12.${ \begin{array}{r} 9 \frac{7}{12} \ +4 \frac{3}{4} \ \hline \end{array} }$Calculate The Sum Of These Mixed Numbers.

Mar 13, 2025 by ADMIN 121 views

12. Calculating the Sum of Mixed Numbers

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Introduction

Mixed numbers are a combination of a whole number and a fraction. They are often used in real-world applications, such as measuring ingredients in cooking or calculating time in sports. In this article, we will explore how to calculate the sum of mixed numbers, using the example of adding 9 $\frac{7}{12}$ and 4 $\frac{3}{4}$ .

Understanding Mixed Numbers

A mixed number is a combination of a whole number and a fraction. It is written in the form of $a \frac{b}{c}$ , where $a$ is the whole number, $b$ is the numerator, and $c$ is the denominator. For example, 3 $\frac{2}{5}$ is a mixed number, where 3 is the whole number, 2 is the numerator, and 5 is the denominator.

Converting Mixed Numbers to Improper Fractions

To add mixed numbers, we need to convert them to improper fractions first. An improper fraction is a fraction where the numerator is greater than or equal to the denominator. To convert a mixed number to an improper fraction, we multiply the whole number by the denominator and add the numerator. The result is the new numerator, and the denominator remains the same.

For example, to convert 9 $\frac{7}{12}$ to an improper fraction, we multiply 9 by 12 and add 7. This gives us 112 + 7 = 119. So, 9 $\frac{7}{12}$ is equal to $\frac{119}{12}$ .

Similarly, to convert 4 $\frac{3}{4}$ to an improper fraction, we multiply 4 by 4 and add 3. This gives us 16 + 3 = 19. So, 4 $\frac{3}{4}$ is equal to $\frac{19}{4}$ .

Adding Improper Fractions

Now that we have converted both mixed numbers to improper fractions, we can add them. To add improper fractions, we need to find a common denominator. The common denominator is the least common multiple (LCM) of the denominators.

In this case, the denominators are 12 and 4. The LCM of 12 and 4 is 12. So, we need to convert $\frac{19}{4}$ to have a denominator of 12.

To do this, we multiply the numerator and denominator of $\frac{19}{4}$ by 3. This gives us $\frac{57}{12}$ .

Now that both fractions have the same denominator, we can add them. We add the numerators and keep the denominator the same. This gives us $\frac{119}{12} + \frac{57}{12} = \frac{176}{12}$ .

Simplifying the Result

The result of the addition is $\frac{176}{12}$ . However, this fraction can be simplified. To simplify a fraction, we divide the numerator and denominator by their greatest common divisor (GCD).

The GCD of 176 and 12 is 4. So, we divide both the numerator and denominator by 4. This gives us $\frac{44}{3}$ .

Conclusion

In this article, we have learned how to calculate the sum of mixed numbers. We started by converting the mixed numbers to improper fractions, then added the improper fractions, and finally simplified the result. The final answer is $\frac{44}{3}$ , which can be written as 14 $\frac{2}{3}$ .

Example Problems

Problem 1

Add 5 $\frac{1}{6}$ and 2 $\frac{3}{4}$ .

Solution

To solve this problem, we need to convert both mixed numbers to improper fractions. 5 $\frac{1}{6}$ is equal to $\frac{31}{6}$ , and 2 $\frac{3}{4}$ is equal to $\frac{11}{4}$ .

Next, we need to find a common denominator. The LCM of 6 and 4 is 12. So, we need to convert both fractions to have a denominator of 12.

To do this, we multiply the numerator and denominator of $\frac{31}{6}$ by 2, and the numerator and denominator of $\frac{11}{4}$ by 3. This gives us $\frac{62}{12}$ and $\frac{33}{12}$ .

Now that both fractions have the same denominator, we can add them. We add the numerators and keep the denominator the same. This gives us $\frac{62}{12} + \frac{33}{12} = \frac{95}{12}$ .

Finally, we can simplify the result by dividing the numerator and denominator by their GCD. The GCD of 95 and 12 is 1. So, the result is $\frac{95}{12}$ , which can be written as 7 $\frac{11}{12}$ .

Problem 2

Add 3 $\frac{2}{3}$ and 1 $\frac{1}{6}$ .

Solution

To solve this problem, we need to convert both mixed numbers to improper fractions. 3 $\frac{2}{3}$ is equal to $\frac{11}{3}$ , and 1 $\frac{1}{6}$ is equal to $\frac{7}{6}$ .

Next, we need to find a common denominator. The LCM of 3 and 6 is 6. So, we need to convert both fractions to have a denominator of 6.

To do this, we multiply the numerator and denominator of $\frac{11}{3}$ by 2, and the numerator and denominator of $\frac{7}{6}$ by 1. This gives us $\frac{22}{6}$ and $\frac{7}{6}$ .

Now that both fractions have the same denominator, we can add them. We add the numerators and keep the denominator the same. This gives us $\frac{22}{6} + \frac{7}{6} = \frac{29}{6}$ .

