Problem 69 - Algebra 1: Concepts and Skills

Question

Add. Write the answer as a fraction or as a mixed number in simplest form. $$ 8 \frac{1}{3}+2 \frac{2}{3} $$

Step-by-Step Solution

Answer

The answer is 11.

Step 1: Add the Whole Numbers

Begin by adding together the whole numbers. For this problem, that's 8 and 2. This yields a result of 10.

Step 2: Add the Fractions

Now, add together the fractional parts of the mixed numbers. In this case, that includes \(\frac{1}{3}\) and \(\frac{2}{3}\). The sum of these fractions is \(\frac{3}{3}\).

Step 3: Combine the Results

Combine the results from steps 1 and 2 to yield the final result. Add the result of the fraction \(\frac{3}{3}\) which simplifies to 1 to the result from the sum of the whole numbers. This gives us a final answer of 10 + 1.

Step 4: Simplify the Result

Simplify the final result. For the given problem, the result is already in its simplest form as a mixed number.

Key Concepts

Understanding Fractions

Fractions can be a bit tricky at first, but once you understand the parts and how they work together, they become much simpler. A fraction represents a part of a whole. It is written as a quotient of two numbers, the numerator and the denominator.

• The **numerator** is the top number. It signifies how many parts we have.
• The **denominator** is the bottom number. It tells us how many equal parts the whole is divided into.

For example, in the fraction \( \frac{1}{3} \), '1' is the numerator, and '3' is the denominator. This fraction tells us that we have one out of three equal parts of a whole. When you're adding fractions, it's important that they have the same denominator, also known as like fractions. This allows us to directly add the numerators together, simplifying our calculations.

Simplifying Fractions

Once you've added or subtracted fractions, it's crucial to simplify the result. Simplifying a fraction means reducing it to its simplest form. This is done by dividing both the numerator and the denominator by their greatest common divisor (GCD).

• Identify the GCD of the numerator and denominator.
• Divide both numbers by this GCD.

For instance, if you end up with the fraction \( \frac{6}{9} \), you would find that the GCD is 3. Dividing the numerator and the denominator by 3, you get \( \frac{2}{3} \). Simplifying ensures your fraction is as straightforward as possible, which makes it easier to understand and utilize in further calculations.

Working with Whole Numbers

Whole numbers are the friendly numbers we often start learning in mathematics. They are the set of numbers without fractions or decimals, starting from 0, 1, 2, 3, and so on. When dealing with mixed numbers, whole numbers play a significant role as they represent the part you deal with directly outside fractional parts.

• Whole numbers are easy to add because we simply sum them like regular counting.
• When combined with fractions in mixed numbers, solve them separately for clarity.

For example, when you add the mixed numbers \(8\frac{1}{3} + 2\frac{2}{3}\), begin by adding the whole numbers (8 + 2 = 10). Managing whole numbers separately helps simplify your calculations. This ensures that when you add the fractional parts, combining them back together with the whole number is an easy step that makes everything intuitively fall into place.