Fractions
Definitions
Definition Fraction
A fraction consists of two numbers: the numerator \(a\) and the denominator \(b \neq 0\), separated by a horizontal bar: 
A fraction can be represented in several ways:
- Symbol: \(\dfrac{2}{3}\)
- Words: two thirds or two over three
- Linear model:
- Number line:
Equivalent Fractions
Definition Equivalent Fractions
- When you multiply the numerator and denominator by the same number, the fractions are equal:
- When you divide the numerator and denominator by the same number, the fractions are equal:
Example
Example
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: "Why is \( \dfrac{1}{2} \) easier?"
- Hugo: "Because \( \dfrac{1}{2} \) is the simplest 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 simpler to understand!"
Definition Simplest form
A fraction is in simplest form if it is written with the smallest possible whole number numerator and denominator, that is, if its numerator and denominator have no common factors other than 1.
Example
- \(\frac{2}{3}\) is in simplest form.
- \(\frac{4}{6}\) is not in simplest form because we can write \(\frac{4}{6} = \frac{2}{3}\).
Method Simplifying a fraction
To simplify a fraction (or write a fraction in its simplest form), divide the numerator and the denominator by their greatest common factor (GCF).
Example
Simplify \(\dfrac{4}{6}\).
$$\begin{aligned}[t]\dfrac{4}{6} &= \dfrac{2 \times \textcolor{olive}{\cancel{2}}}{3 \times \textcolor{olive}{\cancel{2}}} \\
&= \dfrac{2}{3}\end{aligned}$$
Cross Multiplication
We have learned that two fractions are equal if we multiply both the numerator and the denominator by the same number. For example:$$\dfrac{2}{3} = \dfrac{\textcolor{olive}{5} \times 2}{\textcolor{olive}{5} \times 3} = \dfrac{10}{15}$$We can also look at the relationship between the numerators and denominators in another way:$$\begin{aligned}2 \times (\textcolor{olive}{5} \times 3) &= 3 \times (\textcolor{olive}{5} \times 2) \\
2 \times 15 &= 3 \times 10 \\
\end{aligned}$$This leads us to a new way of checking if two fractions are equal: by cross multiplying and comparing the products.
Proposition Cross Multiplication Property
Example
Solve \(x\) for \(\dfrac{10}{5}=\dfrac{x}{8}\).
Addition and Subtraction
Definition Addition and Subtraction of Fractions with Common Denominators
- When we add fractions with the same denominator, we keep the denominator and add the numerators:$$\dfrac{a}{b} + \dfrac{c}{b} = \dfrac{a + c}{b}$$
- When we subtract fractions with the same denominator, we keep the denominator and subtract the numerators:$$\dfrac{a}{b} - \dfrac{c}{b} = \dfrac{a - c}{b}$$
Example
Calculate \(\dfrac{1}{4} + \dfrac{2}{4}\).
- $$\begin{aligned}[t]\dfrac{1}{4} + \dfrac{2}{4} &= \dfrac{1 + 2}{4} \\ &= \dfrac{3}{4}\end{aligned}$$
\(+\) 
\(=\) 
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 unchanged.
Example
Calculate \(\dfrac{3}{4} + \dfrac{5}{6}\).
- Find a common denominator: To add fractions, they must have the same denominator.
- Multiples of 4: 4, 8, 12, 16, 20, \(\dots\)
- Multiples of 6: 6, 12, 18, 24, \(\dots\)
- The smallest common denominator is 12.
- $$\begin{aligned}[t]\dfrac{3}{4}+\dfrac{5}{6} &= \frac{3 \times \textcolor{olive}{3}}{4 \times \textcolor{olive}{3}} + \frac{5 \times \textcolor{olive}{2}}{6 \times \textcolor{olive}{2}}\\ &= \dfrac{9}{12} + \dfrac{10}{12} \qquad\text{(common denominator } = 12) \\ &= \dfrac{9+10}{12} \qquad\text{(add numerators)} \\ &= \dfrac{19}{12}\end{aligned}$$
- Visual representation:
\(+\)
\(=\) 
\(+\)
\(=\) 
Multiplying a Fraction by a Number
Hugo has a cake. He eats \(\dfrac{1}{4}\) of the cake each day.
How much of the cake will he have eaten after 3 days?
