Problem 42 - Algebra 1: Concepts and Skills

Question

Add. Write the answer as a fraction or a mixed number in simplest form. $$ \frac{5}{6}+\frac{7}{6} $$

Step-by-Step Solution

Answer

The simplified form of \( \frac{5}{6} + \frac{7}{6} \) is 2.

Step 1: Understand the Fraction Addition Rule

When adding fractions with the same denominator, keep the denominator the same and add the numerators. Here, both fractions have the same denominator 6.

Step 2: Perform the Addition

Now, add the two numerators while keeping the denominator the same. So, \( \frac{5}{6} + \frac{7}{6} = \frac{5 + 7}{6} = \frac{12}{6} \).

Step 3: Simplify the Fraction (optional)

The fraction \( \frac{12}{6} \) simplifies to 2, because 12 divided by 6 equals 2.

Key Concepts

Adding Fractions

Adding fractions can be easy, especially when the fractions already have the same denominator, like in our example: \( \frac{5}{6} \) and \( \frac{7}{6} \). To add fractions with the same denominator:

• Keep the denominator the same. This means they share the same 'whole' or 'base' number.
• Only add the numerators (the top numbers of the fractions).

Here, you add 5 and 7 to get 12, while the denominator remains as 6. This works because both fractions are talking about the same-sized pieces, making the addition straightforward.

Simplifying Fractions

Simplifying fractions is all about making them as simple as possible. It's like tidying up your desk so it's easy to understand what's there. Once you have added the fractions and ended up with \( \frac{12}{6} \), you should simplify because:

• A fraction is in simplest form when the numerator and denominator are as small as possible, yet retain the same value once divided.
• We simplify by finding the greatest common factor and dividing both the numerator and denominator by it.

For \( \frac{12}{6} \), both 12 and 6 have a common factor of 6. Dividing them both by 6 gives us 2. So, \( \frac{12}{6} = 2 \), a much simpler form!

Mixed Numbers

Mixed numbers consist of a whole number and a proper fraction. They are particularly useful when dealing with improper fractions like \( \frac{12}{6} \) which can be simplified to a whole number. If your resulting fraction after addition extends beyond a whole number, turning it into a mixed number can make it clearer:

• First, divide the numerator by the denominator to find the whole number part.
• The remainder becomes the new numerator of the proper fraction.

In our example, however, when adding \( \frac{5}{6} \) and \( \frac{7}{6} \), you don’t need to create a mixed number because the outcome exactly simplifies to 2. But when the result doesn’t perfectly simplify to a whole number, you'd express it as a mixed number instead.