Types of Fractions

Proper Fractions

Less than the whole.

A proper fraction is a fraction where the top number is smaller than the bottom number. For example, 34\frac{3}{4} is proper because 33 is smaller than 44.
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Key Points

  1. It always represents an amount that is less than one whole.
  2. If you look at a number line, proper fractions always sit between 00 and 11.
  3. It means you haven't collected enough pieces to make a full object yet.

Olympiad Fact

Think of a proper fraction as 'top-light'. Since the denominator is larger, it acts as a sturdy base keeping the fraction safely between 00 and 11.

Improper Fractions

Equal to or greater than one.

An improper fraction is a fraction where the top number is equal to or bigger than the bottom number, like 54\frac{5}{4}.
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Key Points

  1. Because the top number is bigger, you actually have more than one whole object.
  2. If the top and bottom numbers are the exact same (like 44\frac{4}{4}), it equals exactly 11 whole.
  3. On a number line, these fractions sit exactly at 11 or further to the right.

Olympiad Fact

Every whole number nn can be written as an improper fraction n1\frac{n}{1}. This conversion makes it mathematically simple to add or multiply whole numbers with fractions.

Unit Fractions

One single part.

A unit fraction is a simple fraction where the top number is exactly 11. It represents just one single piece of a divided whole, like 13\frac{1}{3} or 15\frac{1}{5}.
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Key Points

  1. The top number is always 11.
  2. The bigger the bottom number gets, the smaller the piece actually is. Sharing a pizza with 1010 people gives you a smaller slice than sharing with 22.
  3. Every other fraction is just a group of unit fractions added together! For instance, 34\frac{3}{4} is just three 14\frac{1}{4}s.

Olympiad Fact

For unit fractions, the relationship is inverse: as the denominator grows, the value of the fraction shrinks. 12\frac{1}{2} is significantly larger than 1100\frac{1}{100}.

Like Fractions

Shared denominators.

Like fractions are a group of fractions that share the exact same bottom number. This means their pieces are the exact same size.
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Key Points

  1. Examples are 27\frac{2}{7}, 47\frac{4}{7}, and 57\frac{5}{7}. Notice how they all end in 'sevenths'.
  2. Because the piece sizes are identical, you can easily compare them just by looking at the top numbers.
  3. You can add or subtract them directly without doing any extra math steps.

Olympiad Fact

Think of the denominator as a 'unit label' (like centimeters or inches). You can only add or subtract quantities if they carry the same unit label!

Unlike Fractions

Different denominators.

Unlike fractions are fractions that have different bottom numbers, like 12\frac{1}{2} and 13\frac{1}{3}. This means their pieces are different sizes.
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Key Points

  1. You cannot add or subtract them directly because the pieces are different sizes, like comparing apples and oranges.
  2. To do any math with them, you first have to make their bottom numbers match perfectly.
  3. You do this by finding the Least Common Multiple (LCM) of the two bottom numbers.
  4. This LCM will act as your new shared bottom number so the piece sizes are finally identical.

Olympiad Fact

Always look for the LCM (Least Common Multiple) for your common denominator. While multiplying the denominators together works, it often creates needlessly large numbers.

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