A, B and C, working alone, can complete a job in 16, 24 and 36 days, respectively. In how many days can they complete the job if they work together?
Understanding Work and Time Problems
Work and time problems often involve calculating how long it takes individuals or groups to complete a task based on their individual work rates. The fundamental principle is that the amount of work done is proportional to the rate of work and the time spent working.
If a person can complete a job in \(d\) days, their work rate is \( \frac{1}{d} \) of the job per day. When multiple people work together, their individual work rates are added to find the combined work rate.
Calculating Individual Work Rates
We are given the time each person takes to complete the job working alone:
- A can complete the job in 16 days.
- B can complete the job in 24 days.
- C can complete the job in 36 days.
Based on this information, we can find their individual work rates per day:
- A's work rate per day = \( \frac{1}{16} \) of the job.
- B's work rate per day = \( \frac{1}{24} \) of the job.
- C's work rate per day = \( \frac{1}{36} \) of the job.
Calculating Combined Work Rate
When A, B, and C work together, their work rates add up. The combined work rate per day is the sum of their individual work rates:
Combined work rate per day = A's work rate + B's work rate + C's work rate
Combined work rate per day = \( \frac{1}{16} + \frac{1}{24} + \frac{1}{36} \)
To add these fractions, we need to find a common denominator, which is the Least Common Multiple (LCM) of 16, 24, and 36.
Finding the LCM of 16, 24, and 36
- Prime factorization of \(16 = 2 \times 2 \times 2 \times 2 = 2^4\)
- Prime factorization of \(24 = 2 \times 2 \times 2 \times 3 = 2^3 \times 3^1\)
- Prime factorization of \(36 = 2 \times 2 \times 3 \times 3 = 2^2 \times 3^2\)
The LCM is found by taking the highest power of all prime factors present:
\( \text{LCM}(16, 24, 36) = 2^4 \times 3^2 = 16 \times 9 = 144 \)
The least common denominator is 144.
Adding the Fractions
Now, we convert each fraction to have a denominator of 144:
- \( \frac{1}{16} = \frac{1 \times 9}{16 \times 9} = \frac{9}{144} \)
- \( \frac{1}{24} = \frac{1 \times 6}{24 \times 6} = \frac{6}{144} \)
- \( \frac{1}{36} = \frac{1 \times 4}{36 \times 4} = \frac{4}{144} \)
Combined work rate per day = \( \frac{9}{144} + \frac{6}{144} + \frac{4}{144} = \frac{9+6+4}{144} = \frac{19}{144} \)
So, working together, A, B, and C can complete \( \frac{19}{144} \) of the job in one day.
| Person | Time to Complete Job (days) | Work Rate (Job per day) |
|---|---|---|
| A | 16 | \( \frac{1}{16} \) |
| B | 24 | \( \frac{1}{24} \) |
| C | 36 | \( \frac{1}{36} \) |
| A, B, & C (Together) | ? | \( \frac{19}{144} \) |
Calculating Time to Complete Job Together
If the combined work rate is \( \frac{19}{144} \) of the job per day, the time taken to complete the entire job (which is 1 unit of work) is the reciprocal of the combined work rate.
Time taken = \( \frac{1}{\text{Combined work rate per day}} = \frac{1}{\frac{19}{144}} = \frac{144}{19} \) days.
Converting to Mixed Number
The time taken is \( \frac{144}{19} \) days. We convert this improper fraction to a mixed number:
\( 144 \div 19 \)
We find how many times 19 goes into 144:
\( 19 \times 7 = 133 \)
The remainder is \( 144 - 133 = 11 \)
So, \( \frac{144}{19} = 7 \frac{11}{19} \)
Therefore, A, B, and C working together can complete the job in \( 7 \frac{11}{19} \) days.
Summary of Steps to Solve Work and Time Problems Together
- Find the individual work rate (fraction of job per day) for each person.
- Add the individual work rates to find the combined work rate per day.
- The time taken to complete the job together is the reciprocal of the combined work rate.
- Convert the result to a mixed number if necessary.
Revision Table: Work and Time Concepts
| Concept | Description | Formula |
|---|---|---|
| Individual Work Rate | Fraction of job completed by one person in one unit of time. | \( \text{Rate} = \frac{1}{\text{Time taken}} \) |
| Combined Work Rate | Sum of individual work rates when people work together. | \( R_{\text{combined}} = R_1 + R_2 + R_3 + \dots \) |
| Time Taken Together | Time required to complete the whole job (1 unit) at the combined rate. | \( \text{Time} = \frac{1}{\text{Combined Rate}} \) |
| Work Done | Rate multiplied by Time. | \( \text{Work} = \text{Rate} \times \text{Time} \) |
Additional Information on Work and Time Calculations
Work and time questions can sometimes involve scenarios where people work for different durations or start/stop at different times. In such cases, you calculate the work done by each person during the time they worked and sum them up to see if the total work is completed.
- If a person works for \(t\) days at a rate of \(R\) per day, the work done is \(R \times t\).
- If a job requires \(W\) units of work, and the rate is \(R\), the time taken is \(T = \frac{W}{R}\). In most basic problems, \(W=1\) (representing 1 whole job).
- The concept of LCM is crucial when adding work rates expressed as fractions. It helps find the total 'units' of work and make calculations easier.
Understanding these core concepts helps solve a variety of work and time problems efficiently.
Similar Questions
X and Y can complete a work in 9 days and 36 days, respectively. X begins to do the work and they work alternately one at a time for one day each. The whole work will be complete in:
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(Round off to the nearest integer)
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Important Questions from Work Efficiency
A and B working together can complete a job in 30 days. The ratio of their efficiencies is 3 : 2. In how many days can the faster person complete the job?
A takes 15 days to complete \(\frac{5}{7} \) of a work. With the help of B, they finish the whole work in 12 days. In how many days, B alone will complete the same work
A alone can complete a work in 14 days and B alone can complete the same work in 21 days. A and B start the work together but A leaves the work after 4 days of the starting of work. In how many days B will complete the remaining work?
For completing a certain work, A is 50% less efficient than B and B is 50% more efficient than C. Working together A, B and C can complete the work in 48 days. A alone can complete the same work in:
30 persons can do a piece of work in 24 days. How many more persons are required to complete the work in 20 days?
