Anjali can complete a job in 10 days. Banu can do it in 5 days. In how many days can the job be done, if they work together?
3 days 8 hours
Understanding the Work and Time Problem
This question asks us to find out how long it takes for two people, Anjali and Banu, to complete a job when they work together, given their individual times to complete the same job. These types of problems are common in the topic of work and time.
Calculating Individual Work Rates
The key to solving work and time problems is to determine the rate at which each person works. The work rate is usually expressed as the fraction of the job completed per unit of time (in this case, per day).
- Anjali can complete the job in 10 days. This means Anjali completes \(\frac{1}{10}\) of the job per day.
- Banu can complete the job in 5 days. This means Banu completes \(\frac{1}{5}\) of the job per day.
Calculating Combined Work Rate
When Anjali and Banu work together, their work rates add up. To find their combined daily work rate, we add their individual daily rates:
\text{Combined daily work rate} = \text{Anjali's daily rate} + \text{Banu's daily rate}
\text{Combined daily work rate} = \frac{1}{10} + \frac{1}{5}
To add these fractions, we need a common denominator, which is 10.
\frac{1}{10} + \frac{1 \times 2}{5 \times 2} = \frac{1}{10} + \frac{2}{10} = \frac{1+2}{10} = \frac{3}{10}
So, when Anjali and Banu work together, they complete \(\frac{3}{10}\) of the job per day.
Calculating Time Taken When Working Together
If the combined daily work rate is \(\frac{3}{10}\) of the job per day, the total time taken to complete the entire job (which is 1 whole job) is the reciprocal of the combined work rate.
\text{Time taken together} = \frac{1}{\text{Combined daily work rate}}
\text{Time taken together} = \frac{1}{\frac{3}{10}} = \frac{10}{3} \text{ days}
Converting Fractional Days to Hours
The time taken is \(\frac{10}{3}\) days. Let's convert this improper fraction into a mixed number:
\frac{10}{3} = 3 \frac{1}{3} \text{ days}
This means it takes 3 full days and \(\frac{1}{3}\) of a day. To convert the fractional part of a day into hours, we multiply the fraction by the number of hours in a day (24 hours).
\text{Fractional part in hours} = \frac{1}{3} \times 24 \text{ hours}
\text{Fractional part in hours} = \frac{24}{3} \text{ hours} = 8 \text{ hours}
Therefore, the total time taken when Anjali and Banu work together is 3 days and 8 hours.
| Person | Time to complete job (days) | Daily Work Rate (Job/Day) |
|---|---|---|
| Anjali | 10 | \(\frac{1}{10}\) |
| Banu | 5 | \(\frac{1}{5}\) |
Combined Daily Work Rate = \(\frac{1}{10} + \frac{1}{5} = \frac{1+2}{10} = \frac{3}{10}\) job/day.
Time taken together = \(\frac{1}{\text{Combined Rate}} = \frac{1}{\frac{3}{10}} = \frac{10}{3}\) days.
\(\frac{10}{3}\) days = \(3 \frac{1}{3}\) days = 3 days + \(\frac{1}{3} \times 24\) hours = 3 days + 8 hours.
Summary of Steps
- Find the individual daily work rate for each person.
- Add the individual daily work rates to find the combined daily work rate.
- Take the reciprocal of the combined daily work rate to find the total time taken when working together.
- Convert the fractional part of the day into hours if needed.
Revision Table: Work and Time Basics
| Concept | Formula/Relation | Explanation |
|---|---|---|
| Work Rate | \(\text{Rate} = \frac{\text{Amount of Work}}{\text{Time Taken}}\) | The amount of work done per unit of time. If 1 job is done in T days, the rate is \(\frac{1}{T}\) job/day. |
| Time Taken | \(\text{Time} = \frac{\text{Amount of Work}}{\text{Rate}}\) | The total duration to complete a certain amount of work at a given rate. To complete 1 job with rate R, time is \(\frac{1}{R}\). |
| Combined Rate (for people A & B) | \(\text{Rate}_{\text{A+B}} = \text{Rate}_{\text{A}} + \text{Rate}_{\text{B}}\) | When working together, their rates add up to complete the work faster. |
| Time Together (for people A & B) | \(\text{Time}_{\text{A+B}} = \frac{1}{\text{Rate}_{\text{A}} + \text{Rate}_{\text{B}}}\) | The total time taken when A and B work together. |
Additional Information: Units and Conversions
It's important to maintain consistent units when solving work and time problems. If rates are per day, the total time will be in days. If the answer needs to be in a combination of days and hours, remember the conversion factor:
- 1 day = 24 hours
If you have a fractional part of a day, say \(\frac{x}{y}\) days, you can convert it to hours by multiplying by 24: \(\frac{x}{y} \times 24\) hours.
In this problem, the fractional part was \(\frac{1}{3}\) of a day, so \(\frac{1}{3} \times 24 = 8\) hours.
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?
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?
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?
