José Vs Mario: Efficiency Math Problem Solved!

by Mei Lin 47 views

Hey guys! Ever wrestled with a word problem that just seems to twist your brain into knots? We've got one here that mixes efficiency, teamwork, and a little bit of time calculation. Let's break it down together and make sure we not only get to the answer but also understand the why behind it. Buckle up, it's math time!

The Efficiency Puzzle: José vs. Mario

The core of this problem revolves around comparing the work rates of José and Mario. We know José is 20% more efficient than Mario. What does this actually mean? In simple terms, if Mario does 100 units of work in an hour, José does 120 units in the same hour. That's the essence of efficiency! Let's dive into how we can use this information to solve our time-related conundrum.

Deciphering the Efficiency Difference

  • To kick things off, let's represent Mario's work rate. Think of it as how fast he gets things done. If we call Mario's work rate 'M,' then José's work rate, being 20% more efficient, is '1.2M.' This is crucial because it sets the stage for comparing their individual contributions to the job when they work together.*
  • Efficiency is key here. Understanding that José's 20% extra efficiency translates to a rate of 1.2 times Mario's is the secret sauce to cracking this problem. It's like saying José has a turbo boost compared to Mario's standard speed. This forms the foundational ratio that we will use to compare their work outputs.
  • But why is this percentage comparison so important? It allows us to convert the qualitative statement (José is more efficient) into a quantitative relationship that we can use in our calculations. Instead of just knowing José is faster, we now know how much faster, which is a game-changer for solving the problem.

The Power of Teamwork: Combined Efforts

When José and Mario team up, magic happens—or rather, work gets done faster! We know they complete a task together in 3 hours and 40 minutes. But how do we translate this combined time into something usable for our calculations? That's where converting everything into minutes comes in handy. Trust me, sticking to one unit of time makes everything smoother. Let's see how we can leverage their teamwork to understand their individual capabilities.

  • First, we need to convert 3 hours and 40 minutes into total minutes. We've got 3 hours * 60 minutes/hour = 180 minutes, plus the additional 40 minutes, giving us a grand total of 220 minutes. This is the timeframe in which they accomplish the work together, and it's a critical number for figuring out how much each person contributes.*
  • Next, think about their combined work rate. If Mario's work rate is 'M' and José's is '1.2M,' their combined rate is M + 1.2M = 2.2M. This is like saying they're not just adding their speeds, but working as a super-efficient duo that maximizes their output. This combined rate is what gets the job done in those 220 minutes.
  • The total work done can be expressed as their combined rate multiplied by the time they worked together. This is a fundamental concept in work-rate problems: Work = Rate * Time. In our case, the total work is 2.2M * 220 minutes. This gives us a numerical representation of the amount of effort required to complete the task, which we can then use to figure out how long it would take Mario alone.

Unraveling the Solo Performance: Mario's Time

Now comes the crucial part: figuring out how much time Mario would need if he were flying solo. This involves using what we've already established about their combined work and teasing out Mario's individual contribution. It's like watching a detective solve a mystery, where every clue we've gathered so far comes together to reveal the final answer.

  • We know the total work done is 2.2M * 220 minutes. Now, to find out how long it would take Mario alone, we use the same formula: Time = Work / Rate. For Mario, this means Time = (2.2M * 220) / M. Notice how Mario's rate 'M' appears in both the numerator and the denominator? This is where the magic happens!.
  • The 'M's cancel each other out! This simplifies the equation significantly, leaving us with Time = 2.2 * 220 minutes. This elegant cancellation is what makes the problem solvable, allowing us to isolate the numerical answer without getting bogged down in algebraic complexities. This is why setting up the problem correctly from the start is so crucial.
  • Calculating this, we get Mario's solo time: 2.2 * 220 = 484 minutes. This is the total time Mario would take to complete the work by himself. But wait, we're not quite done yet! The question asks for the additional time Mario would take compared to when he worked with José.

The Final Showdown: Calculating the Time Difference

We've got Mario's solo time, and we know their combined time. The final step is to find the difference, which will tell us how much extra time Mario needs when he's not teaming up with the super-efficient José. This is like comparing the time it takes to run a race solo versus running with a relay team—it's all about that time gap!

  • We've established that Mario takes 484 minutes alone, and together they take 220 minutes. The additional time Mario needs is simply the difference: 484 minutes - 220 minutes = 264 minutes. This is the extra effort Mario has to put in when he doesn't have José boosting the team's performance.*
  • But let's not jump to conclusions just yet! Our answer is in minutes, but the options given might be in hours and minutes. Time for a little conversion! Dividing 264 minutes by 60 minutes/hour gives us 4 hours and 24 minutes. This is the final piece of the puzzle, the time difference expressed in a format that matches our answer choices.
  • *This conversion is a critical step, and it's easy to overlook. Always double-check the units in your answer against the options provided. Making sure everything aligns is a hallmark of careful problem-solving and can prevent those frustrating