Adding Fractions

Study Notes

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 )

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} )

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} )

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} )

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.

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 ).

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} ).

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} ).

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.