Minimum Steps To Meet: Aziz And Bülent's Walk

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Minimum Steps to Meet: Aziz and Bülent's Walk

Let's dive into a fun math problem involving Aziz and Bülent, who are walking towards each other! This problem combines a bit of distance, steps, and finding the least common multiple. Ready to break it down? Let’s get started!

Understanding the Problem

Okay, so here’s what we know:

  • Aziz starts at point A, and each of his steps is 52 cm long.
  • Bülent starts at point B, and each of his steps is 48 cm long.
  • They walk towards each other and meet at point C.
  • Point C is the midpoint of the total distance between A and B.

Our mission is to find the minimum total number of steps Aziz and Bülent take to meet at point C.

Setting Up the Solution

Since C is the midpoint of the path AB, the distance from A to C is equal to the distance from B to C. Let's call this distance D. Therefore, Aziz covers a distance D with steps of 52 cm each, and Bülent covers the same distance D with steps of 48 cm each. This means that D must be a multiple of both 52 and 48. To minimize the number of steps, we need to find the least common multiple (LCM) of 52 and 48.

Finding the Least Common Multiple (LCM)

First, let’s find the prime factorization of 52 and 48:

  • 52 = 2^2 * 13
  • 48 = 2^4 * 3

To find the LCM, we take the highest power of each prime factor that appears in either factorization:

  • LCM(52, 48) = 2^4 * 3 * 13 = 16 * 3 * 13 = 48 * 13 = 624

So, the least common multiple of 52 and 48 is 624 cm. This means the distance D from A to C (and from B to C) is 624 cm.

Calculating the Number of Steps

Now that we know the distance D, we can calculate how many steps each person takes:

  • Aziz's steps: Distance / Step length = 624 cm / 52 cm/step = 12 steps
  • Bülent's steps: Distance / Step length = 624 cm / 48 cm/step = 13 steps

To find the minimum total number of steps, we add the number of steps Aziz and Bülent took:

  • Total steps = Aziz's steps + Bülent's steps = 12 + 13 = 25 steps

Therefore, the minimum total number of steps they took is 25.

Key Strategies and Considerations

Understanding the Midpoint

The fact that point C is the midpoint is crucial. It tells us that both Aziz and Bülent cover the same distance. This simplifies the problem because we only need to find one distance that is a multiple of both step lengths.

Using the Least Common Multiple (LCM)

The LCM is essential for minimizing the number of steps. If we used any other common multiple, the number of steps would be larger. For example, if we used 2 * LCM(52, 48) = 1248, Aziz would take 24 steps and Bülent would take 26 steps, totaling 50 steps, which is not the minimum.

Prime Factorization

Prime factorization is a reliable method for finding the LCM, especially when dealing with larger numbers. It ensures you account for all necessary factors to find the smallest common multiple.

Check Your Work

Always double-check your calculations to avoid simple arithmetic errors. Make sure the distance you calculate is indeed divisible by both step lengths, and that your final answer makes sense in the context of the problem.

Real-World Applications

Route Optimization

Understanding how to minimize steps or movements can be applied in route optimization for delivery services or even in robotics, where efficient movement is crucial for completing tasks quickly.

Resource Allocation

In resource allocation, finding the LCM helps in dividing resources efficiently among different units or tasks, ensuring minimal waste and optimal utilization.

Event Planning

When organizing events, knowing how to arrange items or activities in the most efficient manner can save time and effort. For example, optimizing the layout of booths at a fair to minimize walking distance for attendees.

Common Pitfalls

Using Any Common Multiple Instead of LCM

A common mistake is to find any common multiple instead of the least common multiple. This will give you a valid solution, but not the minimum number of steps.

Arithmetic Errors

Simple calculation mistakes can lead to incorrect answers. Always double-check your arithmetic, especially when dividing or multiplying larger numbers.

Misunderstanding the Problem

Sometimes, not fully grasping the problem's conditions can lead to wrong assumptions. Make sure you understand every detail, such as the significance of the midpoint, before attempting to solve the problem.

Practice Problems

Problem 1

Alice starts from point X with each step being 60 cm, and Bob starts from point Y with each step being 75 cm, walking towards each other. They meet at point Z, which is the midpoint of the path XY. What is the minimum total number of steps they took?

Problem 2

Two robots, Alpha and Beta, start moving towards each other from opposite ends of a conveyor belt. Alpha's movement unit is 45 mm, and Beta's movement unit is 60 mm. They meet at the exact center of the conveyor belt. What is the minimum total number of movement units the two robots made?

Solutions

Solution 1

  • LCM(60, 75) = 300
  • Alice's steps: 300 / 60 = 5
  • Bob's steps: 300 / 75 = 4
  • Total steps: 5 + 4 = 9

Solution 2

  • LCM(45, 60) = 180
  • Alpha's units: 180 / 45 = 4
  • Beta's units: 180 / 60 = 3
  • Total units: 4 + 3 = 7

Conclusion

So, there you have it! By understanding the problem, using the least common multiple, and being careful with our calculations, we can solve these types of problems efficiently. Remember to break down the problem into smaller parts and always double-check your work. Keep practicing, and you'll become a pro at solving these minimum step problems! And Hey Guys, now you know the strategy. Keep up the great work!