गणित: समय, काम और दूरी

• Mathematics: Time, Work, and Distance

Time, Work, and Distance Overview

• Problems in this category involve relationships between time, work done, and the distances covered.
• Key formulas and concepts are used to solve these problems.
• These problems frequently appear in quantitative aptitude tests.

Basic Formulas

• Speed = Distance / Time
• Distance = Speed x Time
• Time = Distance / Speed
• These formulas are fundamental and apply to various problem types.

Conversions

• km/hr to m/sec conversion: x km/hr = (x * 5/18) m/sec
• m/sec to km/hr conversion: y m/sec = (y * 18/5) km/hr
• Conversions are essential for maintaining consistent units in calculations.

Average Speed

• Average Speed = Total Distance / Total Time
• If a body travels a distance at speed x km/hr and the same distance at speed y km/hr, then the average speed during the whole journey is 2xy / (x + y) km/hr
• Average speed calculations are important when speeds vary over a journey.

Relative Speed

• If two bodies are moving in the same direction at speeds x and y, where x > y, their relative speed is (x - y)
• If two bodies are moving in opposite directions at speeds x and y, their relative speed is (x + y)
• Relative speed is crucial for problems involving two moving objects.

Trains

• Time taken by a train of length L meters to pass a pole or a standing person is equal to the time taken to cover L meters
• Time taken by a train of length L meters to pass a stationary object of length b meters is the time taken to cover (L + b) meters
• If two trains of length a meters and b meters are moving in the same direction at u m/s and v m/s, where u > v, then the time taken by the faster train to pass the slower train is (a + b) / (u - v) sec
• If two trains of length a meters and b meters are moving in opposite directions at u m/s and v m/s, then the time taken by the trains to pass each other is (a + b) / (u + v) sec
• Train-related problems often involve calculating times and distances with consideration of train lengths.

Boats and Streams

• If the speed of a boat in still water is x km/hr and the speed of the stream is y km/hr, then:
  • Speed downstream: (x + y) km/hr
  • Speed upstream: (x - y) km/hr
• If the speed downstream is a km/hr and the speed upstream is b km/hr, then:
  • Speed in still water = 1/2 * (a + b) km/hr
  • Rate of stream = 1/2 * (a - b) km/hr
• These formulas help determine speeds in scenarios with water currents.

Time and Work

• If A can do a piece of work in n days, then A's 1 day's work = 1/n
• If A's 1 day's work is 1/n, then A can finish the work in n days
• If A is twice as good a workman as B, then the ratio of work done by A and B is 2:1
• If A is twice as good a workman as B, then the ratio of times taken by A and B to finish a work is 1:2
• Time and work problems involve calculating individual and combined work rates.

Work and Wages

• If A can do a piece of work in x days and B can do it in y days, then the share of work done by A when working together is y / (x + y)
• The work done is proportional to the wages
• Problems of this type involve dividing earnings based on individual contributions to work.

Chain Rule

• If M1 people can do W1 work in D1 days and M2 people can do W2 work in D2 days, then M1D1/W1 = M2D2/W2
• This rule provides the relationship between the number of workers, work done, and the time taken.

Pipes and Cisterns

• Similar to time and work problems, but involve filling or emptying tanks.
• If a pipe can fill a tank in x hours, then part filled in 1 hour = 1/x
• If a pipe can empty a tank in y hours, then part emptied in 1 hour = 1/y
• If a pipe can fill a tank in x hours and another pipe can empty the same tank in y hours, then the net part filled in 1 hour when both pipes are open = (1/x) - (1/y)
• Inlet: A pipe connected to a tank that fills it.
• Outlet: A pipe connected to a tank that empties it.

Problem-Solving Strategies

• Understand the problem statement to identify given parameters and what needs to be calculated.
• Convert all quantities into consistent units (e.g., meters and seconds, or kilometers and hours).
• Use the appropriate formulas based on the problem type (speed, average speed, relative speed).
• For time and work problems, determine individual work rates and combine them as needed.
• Draw diagrams or visualize scenarios, especially in relative motion and trains problems.
• Check the feasibility and reasonableness of your answer.