Steps Challenge: Who Walked Further?

by Kenji Nakamura 37 views

The Step-by-Step Breakdown

Okay, so here's the situation: Two people walked a total of 64 meters, which is equivalent to 6400 centimeters (since 1 meter = 100 centimeters). Together, they took 100 steps. Now, the first person's steps are longer – each one measures 70 centimeters. The second person's steps are a bit shorter, measuring 50 centimeters each. Our mission, should we choose to accept it, is to find out how many more steps the first person took than the second person. Sounds like a fun challenge, right?

Setting Up the Equations

To crack this, we'll use some good ol' algebra. Let's say the first person took 'x' steps and the second person took 'y' steps. We know two crucial things:

  1. Total Steps: x + y = 100 (Together, they took 100 steps)
  2. Total Distance: 70x + 50y = 6400 (The total distance they walked in centimeters)

See? We've turned a word problem into mathematical equations! Now, we have a system of two equations with two unknowns. This is totally solvable, and we're going to conquer it!

Solving the System

There are a couple of ways we can solve this. One popular method is substitution. Let's solve the first equation (x + y = 100) for x:

x = 100 - y

Now, we'll substitute this value of x into the second equation (70x + 50y = 6400):

70(100 - y) + 50y = 6400

Let's simplify this bad boy:

7000 - 70y + 50y = 6400

Combine those 'y' terms:

7000 - 20y = 6400

Now, let's isolate 'y'. Subtract 7000 from both sides:

-20y = -600

Divide both sides by -20:

y = 30

Woohoo! We've found 'y', which is the number of steps the second person took. Now, let's plug this value back into our equation x = 100 - y to find 'x':

x = 100 - 30

x = 70

Awesome! The first person took 70 steps.

Finding the Difference

We're almost there! The question asks how many more steps the first person took than the second. So, we simply subtract the number of steps the second person took from the number of steps the first person took:

Difference = x - y = 70 - 30 = 40

The Grand Finale: The Answer Revealed

Alright, guys, we did it! The first person took 40 more steps than the second person. That's our final answer! Isn't it satisfying to solve a good math puzzle? We broke down the problem, set up equations, solved them, and found our solution. You're all mathematical masterminds!

Why This Matters: Real-World Applications

Now, you might be thinking, "Okay, cool, but when will I ever use this in real life?" Well, this kind of problem-solving is everywhere! It's not just about steps and distances; it's about understanding relationships between different pieces of information and using logic to find a solution. Think about things like:

  • Budgeting: Figuring out how to allocate money across different categories based on needs and constraints.
  • Project Management: Estimating timelines and resources needed for different tasks.
  • Data Analysis: Identifying patterns and trends in data to make informed decisions.

The skills we used to solve this step problem – setting up equations, solving for unknowns, and interpreting results – are valuable in all these areas and more. So, keep those math muscles flexed! You never know when they'll come in handy.

Let's Talk About Problem-Solving Strategies

Beyond the specific math skills, let's chat a bit about the process we used to solve this problem. This is just as important as the math itself!

  • Read Carefully: The very first step is always to understand the problem completely. What information are you given? What are you trying to find? Underlining key facts and re-reading the problem are great strategies.
  • Break It Down: Complex problems can feel overwhelming. Break them into smaller, more manageable pieces. That's what we did by identifying the total steps and total distance as separate equations.
  • Visualize: Sometimes, drawing a diagram or picture can help you see the relationships between different elements of the problem. For this problem, you could imagine the two people walking and visualize their different stride lengths.
  • Translate into Math: Word problems are written in language, but we need to translate them into the language of mathematics – equations! Identifying the variables (x and y in our case) and setting up the equations is crucial.
  • Solve Systematically: Once you have the equations, use a systematic method like substitution or elimination to solve for the unknowns. This helps avoid errors and keeps your work organized.
  • Check Your Answer: Does your answer make sense in the context of the problem? If the first person took 70 steps and the second person took 30, does that add up to 100 total steps? Does the total distance they walked match the given information? Always double-check!

Practice Makes Perfect: More Problems to Ponder

Just like any skill, problem-solving gets easier with practice. The more you challenge yourself with different types of problems, the better you'll become at identifying patterns, applying strategies, and finding solutions. So, don't be afraid to tackle new challenges!

Here are a few similar problems you might want to try:

  1. Two trains leave stations 300 miles apart at the same time, traveling towards each other. One train travels at 60 mph, and the other travels at 40 mph. How long will it take them to meet?
  2. A store sells apples for $1 each and oranges for $1.50 each. If someone buys 10 pieces of fruit for $12, how many of each fruit did they buy?
  3. A rectangle has a perimeter of 24 inches. If the length is twice the width, what are the dimensions of the rectangle?

These problems might seem different on the surface, but they all involve similar problem-solving strategies. Give them a try, and see what you can discover!

Wrapping Up: You're a Problem-Solving Superstar!

We've journeyed through a fascinating math problem today, guys. We've seen how to break down a complex scenario, translate it into equations, solve those equations, and interpret the results. More importantly, we've explored the broader skills of problem-solving – reading carefully, breaking down problems, visualizing, and checking our work. These are skills that will serve you well in all aspects of life, from academics to careers to everyday decision-making. So, keep practicing, keep exploring, and keep that problem-solving spirit alive! You've got this!