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In aptitude tests, problems involving Time, Work, and Distance are very common. These problems test your ability to understand and manipulate the relationships between how long something takes, how fast it moves, and how much work is done. Mastering these concepts helps you solve real-world problems like calculating travel times, completing tasks efficiently, or determining wages based on work done.
We will use the metric system throughout this section, with distances in kilometres (km), speeds in kilometres per hour (km/h), and time in hours (h) or minutes (min). When discussing wages or costs related to work, we will use Indian Rupees (INR) to keep examples relevant and practical.
Understanding these concepts from first principles will build a strong foundation for solving a wide variety of problems efficiently and accurately.
Before solving problems, it is essential to know the key formulas that connect time, work, distance, and speed. These formulas form the backbone of all calculations in this topic.
| Concept | Formula | Explanation | Variables |
|---|---|---|---|
| Distance | \( D = S \times T \) | Distance equals speed multiplied by time | \(D\): Distance (km), \(S\): Speed (km/h), \(T\): Time (h) |
| Speed | \( S = \frac{D}{T} \) | Speed equals distance divided by time | \(S\): Speed (km/h), \(D\): Distance (km), \(T\): Time (h) |
| Time | \( T = \frac{D}{S} \) | Time equals distance divided by speed | \(T\): Time (h), \(D\): Distance (km), \(S\): Speed (km/h) |
| Work | \( \text{Work} = \text{Rate} \times \text{Time} \) | Work done equals rate of work multiplied by time taken | Work: Total work done, Rate: Work per unit time, Time: Time taken |
| Efficiency | \( \text{Efficiency} = \frac{\text{Work}}{\text{Time}} \) | Efficiency is work done per unit time | Efficiency: Work per unit time, Work: Total work done, Time: Time taken |
| Combined Work | \( \frac{1}{T} = \frac{1}{T_1} + \frac{1}{T_2} + \cdots \) | Time taken when multiple workers work together | \(T\): Total time, \(T_1, T_2\): Individual times |
| Relative Speed (Same Direction) | \( S_{rel} = |S_1 - S_2| \) | Relative speed is difference of speeds when moving in same direction | \(S_1, S_2\): Speeds of two objects |
| Relative Speed (Opposite Direction) | \( S_{rel} = S_1 + S_2 \) | Relative speed is sum of speeds when moving towards each other | \(S_1, S_2\): Speeds of two objects |
| Average Speed | \( S_{avg} = \frac{\text{Total Distance}}{\text{Total Time}} \) | Average speed over multiple segments | \(S_{avg}\): Average speed |
When two objects move, their speeds relative to each other affect how quickly they meet or move apart. This is called relative speed.
There are two main cases:
Understanding relative speed helps solve problems like two trains crossing each other or two people walking towards or away from each other.
Work refers to a task or job that needs to be completed. The amount of work done depends on the rate at which it is done and the time taken.
Efficiency is a measure of how quickly or effectively work is done. It is defined as the amount of work done per unit time.
When multiple workers or machines work together, their combined efficiency is the sum of their individual efficiencies.
graph TD A[Start: Given individual times] --> B[Calculate individual efficiencies as 1 / Time] B --> C[Add efficiencies to get combined efficiency] C --> D[Calculate total time as 1 / combined efficiency] D --> E[End: Total time taken by combined workers]
Step 1: Find the efficiency of each worker.
Efficiency of A = \( \frac{1}{12} \) (job per hour)
Efficiency of B = \( \frac{1}{16} \) (job per hour)
Step 2: Find combined efficiency by adding individual efficiencies.
Combined efficiency = \( \frac{1}{12} + \frac{1}{16} = \frac{4}{48} + \frac{3}{48} = \frac{7}{48} \) jobs per hour
Step 3: Calculate total time taken to complete one job together.
Total time = \( \frac{1}{\text{Combined efficiency}} = \frac{1}{\frac{7}{48}} = \frac{48}{7} \approx 6.86 \) hours
Answer: They will complete the job together in approximately 6 hours and 52 minutes.
Step 1: Convert lengths from meters to kilometers.
Length of first train = 120 m = 0.12 km
Length of second train = 80 m = 0.08 km
Step 2: Calculate relative speed since they move towards each other.
Relative speed = 60 + 40 = 100 km/h
Step 3: Total distance to be covered to cross each other = sum of lengths.
Total distance = 0.12 + 0.08 = 0.20 km
Step 4: Calculate time taken to cross each other.
Time = \( \frac{\text{Distance}}{\text{Speed}} = \frac{0.20}{100} = 0.002 \) hours
Convert time to seconds: \(0.002 \times 3600 = 7.2\) seconds
Answer: The trains will take 7.2 seconds to cross each other completely.
Step 1: Calculate upstream and downstream speeds.
Upstream speed = \( \frac{12}{2} = 6 \) km/h
Downstream speed = \( \frac{12}{1} = 12 \) km/h
Step 2: Use formulas:
Speed of boat in still water, \( b = \frac{\text{Downstream speed} + \text{Upstream speed}}{2} = \frac{12 + 6}{2} = 9 \) km/h
Speed of stream, \( s = \frac{\text{Downstream speed} - \text{Upstream speed}}{2} = \frac{12 - 6}{2} = 3 \) km/h
Answer: Speed of boat in still water is 9 km/h and speed of stream is 3 km/h.
Step 1: Calculate daily wages of the first worker.
Daily wage of first worker = \( \frac{900}{15} = Rs.60 \) per day
Step 2: Calculate work rate (efficiency) of each worker.
Work rate of first worker = \( \frac{1}{15} \) work/day
Work rate of second worker = \( \frac{1}{20} \) work/day
Step 3: Calculate total work done by second worker in 10 days.
Work done = \( \frac{1}{20} \times 10 = \frac{10}{20} = \frac{1}{2} \) of the job
Step 4: Calculate total wages for the job based on first worker's rate.
Total wages for full job = Rs.900
Step 5: Calculate wages for half the job done by second worker.
Wages = \( \frac{1}{2} \times 900 = Rs.450 \)
Answer: The second worker should be paid Rs.450 for 10 days of work.
Step 1: Calculate time taken for each segment.
Time for first segment = \( \frac{60}{40} = 1.5 \) hours
Time for second segment = \( \frac{90}{30} = 3 \) hours
Step 2: Calculate total distance and total time.
Total distance = 60 + 90 = 150 km
Total time = 1.5 + 3 = 4.5 hours
Step 3: Calculate average speed.
Average speed = \( \frac{150}{4.5} = 33.\overline{3} \) km/h
Answer: The average speed for the entire journey is approximately 33.33 km/h.
When to use: At the start of any distance or speed problem to avoid unit mismatch errors.
When to use: When multiple workers or machines work together to find combined efficiency easily.
When to use: In problems involving two moving objects like trains or boats.
When to use: When speed varies over different parts of a journey.
When to use: In problems involving relative motion and streams to avoid confusion.
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