Question

Two persons \(A\) and \(B\) started from two different \(p_{i}\) towards each other. If the ratio of their speeds be \(3: 5\), then what is the ratio of distance covered by \(A\) and \(B\) respectively till the point of meeting? (a) \(1: 2\) (b) \(3: 4\) (c) \(3: 5\) (d) \(5: 3\)

Short answer

Answer: The ratio of the distances covered by person A and person B until they meet is 3:5.

Step-by-step solution

Understand the problem statement and given information

In the problem, we are given the ratio of the speeds between person A and person B as 3:5. They move towards each other, starting from two different points, and they meet at some point in the middle. Our goal is to find the ratio of the distances covered by A and B until they meet.

Write the relative speed formula

When two persons are moving towards each other, their relative speed is the sum of their individual speeds. In this case, the relative speed is 3 + 5 = 8.

Calculate the ratio of distances

Since the persons A and B are moving towards each other, the ratio of their distances covered will be the same as the ratio of their speeds. This is because the time taken for them to meet is the same. So, if person A has covered a distance in the same time as person B, the ratio of their distances will be the same as the ratio of their speeds. Therefore, the ratio of the distances covered by A and B is \[\frac{\text{distance covered by A}}{\text{distance covered by B}} = \frac{\text{speed of A}}{\text{speed of B}} = \frac{3}{5}.\] The answer is (c) 3:5.

Key concepts

Relative Speed

When two objects move towards each other, their relative speed is crucial in determining how quickly they will meet. Relative speed is essentially the sum of their individual speeds when they are heading towards each other.
For example, if two people, A and B, move towards each other with speeds in a ratio of 3:5, the relative speed is not 3 or 5, but rather the sum, which is 8. Relative speed helps us understand how the two entities approach one another:

• It combines the speed of both moving objects.
• Possesses the effect of movement from both sides, therefore it is faster than either entity moving alone.

Easily connecting this to real-world scenarios is useful. Imagine two people running towards each other, the rate at which the distance between them decreases is the relative speed.
Understanding relative speed is a key step in dealing with problems involving two moving objects.

Distance-Time Relationship

The connection between distance and time is explained elegantly by the formula: \[ \text{Distance} = \text{Speed} \times \text{Time} \]This formula is the bedrock of many motion-related problems. It tells us that the distance a moving object covers is a product of its speed and the time it travels at that speed.

• If speed increases while time remains constant, the distance covered will increase too.
• Similarly, for constant speed, a longer time leads to covering more distance.

For the scenario where two parties move towards each other at constant, but different speeds, they meet after the same time span. This shared time allows us to directly relate their distance covered to their speeds.Grasping this relationship helps solve many related problems, especially when determining how far each entity has traveled or how soon they'll meet.

Problem Solving in Algebra

Algebra frequently provides solutions to real-life problems by employing variables and equations. In motion problems involving ratio and proportion, algebra simplifies the understanding of speed, distance, and time.
By setting up equations, we can easily solve for unknowns:

• Use given ratios like speed presets. For example, a speed ratio of 3:5 becomes a useful component in algebraic equations.
• Determine the scale or factor by using variables to represent quantities related through given conditions.

An example of solving for distance traveled involves recognizing that distances are proportional to speeds when time is constant. Therefore, if A and B have speed ratio 3:5, the distance covered by each will share this same ratio when moving towards each other. The mastery of these problem-solving skills in algebra leads to the efficient resolution of real-world problems, not just limited to simple classroom exercises.