Fractions

Definitions

Finding Fractions

Exercise

A bar represents 1. Find the fraction that represents the shaded part:

Answer

A bar (1) is divided into $\textcolor{colorprop}{4}$ equal parts:
$\textcolor{colordef}{5}$ parts are shaded.
So, $\dfrac{\textcolor{colordef}{5}}{\textcolor{colorprop}{4}}$ =

Exercise

A bar represents 1. Find the fraction that represents the shaded part:

Answer

A bar (1) is divided into $\textcolor{colorprop}{3}$ equal parts:
$\textcolor{colordef}{5}$ parts are shaded.
So, $\dfrac{\textcolor{colordef}{5}}{\textcolor{colorprop}{3}}$ =

Exercise

A bar represents 1. Find the fraction that represents the shaded part:

Answer

A bar (1) is divided into $\textcolor{colorprop}{3}$ equal parts:
$\textcolor{colordef}{8}$ parts are shaded.
So, $\dfrac{\textcolor{colordef}{8}}{\textcolor{colorprop}{3}}$ =

Exercise

A circle represents 1. Find the fraction that represents the shaded part:

Answer

A circle (1) is divided into $\textcolor{colorprop}{4}$ equal parts.
$\textcolor{colordef}{7}$ parts are shaded.
So, $\dfrac{\textcolor{colordef}{7}}{\textcolor{colorprop}{4}}$ =

Exercise

A circle represents 1. Find the fraction that represents the shaded part:

Answer

A circle (1) is divided into $\textcolor{colorprop}{3}$ equal parts.
$\textcolor{colordef}{7}$ parts are shaded.
So, $\dfrac{\textcolor{colordef}{7}}{\textcolor{colorprop}{3}}$ =

Finding Fractions in Word Problems

Exercise

Hugo eats 3 parts of a cake that is divided into 4 equal parts. What fraction of the cake does Hugo eat? of the cake

Answer

Hugo eats 3 of the 4 equal parts.
So, Hugo eats $\dfrac{3}{4}$ of the cake.

Exercise

Liam reads 5 chapters of a book that has 8 chapters. What fraction of the book does Liam read? of the book

Answer

Liam reads 5 out of the 8 chapters.
So, Liam reads $\dfrac{5}{8}$ of the book.

Exercise

Vanessa paints 5 squares on a window that has 6 equal squares. What fraction of the window did she paint? of the window

Answer

Vanessa paints 5 out of the 6 equal parts.
So, Vanessa paints $\dfrac{5}{6}$ of the window.

Exercise

Sophia cuts her loaf of bread into 8 equal slices. She uses 2 slices to make sandwiches. What fraction of the bread did Sophia use to make the sandwiches? of the bread

Answer

Sophia used 2 out of the 8 equal slices of bread.
So, Sophia used $\dfrac{2}{8}$ of the bread to make sandwiches.

Writing Fractions from Words

Exercise

Write as fraction:one over four=

Answer

one over four = $\dfrac{1}{4}=$

Exercise

Write as fraction:three over five =

Answer

three over five = $\dfrac{3}{5}=$

Exercise

Write as fraction:three quarters =

Answer

three quarters = $\dfrac{3}{4}=$

Exercise

Write as fraction:six over hundred =

Answer

six over hundred (six hundredths) = $\dfrac{6}{100}$

On the Number Line

Finding Fractions with the Bar Model

Exercise

Find the value of $\textcolor{colordef}{x}$

$\textcolor{colordef}{x}=$

Answer

1 is divided into 3 equals parts.
$\textcolor{colordef}{x}$ is at 1 part from $0$.
So, $\textcolor{colordef}{x}=\dfrac{1}{3}$.

Exercise

Find the value of $\textcolor{colordef}{x}$

$\textcolor{colordef}{x}=$

Answer

1 is divided into 4 equals parts.
$\textcolor{colordef}{x}$ is at 2 parts from $0$.
So, $\textcolor{colordef}{x}=\dfrac{2}{4}$.

Exercise

Find the value of $\textcolor{colordef}{x}$

$\textcolor{colordef}{x}=$

Answer

1 is divided into 5 equals parts.
$\textcolor{colordef}{x}$ is at 3 parts from $0$.
So, $\textcolor{colordef}{x}=\dfrac{3}{5}$.

Exercise

Find the value of $\textcolor{colordef}{x}$

$\textcolor{colordef}{x}=$

Answer

1 is divided into 6 equals parts.
$\textcolor{colordef}{x}$ is at 4 parts from $0$.
So, $\textcolor{colordef}{x}=\dfrac{4}{6}$.

Exercise

Find the value of $\textcolor{colordef}{x}$

$\textcolor{colordef}{x}=$

Answer

1 is divided into 7 equals parts.
$\textcolor{colordef}{x}$ is at 3 parts from $0$.
So, $\textcolor{colordef}{x}=\dfrac{3}{7}$.

Finding Fractions

Exercise

Find the value of $\textcolor{colordef}{x}$

$\textcolor{colordef}{x}=$

Answer

1 is divided into 3 equals parts.
$\textcolor{colordef}{x}$ is at 1 part from 0.
So, $\textcolor{colordef}{x}=\dfrac{1}{3}$.

Exercise

Find the value of $\textcolor{colordef}{x}$

$\textcolor{colordef}{x}=$

Answer

1 is divided into 4 equals parts.
$\textcolor{colordef}{x}$ is at 3 parts from 0.
So, $\textcolor{colordef}{x}=\dfrac{3}{4}$.

Exercise

Find the value of $\textcolor{colordef}{x}$

$\textcolor{colordef}{x}=$

Answer

1 is divided into 6 equals parts.
$\textcolor{colordef}{x}$ is at 2 parts from 0.
So, $\textcolor{colordef}{x}=\dfrac{2}{6}$.

Exercise

Find the value of $\textcolor{colordef}{x}$

$\textcolor{colordef}{x}=$

Answer

1 is divided into 5 equals parts.
$\textcolor{colordef}{x}$ is at 3 parts from 0.
So, $\textcolor{colordef}{x}=\dfrac{3}{5}$.

