Problem 76 - Algebra 1: Concepts and Skills

Question

Add. Write the answer as a mixed number in simplest form. $$ \frac{11}{3}+5 \frac{5}{6} $$

Step-by-Step Solution

Answer

The answer to \( \frac{11}{3} + 5 \frac{5}{6} \) is \( 9 \frac{1}{2} \).

Step 1: Convert the improper fraction to a mixed number

An improper fraction is a fraction where the numerator is bigger than the denominator. To convert it to a mixed number, you divide the numerator by the denominator. In this case, divide 11 by 3, which gives a quotient of 3 and a remainder of 2. Thus, \( \frac{11}{3} \) can be written as 3 \( \frac{2}{3} \).

Step 2: Add the fractions

Next, add the fraction parts of the two mixed numbers together. \( \frac{2}{3} + \frac{5}{6} = \frac{4}{6} + \frac{5}{6} = \frac{9}{6} \).

Step 3: Simplify the resulting fraction to a mixed number

The resulting fraction from step 2 is \( \frac{9}{6} \), which is again an improper fraction and can be simplified to 1 \( \frac{1}{2} \).

Step 4: Add the whole numbers

Add the whole numbers from the original mixed numbers and the whole number from the simplified fraction: 3 (from step 1) + 5 (from the problem) + 1 (from step 3) = 9.

Key Concepts

Improper Fraction to Mixed Number

Understanding how to convert an improper fraction to a mixed number is essential when working with fractions. An improper fraction is characterized by having a numerator (the top number) larger than its denominator (the bottom number), which means it represents a value greater than one whole.

To convert an improper fraction to a mixed number, you divide the numerator by the denominator. The quotient becomes the whole number part of the mixed number, and the remainder becomes the new numerator over the original denominator. Let's see this in action with an example from the exercise:

• Divide 11 by 3 to get a quotient of 3 and a remainder of 2.
• Thus, the mixed number is 3 with the fraction \( \frac{2}{3} \) remaining.

So, \( \frac{11}{3} \) is equivalent to 3 \( \frac{2}{3} \). It represents 3 whole parts and another \( \frac{2}{3} \) of a whole.

Adding Fractions

Adding fractions is another fundamental aspect when working with numbers. To add fractions, they must have a common denominator (the bottom part of the fraction). If they don't, you'll need to find an equivalent fraction for each so that they do.

In our example, we want to add \( \frac{2}{3} \) and \( \frac{5}{6} \). The least common denominator (LCD) for 3 and 6 is 6. So, we convert \( \frac{2}{3} \) to have a denominator of 6:

• Multiply both the numerator and the denominator by 2 to get \( \frac{4}{6} \) from \( \frac{2}{3} \).
• Now, simply add \( \frac{4}{6} \) to \( \frac{5}{6} \) to get \( \frac{9}{6} \).

This process ensures that you accurately combine the fractional parts.

Simplifying Fractions

Simplifying fractions, also known as reducing fractions, involves rewriting a fraction in its simplest form. This is done by finding the greatest common divisor (GCD) of the numerator and denominator and then dividing both by that number.

For example, in the given solution, the fraction \( \frac{9}{6} \) can be simplified. The GCD of 9 and 6 is 3, so by dividing both the numerator and the denominator by 3, we get:

• \( \frac{9 \div 3}{6 \div 3} = \frac{3}{2} \)

Since \( \frac{3}{2} \) is still an improper fraction, we convert it to the mixed number 1 \( \frac{1}{2} \). Simplifying fractions allows us to express them in a more concise and understandable form, making it easier to perform further arithmetic operations.