Solve: 4 ÷ 1/5 + 3/5 × 1/10 + 1/25 + (-1/10) - Step-by-Step

by Rajiv Sharma 60 views

Hey there, math enthusiasts! Today, we're diving into a fascinating mathematical problem that combines division, multiplication, addition, and subtraction of fractions. It might look a bit intimidating at first glance, but don't worry, we'll break it down step by step and make it super easy to understand. Our mission is to solve the expression: 4 ÷ 1/5 + 3/5 × 1/10 + 1/25 + -1/10. So, grab your pencils and let's get started on this mathematical adventure!

Order of Operations: The Key to Success

Before we jump into the calculations, let's quickly recap the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). This set of rules tells us the sequence in which we should perform operations to ensure we arrive at the correct answer. In our expression, we have division, multiplication, addition, and subtraction, so we'll tackle them in the following order:

  1. Division: We'll start with 4 ÷ 1/5.
  2. Multiplication: Next up is 3/5 × 1/10.
  3. Addition and Subtraction: Finally, we'll handle the addition and subtraction from left to right: + 1/25 + -1/10.

Following this order is crucial to avoid any confusion and ensure accuracy. Think of it as the roadmap that guides us through the mathematical landscape. Now that we have our roadmap, let's begin the journey!

Step 1: Diving into Division – 4 ÷ 1/5

Our first task is to tackle the division: 4 ÷ 1/5. Dividing by a fraction can seem a bit tricky, but there's a simple rule to remember: dividing by a fraction is the same as multiplying by its reciprocal. The reciprocal of a fraction is obtained by swapping the numerator and the denominator. So, the reciprocal of 1/5 is 5/1, which is simply 5. Therefore, our division problem transforms into a multiplication problem:

4 ÷ 1/5 = 4 × 5

Now, this is much easier to handle! Multiplying 4 by 5 gives us 20. So, the first part of our expression simplifies to 20. We've successfully conquered the division, and we're one step closer to solving the entire problem. Feels good, right? Let's keep the momentum going!

Step 2: Mastering Multiplication – 3/5 × 1/10

Next up, we have the multiplication: 3/5 × 1/10. Multiplying fractions is pretty straightforward. We simply multiply the numerators (the top numbers) together and the denominators (the bottom numbers) together. So, we have:

(3 × 1) / (5 × 10) = 3 / 50

Therefore, 3/5 × 1/10 equals 3/50. We've successfully navigated the multiplication part of our expression. Notice how we're breaking down the problem into smaller, more manageable chunks. This is a powerful strategy for tackling any complex problem, whether it's in math or in life!

Step 3: Adding and Subtracting Fractions

Now comes the part where we add and subtract fractions. Remember, to add or subtract fractions, they need to have a common denominator. Our expression now looks like this:

20 + 3/50 + 1/25 + -1/10

We have fractions with denominators of 50, 25, and 10. To find a common denominator, we need to find the least common multiple (LCM) of these numbers. The LCM of 50, 25, and 10 is 50. This means we'll rewrite each fraction with a denominator of 50.

  • 3/50 already has the desired denominator.
  • To convert 1/25 to a fraction with a denominator of 50, we multiply both the numerator and the denominator by 2: (1 × 2) / (25 × 2) = 2/50
  • To convert -1/10 to a fraction with a denominator of 50, we multiply both the numerator and the denominator by 5: (-1 × 5) / (10 × 5) = -5/50

Now our expression looks like this:

20 + 3/50 + 2/50 + -5/50

We also need to convert the whole number 20 into a fraction with a denominator of 50. To do this, we multiply 20 by 50/50:

20 × 50/50 = 1000/50

So our expression now becomes:

1000/50 + 3/50 + 2/50 + -5/50

Now that all the fractions have a common denominator, we can simply add and subtract the numerators:

(1000 + 3 + 2 + -5) / 50 = 1000/50

Step 4: Simplifying the Result

We've arrived at the fraction 1000/50. To simplify this, we divide both the numerator and the denominator by their greatest common divisor (GCD), which is 50:

1000 / 50 = 20 50 / 50 = 1

So, our simplified fraction is 20/1, which is simply 20. But wait! We made a mistake in the addition earlier. Let's correct that.

(1000 + 3 + 2 - 5) / 50 = 1000 / 50

So, the correct calculation is:

(1000 + 3 + 2 - 5) / 50 = 1000/50

Now, let's simplify 1000/50 by dividing both the numerator and the denominator by their greatest common divisor (GCD), which is 50:

1000 / 50 = 20 50 / 50 = 1

So, our simplified fraction is 20/1, which simplifies to 20. We're almost there! It seems we had a full circle moment here, but it's always good to double-check our work.

Step 5: The Final Answer

After carefully working through each step, we've arrived at our final answer. The solution to the expression 4 ÷ 1/5 + 3/5 × 1/10 + 1/25 + -1/10 is:

20

Congratulations, mathletes! We've successfully conquered this mathematical challenge. Give yourselves a pat on the back for your hard work and perseverance. Remember, the key to solving complex problems is to break them down into smaller, more manageable steps, and to follow the order of operations. And of course, always double-check your work to avoid any silly mistakes.

Key Takeaways

  • Order of Operations (PEMDAS): Always follow the correct order of operations to ensure accuracy.
  • Dividing by a Fraction: Dividing by a fraction is the same as multiplying by its reciprocal.
  • Multiplying Fractions: Multiply the numerators and the denominators.
  • Adding and Subtracting Fractions: Fractions must have a common denominator before they can be added or subtracted.
  • Simplifying Fractions: Divide the numerator and denominator by their greatest common divisor (GCD).
  • Double-Check Your Work: Always review your calculations to catch any errors.

Practice Makes Perfect

Now that we've solved this problem together, why not try tackling some similar ones on your own? The more you practice, the more confident and skilled you'll become in math. Remember, math is like a muscle – the more you exercise it, the stronger it gets. So, keep practicing, keep exploring, and keep having fun with math!

Conclusion

We've journeyed through the world of fractions, division, multiplication, addition, and subtraction, and emerged victorious! Math can be challenging, but it's also incredibly rewarding. By understanding the fundamental principles and practicing regularly, you can conquer any mathematical obstacle that comes your way. So, keep shining those math skills, and never stop learning!

I hope you guys found this breakdown helpful and insightful. If you have any questions or want to explore other math topics, feel free to ask. Until next time, happy calculating!