Finding Fractions Greater than 1

Exercise

Find the value of $\textcolor{olive}{x}$

$\textcolor{olive}{x}=$

Answer

1 is divided into $\textcolor{colorprop}{3}$ equals parts.
$\textcolor{olive}{x}$ is located at $\textcolor{colordef}{4}$ parts from 0.
So, $\textcolor{olive}{x}=\dfrac{\textcolor{colordef}{4}}{\textcolor{colorprop}{3}}$.

Exercise

Find the value of $\textcolor{olive}{x}$

$\textcolor{olive}{x}=$

Answer

1 is divided into $\textcolor{colorprop}{3}$ equal parts.
$\textcolor{olive}{x}$ is located at $\textcolor{colordef}{7}$ parts from 0.
So, $\textcolor{olive}{x}=\dfrac{\textcolor{colordef}{7}}{\textcolor{colorprop}{3}}$.

Exercise

Find the value of $\textcolor{olive}{x}$

$\textcolor{olive}{x}=$

Answer

1 is divided into $\textcolor{colorprop}{4}$ equal parts.
$\textcolor{olive}{x}$ is located at $\textcolor{colordef}{7}$ parts from 0.
So, $\textcolor{olive}{x}=\dfrac{\textcolor{colordef}{7}}{\textcolor{colorprop}{4}}$.

Exercise

Find the value of $\textcolor{olive}{x}$

$\textcolor{olive}{x}=$

Answer

1 is divided into $\textcolor{colorprop}{6}$ equal parts.
$\textcolor{olive}{x}$ is located at $\textcolor{colordef}{7}$ parts from 0.
So, $\textcolor{olive}{x}=\dfrac{\textcolor{colordef}{7}}{\textcolor{colorprop}{6}}$.

Exercise

Find the value of $\textcolor{olive}{x}$

$\textcolor{olive}{x}=$

Answer

1 is divided into $\textcolor{colorprop}{5}$ equal parts.
$\textcolor{olive}{x}$ is located at $\textcolor{colordef}{8}$ parts from 0.
So, $\textcolor{olive}{x}=\dfrac{\textcolor{colordef}{8}}{\textcolor{colorprop}{5}}$.

Exercise

Find the value of $\textcolor{olive}{x}$

$\textcolor{olive}{x}=$

Answer

1 is divided into $\textcolor{colorprop}{3}$ equal parts.
$\textcolor{olive}{x}$ is located at $\textcolor{colordef}{10}$ parts from 0.
So, $\textcolor{olive}{x}=\dfrac{\textcolor{colordef}{10}}{\textcolor{colorprop}{3}}$.

Equivalent Fractions

Finding the Missing Numerator

Exercise

$\dfrac{2}{4} =$

Answer

= =
The second denominator $2$ is the first denominator $4$ divided by 2 : $4 \times 2 = 2$.
To keep the fractions equivalent, the numerator must also be divided by 2.
This means: $2 \div 2 = 1$, so the missing numerator is 1.

Exercise

$\dfrac{9}{6} =$

Answer

= =
The second denominator $ 2 $ is the first denominator $ 6 $ divided by 3 : $ 6 \div 3 = 2 $.
To keep the fractions equivalent, the numerator must also be divided by 3.
This means: $ 9 \div 3 = 3 $, so the missing numerator is 3.

Exercise

$\dfrac{5}{10} =$

Answer

= =
The second denominator $ 2 $ is the first denominator $ 10 $ divided by 5 : $ 10 \div 5 = 2 $.
To keep the fractions equivalent, the numerator must also be divided by 5.
This means: $ 5 \div 5 = 1 $, so the missing numerator is 1.

Exercise

$\dfrac{16}{12} =$

Answer

= =
The second denominator $ 3 $ is the first denominator $ 12 $ divided by 4 : $ 12 \div 4 = 3 $.
To keep the fractions equivalent, the numerator must also be divided by 4.
This means: $ 16 \div 4 = 4 $, so the missing numerator is 4.

Exercise

$\dfrac{4}{10} =$

Answer

= =
The second denominator $ 5 $ is the first denominator $ 10 $ divided by 2 : $ 10 \div 2 = 5 $.
To keep the fractions equivalent, the numerator must also be divided by 2.
This means: $ 4 \div 2 = 2 $, so the missing numerator is 2.

Finding the Missing Denominator

Exercise

$\dfrac{4}{10} =$

Answer

= =
The second numerator $ 2 $ is the first numerator $ 4 $ divided by 2 : $ 4 \div 2 = 2 $.
To keep the fractions equivalent, the denominator must also be divided by 2.
This means: $ 10 \div 2 = 5 $, so the missing denominator is 5.

Exercise

$\dfrac{6}{12} =$

Answer

= =
The second numerator $ 1 $ is the first numerator $ 6 $ divided by 6 : $ 6 \div 6 = 1 $.
To keep the fractions equivalent, the denominator must also be divided by 6.
This means: $ 12 \div 6 = 2 $, so the missing denominator is 2.

Exercise

$\dfrac{9}{6} =$

Answer

= =
The second numerator $ 3 $ is the first numerator $ 9 $ divided by 3 : $ 9 \div 3 = 3 $.
To keep the fractions equivalent, the denominator must also be divided by 3.
This means: $ 6 \div 3 = 2 $, so the missing denominator is 2.

Exercise

$\dfrac{12}{10} =$

Answer

= =
The second numerator $ 6 $ is the first numerator $ 12 $ divided by 2 : $ 12 \div 2 = 6 $.
To keep the fractions equivalent, the denominator must also be divided by 2.
This means: $ 10 \div 2 = 5 $, so the missing denominator is 5.

