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?
13
Solving the Work and Time Problem
This question involves the concept of work and time, specifically how individuals or groups with different efficiencies can complete a task. The key idea is to calculate the work done per unit of time (usually per day) by each person.
Calculating Individual Work Rates
The work rate of a person is the fraction of the total work they can complete in one day. If a person takes 'd' days to complete the entire work, their 1-day work rate is \( \frac{1}{d} \).
- A man can do the work in 6 days. So, the man's 1-day work rate is \( \frac{1}{6} \).
- A woman can do the work in 9 days. So, the woman's 1-day work rate is \( \frac{1}{9} \).
- A boy can do the work in 18 days. So, the boy's 1-day work rate is \( \frac{1}{18} \).
Combined Work Rate of One Man and One Woman
We need to find the amount of work done by one man and one woman together in one day. This is the sum of their individual 1-day work rates.
Combined work rate of 1 man and 1 woman in 1 day \( = \) Man's rate \( + \) Woman's rate
\( = \frac{1}{6} + \frac{1}{9} \)
To add these fractions, we find a common denominator, which is the least common multiple (LCM) of 6 and 9. The LCM of 6 and 9 is 18.
\( = \frac{1 \times 3}{6 \times 3} + \frac{1 \times 2}{9 \times 2} = \frac{3}{18} + \frac{2}{18} \)
\( = \frac{3 + 2}{18} = \frac{5}{18} \)
So, one man and one woman together complete \( \frac{5}{18} \) of the total work in 1 day.
Work Remaining for the Boys
The goal is to complete the entire work (which is represented as 1 unit of work) in 1 day. One man and one woman complete \( \frac{5}{18} \) of the work in that day. The remaining work must be done by the boys in the same day.
Remaining work \( = \) Total work \( - \) Work done by 1 man and 1 woman
\( = 1 - \frac{5}{18} \)
\( = \frac{18}{18} - \frac{5}{18} = \frac{18 - 5}{18} = \frac{13}{18} \)
So, the boys must complete \( \frac{13}{18} \) of the total work in 1 day.
Calculating the Number of Boys Needed
Each boy completes \( \frac{1}{18} \) of the work in 1 day. To find out how many boys are needed to complete the remaining \( \frac{13}{18} \) of the work in 1 day, we divide the remaining work by the work rate of a single boy.
Number of boys \( = \frac{\text{Remaining work}}{\text{Work rate of one boy}} \)
\( = \frac{13/18}{1/18} \)
\( = \frac{13}{18} \times \frac{18}{1} \)
\( = 13 \)
Therefore, 13 boys must assist one man and one woman to do the work in 1 day.
The final answer is 13.
Revision Table: Work and Time Concepts
| Concept | Explanation | Formula |
|---|---|---|
| Work Rate | Amount of work done per unit of time (e.g., per day). | If a person finishes work in \(d\) days, rate \( = \frac{1}{d}\) per day. |
| Total Work | Usually considered as 1 unit or the LCM of the days taken by individuals. | Total Work \( = \) Rate \( \times \) Time |
| Combined Rate | Sum of individual rates when people work together. | Rate\(_{\text{Total}}\) \( = \) Rate\(_{1}\) \( + \) Rate\(_{2}\) \( + \) ... |
| Time Taken Together | Time to complete the total work (1 unit) at the combined rate. | Time \( = \frac{1}{\text{Combined Rate}}\) |
Additional Information on Work Problems
Work and time problems often involve calculating efficiency and combining the efforts of multiple individuals or machines. Here are some points to remember:
- Work is often considered constant, represented by '1'.
- Efficiency is inversely proportional to the time taken to complete the work. More efficient people take less time.
- When multiple people work together, their rates add up.
- If someone leaves or joins, the combined rate changes for the respective periods.
- Sometimes, the total work is assumed to be the LCM of the days taken by individuals, which helps in avoiding fractions during calculations. In this case, total work would be 18 units (LCM of 6, 9, 18).
- Using LCM of days as total work:
- Man's efficiency: \( \frac{18}{6} = 3 \) units/day
- Woman's efficiency: \( \frac{18}{9} = 2 \) units/day
- Boy's efficiency: \( \frac{18}{18} = 1 \) unit/day
- 1 man + 1 woman efficiency in 1 day: \( 3 + 2 = 5 \) units
- Work to be done in 1 day: 18 units
- Work done by 1 man and 1 woman in 1 day: 5 units
- Remaining work: \( 18 - 5 = 13 \) units
- Number of boys needed: \( \frac{\text{Remaining work}}{\text{Efficiency of one boy}} = \frac{13}{1} = 13 \) boys. This method gives the same result and can be easier for some.
Important Questions from Time and Work
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?
By looking in a mirror, it appears that it is 6:30 in the clock. What is the real time?
