Fractions: Types and Conversion

• A fraction shows a part of a whole, representing equal portions.
• It is written as one number over another, separated by a line.
• The numerator is the number above the line and represents the number of parts described.
• The denominator is the number below the line indicates the total number of equal parts.
• In the fraction 3/4, 3 is the numerator and 4 is the denominator, showing 3 parts out of 4.

Types of Fractions

• Proper fractions have a numerator less than the denominator, such as 1/2 and 3/4.
• Improper fractions have a numerator greater than or equal to the denominator, such as 5/3 and 7/7.
• Mixed numbers combine a whole number and a proper fraction, like 1 1/2 and 2 3/4.
• Equivalent fractions represent the same value, even with different numerators and denominators, such as 1/2 and 2/4.

Converting Between Improper Fractions and Mixed Numbers

• To convert an improper fraction to a mixed number, divide the numerator by the denominator.
• The quotient becomes the whole number.
• The remainder becomes the new numerator, keeping the same denominator.
• For example: 5/3 = 1 2/3 (5 ÷ 3 = 1 with a remainder of 2).
• To convert a mixed number to an improper fraction:
• Multiply the whole number by the denominator.
• Add the numerator to the result.
• Place the sum over the original denominator.
• For example: 2 1/4 = (2 * 4 + 1) / 4 = 9/4.

Simplifying Fractions

• Simplifying means reducing a fraction to its lowest terms.
• Find the greatest common factor (GCF) of the numerator and denominator.
• Divide both by their GCF.
• For example: Simplify 4/6. The GCF of 4 and 6 is 2. Thus, 4/6 = (4 ÷ 2) / (6 ÷ 2) = 2/3.

Adding and Subtracting Fractions

• Fractions must have the same denominator (common denominator) to be added or subtracted.
• If not, find the least common multiple (LCM) of the denominators to use as the common denominator.
• Convert each fraction to an equivalent fraction using the common denominator.
• Add or subtract the numerators, keeping the denominator the same.
• Simplify the resulting fraction if needed.

Adding Fractions

• Example: 1/4 + 2/4 = (1+2)/4 = 3/4
• Example with different denominators: 1/3 + 1/4. The LCM of 3 and 4 is 12. Thus, 1/3 = 4/12 and 1/4 = 3/12, so 4/12 + 3/12 = 7/12.

Subtracting Fractions

• Example: 3/5 - 1/5 = (3-1)/5 = 2/5
• Example with different denominators: 1/2 - 1/3. The LCM of 2 and 3 is 6. Thus 1/2 = 3/6 and 1/3 = 2/6, so 3/6 - 2/6 = 1/6

Multiplying Fractions

• Multiply the numerators together.
• Multiply the denominators together.
• Simplify if needed.
• Example: 2/3 * 3/4 = (2 * 3) / (3 * 4) = 6/12 = 1/2.
• Convert mixed numbers to improper fractions before multiplying.

Dividing Fractions

• Multiply the first fraction by the reciprocal of the second.
• The reciprocal is found by swapping the numerator and denominator.
• Simplify if needed.
• Example: 1/2 ÷ 3/4 = 1/2 * 4/3 = (1 * 4) / (2 * 3) = 4/6 = 2/3.
• Convert mixed numbers to improper fractions before dividing.

Comparing Fractions

• With the same denominator, compare numerators; the larger numerator indicates the greater fraction.
• With different denominators, find a common denominator, convert to equivalent fractions, then compare numerators.
• Alternatively, convert each fraction to a decimal for comparison.

Fractions in Real-World Applications

• Cooking: Used in recipes for ingredient quantities.
• Measurement: Describes length, weight, and volume.
• Finance: Represents portions of money or investments.
• Time: Represents hours, minutes, and seconds as parts of a day or hour.

Understand fractions as parts of a whole, represented by a numerator and denominator. Explore proper, improper, and equivalent fractions. Learn to convert between improper fractions and mixed numbers.