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Fractions

Definitions


Hugo is very hungry after playing soccer. His dad baked two identical cakes.
Hugo eats one whole cake:
Then, Hugo is still hungry, so he eats half of the second cake:
How much cake does Hugo eat in total? Write your answer as a fraction.

  • Hugo eats one whole cake and half of another cake.
  • The numerator (top number) shows how many parts Hugo eats: \(3\).
  • The denominator (bottom number) shows how many equal parts make one cake: \(2\).
  • So Hugo eats \(\dfrac{3}{2}\) cakes in total.


Definition Fraction
A fraction includes two numbers: the numerator and the denominator, separated by a bar.

On the number line


  • Hugo is walking along a path.
  • He stops and asks himself, "Where am I?"
  • His father says, "You are at half of the way that is \(\dfrac{1}{2}\)."

Method Representing a Fraction on the Number Line
To represent the fraction \(\dfrac{\textcolor{colordef}{2}}{\textcolor{colorprop}{3}}\) on a number line.
  1. Draw a straight line and mark the points 0 and 1.
  2. Divide the line between 0 and 1 into \(\textcolor{colorprop}{3}\) equal parts.
  3. Count \(\textcolor{colordef}{2}\) parts from 0 and mark the point.

Equivalent Fractions


Mr. Tariel has a cake that he cuts into 3 equal parts. He plans to give 1 part to his son, Louis.
Louis says, "I want 2 pieces!"
His dad replies, "Alright," and cuts each of the 3 parts in half, making 6 smaller equal parts. He then gives Louis 2 of these smaller pieces.
Louis looks at his plate and feels disappointed.
Why is Louis still not happy?

Even though Louis got 2 pieces instead of 1, the total amount of cake he received is the same as before. His dad just cut the cake into smaller pieces.
\(\quad =\quad \)
In fractions: $$\frac{1}{3} = \frac{2}{6}$$


Definition Equivalent Fractions
  • When you multiply the numerator and the denominator by the same number, the fractions are equals.
  • When you divide the numerator and the denominator by the same number, the fractions are equals.

Simplification


Louis eats \( \dfrac{6}{12} \) of a pizza. Hugo says, "Hey, \( \dfrac{6}{12} \) is the same as \( \dfrac{1}{2} \). It's easier to understand if you simplify the fraction!".
\(=\)
  • Louis: "How is \( \dfrac{1}{2} \) easier?"
  • Hugo: "Because \( \dfrac{1}{2} \) is the simplified form of \( \dfrac{6}{12} \). It means you ate 1 out of 2 slices instead of 6 out of 12 slices. It’s the same amount of pizza, but it’s simpler to understand!"

Method Simplifying a fraction
To simplify a fraction, we find an equivalent fraction with the smallest possible numerator and denominator.
Example
Simplify \(\dfrac{4}{6}\)


Ordering Fractions


  • Hugo eats \(\dfrac{3}{4}\) of a cake.
  • Louis eats \(\dfrac{5}{8}\) of the same cake.
Who eats more cake?

  • We need to compare the fractions \(\dfrac{3}{4}\) and \(\dfrac{5}{8}\).
  • To compare fractions, the pieces must be the same size. We do this by finding a common denominator.
  • Convert \(\dfrac{3}{4}\) to an equivalent fraction with denominator 8:
    \quad \quad
  • Now, Hugo eats \(\dfrac{6}{8}\) of the cake and Louis eats \(\dfrac{5}{8}\).
  • Since \(\dfrac{6}{8} > \dfrac{5}{8}\), Hugo eats more cake.


Definition Ordering Fractions with the Same Denominator
For two fractions with the same denominator, the fraction with the larger numerator is larger.
Example
Compare \(\dfrac{3}{4}\) and \(\dfrac{2}{4}\).

\( = \dfrac{3}{4} \lt \dfrac{2}{4} \)

Method Comparing Fractions with Different Denominators
To compare two fractions with different denominators:
  • Find a common denominator.
  • Convert each fraction to an equivalent fraction with that denominator.
  • Compare the numerators.
Example
Compare \(\dfrac{1}{2}\) and \(\dfrac{3}{4}\).

  • Since \(\dfrac{1}{2}\) and \(\dfrac{3}{4}\) have different denominators, we change \(\dfrac{1}{2}\) into an equivalent fraction with denominator 4:
    \quad \quad
  • Compare the numerators: \[\dfrac{2}{4} < \dfrac{3}{4}\]
  • Therefore, \[\dfrac{1}{2} < \dfrac{3}{4}\]
  • In pictures:
    \quad = \quad \quad < \quad

Addition and Subtraction with Common Denominators


Hugo eats \(\dfrac{2}{4}\) of a cake: and Louis eats \(\dfrac{1}{4}\) of the same cake:
Which fraction of the cake have Hugo and Louis eaten together?

So Hugo and Louis eat \(\dfrac{3}{4}\) of the cake together:


Definition Addition of Fractions with Common Denominators
When we add fractions with common denominators, we keep the denominator the same and add the numerators:
Definition Subtraction of Fractions with Common Denominators
When we subtract fractions with common denominators, we keep the denominator the same and subtract the numerators:

Addition and Subtraction with Different Denominators


Hugo eats \(\dfrac{1}{2}\) of a cake: and Louis eats \(\dfrac{1}{4}\) of the same cake: .
What fraction of the cake have Hugo and Louis eaten together?

  • Step 1: Find a common denominator: To add the fractions, we need equal-sized parts. Divide each of Hugo's parts into two smaller parts:
    \quad \(=\) \quad
    So, Hugo eats \(\dfrac{1}{2} = \dfrac{2}{4}\) of the cake.
  • Step 2: Add the fractions using the common denominator: Now, we can add the two fractions:
    \(+\) \(=\) \(+\)
    \(=\)
    \(\dfrac{1}{2}+\dfrac{1}{4}\) \(=\) \(\dfrac{2}{4}+\dfrac{1}{4}\)
    \(=\) \(\dfrac{3}{4}\)
  • Step 3: Final Answer: Hugo and Louis eat \(\dfrac{3}{4}\) of the cake together:


Method Addition or Subtraction of Fractions with Different Denominators
To add or subtract fractions with different denominators:
  • Find a common denominator: Choose a common multiple of the denominators.
  • Convert each fraction: Rewrite each fraction so it has the common denominator.
  • Add or subtract the numerators: Add or subtract the numerators and keep the denominator the same.