Problem 68 - Algebra 1: Concepts and Skills

Question

Add. Write the answer as a fraction or as a mixed number in simplest form. $$ 3 \frac{8}{11}+5 \frac{3}{11} $$

Step-by-Step Solution

Answer

The answer to the addition is \(9\)

Step 1: Break down the numbers

First, recognize that \(3 \frac{8}{11}\) and \(5 \frac{3}{11}\) are mixed numbers, meaning each has a whole number part and a fraction part. So, we will separate them and treat them individually during addition.

Step 2: Add whole numbers

Next, add the whole numbers part. So, \(3+5=8\)

Step 3: Add fractions

Then, add the fraction parts which are \(\frac{8}{11}+ \frac{3}{11}=\frac{11}{11}\)

Step 4: Simplify the fractions

Now, simplify the resultant fraction which is \(\frac{11}{11}=1\)

Step 5: Add simplified fractions to whole numbers

Finally add the simplified fraction obtained in Step 4 to the result of step 2, \(8+1=9\)

Key Concepts

Mixed Numbers Addition

Understanding mixed numbers addition is critical when working with fractions that include whole numbers. A mixed number is a combination of a whole number and a proper fraction. For example, in the mixed number \(3 \frac{8}{11}\), 3 is the whole number and \(\frac{8}{11}\) is the fraction.

When adding mixed numbers, such as \(3 \frac{8}{11}\) and \(5 \frac{3}{11}\), it's necessary to add both the whole numbers and fractions separately. This approach simplifies the process and helps avoid mistakes. Properly aligning the fractions ensures they have common denominators, making addition straightforward. By handling the whole and fractional parts individually, students can easily track the calculation steps and arrive at the correct answer.

Simplifying Fractions

Simplifying fractions is a crucial step in arithmetic, particularly when dealing with mixed numbers. To simplify a fraction means to reduce it to its simplest form, where the numerator and denominator have no common factors other than 1. In the exercise, when adding the fractions \(\frac{8}{11}\) and \(\frac{3}{11}\), the result is \(\frac{11}{11}\), which is equivalent to 1.

Fractions simplified to whole numbers, like this example, can then be easily added to the whole number portion of a mixed number. It is essential to simplify whenever possible to make the numbers easier to work with and the result clearer to understand. This can involve dividing both parts of the fraction by their greatest common divisor or recognizing when the numerator and denominator are equal, indicating a whole number.

Arithmetic Operations with Fractions

Arithmetic operations with fractions may initially seem complex, but they follow a set of straightforward rules. When adding fractions, as seen in the given mixed numbers, ensure that they have the same denominator. With the same denominators, the numerators can simply be added together.

In the example \(\frac{8}{11}+ \frac{3}{11}\), both fractions have 11 as their common denominator. Thus, the numerators 8 and 3 are added to get 11, giving the result \(\frac{11}{11}\), which can be further simplified to 1. Always pay attention to simplification, as it can turn a complex-looking fraction into a simpler form or even a whole number, thereby making subsequent operations much more manageable.