Question

A person goes to his office at \(1 /\) 3rd of the speed at which he renurs from his office. If the average speed during the whole trip (i. \(e\), one round) is \(12 \mathrm{~km} / \mathrm{h}\). What is the speed of the person while he was going to his office? (a) 10 (b) 6 (c) 8 (d) can't be determined

Short answer

Answer: (c) 8 km/h.

Step-by-step solution

Determine Average Speed Formula

The formula for average speed is given by: Average Speed = (Total Distance) / (Total Time)

Identify Total Distance and Total Time

The total distance for the round trip will be twice the one-way distance, which we can denote as (2D). The time taken to go to the office is denoted as t1, and the time taken for the return trip is denoted as t2.

Use the Data given in the Problem

We are given the average speed for the entire trip as \((12 \mathrm{~km} / \mathrm{h})\). Also, we know that the speed while going to the office is one-third of the return speed, which can be represented as \(s = \frac{1}{3}s'\), where \(s\) is the speed while going to the office, and \(s'\) is the return speed. Time going: t1 = D/s Time returning: t2 = D/s' Using the average speed formula for the entire trip, we have: \[ \frac{2D}{t1 + t2} = 12 \]

Substitute Expressions for t1 and t2

We can rewrite the time expressions as follows: \[t1 = \frac{D}{s}\] \[t2 = \frac{D}{s'}\] Now, we have: \[ \frac{2D}{\frac{D}{s} + \frac{D}{s'}} = 12 \]

Simplify and Solve for s

First, clear the denominators by multiplying both sides by \(s \cdot s'\): \[2D(s \cdot s') = 12 ([s + s']D) \] \[2D(ss') = 12(s + s')D \] Divide both sides by 2D to eliminate D: \[ss' = 6(s + s')\] Now we are given the ratio of speeds: \(s = \frac{1}{3}s'\). Substitute this into the equation, and solve for s. \[s(\frac{3s}{1}) = 6(s + \frac{3s}{1}) \] \[3s^2 = 6(4s) \] \[3s^2 = 24s \] Divide both sides by 3s: \[s = 8\] So, the speed of the person while going to the office is \(\boxed{8 \mathrm{~km} / \mathrm{h}}\), corresponding to answer choice (c).

Key concepts

Average Speed

Average speed is a crucial concept in motion and transportation calculations. It enables us to understand how fast something moves, overall, during a journey that involves different speeds and/or varying distances. Average speed is defined as the total distance traveled divided by the total time taken.

This means if you go on a round trip, like in the given exercise, the formula for average speed is:

• Average Speed = \( \frac{\text{Total Distance}}{\text{Total Time}} \)

In the exercise, the total distance is traveled twice because it’s a round trip (to the office and back).
This approach helps to obtain an average even when traveling at different speeds. Keep in mind, the average speed may not match any of the actual speeds used but rather offers a balanced view over the whole journey.

Round Trip

The concept of a round trip is quite simple yet very significant in the context of speed and distance education. A round trip involves traveling from one point to another and then returning to the starting point.
In this scenario, the total distance traveled is twice the one-way distance. For example, if the person travels from home to office and then back, they cover double the distance they would if traveling only one way.

For problems involving round trips, it's often essential to track the distinct speeds for each phase of the trip - like the journey to work and back home. Understanding this helps break down the problem systematically, ensuring clear calculations and logical conclusions based on given data.

Speed Ratio

Understanding speed ratios helps evaluate how one speed compares to another, as part of analyzing motion problems like the one described in the exercise.

The exercise states that the speed while going to the office is \( \frac{1}{3} \) of the return speed.

• This ratio provides crucial information about the relative speeds during different sections of the journey.
• By using ratios, you don’t need to nail down exact speeds at first but rather focus on how these speeds relate to each other.

You can set up an equation to work out these relationships, which contributes to finding specific speed values. Using ratios simplifies complexity, especially when you can’t instantly determine each speed independently.

Algebraic Equations

Algebraic equations form the backbone of solving mathematical problems such as calculating speed and determining distances. They allow you to express relationships between different quantities precisely.

In situations similar to the exercise, setting up an algebraic equation helps utilize known speed ratios and average speed to find unknown variables.

The key steps include:

• Identify the variables, such as the speed going (\(s\)) and returning (\(s'\)).
• Utilize all given data to build equations involving these variables.
• Substitute known values and ratios into the equations.
• Simplify and solve the equations step-by-step.

For this specific problem, you're encouraged to utilize equations to determine how different sections of the trip relate to each other based on their speeds and distances. Algebra simplifies finding the unknowns through structured calculation.