Simplification

Simplifying Fractions

Exercise

Simplify: $\dfrac{4}{6}=$

Answer

Exercise

Simplify: $\dfrac{2}{4}=$

Answer

Exercise

Simplify: $\dfrac{10}{8}=$

Answer

Exercise

Simplify: $\dfrac{6}{9}=$

Answer

Ordering Fractions

Comparing with Same Denominator with Bar Models

Exercise

Compare using $\gt,\lt,=$:

$=\dfrac{1}{4}$ $\dfrac{2}{4}=$

Answer

$\gt$ means greater than.
$\lt$ means less than.
$=$ means equal to.
$=\dfrac{1}{4}$ less than $\dfrac{2}{4}=$
$=\dfrac{1}{4}\lt \dfrac{2}{4}=$

Exercise

Compare using $\gt,\lt,=$:

$=\dfrac{3}{5}$ $\dfrac{2}{5}=$

Answer

$\gt$ means greater than.
$\lt$ means less than.
$=$ means equal to.
$=\dfrac{3}{5}$ greater than $\dfrac{2}{5}=$
$=\dfrac{3}{5}\gt \dfrac{2}{5}=$

Exercise

Compare using $\gt,\lt,=$:

$=\dfrac{4}{7}$ $\dfrac{3}{7}=$

Answer

$\gt$ means greater than.
$\lt$ means less than.
$=$ means equal to.
$=\dfrac{4}{7}$ greater than $\dfrac{3}{7}=$
$=\dfrac{4}{7}\gt \dfrac{3}{7}=$

Exercise

Compare using $\gt,\lt,=$:

$=\dfrac{7}{4}$ $\dfrac{5}{4}=$

Answer

$\gt$ means greater than.
$\lt$ means less than.
$=$ means equal to.
$=\dfrac{7}{4}$ greater than $\dfrac{5}{4}=$
$=\dfrac{7}{4}\gt \dfrac{5}{4}=$

Exercise

Compare using $\gt,\lt,=$:

$\dfrac{5}{3}$ $\dfrac{6}{3}=$

Answer

$\gt$ means greater than.
$\lt$ means less than.
$=$ means equal to.
$=\dfrac{5}{3}$ less than $\dfrac{6}{3}=$
$=\dfrac{5}{3}\lt \dfrac{6}{3}=$

Comparing With Same Denominator

Exercise

Compare using $\gt,\lt,=$:$\dfrac{7}{3}$ $\dfrac{6}{3}$

Answer

$\gt$ means greater than.
$\lt$ means less than.
$=$ means equal to.
$=\dfrac{7}{3}$ less than $\dfrac{6}{3}=$
So $\dfrac{7}{3}\gt\dfrac{6}{3}$

Exercise

Compare using $\gt,\lt,=$:$\dfrac{5}{4}$ $\dfrac{3}{4}$

Answer

$\gt$ means greater than.
$\lt$ means less than.
$=$ means equal to.
$=\dfrac{5}{4}$ greater than $\dfrac{3}{4}=$
So $\dfrac{5}{4}\gt\dfrac{3}{4}$

Exercise

Compare using $\gt,\lt,=$:$\dfrac{2}{6}$ $\dfrac{4}{6}$

Answer

$\gt$ means greater than.
$\lt$ means less than.
$=$ means equal to.
$=\dfrac{2}{6}$ less than $\dfrac{4}{6}=$
So $\dfrac{2}{6}\lt\dfrac{4}{6}$

Exercise

Compare using $\gt,\lt,=$:$\dfrac{7}{5}$ $\dfrac{3}{5}$

Answer

$\gt$ means greater than.
$\lt$ means less than.
$=$ means equal to.
$=\dfrac{7}{5}$ greater than $\dfrac{3}{5}=$
So $\dfrac{7}{5}\gt\dfrac{3}{5}$

Exercise

Compare using $\gt,\lt,=$:$\dfrac{3}{8}$ $\dfrac{6}{8}$

Answer

$\gt$ means greater than.
$\lt$ means less than.
$=$ means equal to.
$=\dfrac{3}{8}$ less than $\dfrac{6}{8}=$
So $\dfrac{3}{8}\lt\dfrac{6}{8}$

Comparing Fractions with Different Denominators

Exercise

Compare using $\gt,\lt,=$:$\dfrac{3}{4}$ $\dfrac{1}{2}$Hint: color the bars below to help you compare the fractions.$\dfrac{3}{4}=$

and $\dfrac{1}{2}$=

Answer

Find equivalent fractions with the same denominator:
Compare with same denominator: $=\textcolor{colordef}{\dfrac{3}{4}} \gt \dfrac{\textcolor{colorprop}{2}}{\textcolor{colorprop}{4}}=$
So $=\textcolor{colordef}{\dfrac{3}{4}} \gt \textcolor{colorprop}{\dfrac{1}{2}}=$

Exercise

Compare using $\gt,\lt,=$:$\dfrac{5}{6}$ $\dfrac{2}{3}$Hint: color the bars below to help you compare the fractions.$\dfrac{5}{6}=$

and $\dfrac{2}{3}$=

Answer

Find equivalent fractions with the same denominator:
Compare with same denominator: $=\textcolor{colordef}{\dfrac{5}{6}} \gt \dfrac{\textcolor{colorprop}{4}}{\textcolor{colorprop}{6}}=$
So $=\textcolor{colordef}{\dfrac{5}{6}} \gt \textcolor{colorprop}{\dfrac{2}{3}}=$

Exercise

Compare using $\gt,\lt,=$:$\dfrac{5}{6}$ $\dfrac{1}{2}$Hint: color the bars below to help you compare the fractions.$\dfrac{5}{6}=$

and $\dfrac{1}{2}$=

Answer

Find equivalent fractions with the same denominator:
Compare with same denominator: $=\textcolor{colordef}{\dfrac{5}{6}} \gt \dfrac{\textcolor{colorprop}{3}}{\textcolor{colorprop}{6}}=$
So $=\textcolor{colordef}{\dfrac{5}{6}} \gt \textcolor{colorprop}{\dfrac{1}{2}}=$

