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Combined work

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Introduction to Combined Work

In many real-life situations, two or more people or machines work together to complete a task faster. This idea is called combined work. For example, imagine painting a wall: if one painter can finish the job in 4 hours, and another in 6 hours, how long will it take if both work together? This kind of question involves understanding how individual work rates combine to complete the total work.

Understanding combined work is vital for competitive exams like engineering or medical entrances, as it tests logical reasoning and mathematical skills. It also appears in everyday scenarios such as calculating how long a group of people will take to complete a project, or how machines sharing a load affect overall efficiency.

To solve these problems, we must first learn the relationship between work, time, and work rate, and how multiple workers' rates add up.

Work, Time, and Work Rate Relationship

Let's define the three key terms:

  • Work: The total task to be completed. For simplicity, usually considered as 1 complete job or unit.
  • Time: The duration taken to complete the work.
  • Work Rate: The amount of work done per unit time. Think of it as speed of work.

These are connected by simple formulas:

Formula Interpretation Example
Work = Rate x Time The total work is the product of rate and time. If a worker paints 2 m² per hour and works for 3 hours, work done = 2 x 3 = 6 m².
Rate = Work / Time Rate is how much work is done per unit time. If a job requires 1 unit of work and is done in 4 hours, rate = 1 / 4 = 0.25 units/hour.
Time = Work / Rate Time taken to complete work at a given rate. To complete 1 unit of work at rate 0.5 units/hour takes 1 / 0.5 = 2 hours.
Key Concept

Work, Time, and Rate Relationship

Work done equals the product of rate and time. Rate is work per unit time. Time equals total work divided by rate.

Combined Work Rate

When two or more workers work together, their individual rates add up to form the combined work rate.

Why does this happen? Because the amount of work done in one hour by both working simultaneously is the sum of work done by each individually in that hour.

For example, if Worker A can finish a job in 5 hours, their rate is \( \frac{1}{5} \) work units per hour. If Worker B can finish the same job in 10 hours, their rate is \( \frac{1}{10} \). Working together, their combined rate is:

\[ \frac{1}{5} + \frac{1}{10} = \frac{2}{10} + \frac{1}{10} = \frac{3}{10} \text{ work units per hour} \]

So together, they can complete \( \frac{3}{10} \) of the job in 1 hour, taking:

\[ \frac{1}{\frac{3}{10}} = \frac{10}{3} = 3\frac{1}{3} \text{ hours} \]

This is faster than either working alone.

graph TD    A[Find individual times or work rates]    B[Convert times to rates: R = 1 / T]    C[Add individual rates to get combined rate]    D[Calculate total time: T = 1 / (combined rate)]    A --> B --> C --> D

Special cases:

  • If workers have different efficiencies, just convert their times individually, add rates, then invert.
  • If one worker works alone part of the time, calculate work done in that period separately, then for the remaining work use combined rates.

Worked Example 1: Two Workers Completing a Job Together

Example 1: Two Workers Together Easy
A and B can complete a painting job in 6 and 8 hours respectively. If they work together, how long will they take to complete the job?

Step 1: Calculate individual work rates.

Rate of A = \( \frac{1}{6} \) job/hour

Rate of B = \( \frac{1}{8} \) job/hour

Step 2: Find combined work rate.

Combined rate = \( \frac{1}{6} + \frac{1}{8} = \frac{4}{24} + \frac{3}{24} = \frac{7}{24} \) job/hour

Step 3: Calculate time taken by both together.

Time = \( \frac{1}{\frac{7}{24}} = \frac{24}{7} = 3.43 \) hours, or 3 hours 26 minutes approximately.

Answer: Working together, A and B will complete the job in approximately 3 hours 26 minutes.

Worked Example 2: Three Workers with Different Efficiencies

Example 2: Three Workers with Different Efficiencies Medium
Workers X, Y, and Z can finish a task in 12, 15, and 20 hours respectively. If all work together, how long will they take to finish the task?

Step 1: Find individual rates.

X's rate = \( \frac{1}{12} \) job/hour

Y's rate = \( \frac{1}{15} \) job/hour

Z's rate = \( \frac{1}{20} \) job/hour

Step 2: Calculate combined work rate.

Combined rate = \( \frac{1}{12} + \frac{1}{15} + \frac{1}{20} \) = \( \frac{5}{60} + \frac{4}{60} + \frac{3}{60} = \frac{12}{60} = \frac{1}{5} \) job/hour.

Step 3: Calculate total time required.

Time = \( \frac{1}{\frac{1}{5}} = 5 \) hours.

Answer: All three working together complete the task in 5 hours.

Worked Example 3: Work with Pipes Filling a Tank

Example 3: Pipes Filling a Tank Medium
Two pipes A and B can fill a tank in 4 hours and 6 hours respectively. A third pipe C can empty the tank in 12 hours. If all three pipes are open together, how long will it take to fill the tank?

Step 1: Calculate individual rates (consider filling as positive, emptying as negative).

Rate of A = \( \frac{1}{4} \) tank/hour (filling)

Rate of B = \( \frac{1}{6} \) tank/hour (filling)

Rate of C = \( -\frac{1}{12} \) tank/hour (emptying)

Step 2: Find combined rate.

Combined rate = \( \frac{1}{4} + \frac{1}{6} - \frac{1}{12} \)

Convert to common denominator 12:

\( \frac{3}{12} + \frac{2}{12} - \frac{1}{12} = \frac{4}{12} = \frac{1}{3} \) tank/hour

Step 3: Calculate total time to fill the tank.

Time = \( \frac{1}{\frac{1}{3}} = 3 \) hours.

Answer: It will take 3 hours to fill the tank when all pipes are open together.

