# Key Concepts

### 4.1 Visualize Fractions

• Property of One
• Any number, except zero, divided by itself is one.
aa=1aa=1, where a≠0a≠0.
• Mixed Numbers
• mixed number consists of a whole number aa and a fraction bcbc where c≠0c≠0.
• It is written as follows: abcc≠0abcc≠0
• Proper and Improper Fractions
• The fraction abab is a proper fraction if a<ba<b and an improper fraction if a≥ba≥b.
• Convert an improper fraction to a mixed number.
1. Step 1. Divide the denominator into the numerator.
2. Step 2. Identify the quotient, remainder, and divisor.
3. Step 3. Write the mixed number as quotientremainderdivisorremainderdivisor.
• Convert a mixed number to an improper fraction.
1. Step 1. Multiply the whole number by the denominator.
2. Step 2. Add the numerator to the product found in Step 1.
3. Step 3. Write the final sum over the original denominator.
• Equivalent Fractions Property
• If a, b,a, b, and cc are numbers where b≠0b≠0, c≠0c≠0, then ab=a⋅cb⋅cab=a⋅cb⋅c.

• .

### 4.3 Multiply and Divide Mixed Numbers and Complex Fractions

• Multiply or divide mixed numbers.
1. Step 1. Convert the mixed numbers to improper fractions.
2. Step 2. Follow the rules for fraction multiplication or division.
3. Step 3. Simplify if possible.
• Simplify a complex fraction.
1. Step 1. Rewrite the complex fraction as a division problem.
2. Step 2. Follow the rules for dividing fractions.
3. Step 3. Simplify if possible.
• For any positive numbers aa and bb, −ab=a−b=−ab-ab=a-b=-ab.
• Simplify an expression with a fraction bar.
1. Step 1. Simplify the numerator.
2. Step 2. Simplify the denominator.
3. Step 3. Simplify the fraction.

### 4.4 Add and Subtract Fractions with Common Denominators

• If a,b,a, b, and cc are numbers where c≠0c≠0, then ac+bc=a+ccac+bc=a+cc.
• To add fractions, add the numerators and place the sum over the common denominator.
• Fraction Subtraction
• If a,b,a, b, and cc are numbers where c≠0c≠0, then ac−bc=a−ccac-bc=a-cc.
• To subtract fractions, subtract the numerators and place the difference over the common denominator.

### 4.5 Add and Subtract Fractions with Different Denominators

• Find the least common denominator (LCD) of two fractions.
1. Step 1. Factor each denominator into its primes.
2. Step 2. List the primes, matching primes in columns when possible.
3. Step 3. Bring down the columns.
4. Step 4. Multiply the factors. The product is the LCM of the denominators.
5. Step 5. The LCM of the denominators is the LCD of the fractions.
• Equivalent Fractions Property
• If a,ba, b, and cc are whole numbers where b≠0b≠0, c≠0c≠0 then
ab=a⋅cb⋅cab=a⋅cb⋅c and a⋅cb⋅c=aba⋅cb⋅c=ab
• Convert two fractions to equivalent fractions with their LCD as the common denominator.
1. Step 1. Find the LCD.
2. Step 2. For each fraction, determine the number needed to multiply the denominator to get the LCD.
3. Step 3. Use the Equivalent Fractions Property to multiply the numerator and denominator by the number from Step 2.
4. Step 4. Simplify the numerator and denominator.
• Add or subtract fractions with different denominators.
1. Step 1. Find the LCD.
2. Step 2. Convert each fraction to an equivalent form with the LCD as the denominator.
3. Step 3. Add or subtract the fractions.
4. Step 4. Write the result in simplified form.
• Simplify complex fractions.
1. Step 1. Simplify the numerator.
2. Step 2. Simplify the denominator.
3. Step 3. Divide the numerator by the denominator.
4. Step 4. Simplify if possible.

### 4.6 Add and Subtract Mixed Numbers

• Add mixed numbers with a common denominator.
1. Step 1. Add the whole numbers.
2. Step 2. Add the fractions.
3. Step 3. Simplify, if possible.
• Subtract mixed numbers with common denominators.
1. Step 1. Rewrite the problem in vertical form.
2. Step 2. Compare the two fractions.
If the top fraction is larger than the bottom fraction, go to Step 3.
If not, in the top mixed number, take one whole and add it to the fraction part, making a mixed number with an improper fraction.
3. Step 3. Subtract the fractions.
4. Step 4. Subtract the whole numbers.
5. Step 5. Simplify, if possible.
• Subtract mixed numbers with common denominators as improper fractions.
1. Step 1. Rewrite the mixed numbers as improper fractions.
2. Step 2. Subtract the numerators.
3. Step 3. Write the answer as a mixed number, simplifying the fraction part, if possible.