Solving: 5/18 + 1/6 + 5/3. Step-by-step!

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Hey guys! Let's break down this math problem together. We need to find the result of adding 5/18, 1/6, and 5/3. Sounds simple, right? But to get there, we need to go through a few steps to make sure we get the correct answer. So, grab your pencils, and let’s dive in!

Finding a Common Denominator

First things first, to add these fractions, we need a common denominator. A common denominator is a number that all the denominators (the bottom numbers) can divide into evenly. In our case, the denominators are 18, 6, and 3. What number can they all go into? The smallest one is 18. So, we'll use 18 as our common denominator.

Now, we need to convert each fraction to have this new denominator. For 5/18, it already has 18 as the denominator, so we don’t need to change it. Easy peasy! For 1/6, we need to multiply both the numerator (the top number) and the denominator by the same number to get 18 as the new denominator. Since 6 times 3 is 18, we multiply 1 by 3 to get 3. So, 1/6 becomes 3/18. Finally, for 5/3, we need to multiply both the numerator and the denominator by a number to get 18 as the new denominator. Since 3 times 6 is 18, we multiply 5 by 6 to get 30. So, 5/3 becomes 30/18.

Now we have our new fractions: 5/18, 3/18, and 30/18. See? It’s all coming together!

Adding the Fractions

Okay, with our common denominator in place, we can add the fractions. This part is super straightforward. We simply add the numerators together and keep the denominator the same. So, we have:

5/18 + 3/18 + 30/18 = (5 + 3 + 30) / 18

Adding the numerators gives us:

5 + 3 + 30 = 38

So, our fraction is now 38/18.

Simplifying the Fraction

Now that we have our answer, 38/18, it's good practice to simplify it. Simplifying a fraction means reducing it to its lowest terms. To do this, we need to find the greatest common divisor (GCD) of the numerator and the denominator. The GCD is the largest number that divides both numbers without leaving a remainder. For 38 and 18, the GCD is 2.

To simplify, we divide both the numerator and the denominator by the GCD:

38 ÷ 2 = 19 18 ÷ 2 = 9

So, our simplified fraction is 19/9.

Converting to a Mixed Number

Sometimes, it's helpful to convert an improper fraction (where the numerator is larger than the denominator) into a mixed number. A mixed number has a whole number part and a fraction part. To convert 19/9 to a mixed number, we divide 19 by 9.

19 ÷ 9 = 2 with a remainder of 1.

This means that 19/9 is equal to 2 whole numbers and 1/9. So, our mixed number is 2 1/9.

Analyzing the Alternatives

Now, let's look at the alternatives provided:

A) 1 B) 2 C) 3 D) 4

Our result, 2 1/9, is closest to 2. So, the correct answer is B) 2.

Conclusion

So, to recap, we added 5/18, 1/6, and 5/3 by finding a common denominator, adding the numerators, simplifying the fraction, and converting it to a mixed number. The result is approximately 2. Therefore, the correct answer is B) 2.

I hope this step-by-step explanation helps you understand how to solve this type of problem. Keep practicing, and you'll become a fraction master in no time! Remember, math can be fun when you break it down into smaller, manageable steps.

So, you want to become a fraction superstar? Awesome! Understanding how to add fractions is a fundamental skill in mathematics, and this guide will walk you through it step by step, making it super easy and, dare I say, enjoyable. Let's tackle the question: What is the result of the sum 5/18 + 1/6 + 5/3? We'll consider the alternatives A) 1, B) 2, C) 3, D) 4, and justify our answer with a detailed, step-by-step resolution. Get ready to level up your math game!

Why Fractions Matter

Before we dive into the problem, let's quickly touch on why fractions are important. Fractions are everywhere in real life! Think about slicing a pizza, measuring ingredients for a recipe, or even understanding percentages. Mastering fractions opens up a world of mathematical possibilities and helps you make sense of the world around you.

