Time & Distance Area

Time and Distance

Understanding the concepts of time and distance is essential for solving various problems in arithmetic aptitude, especially for BBA students. Here's a detailed explanation of these concepts:

Basic Concepts

Speed:

• Definition: Speed is the rate at which an object covers a distance.
• Formula: $\text{Speed}=\frac{\text{Distance}}{\text{Time}}$
• Units: Common units are meters per second (m/s) and kilometers per hour (km/h).

Distance:

• Definition: Distance is the total path covered by an object.
• Formula: $\text{Distance}=\text{Speed}\times \text{Time}$

Time:

• Definition: Time is the duration during which the distance is covered.
• Formula: $\text{Time}=\frac{\text{Distance}}{\text{Speed}}$

Unit Conversions

Speed from km/h to m/s:

$$\text{Speed (m/s)}=\text{Speed (km/h)}\times \frac{5}{18}$$

Speed from m/s to km/h:

$$\text{Speed (km/h)}=\text{Speed (m/s)}\times \frac{18}{5}$$

Formulas and Examples

Average Speed:

• If an object covers different distances $d_1,d_2,d_3,\dots$ in different times $t_1,t_2,t_3,\dots$

$$\text{Average Speed}=\frac{\text{Total Distance}}{\text{Total Time}}$$

• If the object travels the same distance at different speeds $s_1$ and $s_2$:

$$\text{Average Speed}=\frac{2\times s_1\times s_2}{s_1+s_2}$$

Example:

A car travels 100 km at 50 km/h and another 100 km at 100 km/h. Find the average speed.

• Total Distance = $100\text{km}+100\text{km}=200\text{km}$
• Time for first 100 km = $\frac{100}{50}=2\text{hours}$
• Time for second 100 km = $\frac{100}{100}=1\text{hour}$
• Total Time = $2+1=3\text{hours}$
• Average Speed = $\frac{200}{3}\approx 66.67\text{km/h}$

Relative Speed:

• Same Direction: When two objects move in the same direction, their relative speed is the difference of their speeds.

$$\text{Relative Speed}=\mathrm{\mid }{s}_1-s_2\mathrm{\mid }$$

• Opposite Directions: When two objects move in opposite directions, their relative speed is the sum of their speeds.

$$\text{Relative Speed}=s_1+s_2$$

Example:

Two trains are moving towards each other at speeds of 80 km/h and 120 km/h. If they are 400 km apart, find the time until they meet.

• Relative Speed = $80+120=200\text{km/h}$
• Time = $\frac{400}{200}=2\text{hours}$

Applications

Travel Planning:

• Estimating travel time based on different speeds and distances.
• Planning routes for minimum travel time or maximum efficiency.

Logistics and Supply Chain Management:

• Scheduling deliveries and managing the movement of goods.
• Coordinating the arrival and departure times of vehicles.

Business Operations:

• Optimizing commute times for employees.
• Planning schedules for business meetings and trips.

Tips for Solving Problems

1. Unit Consistency: Ensure all units are consistent before applying formulas.
2. Draw Diagrams: Visualizing the problem can help in understanding and solving it.
3. Use Formulas Wisely: Apply the appropriate formula based on what is given and what needs to be found.
4. Practice: Regular practice of different types of problems enhances problem-solving speed and accuracy.

Example Problems

Simple Distance Calculation:

A car travels at a speed of 60 km/h for 2 hours. Find the distance covered.

$$\text{Distance}=\text{Speed}\times \text{Time}=60\times 2=120\text{km}$$

Time Calculation:

How much time will it take to cover 150 km at a speed of 50 km/h?

$$\text{Time}=\frac{\text{Distance}}{\text{Speed}}=\frac{150}{50}=3\text{hours}$$

Average Speed Calculation:

A car travels 200 km in 4 hours. Calculate the average speed.

$$\text{Average Speed}=\frac{\text{Total Distance}}{\text{Total Time}}=\frac{200\text{km}}{4\text{hours}}=$$

$$\mathrm{<br>}$$

$$50\text{km/h}$$

Relative Speed Calculation:

Two cyclists start from the same point and ride in opposite directions. One rides at 15 km/h and the other at 20 km/h. How far apart will they be in 2 hours?

• Relative Speed = $15+20=35\text{km/h}$
• Distance = $\text{Relative Speed}\times \text{Time}=35\times 2=70\text{km}$

By mastering these concepts, a BBA student can efficiently solve various time and distance problems, which are crucial in business applications such as logistics, supply chain management, and operational planning.