Exercise

Compare using $\gt,\lt,=$:$\dfrac{7}{6}$ $\dfrac{4}{3}$Hint: color the bars below to help you compare the fractions.$\dfrac{7}{6}=$

and $\dfrac{4}{3}$=

Answer

Find equivalent fractions with the same denominator:
Compare with same denominator: $=\textcolor{colordef}{\dfrac{7}{6}} \lt \dfrac{\textcolor{colorprop}{8}}{\textcolor{colorprop}{6}}=$
So $=\textcolor{colordef}{\dfrac{7}{6}} \lt \textcolor{colorprop}{\dfrac{4}{3}}=$

Exercise

Compare using $\gt,\lt,=$:$\dfrac{3}{4}$ $\dfrac{7}{8}$Hint: color the bars below to help you compare the fractions.$\dfrac{3}{4}=$

and $\dfrac{7}{8}$=

Answer

Find equivalent fractions with the same denominator:
Compare with same denominator: $=\textcolor{colordef}{\dfrac{6}{8}} \lt \dfrac{\textcolor{colorprop}{7}}{\textcolor{colorprop}{8}}=$
So $=\textcolor{colordef}{\dfrac{3}{4}} \lt \textcolor{colorprop}{\dfrac{7}{8}}=$

Comparing Fractions to Real-World Problems

Mcq

Hugo spends $\frac{3}{8}$ of his money on Pokemon cards and $\frac{1}{4}$ of his money to buy a tennis racket. On which does he spend more money?

Pokemon cards
Tennis racquet

Answer

Convert to a common denominator:
Since $\dfrac{2}{8} \lt \dfrac{3}{8}$, $\dfrac{1}{4}\lt\dfrac{3}{8}$
So, Hugo spends more money on Pokemon cards than on a tennis racquet.

Mcq

Sophie spends $\frac{1}{2}$ of her money on clothes and $\frac{3}{8}$ of her money on books. On which does she spend more money?

Clothes
Books

Answer

Convert to a common denominator:
Since $\dfrac{4}{8} \gt \dfrac{3}{8}$, $\dfrac{1}{2}\gt\dfrac{3}{8}$
So, Sophie spends more money on clothes than on books.

Mcq

For her cake recipe, Sarah uses $\frac{2}{5}$ of a cup of butter and $\frac{3}{10}$ of a cup of sugar. Which ingredient does she use more of?

Butter
Sugar

Answer

Convert to a common denominator:
Since $\dfrac{4}{10} \gt \dfrac{3}{10}$, Sarah uses more butter than sugar.

Mcq

In Class A, $\frac{6}{10}$ of the students are girls, and in Class B, $\frac{13}{20}$ of the students are girls. In which class is the proportion of girls higher?

Class A
Class B

Answer

Convert to a common denominator:
Since $\dfrac{12}{20} \lt \dfrac{13}{20}$, the proportion of girls is higher in Class B.

Addition and Subtraction with Common Denominators

Adding Fractions with Common Denominators

Exercise

$+$

$=$
$\dfrac{1}{4} + \dfrac{2}{4} =$

Answer

$+$ $=$
$\begin{aligned}[t]\dfrac{1}{4} + \dfrac{2}{4} &= \dfrac{1 + 2}{4} \\&= \dfrac{3}{4}\end{aligned}$

Exercise

$+$

$=$
$\dfrac{3}{5} + \dfrac{1}{5} =$

Answer

$+$ $=$
$\begin{aligned}[t]\dfrac{3}{5} + \dfrac{1}{5} &= \dfrac{3 + 1}{5} \\&= \dfrac{4}{5}\end{aligned}$

Exercise

$+$

$=$
$\dfrac{2}{6} + \dfrac{3}{6} =$

Answer

$+$ $=$
$\begin{aligned}[t]\dfrac{2}{6} + \dfrac{3}{6} &= \dfrac{2 + 3}{6} \\&= \dfrac{5}{6}\end{aligned}$

Exercise

$+$

$=$
$\dfrac{2}{3} + \dfrac{2}{3} =$

Answer

$+$ $=$
$\begin{aligned}[t]\dfrac{2}{3} + \dfrac{2}{3} &= \dfrac{2 + 2}{3} \\&= \dfrac{4}{3}\end{aligned}$

Exercise

$+$

$=$
$\dfrac{4}{5} + \dfrac{2}{5} =$

Answer

$+$ $=$
$\begin{aligned}[t]\dfrac{4}{5} + \dfrac{2}{5} &= \dfrac{4 + 2}{5} \\&= \dfrac{6}{5}\end{aligned}$

Adding Fractions with Common Denominators

Exercise

$\dfrac{1}{4} + \dfrac{2}{4} =$

Answer

$+$ $=$
$\begin{aligned}[t]\dfrac{1}{4} + \dfrac{2}{4} &= \dfrac{1 + 2}{4} \\&= \dfrac{3}{4}\end{aligned}$

Exercise

$\dfrac{3}{5} + \dfrac{1}{5} =$

Answer

$+$ $=$
$\begin{aligned}[t]\dfrac{3}{5} + \dfrac{1}{5} &= \dfrac{3 + 1}{5} \\&= \dfrac{4}{5}\end{aligned}$

Exercise

$\dfrac{2}{6} + \dfrac{3}{6} =$

Answer

$+$ $=$
$\begin{aligned}[t]\dfrac{2}{6} + \dfrac{3}{6} &= \dfrac{2 + 3}{6} \\&= \dfrac{5}{6}\end{aligned}$

Exercise

$\dfrac{2}{3} + \dfrac{2}{3} =$

Answer

$+$ $=$
$\begin{aligned}[t]\dfrac{2}{3} + \dfrac{2}{3} &= \dfrac{2 + 2}{3} \\&= \dfrac{4}{3}\end{aligned}$

Exercise

$\dfrac{4}{5} + \dfrac{2}{5} =$

Answer

$+$ $=$
$\begin{aligned}[t]\dfrac{4}{5} + \dfrac{2}{5} &= \dfrac{4 + 2}{5} \\&= \dfrac{6}{5}\end{aligned}$