Step 1: Finding the Least Common Denominator (LCD)

The first and most crucial step in adding fractions is finding the Least Common Denominator (LCD). The LCD is the smallest number that all the denominators of the fractions can divide into evenly. In our problem, the denominators are 18, 6, and 3. To find the LCD, you can list the multiples of each number until you find a common one:

  • Multiples of 18: 18, 36, 54, ...
  • Multiples of 6: 6, 12, 18, 24, ...
  • Multiples of 3: 3, 6, 9, 12, 15, 18, ...

As you can see, the smallest number that appears in all three lists is 18. Therefore, the LCD is 18. Using the least common denominator simplifies our calculations and ensures we get to the correct solution with minimal fuss.

Step 2: Converting Fractions to Equivalent Fractions

Now that we have the LCD, we need to convert each fraction into an equivalent fraction with a denominator of 18. An equivalent fraction is a fraction that has the same value as another fraction, but with a different denominator. Here’s how we do it:

  • 5/18: This fraction already has a denominator of 18, so we don't need to change it. It remains 5/18.

  • 1/6: To convert this fraction to have a denominator of 18, we need to multiply both the numerator and the denominator by the same number. Since 6 x 3 = 18, we multiply both the numerator and the denominator by 3:

    (1 x 3) / (6 x 3) = 3/18

  • 5/3: Similarly, to convert this fraction, we need to multiply both the numerator and the denominator by a number that will give us a denominator of 18. Since 3 x 6 = 18, we multiply both the numerator and the denominator by 6:

    (5 x 6) / (3 x 6) = 30/18

Now we have three equivalent fractions: 5/18, 3/18, and 30/18. By converting to equivalent fractions, we ensure that we are adding comparable quantities, which is essential for an accurate sum.

Step 3: Adding the Fractions

With all fractions having the same denominator, we can now add them together. To add fractions with a common denominator, simply add the numerators and keep the denominator the same:

5/18 + 3/18 + 30/18 = (5 + 3 + 30) / 18

Now, let's add the numerators:

5 + 3 + 30 = 38

So, the sum of the fractions is 38/18.

Step 4: Simplifying the Fraction

The fraction 38/18 can be simplified. Simplifying a fraction means reducing it to its lowest terms. To do this, we need to find the greatest common divisor (GCD) of the numerator and the denominator. The GCD is the largest number that divides both numbers without leaving a remainder. For 38 and 18, the GCD is 2.

Divide both the numerator and the denominator by 2:

  • 38 ÷ 2 = 19
  • 18 ÷ 2 = 9

So, the simplified fraction is 19/9. Simplifying fractions gives us the most concise representation of the sum, making it easier to interpret and work with.

Step 5: Converting to a Mixed Number (Optional)

While 19/9 is a perfectly valid answer, it's an improper fraction (the numerator is greater than the denominator). We can convert it to a mixed number, which is a whole number and a fraction combined. To do this, divide the numerator by the denominator:

19 ÷ 9 = 2 with a remainder of 1

This means that 19/9 is equal to 2 whole numbers and 1/9. So, the mixed number is 2 1/9. Converting to a mixed number can sometimes provide a clearer understanding of the quantity, especially when comparing it to whole numbers.

Step 6: Choosing the Correct Alternative

Now, let's look at the alternatives provided:

  • A) 1
  • B) 2
  • C) 3
  • D) 4

Our result, 2 1/9, is closest to 2. Therefore, the correct answer is B) 2.

Final Thoughts

So, there you have it! By following these steps, we've successfully added the fractions 5/18, 1/6, and 5/3. The key takeaways are finding the LCD, converting to equivalent fractions, adding the numerators, and simplifying the result. Keep practicing, and you'll be a fraction master in no time! Understanding fraction sums is a stepping stone to more advanced mathematical concepts, so keep building that foundation!

Alright, let's get down to business and solve this fraction problem together! We're tasked with finding the sum of 5/18, 1/6, and 5/3. We've got some answer choices to consider: A) 1, B) 2, C) 3, and D) 4. I'm going to walk you through each step, so you'll not only get the right answer but also understand exactly why it's the right answer. Let's make fraction arithmetic feel like a breeze!