Tank A: 1/4 B: 1/6 C: 1/12

Worked Example 4: Worker Working Alone then Joined by Another

Example 4: Worker Joins Midway Hard
A can complete a job in 10 hours. B can do the same job in 15 hours. A starts working alone and works for 3 hours. Then B joins A. How long will they take together to finish the remaining work?

Step 1: Calculate work done by A in 3 hours.

A's rate = \( \frac{1}{10} \) job/hour

Work done by A in 3 hours = \( 3 \times \frac{1}{10} = \frac{3}{10} \) of the job

Step 2: Find remaining work.

Remaining work = \( 1 - \frac{3}{10} = \frac{7}{10} \) of the job

Step 3: Find combined rate of A and B.

B's rate = \( \frac{1}{15} \) job/hour

Combined rate = \( \frac{1}{10} + \frac{1}{15} = \frac{3}{30} + \frac{2}{30} = \frac{5}{30} = \frac{1}{6} \) job/hour

Step 4: Calculate time for A and B together to finish remaining work.

Time = Work / Rate = \( \frac{7}{10} \div \frac{1}{6} = \frac{7}{10} \times 6 = \frac{42}{10} = 4.2 \) hours

Step 5: Calculate total time to complete the job.

Total time = 3 hours (A alone) + 4.2 hours (A and B together) = 7.2 hours

Answer: They will finish the job in 7.2 hours, i.e., 7 hours 12 minutes.

Worked Example 5: Payment Distribution Based on Work Done

Example 5: Payment Division Based on Work Done Hard
Three workers P, Q, and R do a job and are paid Rs.6000 in total. P can do the job in 8 hours, Q in 12 hours, and R in 6 hours. They work together for 4 hours. Find how much each should be paid?

Step 1: Find individual rates.

P's rate = \( \frac{1}{8} \) job/hour

Q's rate = \( \frac{1}{12} \) job/hour

R's rate = \( \frac{1}{6} \) job/hour

Step 2: Find amount of work done by each in 4 hours.

Work by P = \( 4 \times \frac{1}{8} = \frac{1}{2} \)

Work by Q = \( 4 \times \frac{1}{12} = \frac{1}{3} \)

Work by R = \( 4 \times \frac{1}{6} = \frac{2}{3} \)

Step 3: Total work done = \( \frac{1}{2} + \frac{1}{3} + \frac{2}{3} = \frac{1}{2} + 1 = \frac{3}{2} \)

Note: total work done is more than 1 full job because work done is considered cumulatively.

Step 4: Calculate share of each worker in the total work.

P's share = \( \frac{1/2}{3/2} = \frac{1/2}{3/2} = \frac{1}{3} \)

Q's share = \( \frac{1/3}{3/2} = \frac{1/3}{3/2} = \frac{2}{9} \)

R's share = \( \frac{2/3}{3/2} = \frac{2/3}{3/2} = \frac{4}{9} \)

Step 5: Find payment amount for each.

Total payment = Rs.6000

P's payment = \( \frac{1}{3} \times 6000 = Rs.2000 \)

Q's payment = \( \frac{2}{9} \times 6000 = Rs.1333.33 \)

R's payment = \( \frac{4}{9} \times 6000 = Rs.2666.67 \)

Answer: The payments are Rs.2000 to P, Rs.1333.33 to Q, and Rs.2666.67 to R.

Formula Bank

Formula Bank

Basic Work-Time-Rate Formula
\[ \text{Work} = \text{Rate} \times \text{Time} \]
where: Work = total work done (units), Rate = work done per unit time, Time = time taken
Work Rate of Combined Workers
\[ \text{Combined Rate} = R_1 + R_2 + \dots + R_n \]
where: \( R_1, R_2, \dots, R_n \) = individual work rates
Time Taken by Multiple Workers
\[ T = \frac{\text{Total Work}}{\sum_{i=1}^n R_i} \]
where: T = total time, Total Work = total units of work, \( R_i \) = individual work rates
Work Rate Conversion
\[ R = \frac{1}{T} \]
where: R = work rate (units/hour), T = time taken to complete 1 unit of work

Tips & Tricks

Tip: Convert the entire job into 1 unit to make calculations straightforward.

When to use: Whenever dealing with total work, assume it equals 1 unit regardless of complexity.

Tip: Always find work rates by taking the reciprocal of time before combining.

When to use: When multiple workers or machines are working simultaneously.

Tip: Treat filling pipes as positive work rates and emptying pipes as negative to avoid confusion.

When to use: Problems involving pipes filling and draining tanks.

Tip: For staggered work timings, split the problem into parts based on who is working when.

When to use: Workers start or stop at different times within the task.

Tip: When dividing payments based on work done, use the ratio of work (rate x time), not just the time spent.

When to use: Payment or cost sharing problems in partnerships or teamwork.

Common Mistakes to Avoid

❌ Adding times of workers instead of adding their work rates.
✓ Add individual work rates (reciprocals of times) to find the combined rate.
Why: Time and rate are inversely related; adding times overestimates total work duration.
❌ Ignoring staggered start times and summing all work time together.
✓ Break the problem into segments and calculate work done in each before combining.
Why: Different start times mean varying contributions over time.
❌ Mixing units for time or work without standardizing.
✓ Always convert time units to the same scale (hours/minutes) before calculation.
Why: Unit inconsistency leads to incorrect results.
❌ Treating emptying pipe rates as positive in tank problems.
✓ Use negative signs for emptying rates to calculate net work correctly.
Why: Direction of flow must be considered for accurate net rate.
❌ Dividing payment equally without accounting for work done differences.
✓ Calculate payment shares proportional to actual work done (rate x time).
Why: Efficiency differences affect contribution; equal splits are unfair.
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