Subtracting Fractions with Common Denominators

Exercise

$-$

$=$
$\dfrac{3}{4} - \dfrac{2}{4} =$

Answer

$-$ $=$
$\begin{aligned}[t]\dfrac{3}{4} - \dfrac{2}{4} &= \dfrac{3 - 2}{4} \\&= \dfrac{1}{4}\end{aligned}$

Exercise

$-$

$=$
$\dfrac{4}{5} - \dfrac{3}{5} =$

Answer

$-$ $=$
$\begin{aligned}[t]\dfrac{4}{5} - \dfrac{3}{5} &= \dfrac{4 - 3}{5} \\&= \dfrac{1}{5}\end{aligned}$

Exercise

$-$

$=$
$\dfrac{3}{4} - \dfrac{1}{4} =$

Answer

$-$ $=$
$\begin{aligned}[t]\dfrac{3}{4} - \dfrac{1}{4} &= \dfrac{3 -1}{4} \\&= \dfrac{2}{4}\end{aligned}$

Exercise

$-$

$=$
$\dfrac{4}{3} - \dfrac{2}{3} =$

Answer

$-$ $=$
$\begin{aligned}[t]\dfrac{4}{3} - \dfrac{2}{3} &= \dfrac{4 - 2}{3} \\&= \dfrac{2}{3}\end{aligned}$

Exercise

$-$

$=$
$\dfrac{7}{6} - \dfrac{2}{6} =$

Answer

$-$ $=$
$\begin{aligned}[t]\dfrac{7}{6} - \dfrac{2}{6} &= \dfrac{7 - 2}{6} \\&= \dfrac{5}{6}\end{aligned}$

Subtracting Fractions with Common Denominators

Exercise

$\dfrac{3}{4} - \dfrac{2}{4} =$

Answer

$-$ $=$
$\begin{aligned}[t]\dfrac{3}{4} - \dfrac{2}{4} &= \dfrac{3 - 2}{4} \\&= \dfrac{1}{4}\end{aligned}$

Exercise

$\dfrac{4}{5} - \dfrac{3}{5} =$

Answer

$-$ $=$
$\begin{aligned}[t]\dfrac{4}{5} - \dfrac{3}{5} &= \dfrac{4 - 3}{5} \\&= \dfrac{1}{5}\end{aligned}$

Exercise

$\dfrac{3}{4} - \dfrac{1}{4} =$

Answer

$-$ $=$
$\begin{aligned}[t]\dfrac{3}{4} - \dfrac{1}{4} &= \dfrac{3 -1}{4} \\&= \dfrac{2}{4}\end{aligned}$

Exercise

$\dfrac{4}{3} - \dfrac{2}{3} =$

Answer

$-$ $=$
$\begin{aligned}[t]\dfrac{4}{3} - \dfrac{2}{3} &= \dfrac{4 - 2}{3} \\&= \dfrac{2}{3}\end{aligned}$

Exercise

$\dfrac{7}{6} - \dfrac{2}{6} =$

Answer

$-$ $=$
$\begin{aligned}[t]\dfrac{7}{6} - \dfrac{2}{6} &= \dfrac{7 - 2}{6} \\&= \dfrac{5}{6}\end{aligned}$

Addition and Subtraction with Unlike Denominators

Finding a Common Denominator

Exercise

$\dfrac{1}{2}+\dfrac{1}{4}=$ $+\dfrac{1}{4}$

Answer

Since $ \dfrac{1}{2}$ and $ \dfrac{1}{4} $ have different denominators, rewrite $ \dfrac{1}{2} $ with the denominator $ 4 $: This ensures the fractions have the same denominator.
$$\begin{aligned}[t]\dfrac{1}{2}+\dfrac{1}{4} &= \dfrac{2}{4} + \dfrac{1}{4}\\ \end{aligned}$$
$+$ $=$ $+$

Exercise

$\dfrac{2}{3}+\dfrac{5}{6}=$ $+\dfrac{5}{6}$

Answer

Since $ \dfrac{2}{3} $ and $ \dfrac{5}{6} $ have different denominators, rewrite $ \dfrac{2}{3} $ with the denominator $ 6 $: This ensures the fractions have the same denominator.
$$\begin{aligned}[t]\dfrac{2}{3}+\dfrac{5}{6} &= \dfrac{4}{6} + \dfrac{5}{6}\\ \end{aligned}$$
$+$ $=$ $+$

Exercise

$\dfrac{3}{8}+\dfrac{1}{2}=\dfrac{3}{8}+$

Answer

Since $ \dfrac{3}{8} $ and $ \dfrac{1}{2} $ have different denominators, rewrite $ \dfrac{1}{2} $ with the denominator $ 8 $: This ensures the fractions have the same denominator.
$$\begin{aligned}[t]\dfrac{3}{8}+\dfrac{1}{2} &= \dfrac{3}{8} + \dfrac{4}{8}\\ \end{aligned}$$
$+$ $=$ $+$

Exercise

$\dfrac{5}{6}+\dfrac{3}{2}=\dfrac{5}{6}+$

Answer

Since $ \dfrac{5}{6} $ and $ \dfrac{3}{2} $ have different denominators, rewrite $ \dfrac{3}{2} $ with the denominator $ 6 $: This ensures the fractions have the same denominator.
$$\begin{aligned}[t]\dfrac{5}{6}+\dfrac{3}{2} &= \dfrac{5}{6} + \dfrac{9}{6}\\ \end{aligned}$$
$+$ $=$ $+$

Exercise

$\dfrac{3}{10}+\dfrac{6}{5}=\dfrac{3}{10}+$

Answer

Since $ \dfrac{3}{10} $ and $ \dfrac{6}{5} $ have different denominators, rewrite $ \dfrac{6}{5} $ with the denominator $ 10 $: This ensures the fractions have the same denominator.
$$\begin{aligned}[t]\dfrac{3}{10}+\dfrac{6}{5} &= \dfrac{3}{10} + \dfrac{12}{10}\\ \end{aligned}$$
$+$ $=$ $+$

