International Baccalaureate · MYP 1

Fractions

⌚ ~194 min ▢ 22 blocks ✓ 89 exercises ➣ Prerequisites : Number Properties

I Fractions and Division

I.1 What a Fraction Is

Discover

Consider the model below, which illustrates one whole unit and one half:

. To express this length as a single fraction, we follow two steps:

Identify the denominator: The units are divided into halves, so the denominator is 2.
Count the parts: In total, there are three shaded parts of this size. This count becomes the numerator.

Therefore, this length is represented by the fraction $\dfrac{3}{2}$.

Definition — Fraction

A fraction has two numbers separated by a fraction bar: the numerator (on top) and the denominator (on the bottom).

The numerator and the denominator are whole numbers, and the denominator is never zero: a unit cannot be divided into zero equal parts.

Method — Decomposing a Fraction into a Product

A fraction counts parts that are all the same size. So $\dfrac{4}{3}$ is four parts of one third, and counting four equal parts is a multiplication: $$\dfrac{4}{3} = \dfrac{1}{3} + \dfrac{1}{3} + \dfrac{1}{3} + \dfrac{1}{3} = 4 \times \dfrac{1}{3}$$ The rule is read straight off the writing: the numerator says how many, the denominator says which part. $$\dfrac{n}{b} = n \times \dfrac{1}{b}$$

Example

Write $\dfrac{5}{8}$ as a product.

Answer

$$\begin{aligned}[t] \dfrac{5}{8} &= \dfrac{1}{8} + \dfrac{1}{8} + \dfrac{1}{8} + \dfrac{1}{8} + \dfrac{1}{8} && \text{(five parts of one eighth)} \\ &= 5 \times \dfrac{1}{8} && \text{(counting five equal parts is a multiplication)} \end{aligned}$$

Skills to practice

Decomposing a Fraction into a Product

I.2 Fractions as Division

Discover

Consider a scenario where two identical units (cakes) are to be shared equally among three individuals.

This scenario represents the division problem $2 \div 3$. How can the result of this division be expressed as a fraction?

Answer

To solve this, each unit is divided into three equal parts (thirds). Each of the three individuals receives one part from each of the two units.

Each individual's total share consists of two pieces, where each piece is $\dfrac{1}{3}$ of a unit. Therefore, each person receives a total of $\dfrac{2}{3}$ of a unit. This demonstrates that the division $2 \div 3$ is equal to the fraction $\dfrac{2}{3}$.

Proposition — The Fraction as a Quotient

For any whole numbers $a$ and $b$ (where $b \neq 0$), the division of $a$ by $b$ is represented by the fraction $\dfrac{a}{b}$.$$a \div b = \frac{a}{b}$$ In this context:

The numerator ($a$) corresponds to the dividend.
The denominator ($b$) corresponds to the divisor.

Consequently, the fraction $\dfrac{a}{b}$ is the number which, when multiplied by the divisor $b$, yields the dividend $a$.

Proof

Share $a$ units among $b$ people. Cut every unit into $b$ equal parts: each person receives one part of each unit, that is $a$ parts of size $\frac{1}{b}$, so $$\begin{aligned}[t] a \div b &= a \times \tfrac{1}{b} && \text{($a$ parts of size $\tfrac{1}{b}$)}\\ &= \tfrac{a}{b}. && \text{(a fraction counts its parts)}\end{aligned}$$ Conversely, $b$ people holding $\frac{a}{b}$ each hold together $b \times a$ parts of size $\frac{1}{b}$; $b$ such parts make one unit, so they hold $a$ units: $\frac{a}{b} \times b = a$.

Example

The fraction $\dfrac{2}{3}$ is the same as saying “2 divided by 3”.

