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)