Adding Fractions Step by Step

Exercise

$\dfrac{1}{2}+\dfrac{1}{4}$	$=$	$+$
	$=$

Answer

Since $ \dfrac{1}{2}$ and $ \dfrac{1}{4} $ have different denominators, rewrite $ \dfrac{1}{2} $ with the denominator $ 4 $: This ensures the fractions have the same denominator.
$$\begin{aligned}[t]\dfrac{1}{2}+\dfrac{1}{4} &= \dfrac{2}{4} + \dfrac{1}{4}\\ &= \dfrac{3}{4}\end{aligned}$$
$+$ $=$ $+$

$=$

Exercise

$\dfrac{1}{3}+\dfrac{5}{6}$	$=$	$+$
	$=$

Answer

Since $ \dfrac{1}{3}$ and $ \dfrac{5}{6} $ have different denominators, rewrite $ \dfrac{1}{3} $ with the denominator $ 6 $:
$$\begin{aligned}[t]\dfrac{1}{3}+\dfrac{5}{6} &= \dfrac{2}{6} + \dfrac{5}{6}\\ &= \dfrac{7}{6}\\ \end{aligned}$$
$+$ $=$ $+$

$=$

Exercise

$\dfrac{1}{2}+\dfrac{2}{6}$	$=$	$+$
	$=$

Answer

Since $ \dfrac{1}{2}$ and $ \dfrac{2}{6} $ have different denominators, rewrite $ \dfrac{1}{2} $ with the denominator $ 6 $:
$$\begin{aligned}[t]\dfrac{1}{2}+\dfrac{2}{6} &= \dfrac{3}{6} + \dfrac{2}{6}\\ &= \dfrac{5}{6}\end{aligned}$$
$+$ $=$ $+$

$=$

Exercise

$\dfrac{3}{10}+\dfrac{2}{5}$	$=$	$+$
	$=$

Answer

Since $ \dfrac{3}{10}$ and $ \dfrac{2}{5} $ have different denominators, rewrite $ \dfrac{2}{5} $ with the denominator $ 10 $: This ensures the fractions have the same denominator.
$$\begin{aligned}[t]\dfrac{3}{10}+\dfrac{2}{5} &= \dfrac{3}{10} + \dfrac{4}{10}\\ &= \dfrac{7}{10}\end{aligned}$$
$+$ $=$ $+$

$=$

Exercise

$\dfrac{3}{8}+\dfrac{1}{2}$	$=$	$+$
	$=$

Answer

Since $ \dfrac{3}{8}$ and $ \dfrac{1}{2} $ have different denominators, rewrite $ \dfrac{1}{2} $ with the denominator $ 8 $: This ensures the fractions have the same denominator.
$$\begin{aligned}[t]\dfrac{3}{8}+\dfrac{1}{2} &= \dfrac{3}{8} + \dfrac{4}{8}\\ &= \dfrac{7}{8}\end{aligned}$$
$+$ $=$ $+$

$=$

Adding Fractions

Exercise

$\dfrac{2}{5}+\dfrac{3}{10}$	$=$	$+$
	$=$

Answer

Since $ \dfrac{2}{5}$ and $ \dfrac{3}{10} $ have different denominators, rewrite $ \dfrac{2}{5} $ with the denominator $ 10 $: This ensures the fractions have the same denominator.
$$\begin{aligned}[t]\dfrac{2}{5}+\dfrac{3}{10} &= \dfrac{4}{10} + \dfrac{3}{10}\\ &= \dfrac{7}{10}\end{aligned}$$
$+$ $=$ $+$

$=$

Exercise

$\dfrac{3}{4}+\dfrac{1}{8}$	$=$	$+$
	$=$

Answer

Since $ \dfrac{3}{4}$ and $ \dfrac{1}{8} $ have different denominators, rewrite $ \dfrac{3}{4} $ with the denominator $ 8 $: This ensures the fractions have the same denominator.
$$\begin{aligned}[t]\dfrac{3}{4}+\dfrac{1}{8} &= \dfrac{6}{8} + \dfrac{1}{8}\\ &= \dfrac{7}{8}\end{aligned}$$
$+$ $=$ $+$

$=$

Exercise

$\dfrac{1}{2}+\dfrac{2}{6}$	$=$	$+$
	$=$

Answer

Since $ \dfrac{1}{2}$ and $ \dfrac{2}{6} $ have different denominators, rewrite $ \dfrac{1}{2} $ with the denominator $ 6 $:
$$\begin{aligned}[t]\dfrac{1}{2}+\dfrac{2}{6} &= \dfrac{3}{6} + \dfrac{2}{6}\\ &= \dfrac{5}{6}\end{aligned}$$
$+$ $=$ $+$

$=$

Exercise

$\dfrac{3}{10}+\dfrac{2}{5}$	$=$	$+$
	$=$

Answer

Since $ \dfrac{3}{10}$ and $ \dfrac{2}{5} $ have different denominators, rewrite $ \dfrac{2}{5} $ with the denominator $ 10 $: This ensures the fractions have the same denominator.
$$\begin{aligned}[t]\dfrac{3}{10}+\dfrac{2}{5} &= \dfrac{3}{10} + \dfrac{4}{10}\\ &= \dfrac{7}{10}\end{aligned}$$
$+$ $=$ $+$

$=$

Exercise

$\dfrac{3}{8}+\dfrac{1}{2}$	$=$	$+$
	$=$

Answer

Since $ \dfrac{3}{8}$ and $ \dfrac{1}{2} $ have different denominators, rewrite $ \dfrac{1}{2} $ with the denominator $ 8 $: This ensures the fractions have the same denominator.
$$\begin{aligned}[t]\dfrac{3}{8}+\dfrac{1}{2} &= \dfrac{3}{8} + \dfrac{4}{8}\\ &= \dfrac{7}{8}\end{aligned}$$
$+$ $=$ $+$