The fraction $\dfrac{2}{3}$ is the number which, when multiplied by $3$, gives $2$:$$\dfrac{2}{3}\times 3 = 2 $$

Skills to practice

Converting Division to Fractions
Finding Fractions in Word Problems

I.3 Fractions Greater Than One

Discover

Fractions can represent values greater than one. Consider the fraction $\dfrac{5}{2}$, which represents 5 half-sized parts of a unit.

While this “improper fraction” is a valid mathematical representation, it is often more intuitive to express such numbers as a combination of whole units and a remaining fractional part.

Definition — Proper and Improper Fractions

Fractions are classified based on the relationship between the numerator and the denominator.

A proper fraction is a fraction where the numerator is less than the denominator. Its value is always less than 1. Example: $\dfrac{2}{3}$.
An improper fraction is a fraction where the numerator is greater than or equal to the denominator. Its value is always greater than or equal to 1. Example: $\dfrac{5}{3}$.

Definition — Mixed Number

A mixed number is an alternative way to represent an improper fraction. It consists of an integer part (the number of whole units) and a proper fraction part.
The improper fraction $\dfrac{5}{2}$ can be visualised as two whole units and one half.

This is written as the mixed number $2\dfrac{1}{2}$. By convention, the addition sign is omitted:$$ 2\frac{1}{2} \quad \text{is equivalent to} \quad 2 + \frac{1}{2} $$ Caution: A mixed number like $2\frac{1}{2}$ always means $2+\frac{1}{2}$, not $2\times\frac{1}{2}$. If you mean multiplication, write $2\times\frac{1}{2}$, $2\cdot\frac{1}{2}$, or $(2)\!\left(\frac{1}{2}\right)$.

Proposition — Whole Number Plus a Fraction

Every fraction greater than $1$ can be written as the sum of a whole number and a fraction less than $1$; conversely, the sum of a non-zero whole number and a non-zero fraction less than $1$ is a fraction greater than $1$:

Such a writing, like $2 + \dfrac{1}{3}$, is called a mixed number.

Method — Decomposing a Fraction Greater Than 1

To write $\dfrac{n}{b}$ as a whole number plus a fraction less than $1$:

Count the whole units: each whole unit is made of $b$ parts, so divide with remainder: $n = q \times b + r$, with a remainder $r$ smaller than $b$.
The $r$ remaining parts form the fraction: $\dfrac{n}{b} = q + \dfrac{r}{b}$, and $\dfrac{r}{b} < 1$ because $r < b$.

Conversely, to recompose a single fraction, each whole unit becomes $b$ parts. On the number line, the decomposition also places the fraction between two consecutive whole numbers: $q < \dfrac{n}{b} < q+1$ (when the remainder is not zero — if $r = 0$, the fraction is exactly the whole number $q$).

Example

Write $\dfrac{7}{3}$ as the sum of a whole number and a fraction less than $1$.

Answer

$\begin{aligned}[t]\dfrac{7}{3}&=\dfrac{2\times 3+1}{3}&&(\text{division of 7 by 3: } 7=2\times 3+1)\\&=\dfrac{2\times 3}{3}+\dfrac{1}{3}\\&=2+\dfrac{1}{3}\\\end{aligned}$

Skills to practice

Decomposing Fractions in Everyday Situations

II Fractions on a Number Line

II.1 Placing Fractions on a Number Line

Discover

Fractions do not only represent parts of a shape; they can also represent points on a number line. The segment from 0 to 1 is one unit.
If we divide the unit into 2 equal parts, the point in the middle represents the fraction $\dfrac{1}{2}$.

Method — Representing a Fraction on the Number Line

To represent the fraction $\dfrac{\textcolor{colordef}{2}}{\textcolor{colorprop}{3}}$ on a number line:

Draw a straight line and mark the points $0$ and $1$.
Divide the segment from $0$ to $1$ into $\textcolor{colorprop}{3}$ equal parts (the denominator).
Count $\textcolor{colordef}{2}$ parts from $0$ (the numerator) and mark the point at $\dfrac{\textcolor{colordef}{2}}{\textcolor{colorprop}{3}}$.

