A can do a certain work in 15 days, while B can do the same work in 21 days. If they work together, then in how many days will the same work be completed?
Understanding Work and Time Problems
Work and time problems are common in quantitative aptitude. The core idea is that if a person can complete a piece of work in \(n\) days, their work rate is \(1/n\) of the total work per day. When multiple people work together, their individual work rates are added to find their combined work rate.
Calculating Individual Work Rates
- Person A can do the work in 15 days.
- So, A's daily work rate is \( \frac{1}{15} \) of the total work per day.
- Person B can do the same work in 21 days.
- So, B's daily work rate is \( \frac{1}{21} \) of the total work per day.
Calculating Combined Work Rate
When A and B work together, their combined daily work rate is the sum of their individual daily work rates.
Combined daily work rate \( = \) A's rate \( + \) B's rate
Combined daily work rate \( = \frac{1}{15} + \frac{1}{21} \)
To add these fractions, we need to find a common denominator for 15 and 21. The least common multiple (LCM) of 15 and 21 is 105.
- \(15 = 3 \times 5\)
- \(21 = 3 \times 7\)
- LCM(15, 21) \( = 3 \times 5 \times 7 = 105\)
Now, we convert the fractions to have a denominator of 105:
\( \frac{1}{15} = \frac{1 \times 7}{15 \times 7} = \frac{7}{105} \)
\( \frac{1}{21} = \frac{1 \times 5}{21 \times 5} = \frac{5}{105} \)
Add the fractions:
Combined daily work rate \( = \frac{7}{105} + \frac{5}{105} = \frac{7 + 5}{105} = \frac{12}{105} \)
We can simplify the fraction \( \frac{12}{105} \) by dividing both the numerator and denominator by their greatest common divisor, which is 3.
Combined daily work rate \( = \frac{12 \div 3}{105 \div 3} = \frac{4}{35} \)
So, A and B together complete \( \frac{4}{35} \) of the work each day.
Calculating Time Taken Together
If the combined daily work rate is \( \frac{4}{35} \) of the work per day, then the total number of days required to complete the entire work (which is 1 unit of work) is the reciprocal of the combined daily work rate.
Time taken together \( = \frac{1}{\text{Combined daily work rate}} = \frac{1}{\frac{4}{35}} = \frac{35}{4} \) days.
Converting to Mixed Number
The time taken is \( \frac{35}{4} \) days. We can convert this improper fraction into a mixed number to match the options.
\( \frac{35}{4} = 35 \div 4 \)
Dividing 35 by 4:
- 4 goes into 35 eight times (\(4 \times 8 = 32\)).
- The remainder is \(35 - 32 = 3\).
So, \( \frac{35}{4} \) as a mixed number is \( 8 \frac{3}{4} \).
Conclusion
If A and B work together, the same work will be completed in \( 8 \frac{3}{4} \) days.
Revision Table: Work and Time Concepts
| Concept | Formula/Description |
|---|---|
| Individual Work Rate | If a person completes work in \(n\) days, daily rate \( = \frac{1}{n} \). |
| Combined Work Rate | Sum of individual daily rates. |
| Time Taken Together | \( \frac{1}{\text{Combined daily work rate}} \). |
| Total Work | Usually considered as 1 unit. |
Additional Information: Solving Work Problems
Work and time problems often involve calculating how fast individuals or groups can complete a task. Key principles include:
- Inverse Relationship: Time taken and work rate are inversely proportional. More time means a slower rate, less time means a faster rate.
- Efficiency: Sometimes problems mention efficiency percentages, which relate to the work rate compared to a standard.
- Multiple Workers: When more workers are added (assuming same efficiency), the time taken decreases.
- Work Done: Work done is the product of work rate and time taken. \( \text{Work Done} = \text{Rate} \times \text{Time} \).
Understanding these basic principles allows you to tackle various types of work and time questions effectively.
Important Questions from Work Efficiency
A and B together can complete a certain work in 20 days whereas B and C together can complete it in 24 days. If A is twice as good a workman as C, then in what time will B alone do 40% of the same work?
14 men can complete a work in 15 days. If 21 men are employed, then in how many days will they complete the same work?
To do a certain work, A and B work on alternate days with B beginning the work on the first day. A alone can complete the same work in 24 days. If the work gets completed in \(11 \frac{1}{3}\) days, then B alone can complete \(\rm \frac{7}{9}^{th}\) part of the original work in:
Two men and 7 women can complete a work in 28 days whereas 6 men and 16 women can do the same work in 11 days. In how many days can 7 men complete the same work?
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?
