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Understanding Time and Work is essential in solving many practical and competitive exam problems. The concept allows us to quantify how much work is done over time and how multiple workers can collaborate efficiently. In this chapter, work is treated as a measurable quantity, which can be performed at a specific rate, usually assumed constant for simplicity.
The goal is to learn how work, rate, and time interact and to use this relationship to solve diverse problems-from a single worker completing a task, to multiple workers collaborating, and even calculating costs related to the work done.
Work and time share an important inverse relationship: as the work rate increases, the time taken decreases, and vice versa. Grasping this connection helps resolve many real-life and exam-oriented challenges.
Before solving problems, let's define the key terms:
We assume that the work rate is constant-meaning the worker performs work steadily without breaks or slowdowns.
This diagram shows:
Why is this important? Understanding this lets us solve questions like: "How long will it take to finish a job?" or "If two workers work together, how fast can they complete a task?"
When one worker is involved, the problem typically asks for:
We use the basic formula connecting work, rate, and time:
Or equivalently, to find time:
Here is a stepwise approach to solve such problems:
graph TD A[Identify total work W] B[Determine work rate R = W / T or given] C[Calculate time using T = W / R] D[Find result] A --> B --> C --> D
Often, multiple workers tackle a job together. Their individual work rates add up because they work simultaneously, contributing cumulatively to the total work done per unit time.
For example, if Worker A can do 1 unit of work in 3 hours (rate = 1/3 units/hour) and Worker B can do 1 unit in 6 hours (rate = 1/6 units/hour), together their combined rate is:
\( R_{combined} = R_1 + R_2 = \frac{1}{3} + \frac{1}{6} = \frac{1}{2} \) units per hour.
Thus, they together complete 1 unit of work in 2 hours.
| Worker | Time to complete 1 unit (hours) | Work Rate (units/hour) | Notes |
|---|---|---|---|
| A | 3 | 1/3 | Faster worker |
| B | 6 | 1/6 | Slower worker |
| Combined | 2 | 1/2 | Sum of rates, less time |
The key formulas for combined workers are:
This method reduces total time needed to complete tasks significantly when workers have complementary speeds.
Some problems tie the amount of work done directly to payment or cost in Indian Rupees (INR). Payment is often proportional to the work completed by each worker.
For example, if the total payment for a task is Rs.1200, and a worker completes half the work, that worker earns Rs.600.
Calculations of cost and time require careful consideration of the work share and respective rates.
Step 1: Total work \( W = 1 \) unit (painting the entire fence is considered as 1 unit).
Step 2: Calculate work rate \( R = \frac{W}{T} = \frac{1}{5} \) units/hour.
Step 3: Work done in 2 hours is \( W_{done} = R \times \text{time} = \frac{1}{5} \times 2 = \frac{2}{5} \) units.
Answer: The worker completes \(\frac{2}{5}\) of the work in 2 hours.
Step 1: Find individual work rates:
Worker A's rate, \( R_A = \frac{1}{4} \) units/hour.
Worker B's rate, \( R_B = \frac{1}{6} \) units/hour.
Step 2: Combined work rate is:
\( R_{combined} = R_A + R_B = \frac{1}{4} + \frac{1}{6} = \frac{3}{12} + \frac{2}{12} = \frac{5}{12} \) units/hour.
Step 3: Time taken together to complete one unit of work:
\( T = \frac{1}{R_{combined}} = \frac{1}{5/12} = \frac{12}{5} = 2.4 \) hours.
Answer: Both workers together complete the job in 2.4 hours (2 hours and 24 minutes).
Step 1: Work done in 3 hours = 60% = 0.6 units.
Step 2: Work rate \( R = \frac{0.6}{3} = 0.2 \) units/hour.
Step 3: Total time to complete 1 unit of work:
\( T = \frac{1}{R} = \frac{1}{0.2} = 5 \) hours.
Answer: The worker will take 5 hours to complete the entire job.
Step 1: Calculate individual rates:
Worker A's rate: \( R_A = \frac{1}{10} \) units/hour.
Worker B's rate: \( R_B = \frac{1}{15} \) units/hour.
Step 2: Combined rate:
\( R_{combined} = \frac{1}{10} + \frac{1}{15} = \frac{3}{30} + \frac{2}{30} = \frac{5}{30} = \frac{1}{6} \) units/hour.
Step 3: Total time to complete the job:
\( T = \frac{1}{R_{combined}} = 6 \) hours.
Step 4: Work done by each worker in 6 hours:
Work by A = \( R_A \times T = \frac{1}{10} \times 6 = 0.6 \) units.
Work by B = \( R_B \times T = \frac{1}{15} \times 6 = 0.4 \) units.
Step 5: Payment split in ratio of work done:
Total payment = Rs.3000
Payment to A = \( 3000 \times \frac{0.6}{1} = Rs.1800 \)
Payment to B = \( 3000 \times \frac{0.4}{1} = Rs.1200 \)
Answer: Worker A earns Rs.1800 and Worker B earns Rs.1200.
Step 1: Find individual rates:
\( R_A = \frac{1}{12} \), \( R_B = \frac{1}{15} \), \( R_C = \frac{1}{20} \) units/hour.
Step 2: Calculate combined rate:
\( R_{combined} = \frac{1}{12} + \frac{1}{15} + \frac{1}{20} \)
Find common denominator (60):
\(= \frac{5}{60} + \frac{4}{60} + \frac{3}{60} = \frac{12}{60} = \frac{1}{5} \)
Step 3: Total time working together:
\( T = \frac{1}{R_{combined}} = 5 \) hours.
Answer: All three workers together complete the task in 5 hours.
When to use: When problems give mixed units of time.
When to use: When multiple workers perform the same job simultaneously.
When to use: When calculating work done in partial time or vice versa.
When to use: When solving for missing time or rates.
When to use: When payment depends on share of work completed.
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