Skills to practice

Placing a Fraction on a Graduated Half-Line

II.2 Graduating a Segment

Method — Dividing a Segment into Equal Parts with a Guide-Line Sheet

To share a segment $[AB]$ into $5$ equal parts without measuring, use a guide-line sheet: a sheet of parallel lines, regularly spaced — the lines of an exercise book will do.

Place the segment across the lines, so that $A$ lies on the first line and $B$ on the sixth: $6$ lines make $5$ gaps. Choose a sheet whose lines are close enough for the segment to reach the sixth line.
Mark the points where the four lines in between cross $[AB]$: they cut $[AB]$ into $5$ equal parts (the lines are equally spaced, so the pieces are equal).
Read the marks: each part is $\dfrac{1}{5}$ of $AB$; the marked points are at $\dfrac{1}{5}$, $\dfrac{2}{5}$, $\dfrac{3}{5}$, $\dfrac{4}{5}$ of $AB$ from $A$.

For halves, quarters or eighths, first cut a paper strip to the length of $[AB]$, then fold it in $2$, in $4$, in $8$: this gives the same marks.

Example

A segment $[AB]$ is $10$ cm long. Divide it into thirds with a guide-line sheet, then place the point $C$ such that $AC = \dfrac{2}{3}\,AB$. Check by computing $AC$.

Answer

Place $A$ on the first line and $B$ on the fourth of the guide-line sheet ($4$ lines, $3$ gaps): the two lines in between cut $[AB]$ into $3$ equal parts. $C$ is the second mark from $A$.

Check: $AC$ is $\dfrac{2}{3}$ of $10$ cm, so $AC = \dfrac{20}{3}$ cm, that is about $6.7$ cm: a length the ruler cannot mark exactly — the guide-line sheet gives it exactly, and the ruler only confirms it approximately.

Skills to practice

Reading a Fraction on a Number Line
Graduating a Segment

III Equal Fractions

III.1 Equal Fractions

Discover

A cake is cut into 3 equal parts. One part, representing $\dfrac{1}{3}$ of the cake, is set aside.

Then each of the three parts is cut in half. Now the cake is divided into 6 equal parts. The same amount of cake as before is now 2 of these parts.

Even though the number of pieces changed, did the shaded amount of cake change?

Answer

No—the amount of cake did not change. The models show that the shaded portion is the same in both cases.

$\quad=\quad$

Therefore, the fractions $\dfrac{1}{3}$ and $\dfrac{2}{6}$ represent the same value.

Definition — Equivalent Fractions

Two fractions are equal (also called equivalent) if they represent the same amount: they are two ways of writing the same number. You can make an equivalent fraction by multiplying the numerator and the denominator by the same nonzero whole number, or by dividing them by a common divisor.

Skills to practice

Finding a Missing Numerator by Simplifying

III.2 Lowest Terms

Discover

Consider the fraction $\dfrac{4}{6}$:

. While this is a valid representation of a quantity, it is not the most efficient. Among all possible equivalent fractions, there is one that uses the smallest possible integers for the numerator and denominator (no whole number greater than $1$ divides both).
The process of finding this simplest form is known as simplifying or reducing a fraction.

Method — Simplifying a fraction

To simplify a fraction is to find an equivalent fraction that uses the smallest possible whole numbers for its numerator and denominator.
The procedure is to divide both the numerator and the denominator by the same number (a common factor). This process may need to be repeated until no whole number greater than $1$ divides both the numerator and the denominator.

Example

Simplify $\dfrac{4}{6}$.

Answer

Skills to practice

Simplifying Fractions as Much as Possible

IV Comparing Fractions

IV.1 Comparing Fractions

Discover

Which is larger: $\dfrac{3}{4}$ or $\dfrac{5}{8}$?

$\;$

Visually, it is difficult to compare them because one is divided into quarters and the other into eighths. One sure way to compare them precisely is to first express them with a common denominator.

