Find The Sum.${ \begin{aligned} 2 \frac{3}{4} + 1 \frac{2}{3} & = 2 \frac{9}{12} + 1 \frac{8}{12} \ & = 4 \frac{1}{11} \end{aligned} }$

Introduction

In mathematics, mixed numbers are a combination of a whole number and a fraction. They are often used to represent quantities that are not whole, but can be expressed as a sum of a whole number and a fraction. In this article, we will explore how to find the sum of mixed numbers, using the example of $2 \frac{3}{4} + 1 \frac{2}{3}$.

Understanding Mixed Numbers

A mixed number is a combination of a whole number and a fraction. It is written in the form $a \frac{b}{c}$, where $a$ is the whole number, $b$ is the numerator, and $c$ is the denominator. For example, $2 \frac{3}{4}$ is a mixed number, where $2$ is the whole number, $3$ is the numerator, and $4$ is the denominator.

Adding Mixed Numbers

When adding mixed numbers, we need to follow a specific procedure. The first step is to convert the mixed numbers to improper fractions. 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. For example, to convert $2 \frac{3}{4}$ to an improper fraction, we multiply $2$ by $4$ and add $3$, which gives us $\frac{11}{4}$.

Converting Mixed Numbers to Improper Fractions

To convert a mixed number to an improper fraction, we follow these steps:

1. Multiply the whole number by the denominator.
2. Add the numerator to the result.
3. Write the result as a fraction, with the numerator as the new numerator and the denominator as the original denominator.

For example, to convert $2 \frac{3}{4}$ to an improper fraction, we follow these steps:

1. Multiply $2$ by $4$, which gives us $8$.
2. Add $3$ to $8$, which gives us $11$.
3. Write the result as a fraction, with $11$ as the numerator and $4$ as the denominator, which gives us $\frac{11}{4}$.

Adding Improper Fractions

Once we have converted the mixed numbers to improper fractions, we can add them together. To add improper fractions, we need to follow these steps:

1. Find a common denominator for the fractions.
2. Add the numerators of the fractions.
3. Write the result as a fraction, with the numerator as the new numerator and the denominator as the common denominator.

For example, to add $\frac{11}{4}$ and $\frac{8}{12}$, we follow these steps:

1. Find a common denominator for the fractions, which is $12$.
2. Add the numerators of the fractions, which gives us $\frac{11 \times 3}{4 \times 3} + \frac{8}{12} = \frac{33}{12} + \frac{8}{12}$.
3. Add the fractions, which gives us $\frac{33 + 8}{12} = \frac{41}{12}$.

Simplifying the Result

Once we have added the improper fractions, we need to simplify the result. To simplify a fraction, we need to find the greatest common divisor (GCD) of the numerator and the denominator and divide both numbers by the GCD.

For example, to simplify $\frac{41}{12}$, we need to find the GCD of $41$ and $12$. The GCD of $41$ and $12$ is $1$, so we cannot simplify the fraction further.

Conclusion

In conclusion, finding the sum of mixed numbers involves converting the mixed numbers to improper fractions, adding the improper fractions, and simplifying the result. By following these steps, we can find the sum of mixed numbers and express the result as a simplified fraction.

Example Problem

Find the sum of $2 \frac{3}{4}$ and $1 \frac{2}{3}$.

To solve this problem, we need to follow the steps outlined above:

1. Convert the mixed numbers to improper fractions: $2 \frac{3}{4} = \frac{11}{4}$ $1 \frac{2}{3} = \frac{5}{3}$
2. Find a common denominator for the fractions, which is $12$.
3. Add the numerators of the fractions: $\frac{11 \times 3}{4 \times 3} + \frac{5 \times 4}{3 \times 4} = \frac{33}{12} + \frac{20}{12}$
4. Add the fractions: $\frac{33 + 20}{12} = \frac{53}{12}$
5. Simplify the result: The GCD of $53$ and $12$ is $1$, so we cannot simplify the fraction further.

Therefore, the sum of $2 \frac{3}{4}$ and $1 \frac{2}{3}$ is $\frac{53}{12}$.

Tips and Tricks

• When adding mixed numbers, make sure to convert them to improper fractions first.
• When adding improper fractions, make sure to find a common denominator.
• When simplifying the result, make sure to find the GCD of the numerator and the denominator.

Q: What is a mixed number?

A: A mixed number is a combination of a whole number and a fraction. It is written in the form $a \frac{b}{c}$, where $a$ is the whole number, $b$ is the numerator, and $c$ is 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 need to multiply the whole number by the denominator and add the numerator. For example, to convert $2 \frac{3}{4}$ to an improper fraction, you multiply $2$ by $4$ and add $3$, which gives you $\frac{11}{4}$.

Q: How do I add mixed numbers?

A: To add mixed numbers, you need to follow these steps:

1. Convert the mixed numbers to improper fractions.
2. Find a common denominator for the fractions.
3. Add the numerators of the fractions.
4. Write the result as a fraction, with the numerator as the new numerator and the denominator as the common denominator.

Q: What is a common denominator?

A: A common denominator is a number that is a multiple of both denominators in a fraction. For example, if you have two fractions with denominators of $4$ and $6$, the common denominator would be $12$.

Q: How do I simplify a fraction?

A: To simplify a fraction, you need to find the greatest common divisor (GCD) of the numerator and the denominator and divide both numbers by the GCD.

Q: What is the greatest common divisor (GCD)?

A: The greatest common divisor (GCD) is the largest number that divides both the numerator and the denominator of a fraction without leaving a remainder.

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

A: There are several ways to find the GCD of two numbers, including:

• Listing the factors of each number and finding the greatest common factor.
• Using the Euclidean algorithm.
• Using a calculator or online tool.

Q: Can I add fractions with different denominators?

A: Yes, you can add fractions with different denominators by finding a common denominator and then adding the fractions.

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: Can I subtract mixed numbers?

A: Yes, you can subtract mixed numbers by following the same steps as adding mixed numbers, but with subtraction instead of addition.

Q: Can I multiply mixed numbers?

A: Yes, you can multiply mixed numbers by following the same steps as multiplying fractions, but with mixed numbers instead of fractions.

Q: Can I divide mixed numbers?

A: Yes, you can divide mixed numbers by following the same steps as dividing fractions, but with mixed numbers instead of fractions.

Conclusion

In conclusion, mixed numbers are a combination of a whole number and a fraction, and can be added, subtracted, multiplied, and divided using the same rules as fractions. By following the steps outlined in this article, you can perform operations with mixed numbers with ease.