Solving: 1/5 + [68/21] - A Step-by-Step Guide

by Dimemap Team 46 views

Hey guys! Let's break down this mathematical expression together. We've got 1/5 + [68/21], and it might look a little intimidating at first, but don't worry, we'll get through it step by step. Math can be super fun once you understand the process, so let's dive right in!

Understanding the Basics

Before we even touch the problem, let's brush up on some fundamental concepts. We're dealing with fractions here, and adding fractions requires a common denominator. Think of it like this: you can't add apples and oranges directly, right? You need to find a common unit, like "fruits". Similarly, with fractions, we need a common denominator to add them accurately.

In our expression, we have 1/5 and 68/21. The denominators are 5 and 21. To find a common denominator, we need to find the least common multiple (LCM) of 5 and 21. If you're scratching your head, don't sweat it! The LCM is the smallest number that both 5 and 21 can divide into evenly. This is crucial for correctly adding these fractions.

Finding the Least Common Multiple (LCM)

There are a couple of ways to find the LCM. One method is listing the multiples of each number until you find a common one. Let's try that:

  • Multiples of 5: 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60, 65, 70, 75, 80, 85, 90, 95, 100, 105...
  • Multiples of 21: 21, 42, 63, 84, 105...

Aha! We see that 105 is a common multiple. In fact, it's the least common multiple. This means 105 is the smallest number that both 5 and 21 divide into without leaving a remainder. This is super important for our next steps.

Another method to find the LCM is prime factorization. Break down each number into its prime factors:

  • 5 = 5 (5 is already a prime number)
  • 21 = 3 x 7

Then, take the highest power of each prime factor present in either number and multiply them together. In this case, we have 3, 5, and 7. So, the LCM is 3 x 5 x 7 = 105. See? We got the same answer!

Converting Fractions to a Common Denominator

Now that we know our common denominator is 105, we need to convert both fractions. This involves changing the fractions so that they both have a denominator of 105, without changing their actual value. It's like changing the units but keeping the quantity the same.

For 1/5, we need to figure out what to multiply the denominator (5) by to get 105. We can do this by dividing 105 by 5, which gives us 21. So, we multiply both the numerator and the denominator of 1/5 by 21:

(1 x 21) / (5 x 21) = 21/105

Now, let's do the same for 68/21. We divide 105 by 21, which gives us 5. So, we multiply both the numerator and the denominator of 68/21 by 5:

(68 x 5) / (21 x 5) = 340/105

Awesome! Now we have two fractions with the same denominator: 21/105 and 340/105. This means we can finally add them together!

Adding the Fractions

Adding fractions with a common denominator is actually quite simple. You just add the numerators (the top numbers) and keep the denominator the same. Think of it like adding slices of a pizza. If you have 2 slices out of 10 and add 3 more slices out of 10, you end up with 5 slices out of 10.

So, let's add our fractions:

21/105 + 340/105 = (21 + 340) / 105

21 + 340 = 361

Therefore, our result is 361/105. We're almost there!

Simplifying the Result

Our answer is 361/105, but we should always check if we can simplify the fraction. Simplifying means reducing the fraction to its lowest terms. This involves finding the greatest common divisor (GCD) of the numerator and the denominator and dividing both by it. The greatest common divisor is the largest number that divides both 361 and 105 without leaving a remainder.

Finding the GCD can sometimes be tricky. One way is to list the factors of each number and find the largest one they have in common. Another method is using the Euclidean algorithm, which is a more systematic approach.

Let's try listing the factors:

  • Factors of 361: 1, 19, 361
  • Factors of 105: 1, 3, 5, 7, 15, 21, 35, 105

We see that the only common factor is 1. This means that 361 and 105 are relatively prime, and the fraction 361/105 is already in its simplest form! Sometimes, you will need to simplify, but in this case, it's already done. Lucky us!

Converting to a Mixed Number (Optional)

While 361/105 is a perfectly valid answer, sometimes it's helpful to express it as a mixed number. A mixed number is a whole number and a fraction combined, like 2 1/2. This can sometimes give us a better sense of the quantity.

To convert an improper fraction (where the numerator is greater than the denominator) to a mixed number, we divide the numerator by the denominator. The quotient (the result of the division) becomes the whole number part, the remainder becomes the numerator of the fractional part, and the denominator stays the same.

So, let's divide 361 by 105:

361 ÷ 105 = 3 with a remainder of 46

This means that 361/105 is equal to 3 whole units and 46/105 of another unit. So, our mixed number is 3 46/105.

Final Answer

We've done it! We've successfully solved the mathematical expression 1/5 + [68/21]. Our final answer is:

  • Improper Fraction: 361/105
  • Mixed Number (optional): 3 46/105

Both of these represent the same value, so you can choose whichever format you prefer. The most important thing is that you understand the steps we took to get there. Remember, math is all about building on fundamental concepts and practicing consistently.

Key Takeaways

Let's recap the key steps we followed to solve this problem. This will help you tackle similar problems in the future:

  1. Find the Least Common Multiple (LCM): This is crucial for adding fractions with different denominators.
  2. Convert Fractions to a Common Denominator: Multiply the numerator and denominator of each fraction by the appropriate factor to get the LCM as the new denominator.
  3. Add the Numerators: Once you have a common denominator, simply add the numerators.
  4. Simplify the Result: Check if you can simplify the fraction by finding the Greatest Common Divisor (GCD) and dividing both the numerator and denominator by it.
  5. Convert to a Mixed Number (Optional): If you want, you can convert the improper fraction to a mixed number by dividing the numerator by the denominator.

By mastering these steps, you'll be able to confidently tackle a wide range of fraction problems. Remember, practice makes perfect, so keep at it!

Practice Makes Perfect

Now that we've worked through this example together, try solving some similar problems on your own. Here are a few to get you started:

  1. 1/3 + 2/5
  2. 3/4 + 1/6
  3. 5/8 + 2/3

Work through these problems step by step, using the techniques we discussed. Don't be afraid to make mistakes – that's how we learn! If you get stuck, review the steps we took in this article or ask for help. The key is to keep practicing until you feel comfortable with the process.

Conclusion

So, there you have it! We've successfully solved the mathematical expression 1/5 + [68/21] and learned a lot about adding fractions along the way. Remember, math is a journey, not a destination. There's always more to learn, and the more you practice, the better you'll become. Keep up the great work, guys, and I'll see you in the next math adventure!