$=$

Adding Fractions

Exercise

$\dfrac{2}{5} + \dfrac{3}{10} =$

Answer

Since $ \dfrac{2}{5}$ and $ \dfrac{3}{10} $ have different denominators, rewrite $ \dfrac{2}{5} $ with the denominator $ 10 $: This ensures the fractions have the same denominator.
$$\begin{aligned}[t]\dfrac{2}{5}+\dfrac{3}{10} &= \dfrac{4}{10} + \dfrac{3}{10}\\ &= \dfrac{7}{10}\end{aligned}$$
$+$ $=$ $+$

$=$

Exercise

$\dfrac{1}{4} + \dfrac{3}{8} =$

Answer

Since $ \dfrac{1}{4}$ and $ \dfrac{3}{8} $ have different denominators, rewrite $ \dfrac{1}{4} $ with the denominator $ 8 $: This ensures the fractions have the same denominator.
$$\begin{aligned}[t]\dfrac{1}{4}+\dfrac{3}{8} &= \dfrac{2}{8} + \dfrac{3}{8}\\ &= \dfrac{5}{8}\end{aligned}$$
$+$ $=$ $+$

$=$

Exercise

$\dfrac{2}{3} + \dfrac{1}{6} =$

Answer

Since $ \dfrac{2}{3}$ and $ \dfrac{1}{6} $ have different denominators, rewrite $ \dfrac{2}{3} $ with the denominator $ 6 $: This ensures the fractions have the same denominator.
$$\begin{aligned}[t]\dfrac{2}{3}+\dfrac{1}{6} &= \dfrac{4}{6} + \dfrac{1}{6}\\ &= \dfrac{5}{6}\end{aligned}$$
$+$ $=$ $+$

$=$

Exercise

$\dfrac{3}{5} + \dfrac{2}{15} =$

Answer

Since $ \dfrac{3}{5}$ and $ \dfrac{2}{15} $ have different denominators, rewrite $ \dfrac{3}{5} $ with the denominator $ 15 $: This ensures the fractions have the same denominator.
$$\begin{aligned}[t]\dfrac{3}{5}+\dfrac{2}{15} &= \dfrac{9}{15} + \dfrac{2}{15}\\ &= \dfrac{11}{15}\end{aligned}$$
$+$ $=$ $+$

$=$

Exercise

$\dfrac{3}{10}+\dfrac{2}{5}=$

Answer

Since $ \dfrac{3}{10}$ and $ \dfrac{2}{5} $ have different denominators, rewrite $ \dfrac{2}{5} $ with the denominator $ 10 $: This ensures the fractions have the same denominator.
$$\begin{aligned}[t]\dfrac{3}{10}+\dfrac{2}{5} &= \dfrac{3}{10} + \dfrac{4}{10}\\ &= \dfrac{7}{10}\end{aligned}$$
$+$ $=$ $+$

$=$

Exercise

$\dfrac{3}{8}+\dfrac{1}{2}=$

Answer

Since $ \dfrac{3}{8}$ and $ \dfrac{1}{2} $ have different denominators, rewrite $ \dfrac{1}{2} $ with the denominator $ 8 $: This ensures the fractions have the same denominator.
$$\begin{aligned}[t]\dfrac{3}{8}+\dfrac{1}{2} &= \dfrac{3}{8} + \dfrac{4}{8}\\ &= \dfrac{7}{8}\end{aligned}$$
$+$ $=$ $+$

$=$

Subtracting Fractions

Exercise

$\dfrac{2}{5}-\dfrac{3}{10}$	$=$	$-$
	$=$

Answer

Since $ \dfrac{2}{5}$ and $ \dfrac{3}{10} $ have different denominators, rewrite $ \dfrac{2}{5} $ with the denominator $ 10 $: This ensures the fractions have the same denominator.
$$\begin{aligned}[t]\dfrac{2}{5}-\dfrac{3}{10} &= \dfrac{4}{10} - \dfrac{3}{10}\\ &= \dfrac{4-3}{10} \\ &= \dfrac{1}{10}\end{aligned}$$
$-$ $=$ $-$

$=$

Exercise

$\dfrac{7}{6}-\dfrac{1}{3}$	$=$	$-$
	$=$

Answer

Since $ \dfrac{7}{6} $ and $ \dfrac{1}{3} $ have different denominators, rewrite $ \dfrac{1}{3} $ with the denominator $ 6 $: This ensures the fractions have the same denominator.
$$\begin{aligned}[t]\dfrac{7}{6}-\dfrac{1}{3} &= \dfrac{7}{6} - \dfrac{2}{6}\\ &= \dfrac{7-2}{6} \\ &= \dfrac{5}{6}\end{aligned}$$
$-$ $=$ $-$

$=$

Exercise

$\dfrac{7}{8}-\dfrac{3}{4}$	$=$	$-$
	$=$

Answer

Since $ \dfrac{7}{8} $ and $ \dfrac{3}{4} $ have different denominators, rewrite $ \dfrac{3}{4} $ with the denominator $ 8 $: This ensures the fractions have the same denominator.
$$\begin{aligned}[t]\dfrac{7}{8}-\dfrac{3}{4} &= \dfrac{7}{8} - \dfrac{6}{8}\\ &= \dfrac{7-6}{8} \\ &= \dfrac{1}{8}\end{aligned}$$
$-$ $=$ $-$

$=$

Exercise

$\dfrac{5}{3} - \dfrac{5}{9}$	$=$	$-$
	$=$

Answer

Since $ \dfrac{5}{3} $ and $ \dfrac{5}{9} $ have different denominators, rewrite $ \dfrac{5}{3} $ with the denominator $ 9 $: This ensures the fractions have the same denominator.
$$\begin{aligned}[t]\dfrac{5}{3} - \dfrac{5}{9} &= \dfrac{15}{9} - \dfrac{5}{9}\\ &= \dfrac{15 - 5}{9} \\ &= \dfrac{10}{9}\end{aligned}$$
$-$ $=$ $-$