Laying the Groundwork: Understanding Fractions

Before we jump into the calculations, let’s quickly recap what fractions are all about. A fraction represents a part of a whole. It consists of two parts: the numerator (the number on top) and the denominator (the number on the bottom). The denominator tells us how many equal parts the whole is divided into, and the numerator tells us how many of those parts we have. When adding fractions, it's crucial to have a common denominator. This ensures that we are adding comparable parts.

Step 1: Identifying the Common Denominator

So, what's the first thing we need to do? Find a common denominator! The common denominator is a number that all of the denominators (18, 6, and 3 in our case) can divide into evenly. Ideally, we want to find the least common denominator (LCD) to keep the numbers manageable. Let’s think about this: 18 is divisible by 6 and 3. Therefore, 18 is our LCD. Identifying the common denominator is the cornerstone of adding fractions correctly. It sets the stage for the rest of the calculation.

Step 2: Converting to Equivalent Fractions

Next, we need to convert all our fractions to equivalent fractions with a denominator of 18. Remember, an equivalent fraction is a fraction that represents the same value but has a different denominator. Here's how we do it:

  • 5/18: This one's already good to go! It already has the common denominator of 18, so we don't need to change it.

  • 1/6: To convert 1/6 to an equivalent fraction with a denominator of 18, we need to multiply both the numerator and the denominator by the same number. Since 6 multiplied by 3 equals 18, we'll multiply both the top and bottom of the fraction by 3:

    (1 * 3) / (6 * 3) = 3/18

  • 5/3: Now, let's convert 5/3. We need to multiply both the numerator and the denominator by a number that gives us a denominator of 18. Since 3 multiplied by 6 equals 18, we'll multiply both the top and bottom of the fraction by 6:

    (5 * 6) / (3 * 6) = 30/18

Now we have our equivalent fractions: 5/18, 3/18, and 30/18. By creating equivalent fractions, we've transformed the original problem into one that is much easier to solve.

Step 3: Adding the Numerators

Now comes the fun part: adding the fractions! Since they all have the same denominator, we simply add the numerators and keep the denominator the same:

5/18 + 3/18 + 30/18 = (5 + 3 + 30) / 18

Let’s add those numerators:

5 + 3 + 30 = 38

So, our fraction is now 38/18.

Step 4: Simplifying the Result

Our result is 38/18, but it’s always a good practice to simplify the fraction if possible. Simplifying means reducing the fraction to its lowest terms. To do this, we need to find the greatest common divisor (GCD) of the numerator and the denominator. The GCD is the largest number that divides both numbers without leaving a remainder. In this case, the GCD of 38 and 18 is 2.

Let's divide both the numerator and the denominator by 2:

  • 38 ÷ 2 = 19
  • 18 ÷ 2 = 9

So, the simplified fraction is 19/9. Simplifying the result provides the most accurate and easily understandable form of the fraction.

Step 5: Converting to a Mixed Number (If Necessary)

Although 19/9 is a perfectly valid answer, it's an improper fraction (where the numerator is larger than the denominator). It can be helpful to convert it to a mixed number, which consists of a whole number and a proper fraction. To convert 19/9 to a mixed number, we divide 19 by 9:

19 ÷ 9 = 2 with a remainder of 1

This means that 19/9 is equal to 2 whole numbers and 1/9. So, our mixed number is 2 1/9.

Step 6: Choosing the Right Answer

Now, let's go back to the alternatives:

  • A) 1
  • B) 2
  • C) 3
  • D) 4

Our result is 2 1/9, which is closest to 2. Therefore, the correct answer is B) 2.

Final Thoughts on Fraction Sums

So, we've successfully navigated through this fraction problem! We found the common denominator, converted the fractions, added them up, and simplified the result. The correct answer is B) 2. Remember, understanding fraction sums is a vital skill in math, so keep practicing and you'll become a pro in no time!