Gariba's Journey Calculating Walking Speed And Riding Distance

This article delves into a fascinating mathematical problem involving Gariba's journey, where he combines cycling and walking to cover a total distance. We'll explore how to calculate his walking speed and the distance he covered while cycling, given the time spent on each mode of transport, the total distance, and the relationship between his cycling and walking speeds. This problem perfectly illustrates the application of basic algebraic principles to solve real-world scenarios. Let's embark on this mathematical adventure and unravel the solution step by step. This detailed exploration aims to enhance your understanding of problem-solving techniques and the practical application of mathematical concepts.

Problem Statement: Gariba's Journey

Gariba embarked on a journey covering a total distance of 10.4 km. He spent 45 minutes cycling and an hour walking. The crucial detail is that his riding speed is three times his walking speed. Our mission is to determine, correct to two significant figures:

(a) Gariba's walking speed (b) The distance Gariba traveled by riding his bicycle

This problem presents a classic scenario where we need to use the relationships between distance, speed, and time, along with some algebraic manipulation, to arrive at the solution. We will break down the problem into manageable parts, define variables, formulate equations, and solve them systematically. This approach will not only help us find the answers but also enhance our problem-solving skills in mathematics and related fields.

Setting Up the Equations: Defining Variables and Relationships

To solve this problem effectively, let's first define our variables:

• Let v represent Gariba's walking speed in kilometers per hour (km/h).
• Since his riding speed is three times his walking speed, his riding speed is 3v km/h.

Now, let's convert the given times into hours:

• 45 minutes is equal to 45/60 = 0.75 hours.
• 1 hour remains as 1 hour.

We know that distance is equal to speed multiplied by time. Therefore:

• Distance traveled by riding = Riding speed × Time spent riding = 3v × 0.75 = 2.25v km
• Distance traveled by walking = Walking speed × Time spent walking = v × 1 = v km

The total distance covered is 10.4 km, which is the sum of the distances traveled by riding and walking. This gives us our primary equation:

2. 25v + v = 10.4

This equation is the cornerstone of our solution. It represents the relationship between Gariba's walking speed and the total distance he covered. By solving this equation, we can find his walking speed and subsequently calculate the distance he traveled by riding. The next step involves simplifying and solving this equation to find the value of v.

Solving for Walking Speed: Algebraic Manipulation

Now that we have our equation, 2.25v + v = 10.4, let's solve for v, which represents Gariba's walking speed. First, we combine the terms on the left side of the equation:

3. 25v + v = 3.25v

So, our equation becomes:

4. 25v = 10.4

To isolate v, we divide both sides of the equation by 3.25:

v = 10.4 / 3.25

v = 3.2

Therefore, Gariba's walking speed is 3.2 km/h. However, the question asks for the answer correct to two significant figures. Since 3.2 already has two significant figures, we don't need to round it further. This result gives us a clear understanding of how fast Gariba was walking during his journey. Now that we have the walking speed, we can move on to calculate the distance he traveled by riding his bicycle.

Calculating Riding Distance: Applying the Walking Speed

Having determined Gariba's walking speed to be 3.2 km/h, we can now calculate the distance he traveled by riding his bicycle. We recall that the distance traveled by riding is given by:

Distance traveled by riding = 2.25v km

We substitute the value of v (3.2 km/h) into this equation:

Distance traveled by riding = 2.25 × 3.2

Distance traveled by riding = 7.2 km

This is the exact distance Gariba covered while cycling. Since the question requires the answer to be correct to two significant figures, and 7.2 already has two significant figures, we don't need to round it further. This result provides us with a precise measure of the portion of the journey Gariba completed on his bicycle, complementing our earlier calculation of his walking speed. With both the walking speed and riding distance calculated, we have a complete picture of Gariba's journey.

Final Answers: Walking Speed and Riding Distance

In summary, after carefully analyzing the problem and performing the necessary calculations, we have arrived at the following answers:

(a) Gariba's walking speed: 3.2 km/h (correct to two significant figures) (b) The distance Gariba traveled by riding his bicycle: 7.2 km (correct to two significant figures)

These answers provide a clear and concise solution to the problem posed. We have successfully determined Gariba's walking speed and the distance he covered while cycling by applying fundamental mathematical principles and algebraic techniques. This exercise not only answers the specific questions but also illustrates the broader application of mathematics in solving real-world problems.

Conclusion: Problem-Solving in Action

This problem involving Gariba's journey serves as an excellent example of how mathematical concepts can be applied to solve practical scenarios. By breaking down the problem into smaller, manageable parts, defining variables, setting up equations, and solving them systematically, we were able to determine Gariba's walking speed and the distance he traveled by riding his bicycle. The key takeaway from this exercise is the importance of a structured approach to problem-solving. Whether in mathematics or other fields, the ability to analyze a problem, identify relevant information, formulate a plan, and execute it effectively is crucial for success. We hope this detailed exploration has enhanced your understanding of problem-solving techniques and the practical application of mathematical principles. This example underscores the value of mathematics not just as an academic subject but as a powerful tool for understanding and navigating the world around us.