Question

A certain distance is covered at a certain speed. If half of this distance is covered in double the time, the ratio of the two speed is: a.1:16 b. 4:1 c. 2:1 d.2:8

Short answer

Answer: The ratio between the original speed and the new speed is 4:1.

Step-by-step solution

Define the variables

Let: - D = total distance - T = total amount of time - S₁ = original speed - S₂ = new speed

Step ₂: Write the relationship between distance, time, and speed

The relationship between distance, time, and speed is given by: Distance = Speed × Time For the original speed and the new speed, we can rewrite the relationship as follows: D = S₁ × T (D/2) = S₂ × (2T)

Step ₃: Derive the relationship between the speeds

We now have two equations with two variables. Let's solve them to find the relationship between S₁ and S₂: From the first equation, S₁ = D / T Now, we will substitute this in the second equation: (D/2) = S₂ × (2T) Dividing both sides by 2T, we have: (D/2)/(2T) = S₂ Now, D/(4T) = S₂

Step ₄: Find the ratio between the two speeds

We know the relationship between S₁ and S₂: S₁ = D / T S₂ = D / (4T) To find the ratio S₁ : S₂, we'll divide S₁ by S₂: (S₁ / S₂) = ((D / T) / (D / (4T))) Since D ≠ 0 and T ≠ 0, we can simplify this expression by multiplying the numerator and the denominator by 4T: (S₁ / S₂) = ((D / T) * (4T / D)) The D and T cancel out, leaving: (S₁ / S₂) = 4 / 1 So, the ratio of the two speeds is S₁ : S₂ = 4 : 1. The correct answer is (b) 4:1.

Key concepts

Distance-Time-Speed Relationship

In solving problems related to speed and distance, it's essential to understand their intrinsic relationship. The fundamental formula is: \( \text{Distance} = \text{Speed} \times \text{Time} \). This means that if you know any two of these variables, you can solve for the third.

This concept is useful in many real-world scenarios. Whether you are calculating the time it will take to travel a certain distance, or determining how fast you must go to reach a destination on time, this relationship provides the necessary framework.

For instance, if half of a journey takes double the time it used to, as in the exercise, it directly affects the speed. By manipulating the formula to solve for speed, \( \text{Speed} = \frac{\text{Distance}}{\text{Time}} \), we can effectively determine changes in speed due to alterations in distance or time.

Ratio and Proportion

Ratios and proportions are mathematical tools that help compare quantities. In the context of speed and distance problems, they provide a method to compare different speeds when distances and times change.

For example, the exercise asks to find the ratio of two speeds. By setting up equations for each scenario (original and modified), you can find each corresponding speed.

• Original speed formula: \( S_1 = \frac{D}{T} \)
• Modified speed formula: \( S_2 = \frac{D}{4T} \)

To find out how these speeds compare, compute the ratio \( S_1 : S_2 \). This ratio helps us understand the relative change in speed due to the change in travel conditions.

Problem Solving Techniques

Solving complex word problems can become manageable by following a step-by-step strategy. The critical steps involve defining variables, understanding given information, and applying the correct formulas.

• Start by defining variables clearly, which helps in organizing the information.
• Identify the equations that relate to the problem, such as distance-time-speed relationships.
• Substitute known values into equations and solve for the unknowns.
• Finally, interpret the results, such as finding the ratio of speeds as in the exercise.

Practicing such techniques strengthens your mathematical problem-solving skills and prepares you for a variety of related questions.

Mathematics Concepts

A deep dive into mathematics concepts like the distance-time-speed relationship or ratios not only aids in solving such specific problems but also builds a strong foundation for more advanced mathematics.

Core concepts such as understanding equations, manipulating algebraic expressions, and simplifying ratios are crucial skills to develop.

By engaging with these foundational ideas, students learn to apply mathematical thinking and logical reasoning to a broad range of problems. Emphasizing the 'why' and 'how' behind these concepts ensures that students do not just memorize procedures but also gain insight into mathematical reasoning.