$=$

Exercise

$\dfrac{7}{2} - \dfrac{7}{4}$	$=$	$-$
	$=$

Answer

Since $ \dfrac{7}{2} $ and $ \dfrac{7}{4} $ have different denominators, rewrite $ \dfrac{7}{2} $ with the denominator $ 4 $: This ensures the fractions have the same denominator.
$$\begin{aligned}[t]\dfrac{7}{2} - \dfrac{7}{4} &= \dfrac{14}{4} - \dfrac{7}{4}\\ &= \dfrac{14 - 7}{4} \\ &= \dfrac{7}{4}\end{aligned}$$
$-$ $=$ $-$

$=$

Solving Real-World Problems

Exercise

Louis has a whole cake. He cuts it into 8 equal slices and eats 3 slices. What fraction of the whole cake remains? of the cake

Answer

Represent the cake as a fraction
The whole cake is divided into 8 slices, so the whole cake is $ \dfrac{8}{8} $.
Subtract the slices eaten by Louis
Louis eats 3 slices, which is $ \dfrac{3}{8} $ of the cake. Remaining cake after Louis eats: $$ \dfrac{8}{8} - \dfrac{3}{8} = \dfrac{5}{8}. $$ $-$ $=$
Final Answer:
The fraction of the cake that remains is $ \dfrac{5}{8} $.

Exercise

Today, Louis eats $ \dfrac{1}{2} $ of a croissant. Then, Louis eats $ \dfrac{1}{4} $ of another croissant. How much croissant did Louis eat in total? of a croissant

Answer

Represent the croissants as fractions
Louis eats $ \dfrac{1}{2} $ of the first croissant and $ \dfrac{1}{4} $ of the second croissant. To find the total, add the two fractions: $$ \dfrac{1}{2} + \dfrac{1}{4}. $$
Find a common denominator
The denominators are 2 and 4. The least common denominator is 4. Convert $ \dfrac{1}{2} $ to a fraction with denominator 4:
Add the fractions
$$\begin{aligned}[t]\dfrac{1}{2}+\dfrac{1}{4} &= \dfrac{2}{4} + \dfrac{1}{4}\\ &= \dfrac{3}{4}\end{aligned}$$

$+$ $=$ $+$

$=$
Final Answer:
Louis ate a total of $ \dfrac{3}{4} $ of a croissant.

Exercise

At the beginning, there are $ \dfrac{5}{6} $ of a cake. After eating, there are $ \dfrac{2}{3} $ of the cake. What quantity of cake did Louis eat? of the cake

Answer

Represent the cake as fractions
At the beginning, there is $ \dfrac{5}{6} $ of the cake. After eating, $ \dfrac{2}{3} $ of the cake remains. To find the quantity Louis ate, subtract the remaining cake from the initial amount: $$ \dfrac{5}{6} - \dfrac{2}{3}. $$
Find a common denominator
The denominators are 6 and 3. The least common denominator is 6. Convert $ \dfrac{2}{3} $ to a fraction with denominator 6:
Subtract the fractions
$$\begin{aligned}[t]\dfrac{5}{6} - \dfrac{2}{3} &= \dfrac{5}{6} - \dfrac{4}{6}\\ &= \dfrac{1}{6}\end{aligned}$$

$-$ $=$ $-$

$=$
Final Answer:
Louis ate $ \dfrac{1}{6} $ of the cake.

Exercise

At the beginning, there are $ \dfrac{7}{8} $ of a pizza. After eating, there are $ \dfrac{3}{4} $ of the pizza. What quantity of pizza did Louis eat? of the pizza

Answer

Represent the pizza as fractions
At the beginning, there is $ \dfrac{7}{8} $ of the pizza. After eating, $ \dfrac{3}{4} $ of the pizza remains. To find the quantity Louis ate, subtract the remaining pizza from the initial amount: $$ \dfrac{7}{8} - \dfrac{3}{4}. $$
Find a common denominator
The denominators are 8 and 4. The least common denominator is 8. Convert $ \dfrac{3}{4} $ to a fraction with denominator 8:
Subtract the fractions
$$\begin{aligned}[t]\dfrac{7}{8} - \dfrac{3}{4} &= \dfrac{7}{8} - \dfrac{6}{8}\\ &= \dfrac{1}{8}\end{aligned}$$

$-$ $=$ $-$

$=$
Final Answer:
Louis ate $ \dfrac{1}{8} $ of the pizza.

Exercise

Louis read $ \dfrac{2}{5} $ of his book on Saturday and $ \dfrac{3}{10} $ of his book on Sunday. How much of his book did Louis read in total? of the book

Answer

Represent the book as fractions
Louis read $ \dfrac{2}{5} $ of the book on Saturday and $ \dfrac{3}{10} $ of the book on Sunday. To find the total, add the two fractions: $$ \dfrac{2}{5} + \dfrac{3}{10}. $$
Find a common denominator
The denominators are 5 and 10. The least common denominator is 10. Convert $ \dfrac{2}{5} $ to a fraction with denominator 10:
Add the fractions
$$\begin{aligned}[t]\dfrac{2}{5} + \dfrac{3}{10} &= \dfrac{4}{10} + \dfrac{3}{10}\\ &= \dfrac{7}{10}\end{aligned}$$

$+$ $=$ $+$

$=$
Final Answer:
Louis read a total of $ \dfrac{7}{10} $ of his book.

\(\dfrac{1}{2}+\dfrac{1}{4}\)	\(=\)	\(+\)
	\(=\)

\(+\)	\(=\)	\(+\)
	\(=\)

\(\dfrac{1}{3}+\dfrac{5}{6}\)	\(=\)	\(+\)
	\(=\)

\(+\)	\(=\)	\(+\)
	\(=\)

\(\dfrac{1}{2}+\dfrac{2}{6}\)	\(=\)	\(+\)
	\(=\)

\(\dfrac{2}{5}-\dfrac{3}{10}\)	\(=\)	\(-\)
	\(=\)

\(-\)	\(=\)	\(-\)
	\(=\)

\(\dfrac{7}{6}-\dfrac{1}{3}\)	\(=\)	\(-\)
	\(=\)

\(-\)	\(=\)	\(-\)
	\(=\)