Finally, we can simplify the result by dividing the numerator and denominator by their GCD. The GCD of 29 and 6 is 1. So, the result is $\frac{29}{6}$ , which can be written as 4 $\frac{5}{6}$ .

Final Answer

The final answer is $\boxed{14 \frac{2}{3}}$ .

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Introduction

In our previous article, we explored how to calculate the sum of mixed numbers. We learned how to convert mixed numbers to improper fractions, add improper fractions, and simplify the result. In this article, we will answer some frequently asked questions about calculating the sum of mixed numbers.

Q&A

Q: What is the difference between a mixed number and an improper fraction?

A: A mixed number is a combination of a whole number and a fraction, while an improper fraction is a fraction where the numerator is greater than or equal to the denominator.

Q: How do I convert a mixed number to an improper fraction?

A: To convert a mixed number to an improper fraction, you multiply the whole number by the denominator and add the numerator. The result is the new numerator, and the denominator remains the same.

Q: What is the common denominator?

A: The common denominator is the least common multiple (LCM) of the denominators. It is the smallest number that both denominators can divide into evenly.

Q: How do I find the LCM of two numbers?

A: To find the LCM of two numbers, you can list the multiples of each number and find the smallest number that appears in both lists. Alternatively, you can use the formula: LCM(a, b) = (a × b) / GCD(a, b), where GCD is the greatest common divisor.

Q: Can I simplify an improper fraction?

A: Yes, you can simplify an improper fraction by dividing the numerator and denominator by their greatest common divisor (GCD).

Q: What is the GCD of two numbers?

A: The GCD of two numbers is the largest number that both numbers can divide into evenly.

Q: How do I add mixed numbers?

A: To add mixed numbers, you need to convert them to improper fractions first. Then, you add the improper fractions and simplify the result.

Q: Can I add mixed numbers with different denominators?

A: Yes, you can add mixed numbers with different denominators. However, you need to find a common denominator first.

Q: How do I find a common denominator?

A: To find a common denominator, you can list the multiples of each denominator and find the smallest number that appears in both lists. Alternatively, you can use the formula: LCM(a, b) = (a × b) / GCD(a, b), where GCD is the greatest common divisor.

Q: Can I subtract mixed numbers?

A: Yes, you can subtract mixed numbers. However, you need to convert them to improper fractions first and then subtract the improper fractions.

Q: How do I subtract mixed numbers with different denominators?

A: To subtract mixed numbers with different denominators, you need to find a common denominator first. Then, you convert the mixed numbers to improper fractions and subtract the improper fractions.

Example Problems

Problem 1

Add 3 $\frac{2}{3}$ and 2 $\frac{1}{6}$ .

Solution

To solve this problem, we need to convert both mixed numbers to improper fractions. 3 $\frac{2}{3}$ is equal to $\frac{11}{3}$ , and 2 $\frac{1}{6}$ is equal to $\frac{13}{6}$ .

Next, we need to find a common denominator. The LCM of 3 and 6 is 6. So, we need to convert both fractions to have a denominator of 6.

To do this, we multiply the numerator and denominator of $\frac{11}{3}$ by 2, and the numerator and denominator of $\frac{13}{6}$ by 1. This gives us $\frac{22}{6}$ and $\frac{13}{6}$ .

Now that both fractions have the same denominator, we can add them. We add the numerators and keep the denominator the same. This gives us $\frac{22}{6} + \frac{13}{6} = \frac{35}{6}$ .

Finally, we can simplify the result by dividing the numerator and denominator by their GCD. The GCD of 35 and 6 is 1. So, the result is $\frac{35}{6}$ , which can be written as 5 $\frac{5}{6}$ .

Problem 2

Subtract 4 $\frac{3}{4}$ from 6 $\frac{2}{3}$ .

Solution

To solve this problem, we need to convert both mixed numbers to improper fractions. 4 $\frac{3}{4}$ is equal to $\frac{19}{4}$ , and 6 $\frac{2}{3}$ is equal to $\frac{20}{3}$ .

Next, we need to find a common denominator. The LCM of 4 and 3 is 12. So, we need to convert both fractions to have a denominator of 12.

To do this, we multiply the numerator and denominator of $\frac{19}{4}$ by 3, and the numerator and denominator of $\frac{20}{3}$ by 4. This gives us $\frac{57}{12}$ and $\frac{80}{12}$ .

Now that both fractions have the same denominator, we can subtract them. We subtract the numerators and keep the denominator the same. This gives us $\frac{80}{12} - \frac{57}{12} = \frac{23}{12}$ .

Finally, we can simplify the result by dividing the numerator and denominator by their GCD. The GCD of 23 and 12 is 1. So, the result is $\frac{23}{12}$ , which can be written as 1 $\frac{11}{12}$ .

Final Answer

The final answer is $\boxed{1 \frac{11}{12}}$ .