Question

A covers \(1 / 3 \mathrm{rd}\) of his journey at the speed of \(10 \mathrm{~km} / \mathrm{h}\) and half of the rest at the speed of \(20 \mathrm{~km} / \mathrm{h}\) and rest at the speed of \(30 \mathrm{~km} / \mathrm{h}\). What is the average speed of \(A ?\) (a) \(6 \frac{2}{11} \mathrm{~km} / \mathrm{h}\) (b) \(16 \frac{4}{11} \mathrm{~km} / \mathrm{h}\) (c) \(16 \frac{4}{11} \mathrm{~m} / \mathrm{s}\) (d) none of these

Short answer

#tag_title# Short Answer #tag_content# To find the average speed of A during the entire journey, we determined the fractions of the journey and calculated the time taken for each part. We added the time taken for all parts to get the total time taken. Finally, we divided the total distance (D) by the total time taken to get the average speed: \(D /\) ( \((1/3)D / 10\) + \((1/3)D / 20\) + \((1/3)D / 30\)).

Step-by-step solution

Determine the fractions of the journey

The problem states that A traveled 1/3 of his journey at 10 km/h, half of the rest (which is half of 2/3, or 1/3 of the total distance) at 20 km/h, and the remaining distance at 30 km/h. Now let's take that his entire trip is of distance D. Therefore, we have: - 1/3 of D at 10 km/h - 1/3 of D at 20 km/h - 1/3 of D at 30 km/h

Calculate the time taken for each part

Now, we will calculate the time taken for each part of the journey using the formula time = distance/speed. - Time taken for the first part: \((1/3)D / 10\) - Time taken for the second part: \((1/3)D / 20\) - Time taken for the third part: \((1/3)D / 30\)

Calculate the total time taken

To calculate the total time taken for the journey, we add the time taken for each part: Total time taken = \((1/3)D / 10\) + \((1/3)D / 20\) + \((1/3)D / 30\)

Determine the average speed

To find the average speed, we divide the total distance by the total time taken: Average Speed = \(D /\) Total Time Taken Substitute the values of D and Total Time Taken from the previous steps: Average Speed = \(D /\) ( \((1/3)D / 10\) + \((1/3)D / 20\) + \((1/3)D / 30\))

Key concepts

Fractions of Journey

In this exercise, we start by breaking down the entire journey into fractions.
A covers his journey in three segments, each representing a specific fraction of the whole distance.
Understanding this breakdown is key to solving the problem correctly.

Here's how the journey is divided:

• The first (\(\frac{1}{3}\)) of the entire distance, denoted as (\(D\)), was traveled at a speed of (\(10\) km/h).
• The next segment is half of the remaining distance. Since (\(\frac{2}{3}\) was left, half of it would be another (\(\frac{1}{3}\)) of (\(D\)). This part of the trip was at (\(20\) km/h).
• The last (\(\frac{1}{3}\)) of (\(D\)) was completed at (\(30\) km/h).

This clear division allows us to handle each segment separately when calculating time and speed. Recognizing how the journey is split into different fractions ensures each step of the problem is approached correctly.

Distance and Speed Relationship

The relationship between distance, speed, and time is fundamental to understanding and solving problems involving motion.
This exercise is based on the formula that links these components:

• Speed = Distance / Time
• Distance = Speed (\(\times\)) Time
• Time = Distance / Speed

In our scenario, it's vital to recognize how the speed of travel affects the time taken to cover each fraction of the journey.

For instance, traveling a fixed (\(\frac{1}{3}\) of distance (\(D\)) at 10 km/h will take longer compared to traveling the same fraction at 30 km/h.
This direct relationship allows for calculating how long each part of the trip takes and illustrates why speed influences travel time crucially.
By grasping this relationship, you can apply it to varied problems involving speeds and timings across different travel fractions.

Time Calculation

Calculating time for each section of the journey is done using the fundamental formula (\(\text{Time} = \frac{\text{Distance}}{\text{Speed}}\)).
It is essential to apply this correctly to each segment.

For our problem:

• The time for the first (\(\frac{1}{3}\) of (\(D\)) at (\(10\) km/h) is calculated as: (\(\frac{\left(1/3\right)D}{10}\)).
• For the next (\(\frac{1}{3}\) of (\(D\)) at (\(20\) km/h), the time taken is: (\(\frac{\left(1/3\right)D}{20}\)).
• Finally, the time for covering the last part at (\(30\) km/h which is also (\(\frac{1}{3}\) of (\(D\)) is: (\(\frac{\left(1/3\right)D}{30}\)).

Summing these times gives the total time taken for the journey, crucial for calculating the average speed.
Make sure to compute each section accurately to ensure the final calculations are correct.

Mathematical Problem-Solving

The key to tackling mathematical problems like this is to break them down into manageable steps.
Follow a systematic approach, ensuring each calculation is methodically processed.

Here's how to solve this exercise step-by-step:

• Identify all given data and how it relates to different journey segments.
• Calculate the time taken for each part of the journey using the appropriate speed for each fraction of the distance.
• Add the times together to find the total time taken for the journey.
• Finally, use the total distance and total time to calculate the average speed using the formula: (\(\text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}}\)).

Solutions involve thorough analysis and repeated checking of each step to avoid errors.
Using such a method ensures clarity and confidence in your final answer, which in this case results in calculating the average speed of 16 (\(\frac{4}{11}\) km/h as the correct answer (b).
Practice systematically breaking down complex problems for mastery in mathematical problem-solving.