How much of the cake will he have eaten after 3 days?
After 3 days, Hugo will have eaten:
\(3\times\)  | \(=\) | \(+\) \(+\) |
| \(=\) |
Definition Multiplying a Fraction by a Number
To multiply a fraction by a number:
- Multiply the numerator by the number.
- Keep the denominator the same.
Example
Calculate \(3 \times \dfrac{2}{5}\).
- Calculation: $$ \begin{aligned} 3 \times \dfrac{2}{5} &= \dfrac{3 \times 2}{5} \\ &= \dfrac{6}{5} \end{aligned} $$
- Visual representation:
\(3\times\) 
\(=\) \(+\) \(+\) \(=\) 
Multiplication of Fractions
Find the area of the shaded rectangle with side lengths \(\textcolor{colordef}{\dfrac{2}{3}}\) and \(\textcolor{colorprop}{\dfrac{1}{2}}\). 
- The unit rectangle is divided into \(\textcolor{colordef}{3}\) columns and \(\textcolor{colorprop}{2}\) rows, giving a total of \(3 \times 2 = 6\) equal parts.
- The shaded rectangle covers \(\textcolor{colordef}{2}\) columns and \(\textcolor{colorprop}{1}\) row, so it covers \(2 \times 1 = \textcolor{olive}{2}\) parts.
- Therefore, the area of the shaded rectangle is \(\textcolor{olive}{\dfrac{2}{6}}\).
- As the area of a rectangle is the product of its side lengths: $$ \begin{aligned}[t] \textcolor{colordef}{\dfrac{2}{3}} \times \textcolor{colorprop}{\dfrac{1}{2}} &= \dfrac{\textcolor{colordef}{2} \times \textcolor{colorprop}{1}}{\textcolor{colordef}{3} \times \textcolor{colorprop}{2}} \\ &= \textcolor{olive}{\dfrac{2}{6}} \end{aligned} $$
Definition Multiplying Fractions
To multiply two fractions, multiply the numerators together and multiply the denominators together:$$\textcolor{colordef}{\dfrac{a}{b}} \times \textcolor{colorprop}{\dfrac{c}{d}} = \dfrac{\textcolor{colordef}{a} \times \textcolor{colorprop}{c}}{\textcolor{colordef}{b} \times \textcolor{colorprop}{d}}$$
Example
Calculate \( \dfrac{5}{2} \times \dfrac{3}{4}\).
$$\begin{aligned}[t]\dfrac{5}{2} \times \dfrac{3}{4} &= \dfrac{5 \times 3}{2 \times 4} \\
&= \dfrac{15}{8}\end{aligned}$$
Method Canceling Common Factors
To make multiplication easier, you can cancel any common factors in the numerators and denominators before multiplying.
Example
Calculate \(\dfrac{31}{7} \times \dfrac{12}{31}\).
$$\begin{aligned}[t]\dfrac{31}{7} \times \dfrac{12}{31}&= \dfrac{\textcolor{colordef}{\cancel{31}} \times 12}{7 \times \textcolor{colordef}{\cancel{31}}} \quad \text{(cancel the common factor 31)} \\
&= \dfrac{12}{7}\end{aligned}$$
Division of Fractions
Definition Reciprocal
The reciprocal of a number is the number which, when multiplied by the original number, gives \(1\).
Proposition Reciprocal of a fraction
The reciprocal of the fraction \(\dfrac{a}{b}\) is \(\dfrac{b}{a}\).
$$\begin{aligned}[t]\dfrac{a}{b} \times \dfrac{b}{a} &= \dfrac{a \times b}{b \times a} \\
&= \dfrac{1}{1} \\
&= 1.\end{aligned}$$
Example
State the reciprocal of \(\dfrac{5}{7}\).
The reciprocal of \(\dfrac{5}{7}\) is \(\dfrac{7}{5}\).
