done is called efficiency. If a group completes a work in \(D\) days, their work rate per day is \( \frac{1}{D} \) of the total work. If there are \(N\) workers in the group, the work rate of one worker per day is \( \frac{1}{N \times D} \) of the total work.
Calculating Individual Efficiencies
Let's determine the work rate for one boy, one man, and one woman based on the given information:
- 9 boys complete the work in 360 days. Total boy-days = \( 9 \times 360 = 3240 \) boy-days. Work done by 1 boy in 1 day = \( \frac{1}{3240} \) of the total work.
- 18 men complete the work in 72 days. Total man-days = \( 18 \times 72 = 1296 \) man-days. Work done by 1 man in 1 day = \( \frac{1}{1296} \) of the total work.
- 12 women complete the work in 162 days. Total woman-days = \( 12 \times 162 = 1944 \) woman-days. Work done by 1 woman in 1 day = \( \frac{1}{1944} \) of the total work.
To make calculations easier, we can find a common unit of work or relative efficiencies. A common approach is to assume the total work is the Least Common Multiple (LCM) of the total "person-days" for each category.
Total work units corresponding to:
- Boys: 3240 boy-days
- Men: 1296 man-days
- Women: 1944 woman-days
Let's find the LCM of 3240, 1296, and 1944.
- \( 3240 = 2^3 \times 3^4 \times 5 \)
- \( 1296 = 2^4 \times 3^4 \)
- \( 1944 = 2^3 \times 3^5 \)
LCM(3240, 1296, 1944) = \( 2^{\max(3,4,3)} \times 3^{\max(4,4,5)} \times 5^{\max(1,0,0)} = 2^4 \times 3^5 \times 5^1 = 16 \times 243 \times 5 = 19440 \).
Let the total work be 19440 units.
Now, we can find the efficiency (units of work per day) of each individual:
- Efficiency of 1 boy = \( \frac{19440 \text{ units}}{3240 \text{ boy-days}} = 6 \) units/boy-day
- Efficiency of 1 man = \( \frac{19440 \text{ units}}{1296 \text{ man-days}} = 15 \) units/man-day
- Efficiency of 1 woman = \( \frac{19440 \text{ units}}{1944 \text{ woman-days}} = 10 \) units/woman-day
Calculating Combined Efficiency
The question asks for the time taken by a group of 4 men, 12 women, and 10 boys. Let's calculate their combined work rate per day.
- Work done by 4 men in 1 day = \( 4 \times (\text{Efficiency of 1 man}) = 4 \times 15 = 60 \) units.
- Work done by 12 women in 1 day = \( 12 \times (\text{Efficiency of 1 woman}) = 12 \times 10 = 120 \) units.
- Work done by 10 boys in 1 day = \( 10 \times (\text{Efficiency of 1 boy}) = 10 \times 6 = 60 \) units.
Total work done by 4 men, 12 women, and 10 boys in 1 day (Combined Efficiency) = \( 60 + 120 + 60 = 240 \) units/day.
Calculating Time Taken
The total work is 19440 units, and the combined efficiency of the new group is 240 units per day. The time taken to complete the work is given by:
\( \text{Time} = \frac{\text{Total Work}}{\text{Combined Efficiency}} \)
\( \text{Time} = \frac{19440 \text{ units}}{240 \text{ units/day}} \)
\( \text{Time} = \frac{1944}{24} \text{ days} \)
Performing the division:
\( \frac{1944}{24} = \frac{972}{12} = \frac{486}{6} = 81 \)
So, the group of 4 men, 12 women, and 10 boys can complete the work in 81 days.
Final Answer
The combined group can complete the work in 81 days.
| Group | Number | Days to Complete Work | Total Person-Days | Assumed Total Work (LCM) | Efficiency per Person (Units/Day) |
|---|---|---|---|---|---|
| Boys | 9 | 360 | 3240 | 19440 | \( \frac{19440}{3240} = 6 \) |
| Men | 18 | 72 | 1296 | 19440 | \( \frac{19440}{1296} = 15 \) |
| Women | 12 | 162 | 1944 | 19440 | \( \frac{19440}{1944} = 10 \) |
| New Group Members | Number | Efficiency per Person (Units/Day) | Total Daily Contribution (Units) |
|---|---|---|---|
| Men | 4 | 15 | \( 4 \times 15 = 60 \) |
| Women | 12 | 10 | \( 12 \times 10 = 120 \) |
| Boys | 10 | 6 | \( 10 \times 6 = 60 \) |
| Combined | \( 60 + 120 + 60 = 240 \) |
Time taken by the new group = \( \frac{\text{Total Work}}{\text{Combined Daily Contribution}} = \frac{19440}{240} = 81 \) days.
Revision Table: Work and Time Concepts
| Concept | Explanation | Formula |
|---|---|---|
| Work Rate | Amount of work done per unit of time (e.g., per day). | Work Rate = \( \frac{1}{\text{Time Taken}} \) (for total work) |
| Total Work | The entire task to be completed. Often assumed as 1 unit or an LCM of person-days. | Total Work = Rate \( \times \) Time |
| Efficiency | The rate at which a single person (or unit) does work. | Efficiency per Person = \( \frac{\text{Total Work}}{\text{Total Person-Days}} \) |
| Combined Work Rate | The sum of individual work rates when multiple people/groups work together. | Combined Rate = Sum of Individual Rates |
| Time Taken by Combined Group | Time required for a group to complete the total work. | Time = \( \frac{\text{Total Work}}{\text{Combined Work Rate}} \) |
Additional Information: Solving Work Problems
Work and Time problems are common in competitive exams. The key is to standardize the work rate, usually to a 'per day' rate for an individual worker. Once individual efficiencies are known, you can calculate the combined efficiency of any group and find the time required.
- Inverse Proportionality: The number of workers and the time taken to complete work are inversely proportional, assuming constant efficiency. If you double the workers, the time taken is halved.
- Consistency: Ensure you are using consistent units throughout your calculation (e.g., work per day, total work units).
- "And" vs. "Or": In questions stating "A can do in X days AND B can do in Y days", you add their individual rates to find their combined rate when working together. If it says "A OR B can do in Z days" for the *same* task, it implies A takes Z days and B takes Z days individually (less common phrasing). In this question, "9 boys can complete... AND 18 men can complete... AND 12 women can complete..." means these are separate groups doing the same work, establishing their individual capacities.
- LCM Method Advantage: Using the LCM of total person-days as the total work unit helps avoid fractions until the very final step, simplifying calculations significantly.
Understanding these fundamental principles will help you tackle a wide variety of work and time problems.
Important Questions from Time and Work
A man, a woman, and a boy can do a work in 6, 9, and 18 days respectively. How many boys must assist one man and one woman to do the work in 1 day?
A, B, and C worked together for 5 days to build a wall and then B left the work. A and C together finished the remaining work in 5 days. In how many days can A and C working together finish building the whole wall if B alone can do it in 25 days?
Three men can complete a work in 6 days and five women can complete the same work in 18 days. In how many days can 4 men and 10 women together complete the same work?
Mother, daughter, and son can do a work in 3, 4, and 5 days respectively. How many days will they take to complete the work, if they work together?
Rajan does 7/11 of a work in 21 days. How many more days will he take to complete the work?
