Question

A covers \(1 / 4\) th of his journey at \(20 \mathrm{~km} / \mathrm{h}\) and \(1 / 3 \mathrm{rd}\) of the rest at \(25 \mathrm{~km} / \mathrm{h}\) and half of the rest at \(30 \mathrm{~km} / \mathrm{h}\) and rest at the speed of \(40 \mathrm{~km} / \mathrm{h}\). What is the average speed of \(A\) ? (a) \(13 \frac{78}{89} \mathrm{~km} / \mathrm{h}\) (b) \(12 \mathrm{~km} / \mathrm{h}\) (c) \(26 \frac{86}{89} \mathrm{~km} / \mathrm{h}\) (d) \(28 \mathrm{~km} / \mathrm{h}\)

Short answer

Answer: The average speed of A's entire journey is \(13\frac{78}{89}\) km/h.

Step-by-step solution

Represent the Journey's Parts as Fractions of the Total Journey

Let P be the total length of A's journey. Since A covers 1/4th of his journey at 20 km/h, A covers P/4 at 20 km/h. A covers 1/3rd of the rest at 25 km/h, which means A covers 1/3 of 3P/4 at 25 km/h, or P/4 at 25 km/h. A covers half of the remaining journey at 30 km/h, which means A covers 1/2 of (1/2)P at 30 km/h, or P/4 at 30 km/h. Finally, A covers the rest at 40 km/h, which is again P/4 at 40 km/h.

Calculate the Time Spent in Each Part of the Journey

Using the formula time = distance / speed, we can calculate the time spent in each part of the journey as follows: Time at 20 km/h = \((P/4) / 20\) = \(P / (4 * 20)\) Time at 25 km/h = \((P/4) / 25\) = \(P / (4 * 25)\) Time at 30 km/h = \((P/4) / 30\) = \(P / (4 * 30)\) Time at 40 km/h = \((P/4) / 40\) = \(P / (4 * 40)\)

Calculate the Total Time Spent in the Journey

The total time spent in the journey will be the sum of the times spent at each speed: Total Time (T) = \(P / (4 * 20) + P / (4 * 25) + P / (4 * 30) + P / (4 * 40)\)

Calculate the Total Distance Covered in the Journey

Since we have represented each part of the journey as a fraction of the total distance (P), the total distance covered will be P.

Find the Average Speed for the Entire Journey

The average speed (V) can be found using the formula average speed = total distance / total time: V = P / T Substituting the values calculated in steps 3 and 4, we have: V = \(P / (P / (4 * 20) + P / (4 * 25) + P / (4 * 30) + P / (4 * 40))\) Now we can simplify the expression and solve for V: V = \(\frac{P}{P*(1/(4*20) + 1/(4*25) + 1/(4*30) + 1/(4*40))}\) V = \(\frac{1}{1/(4*20) + 1/(4*25) + 1/(4*30) + 1/(4*40)}\) V = \(\frac{1}{1/80 + 1/100 + 1/120 + 1/160}\) V = \(\frac{1}{(120 + 96 + 80 + 60)/48000}\) = \(\frac{48000}{356}\) = \(13\frac{78}{89}\) km/h So the average speed of A is \(13\frac{78}{89}\) km/h or (a).

Key concepts

Time, Speed and Distance

Understanding how time, speed, and distance interrelate is crucial for solving problems related to motion and travel. The core formula connecting these three elements is:

• Speed = Distance / Time

This equation can be rearranged to find time as Time = Distance / Speed or to find distance as Distance = Speed × Time.

In the given exercise, A's journey is divided into several parts, each with a distinct speed and distance. By expressing the parts of the journey as fractions of the total distance, we create a mathematical relationship that simplifies the calculation process.

When dealing with fractional parts of a journey, it's essential to treat each segment separately and apply the time, speed, and distance relationship to calculate the time taken for each segment.

The concept of dividing the journey into parts also introduces the importance of maintaining uniform units, such as kilometers and hours, to ensure the accuracy of your calculations.

Speed Calculation

Calculating the speed is about determining how fast something moves over a certain distance in a specific amount of time. In this exercise, the average speed is our main target.

Average speed is calculated by dividing the total distance covered by the total time taken for the journey. It considers the variable speeds at which different parts of the journey are traveled.

• Here, A travels four segments of his journey, each at different speeds of 20 km/h, 25 km/h, 30 km/h, and 40 km/h.
• The distance covered in each segment is a fraction of the total distance P, specifically P/4.
• Accurately obtaining the total time involved requires calculating each segment's individual time using the formula: Time = Distance / Speed.

Once you have these individual times, summing them up will give the total time, which is then used with the total distance to compute the average speed.

Journey Analysis

Breaking down a journey into smaller segments simplifies complex problems and makes it easier to analyze the movement. In the scenario presented, the journey analysis involved:

• Dividing the whole journey into four unequal segments, each with specified speeds.
• Using these segments, each expressed as a mathematical fraction of the entire journey, allowed for precise calculation of the time taken for each segment.

Once the journey is broken down into these manageable components, it becomes easier to gather insights. These insights help calculate aggregate measures like average speed, ensuring every aspect is accounted for.

Careful journey analysis like this ensures that each detail is handled, creating a clear path to accurately solving the problem. This process is valuable for any travel problem which needs detailed understanding, and forms the bedrock of many real-world applications in navigation and logistics.