Answer

To compare $\dfrac{3}{4}$ and $\dfrac{5}{8}$, write them with a common denominator. $8$ is a multiple of $4$: each quarter is cut into $2$ eighths, so the numerator and the denominator of $\dfrac{3}{4}$ are both multiplied by $2$. $$\begin{aligned}[t] \dfrac{3}{4} &= \dfrac{3 \times 2}{4 \times 2} && \text{(each quarter cut into 2 eighths)}\\ &= \dfrac{6}{8} \end{aligned}$$

$\;=\;$

$\;>\;$

Both fractions now count eighths. With the same denominator, the larger numerator gives the larger fraction: $6 > 5$, so $\dfrac{6}{8} > \dfrac{5}{8}$. Therefore $\dfrac{3}{4} > \dfrac{5}{8}$: $\dfrac{3}{4}$ is the larger.

Proposition — Ordering Fractions with the Same Denominator

For two fractions with the same denominator, the fraction with the larger numerator is larger.

Example

Compare $\dfrac{6}{8}$ and $\dfrac{5}{8}$.

Answer

$= \dfrac{6}{8} > \dfrac{5}{8} = $

Method — Comparing Fractions with Different Denominators

The standard procedure for comparing two fractions with different denominators is as follows:

Find a common denominator: Identify a common multiple of both denominators. A simple method is to multiply the denominators together.
Create equivalent fractions: Convert each fraction into an equivalent fraction with the chosen common denominator.
Compare the numerators: Once the denominators are the same, the fraction with the larger numerator is the greater fraction.

Example

Compare $\dfrac{3}{4}$ and $\dfrac{5}{8}$.

Answer

We will apply the three-step procedure.

1. Find a common denominator: The denominators are 4 and 8. A common multiple is 8.
2. Create equivalent fractions:
- The fraction $\dfrac{5}{8}$ already has the common denominator.
- Convert $\dfrac{3}{4}$ to an equivalent fraction with a denominator of 8. To change the denominator from 4 to 8, we multiply by 2. Therefore, we must also multiply the numerator by 2.
3. Compare the numerators: We now compare the equivalent fractions. $= \dfrac{6}{8} > \dfrac{5}{8} = $
Conclusion: Therefore, it is concluded that $\dfrac{3}{4} > \dfrac{5}{8}$.

Skills to practice

Comparing Fractions with Different Denominators

IV.2 Comparing to 1 and Ordering Fractions

Proposition — Comparing a Fraction to 1

A fraction is compared to $1$ by comparing its numerator to its denominator:

if the numerator is smaller than the denominator, the fraction is smaller than $1$: $\dfrac{13}{21} < 1$;
if they are equal, the fraction equals $1$: $\dfrac{21}{21} = 1$;
if the numerator is larger, the fraction is larger than $1$: $\dfrac{16}{12} > 1$.

The same idea compares a fraction to $\dfrac{1}{2}$: $\dfrac{5}{12} < \dfrac{6}{12} = \dfrac{1}{2}$, so $\dfrac{5}{12} < \dfrac{1}{2}$.

Proposition — Same Numerator

Two fractions with the same non-zero numerator are ordered the other way round from their denominators: the larger the denominator, the smaller the parts, so the smaller the fraction. $$\dfrac{3}{5} > \dfrac{3}{8} \qquad \text{because a fifth is larger than an eighth, and there are three of each.}$$

Example

Compare, without any computation: $\dfrac{7}{8}$ and $\dfrac{10}{9}$; $\dfrac{4}{7}$ and $\dfrac{4}{11}$; $\dfrac{5}{12}$ and $\dfrac{6}{11}$.

