Adding Fractions A Step By Step Guide To Performing Operations And Expressing Answers In Simplest Form

Fractions are a fundamental concept in mathematics, representing parts of a whole. Operations with fractions, such as addition, are essential skills to master for various mathematical applications. This comprehensive guide will delve into the process of adding fractions, focusing on expressing the answers in their lowest terms. We'll break down each step with detailed explanations and examples to help you confidently tackle fraction addition problems. This article aims to guide you through fraction addition, ensuring you grasp the concepts and techniques needed to solve such problems efficiently and accurately. Whether you're a student learning fractions for the first time or someone looking to refresh your knowledge, this guide will provide you with the tools and understanding necessary to master fraction addition.

1. ${\frac{7}{10} + 2\frac{2}{5} = }$

In this first problem, we're tasked with adding a proper fraction, ${\frac{7}{10}}$, to a mixed number, ${2\frac{2}{5}}$. To effectively add these numbers, we must first convert the mixed number into an improper fraction. This conversion allows us to work with fractions that have a single numerator and denominator, making the addition process straightforward. The mixed number ${2\frac{2}{5}}$ represents 2 whole units plus ${\frac{2}{5}}$ of another unit. To convert it, we multiply the whole number part (2) by the denominator of the fractional part (5) and then add the numerator (2). This result becomes the new numerator, while the denominator remains the same. Thus, ${2\frac{2}{5}}$ becomes ${\frac{(2 \times 5) + 2}{5} = \frac{12}{5}}$. Now that we have both fractions in the form of proper or improper fractions, we can proceed with the addition. The problem is now ${\frac{7}{10} + \frac{12}{5}}$. To add these fractions, they need to have a common denominator. The least common multiple (LCM) of 10 and 5 is 10. The fraction ${\frac{7}{10}}$ already has the desired denominator. To convert ${\frac{12}{5}}$ to a fraction with a denominator of 10, we multiply both the numerator and the denominator by 2, resulting in ${\frac{12 \times 2}{5 \times 2} = \frac{24}{10}}$. Now we can add the fractions: ${\frac{7}{10} + \frac{24}{10}}$. We add the numerators while keeping the denominator the same: ${\frac{7 + 24}{10} = \frac{31}{10}}$. The resulting fraction, ${\frac{31}{10}}$, is an improper fraction, meaning the numerator is greater than the denominator. To express the answer in its simplest form, we convert it back to a mixed number. We divide 31 by 10, which gives us 3 with a remainder of 1. This means ${\frac{31}{10}}$ is equal to 3 whole units and ${\frac{1}{10}}$ of another unit. Therefore, the mixed number is ${3\frac{1}{10}}$. This fraction is already in its lowest terms since 1 and 10 have no common factors other than 1. Thus, the final answer to the first problem is ${3\frac{1}{10}}$, expressed in its lowest terms. This detailed breakdown illustrates the importance of converting mixed numbers to improper fractions, finding common denominators, and simplifying the final result to its lowest terms, ensuring accuracy and clarity in fraction addition.

2. ${2\frac{2}{5} + 1\frac{6}{9} = }$

In this second problem, we are tasked with adding two mixed numbers: ${2\frac{2}{5}}$ and ${1\frac{6}{9}}$. As with the previous problem, the first step is to convert these mixed numbers into improper fractions. This conversion is crucial because it simplifies the addition process by allowing us to work with fractions that have a single numerator and denominator. The mixed number ${2\frac{2}{5}}$ can be converted to an improper fraction by multiplying the whole number part (2) by the denominator (5) and then adding the numerator (2). This result becomes the new numerator, while the denominator remains the same. Thus, ${2\frac{2}{5}}$ becomes ${\frac{(2 \times 5) + 2}{5} = \frac{12}{5}}$. Similarly, we convert the mixed number ${1\frac{6}{9}}$ into an improper fraction. We multiply the whole number part (1) by the denominator (9) and add the numerator (6). This gives us ${\frac{(1 \times 9) + 6}{9} = \frac{15}{9}}$. Before proceeding with the addition, it's beneficial to simplify the fraction ${\frac{15}{9}}$ to its lowest terms. Both 15 and 9 are divisible by 3, so we divide both the numerator and the denominator by 3: ${\frac{15 \div 3}{9 \div 3} = \frac{5}{3}}$. This simplification makes the subsequent steps easier. Now, our problem is ${\frac{12}{5} + \frac{5}{3}}$. To add these fractions, we need to find a common denominator. The least common multiple (LCM) of 5 and 3 is 15. To convert ${\frac{12}{5}}$ to a fraction with a denominator of 15, we multiply both the numerator and the denominator by 3: ${\frac{12 \times 3}{5 \times 3} = \frac{36}{15}}$. Similarly, to convert ${\frac{5}{3}}$ to a fraction with a denominator of 15, we multiply both the numerator and the denominator by 5: ${\frac{5 \times 5}{3 \times 5} = \frac{25}{15}}$. Now we can add the fractions: ${\frac{36}{15} + \frac{25}{15}}$. We add the numerators while keeping the denominator the same: ${\frac{36 + 25}{15} = \frac{61}{15}}$. The resulting fraction, ${\frac{61}{15}}$, is an improper fraction. To express the answer in its simplest form, we convert it back to a mixed number. We divide 61 by 15, which gives us 4 with a remainder of 1. This means ${\frac{61}{15}}$ is equal to 4 whole units and ${\frac{1}{15}}$ of another unit. Therefore, the mixed number is ${4\frac{1}{15}}$. This fraction is already in its lowest terms since 1 and 15 have no common factors other than 1. Thus, the final answer to the second problem is ${4\frac{1}{15}}$, expressed in its lowest terms. This problem highlights the importance of simplifying fractions before adding, finding the least common multiple to establish a common denominator, and converting improper fractions back to mixed numbers for the final answer. Each step is crucial for accuracy and clarity in fraction addition.

