Proper, Improper Fractions and Mixed Numbers

• A fraction represents a part of a whole or, more generally, any number of equal parts
• It is written as a ratio of two numbers: numerator and denominator
• Numerator is the number of parts we are considering
• Denominator is the total number of parts the whole is divided into

Proper Fraction

• A proper fraction is a fraction where the numerator (the top number) is less than the denominator (the bottom number)
• This means the value of the fraction is less than 1
• Examples of proper fractions: 1/2, 3/4, 2/5, 7/10, 15/16
• In each case, the numerator is smaller than the denominator

Improper Fraction

• An improper fraction is a fraction where the numerator is greater than or equal to the denominator
• This means the value of the fraction is greater than or equal to 1
• Examples of improper fractions: 5/3, 7/2, 4/4, 11/4, 8/5
• When the numerator equals the denominator, the fraction is equal to 1 (e.g. 4/4 = 1)

Mixed Number

• A mixed number is a whole number combined with a proper fraction
• It represents the sum of the whole number and the fraction
• Examples of mixed numbers: 1 1/2, 2 3/4, 5 1/3
• Improper fractions can be converted into mixed numbers
• To convert an improper fraction to a mixed number, divide the numerator by the denominator
• The quotient is the whole number part, the remainder is the numerator of the fractional part, and the denominator stays the same
• Example: 7/3 = 2 1/3 (7 divided by 3 is 2 with a remainder of 1)
• To convert a mixed number to an improper fraction, multiply the whole number by the denominator and add the numerator
• Put the result over the original denominator
• Example: 2 1/3 = (2*3 + 1)/3 = 7/3

Comparing Fractions

• To compare fractions, we determine which fraction is larger, smaller, or if they are equal
• Symbols used for comparison:
  • '<' means "less than"
  • '>' means "greater than"
  • '=' means "equal to"

Comparing Fractions with the Same Denominator

• When fractions have the same denominator, the fraction with the larger numerator is the larger fraction
• Example: 3/5 and 2/5
• Since 3 > 2, then 3/5 > 2/5

Comparing Fractions with the Same Numerator

• When fractions have the same numerator, the fraction with the smaller denominator is the larger fraction
• Example: 3/4 and 3/5
• Since 4 < 5, then 3/4 > 3/5
• This is because if you divide something into fewer parts (smaller denominator), each part is larger

Comparing Fractions with Different Numerators and Denominators

• To compare fractions with different numerators and denominators, a common denominator must be found
• This involves finding the least common multiple (LCM) of the denominators
• Once a common denominator is found, rewrite each fraction with the common denominator
• Then, compare the numerators as in the "same denominator" case
• Example: Compare 2/3 and 3/4
  • The least common multiple of 3 and 4 is 12
  • Rewrite 2/3 as 8/12 (multiply numerator and denominator by 4)
  • Rewrite 3/4 as 9/12 (multiply numerator and denominator by 3)
  • Since 8 < 9, then 8/12 < 9/12, so 2/3 < 3/4

Comparing Fractions to 1/2

• This is a useful shortcut for quick comparisons
• If the numerator is more than half of the denominator, the fraction is greater than 1/2
• If the numerator is less than half of the denominator, the fraction is less than 1/2
• If the numerator is equal to half of the denominator, the fraction is equal to 1/2
• Examples:
  • 3/5: Half of 5 is 2.5, and 3 > 2.5, so 3/5 > 1/2
  • 2/7: Half of 7 is 3.5, and 2 < 3.5, so 2/7 < 1/2
  • 4/8: Half of 8 is 4, and 4 = 4, so 4/8 = 1/2

Using the Cross-Multiplication Method

• An alternate method to compare fractions a/b and c/d
• Multiply the numerator of the first fraction by the denominator of the second fraction (a * d)
• Multiply the numerator of the second fraction by the denominator of the first fraction (b * c)
• Compare the results:
  • If a * d > b * c, then a/b > c/d
  • If a * d < b * c, then a/b < c/d
  • If a * d = b * c, then a/b = c/d
• Example: Compare 2/3 and 3/4
  • 2 * 4 = 8
  • 3 * 3 = 9
  • Since 8 < 9, then 2/3 < 3/4

Examples of Comparing Fractions with Symbols

• 1/4 < 1/2 (Because 1/4 is less than 1/2)
• 3/5 > 1/3 (Because 3/5 is greater than 1/3)
• 2/4 = 1/2 (Because 2/4 is equal to 1/2)
• 5/8 > 1/2 (Because 5/8 is greater than 1/2)
• 7/10 > 2/5 (Convert 2/5 to 4/10. Since 7/10 > 4/10)
• 3/7 < 1/2 (Because 3 is less than half of 7 which is 3.5)
• 5/5 = 1 (Because when the numerator and denominator are the same, the fraction equals 1)
• 6/4 > 1 (Because 6/4 is an improper fraction where the numerator is greater than the denominator)

Understand the basics of fractions, including proper fractions (numerator less than denominator), and improper fractions (numerator greater than or equal to denominator). Mixed numbers combine whole numbers and fractions. Examples provided.