Definition Division of fractions
To divide by a fraction, multiply by its reciprocal:$$\dfrac{a}{b} \div \dfrac{c}{d} = \dfrac{a}{b} \times \dfrac{d}{c}$$or equivalently,$$\dfrac{\frac{a}{b}}{\frac{c}{d}} = \dfrac{a}{b} \times \dfrac{d}{c}$$
Example
Calculate \(\dfrac{2}{3} \div \dfrac{5}{7}\).
$$\begin{aligned}[t]\dfrac{2}{3} \div \dfrac{5}{7} &= \dfrac{2}{3} \times \dfrac{7}{5} && \text{(multiply by the reciprocal)} \\
&= \dfrac{2 \times 7}{3 \times 5} && \text{(multiply numerators and denominators)} \\
&= \dfrac{14}{15}.\end{aligned}$$
Sign Rules
Recall from the chapter on negative numbers that dividing a positive by a negative, or a negative by a positive, yields a negative result.
Since the fraction bar represents division, consider the fraction:$$\frac{-3}{2}=\overbrace{(-3)}^{\text{negative}} \div \overbrace{2}^{\text{positive}}=\overbrace{-\left(3\div 2\right)}^{\text{negative}}=-\frac{3}{2}\,.$$Similarly,$$\frac{3}{-2}=\overbrace{3}^{\text{positive}} \div \overbrace{(-2)}^{\text{negative}}=\overbrace{-\left(3\div 2\right)}^{\text{negative}}=-\frac{3}{2}\,.$$So, in general, a negative sign in the numerator or denominator gives a negative fraction.
Since the fraction bar represents division, consider the fraction:$$\frac{-3}{2}=\overbrace{(-3)}^{\text{negative}} \div \overbrace{2}^{\text{positive}}=\overbrace{-\left(3\div 2\right)}^{\text{negative}}=-\frac{3}{2}\,.$$Similarly,$$\frac{3}{-2}=\overbrace{3}^{\text{positive}} \div \overbrace{(-2)}^{\text{negative}}=\overbrace{-\left(3\div 2\right)}^{\text{negative}}=-\frac{3}{2}\,.$$So, in general, a negative sign in the numerator or denominator gives a negative fraction.
Proposition Sign Rules
For any numbers \(a\) and \(b\neq 0\),$$\dfrac{-a}{b} = \dfrac{a}{-b} = -\dfrac{a}{b}\,,$$and$$\dfrac{-a}{-b} = \dfrac{a}{b}\,.$$
Example
Simplify \(\dfrac{-4}{-6}\).
$$\begin{aligned}[t]\dfrac{-4}{-6} &= \dfrac{4}{6} \quad &&\text{(a negative divided by a negative is positive)}\\
&= \dfrac{2\times\cancel{\textcolor{olive}{2}}}{3\times\cancel{\textcolor{olive}{2}}} \quad &&\text{(cancel the common factor 2)}\\
&= \dfrac{2}{3}\,.\end{aligned}$$
Fraction as Quotient
Two cakes are shared equally among three people. 
- Use the figure to determine what fraction of the cakes each person receives.
- Copy and complete: \(\ldots\) cakes \(\div\) \(\ldots\) people \(= \dfrac{\ldots}{\ldots}\) of a cake each.
- Each cake is divided into three equal parts. Each person receives one piece from each cake, totaling two pieces. Since each cake is divided into three parts, each piece represents \(\dfrac{1}{3}\) of a cake. Therefore, each person receives:$$\frac{1}{3} + \frac{1}{3} = \frac{2}{3} \text{ of the cakes.}$$
- \(2\) cakes \(\div\) \(3\) people \(= \dfrac{2}{3}\) of a cake each.
Proposition Fraction as Quotient
A fraction is a quotient that represents the result of division. It tells us how much of something we have when we divide it into equal parts. 
The fraction \(\dfrac{2}{3}\) is the number which, when multiplied by \(3\), gives \(2\):$$\dfrac{2}{3}\times 3 = 2 $$
- The top number (numerator) is the whole.
- The bottom number (denominator) is the number of equal parts the whole is divided into.
Order of Operations
Definition Order of operations
The division line in a fraction acts as a grouping symbol, similar to brackets. Therefore, according to the PEMDAS rule, the numerator and denominator should be evaluated before performing the division.
Example
Simplify \(\frac{1+7}{3\times 4}\).
$$\begin{aligned}[t]\frac{1+7}{3\times 4} &= \frac{8}{12} \quad \text{(evaluate numerator and denominator first)} \\
&= \frac{2\times \textcolor{colordef}{\cancel{4}}}{3\times \textcolor{colordef}{\cancel{4}}} \quad \text{(simplify by canceling common factor)} \\
&= \frac{2}{3}\end{aligned}$$