3. ${3\frac{2}{7} + 4\frac{3}{4} = }$

For this third problem, we are required to add two mixed numbers: ${3\frac{2}{7}}$ and ${4\frac{3}{4}}$. The initial step, as with the previous examples, involves converting these mixed numbers into improper fractions. This conversion is a fundamental technique in adding mixed numbers, as it simplifies the addition process. The mixed number ${3\frac{2}{7}}$ is converted by multiplying the whole number part (3) by the denominator (7) and adding the numerator (2). The result becomes the new numerator, with the denominator remaining the same. Thus, ${3\frac{2}{7}}$ becomes ${\frac{(3 \times 7) + 2}{7} = \frac{23}{7}}$. Similarly, we convert the mixed number ${4\frac{3}{4}}$ into an improper fraction. We multiply the whole number part (4) by the denominator (4) and add the numerator (3). This gives us ${\frac{(4 \times 4) + 3}{4} = \frac{19}{4}}$. Now, we have the problem ${\frac{23}{7} + \frac{19}{4}}$. To add these fractions, we need to find a common denominator. The least common multiple (LCM) of 7 and 4 is 28. To convert ${\frac{23}{7}}$ to a fraction with a denominator of 28, we multiply both the numerator and the denominator by 4: ${\frac{23 \times 4}{7 \times 4} = \frac{92}{28}}$. Similarly, to convert ${\frac{19}{4}}$ to a fraction with a denominator of 28, we multiply both the numerator and the denominator by 7: ${\frac{19 \times 7}{4 \times 7} = \frac{133}{28}}$. Now we can add the fractions: ${\frac{92}{28} + \frac{133}{28}}$. We add the numerators while keeping the denominator the same: ${\frac{92 + 133}{28} = \frac{225}{28}}$. The resulting fraction, ${\frac{225}{28}}$, is an improper fraction. To express the answer in its simplest form, we convert it back to a mixed number. We divide 225 by 28, which gives us 8 with a remainder of 1. This means ${\frac{225}{28}}$ is equal to 8 whole units and ${\frac{1}{28}}$ of another unit. Therefore, the mixed number is ${8\frac{1}{28}}$. This fraction is already in its lowest terms since 1 and 28 have no common factors other than 1. Thus, the final answer to the third problem is ${8\frac{1}{28}}$, expressed in its lowest terms. This problem further illustrates the crucial steps of converting mixed numbers to improper fractions, finding the least common multiple to establish a common denominator, and converting improper fractions back to mixed numbers for the final answer. Each step is vital for accurately adding fractions and expressing the result in its simplest form.

