Adding Fractions

Study Notes

Like Denominators

Definition: Fractions that have the same denominator.
Process:
1. Keep the denominator the same.
2. Add the numerators.
3. Write the sum over the common denominator.
Example:
- ( \frac{2}{5} + \frac{3}{5} = \frac{2 + 3}{5} = \frac{5}{5} = 1 )

Unlike Denominators

Definition: Fractions that have different denominators.
Process:
1. Find the least common denominator (LCD).
2. Convert each fraction to an equivalent fraction with the LCD.
3. Add the numerators.
4. Write the sum over the LCD.
Example:
- ( \frac{1}{4} + \frac{1}{6} )
- LCD = 12
- Convert: ( \frac{3}{12} + \frac{2}{12} = \frac{5}{12} )

Simplifying Fractions

Definition: Reducing a fraction to its lowest terms.
Process:
1. Find the greatest common divisor (GCD) of the numerator and denominator.
2. Divide both by the GCD.
Example:
- ( \frac{8}{12} )
- GCD = 4
- Simplified: ( \frac{8 \div 4}{12 \div 4} = \frac{2}{3} )

Mixed Numbers

Definition: A whole number combined with a fraction.
Addition Process:
1. Convert mixed numbers to improper fractions.
2. Follow the steps for adding fractions (like or unlike denominators).
3. Convert back to a mixed number if necessary.
Example:
- ( 2 \frac{1}{3} + 1 \frac{2}{5} )
- Convert: ( \frac{7}{3} + \frac{7}{5} )
- Find LCD (15): ( \frac{35}{15} + \frac{21}{15} = \frac{56}{15} = 3 \frac{11}{15} )

Word Problems

Approach:
1. Read the problem carefully and identify the fractions involved.
2. Determine if the fractions have like or unlike denominators.
3. Use the appropriate method to add the fractions.
4. Simplify the result if needed and convert to a mixed number if applicable.
Example:
- A recipe calls for ( \frac{1}{4} ) cup of sugar and ( \frac{1}{6} ) cup of brown sugar.
- Determine the total sugar:
  - Find LCD (12):
  - Convert: ( \frac{3}{12} + \frac{2}{12} = \frac{5}{12} ) cup of sugar.

Adding Fractions: Key Concepts

Like Denominators

Fractions with the same denominator allow for straightforward addition.
Keep the denominator constant, add the numerators, and place the result over the same denominator.
Example: ( \frac{2}{5} + \frac{3}{5} = \frac{5}{5} = 1 ).

Unlike Denominators

Fractions with different denominators require more steps for addition.
Start by identifying the least common denominator (LCD).
Convert each fraction using the LCD, then add the numerators, and place the result over the LCD.
Example: For ( \frac{1}{4} + \frac{1}{6} ) with an LCD of 12, convert to ( \frac{3}{12} + \frac{2}{12} = \frac{5}{12} ).

Simplifying Fractions

Simplifying fractions involves reducing them to their simplest form.
Find the greatest common divisor (GCD) for the numerator and denominator.
Divide both parts by the GCD to simplify.
Example: ( \frac{8}{12} ) simplifies to ( \frac{2}{3} ) using a GCD of 4.

Mixed Numbers

A mixed number combines a whole number with a fraction.
To add mixed numbers, convert them into improper fractions first.
Follow the addition process for fractions, then convert the resulting improper fraction back to a mixed number if needed.
Example: Adding ( 2 \frac{1}{3} + 1 \frac{2}{5} ) involves converting to ( \frac{7}{3} + \frac{7}{5} ) and culminating in ( 3 \frac{11}{15} ).

Word Problems

Solve word problems by carefully reading and identifying involved fractions.
Determine if fractions have like or unlike denominators to apply the appropriate addition method.
Simplify the resulting fraction and convert to a mixed number if required.
Example: From a recipe requiring ( \frac{1}{4} ) cup of sugar and ( \frac{1}{6} ) cup of brown sugar, the total is ( \frac{5}{12} ) cups after conversion using the LCD of 12.

Description

Test your understanding of adding fractions with like and unlike denominators. This quiz covers the definitions, processes, and examples needed to combine fractions effectively. Simplifying fractions is also included, helping you reduce them to their lowest terms.