Answer

$\dfrac{7}{8} < 1$ (numerator smaller than denominator) and $\dfrac{10}{9} > 1$: so $\dfrac{7}{8} < \dfrac{10}{9}$ — the number $1$ sits between them.
Same numerator $4$: sevenths are larger than elevenths, so $\dfrac{4}{7} > \dfrac{4}{11}$.
Compare both to $\dfrac{1}{2}$: $\dfrac{5}{12} < \dfrac{6}{12} = \dfrac{1}{2}$, and $\dfrac{6}{11} > \dfrac{6}{12} = \dfrac{1}{2}$ (same numerator, smaller denominator). So $\dfrac{5}{12} < \dfrac{1}{2} < \dfrac{6}{11}$.

Method — Ordering Several Fractions

Write all the fractions with the same denominator (a common multiple of the denominators).
Order the numerators: with equal denominators, the larger the numerator, the larger the fraction.
Write the chain with the original fractions, in increasing or decreasing order.

Example

Order from smallest to largest: $\dfrac{2}{3}$, $\dfrac{3}{4}$ and $\dfrac{7}{12}$.

Answer

A common denominator of $3$, $4$ and $12$ is $12$: $$\dfrac{2}{3} = \dfrac{8}{12}, \qquad \dfrac{3}{4} = \dfrac{9}{12}, \qquad \dfrac{7}{12}.$$ Ordering the numerators: $7 < 8 < 9$, so $$\dfrac{7}{12} < \dfrac{2}{3} < \dfrac{3}{4}.$$

Skills to practice

Comparing a Fraction to 1
Comparing Fractions with the Same Numerator
Ordering Fractions and Justifying

V Adding and Subtracting Fractions

V.1 The Same Denominator

Discover

Hugo's cake is cut into $4$ equal parts (quarters). Hugo eats $\dfrac{2}{4}$ of the cake $=$

, then $\dfrac{1}{4}$ more $=$

.
What fraction of the cake has he eaten?

Answer

The parts are all the same size (quarters), so the eaten parts can simply be counted together: $2+1=3$ quarters.

Hugo has eaten $\dfrac{3}{4}$ of the cake.

Definition — Addition of Fractions with Common Denominators

To add fractions with the same denominator, add the numerators and keep the denominator the same:

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

Definition — Subtraction of Fractions with Common Denominators

To subtract fractions with the same denominator, subtract the numerators and keep the denominator the same:

$$\begin{aligned}\dfrac{3}{4} - \dfrac{1}{4} &= \dfrac{3 - 1}{4}\\ &= \dfrac{2}{4}\end{aligned}$$ Here the first fraction is the larger one, so these parts can be taken away.

Skills to practice

Adding or Subtracting a Whole Number and a Fraction

V.2 Related Denominators

Discover

To add or subtract fractions by combining their numerators, the fractions must first represent parts of the same size, i.e., be written with a common denominator.
Consider the problem of adding $\dfrac{1}{2}$ and $\dfrac{1}{4}$.

$+$

Because the fractions have unlike denominators (2 and 4), the parts are of different sizes. A direct addition of the numerators is not possible. To solve this, we must first express the fractions with a common denominator.

Method — Procedure for Adding or Subtracting Fractions

To add or subtract fractions with unlike denominators, follow this three-step procedure:

Find a Common Denominator: Identify a common multiple of the denominators.
Create Equivalent Fractions: Convert each fraction to an equivalent fraction with the common denominator.
Add or Subtract the Numerators: With the denominators now the same, perform the operation on the numerators and keep the common denominator. For a subtraction, the first fraction must not be smaller than the second.

Example

We will apply the procedure to calculate $\dfrac{1}{2} + \dfrac{1}{4}$.