4. ${5\frac{4}{6} + 1\frac{2}{3} = }$

In this fourth problem, we are tasked with adding two mixed numbers: ${5\frac{4}{6}}$ and ${1\frac{2}{3}}$. Following the established method, our first step is to convert these mixed numbers into improper fractions. This conversion is essential for simplifying the addition process. To convert the mixed number ${5\frac{4}{6}}$, we multiply the whole number part (5) by the denominator (6) and add the numerator (4). This result becomes the new numerator, while the denominator remains the same. Thus, ${5\frac{4}{6}}$ becomes ${\frac{(5 \times 6) + 4}{6} = \frac{34}{6}}$. Similarly, we convert the mixed number ${1\frac{2}{3}}$ into an improper fraction. We multiply the whole number part (1) by the denominator (3) and add the numerator (2). This gives us ${\frac{(1 \times 3) + 2}{3} = \frac{5}{3}}$. Before proceeding with the addition, it's beneficial to simplify the fraction ${\frac{34}{6}}$ to its lowest terms. Both 34 and 6 are divisible by 2, so we divide both the numerator and the denominator by 2: ${\frac{34 \div 2}{6 \div 2} = \frac{17}{3}}$. This simplification makes the subsequent steps easier. Now, our problem is ${\frac{17}{3} + \frac{5}{3}}$. Since both fractions already have a common denominator of 3, we can proceed directly with the addition. We add the numerators while keeping the denominator the same: ${\frac{17 + 5}{3} = \frac{22}{3}}$. The resulting fraction, ${\frac{22}{3}}$, is an improper fraction. To express the answer in its simplest form, we convert it back to a mixed number. We divide 22 by 3, which gives us 7 with a remainder of 1. This means ${\frac{22}{3}}$ is equal to 7 whole units and ${\frac{1}{3}}$ of another unit. Therefore, the mixed number is ${7\frac{1}{3}}$. This fraction is already in its lowest terms since 1 and 3 have no common factors other than 1. Thus, the final answer to the fourth problem is ${7\frac{1}{3}}$, expressed in its lowest terms. This problem highlights the efficiency of simplifying fractions before adding and demonstrates that when fractions share a common denominator, the addition process becomes more straightforward. Each step, from converting mixed numbers to improper fractions to simplifying and adding, is crucial for achieving an accurate and clear final answer.

5. ${\frac{4}{6} + 7\frac{3}{4} = }$

In this fifth and final problem, we are required to add a proper fraction, ${\frac{4}{6}}$, to a mixed number, ${7\frac{3}{4}}$. Consistent with our previous approaches, the first step is to convert the mixed number into an improper fraction. Additionally, it's often beneficial to simplify fractions before proceeding with addition, which we will also do here. Let’s start by simplifying ${\frac{4}{6}}$. Both 4 and 6 are divisible by 2, so we divide both the numerator and the denominator by 2: ${\frac{4 \div 2}{6 \div 2} = \frac{2}{3}}$. Now, we convert the mixed number ${7\frac{3}{4}}$ into an improper fraction. We multiply the whole number part (7) by the denominator (4) and add the numerator (3). This gives us ${\frac{(7 \times 4) + 3}{4} = \frac{31}{4}}$. Our problem now is ${\frac{2}{3} + \frac{31}{4}}$. To add these fractions, we need to find a common denominator. The least common multiple (LCM) of 3 and 4 is 12. To convert ${\frac{2}{3}}$ to a fraction with a denominator of 12, we multiply both the numerator and the denominator by 4: ${\frac{2 \times 4}{3 \times 4} = \frac{8}{12}}$. Similarly, to convert ${\frac{31}{4}}$ to a fraction with a denominator of 12, we multiply both the numerator and the denominator by 3: ${\frac{31 \times 3}{4 \times 3} = \frac{93}{12}}$. Now we can add the fractions: ${\frac{8}{12} + \frac{93}{12}}$. We add the numerators while keeping the denominator the same: ${\frac{8 + 93}{12} = \frac{101}{12}}$. The resulting fraction, ${\frac{101}{12}}$, is an improper fraction. To express the answer in its simplest form, we convert it back to a mixed number. We divide 101 by 12, which gives us 8 with a remainder of 5. This means ${\frac{101}{12}}$ is equal to 8 whole units and ${\frac{5}{12}}$ of another unit. Therefore, the mixed number is ${8\frac{5}{12}}$. This fraction is already in its lowest terms since 5 and 12 have no common factors other than 1. Thus, the final answer to the fifth problem is ${8\frac{5}{12}}$, expressed in its lowest terms. This final problem reinforces the importance of simplifying fractions at the outset, converting mixed numbers to improper fractions, finding the least common multiple for a common denominator, and converting improper fractions back to mixed numbers for the simplest final form. Each step contributes to an accurate and clearly presented solution.

In conclusion, mastering the addition of fractions involves several key steps: converting mixed numbers to improper fractions, finding a common denominator (preferably the least common multiple), adding the numerators while keeping the denominator the same, and simplifying the resulting fraction to its lowest terms, often by converting improper fractions back to mixed numbers. By following these steps meticulously, you can confidently and accurately perform fraction addition, ensuring your answers are both correct and clearly expressed.