Solving time and distance problems without relying on formula memorization involves logical reasoning and breaking down the problem into manageable parts. Here's how to approach it step by step:

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General Approach

1. Understand the Basics:

   • Distance is the length of the path traveled.
   • Speed is how fast you're moving.
   • Time is how long you take to cover a distance.

2. Use Unitary Method:

   • Think of "per unit" changes: If a car covers 60 km in 1 hour, then in 1 minute (1/60 of an hour), it covers 1 km.

3. Ratio and Proportion:

   • Speed and time are inversely proportional: If speed increases, time decreases.
   • For example, if a journey takes 2 hours at 40 km/h, halving the speed to 20 km/h doubles the time to 4 hours.

4. Common-Sense Logic:

   • If you know the distance and time for a journey, you can deduce the speed.
   • If you know speed and time, multiply to find the distance.

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Example Problems

Example 1: A man walks 20 km in 4 hours. What is his speed?

1. In 1 hour, he walks $20 \div 4 = 5 \, \text{km}$.
2. His speed is 5 km/h.

Example 2: A train travels 90 km in 1.5 hours. What is its speed?

1. In 1 hour, it travels $90 \div 1.5 = 60 \, \text{km}$.
2. Speed = 60 km/h.

Example 3: A car travels 240 km at a speed of 60 km/h. How long does it take?

1. For every 60 km, it takes 1 hour.
2. For 240 km, it will take $240 \div 60 = 4 \, \text{hours}$.

Example 4: Two cyclists start at the same point and ride in opposite directions at 15 km/h and 20 km/h, respectively. How far apart will they be after 3 hours?

1. Distance covered by the first cyclist in 3 hours = $15 \times 3 = 45 \, \text{km}$.
2. Distance covered by the second cyclist in 3 hours = $20 \times 3 = 60 \, \text{km}$.
3. Total distance = $45 + 60 = 105 \, \text{km}$.

Example 5: A person increases his speed by 10% and completes a 50 km journey in 45 minutes less. What was his original speed?

1. Assume his original speed is $x \, \text{km/h}$.
2. Time taken initially = $\frac{50}{x}$ hours.
3. After increasing speed by 10%, new speed = $1.1x$.
4. New time taken = $\frac{50}{1.1x}$.
5. Time difference = $\frac{50}{x} - \frac{50}{1.1x} = \frac{45}{60} \, \text{hours}$.
6. Solve the equation logically without plugging into formulas.

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Tips for Competitive Exams

1. Think of real-life analogies: Imagine walking, cycling, or driving to make problems intuitive.
2. Use estimation: Quickly eliminate improbable options by estimating answers.
3. Work backward: For multiple-choice questions, test options to find the solution.
4. Break down problems: Simplify multi-step problems into smaller logical steps.
5. Practice mental math: Familiarize yourself with quick multiplications, divisions, and percentages.

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Would you like a detailed walkthrough of any specific type of problem or more practice examples? 😊

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5 Related Questions to Explore

1. How can you calculate relative speed in cases where two objects move toward each other or away from each other?
2. What are the differences between average speed and instantaneous speed, and how can you calculate them logically?
3. How can you handle time, speed, and distance problems involving circular paths?
4. How do you solve time and distance problems involving trains and platforms without formulas?
5. What is the role of proportional reasoning in solving complex speed-distance problems?

One Tip

Always visualize the problem with a basic diagram or timeline. This helps in simplifying relationships between distance, time, and speed.