Step 1: Find a common denominator. The denominators are 2 and 4. A common multiple is 4.
Step 2: Create equivalent fractions.
- The fraction $\dfrac{1}{4}$ already has the common denominator.
- Convert $\dfrac{1}{2}$ into an equivalent fraction with a denominator of 4:
Step 3: Add the numerators. Now that both fractions have the same denominator, we can add them: $$ \frac{2}{4} + \frac{1}{4} = \frac{2+1}{4} = \frac{3}{4} $$

The visual representation of this process is:

$+$	$=$	$+$
	$=$
$\dfrac{1}{2}+\dfrac{1}{4}$	$=$	$\dfrac{2}{4}+\dfrac{1}{4}$
	$=$	$\dfrac{3}{4}$

The total fraction is $\dfrac{3}{4}$.

Skills to practice

Adding Fractions
Subtracting Fractions
Solving Real-World Problems

VI Fractions and Whole Numbers

VI.1 A Fraction of a Quantity

Discover

How much is $\dfrac{2}{3}$ of $18$? Split $18$ into $3$ equal parts: one part is $18\div 3 = 6$. Two parts make $2\times 6 = 12$. So $\dfrac{2}{3}$ of $18$ is $12$.
Taking $\dfrac{a}{b}$ of a number always works this way: divide by the denominator, multiply by the numerator — which is exactly the multiplication $\dfrac{a}{b}\times N$.

Method — Taking a Unit Fraction of a Quantity

To take $\dfrac{1}{b}$ of a quantity, split the quantity into $b$ equal parts and take one part: $$\frac{1}{b}\,\text{of }N = N \div b$$

Example

How many marbles is one third of $12$ marbles?
How many dollars is one quarter of $\dollar 20$?

Answer

One third of $12$: split $12$ into $3$ equal parts and take one part: $12 \div 3 = 4$ marbles.
One quarter of $20$: split $20$ into $4$ equal parts and take one part: $20 \div 4 = 5$ dollars.

Skills to practice

Calculating Fractions of a Whole

VI.2 Multiplying a Fraction by a Whole Number

Proposition — Multiplying a Fraction by a Number

So far a fraction was taken of a quantity; the same rule applies to any whole number $N$. For a whole number $N$ and a fraction $\dfrac{a}{b}$ (with $b \neq 0$): $$\frac{a}{b} \times N = \frac{a \times N}{b}.$$ Only the numerator is multiplied: the denominator says into how many equal parts the unit is cut, and multiplying does not change that cutting. Two readings, one result:

divide first: $N \div b$ gives one part, then $\times\, a$ takes $a$ of them;
multiply first: $a \times N$, then $\div\, b$.

Example

Compute $\dfrac{3}{4} \times 20$ using each of the two readings.

Answer

Divide first. One quarter of $20$ is $20 \div 4 = 5$; three quarters are $5 \times 3 = \textcolor{colordef}{15}$.
Multiply first. $\dfrac{3}{4} \times 20 = \dfrac{3 \times 20}{4} = \dfrac{60}{4} = \textcolor{colordef}{15}$.

The two readings agree, as they always do. The first is easier when $N$ is a multiple of $b$; the second always works.

Example

Compute $\dfrac{2}{5} \times 7$.

Answer

Here $7$ is not a multiple of $5$, so dividing first would not give a whole number. Multiply first: $$\frac{2}{5} \times 7 = \frac{2 \times 7}{5} = \frac{14}{5}.$$ The result is a fraction, and that is perfectly normal — a fraction of a quantity need not be a whole number.

Method — Taking a Fraction of a Quantity

To find a fraction of a quantity, multiply that quantity by the fraction: $$\textcolor{colorprop}{\frac{a}{b}\,\text{of }N=\frac{a}{b}\times N}$$

Example

In a class of 30 students, $\dfrac{2}{3}$ of the class are girls. How many girls are there?

Answer

The fraction $\dfrac{2}{3}$ represents the part of the whole class that are girls.

Method 1 (unitary method). First find one part: $30\div 3=10$. Then take two parts: $10\times 2=\textcolor{colorprop}{20}$.
Method 2 (formula).$$\begin{aligned}\textcolor{colorprop}{\text{Number of girls}}&=\frac{2}{3}\text{ of }\textcolor{colordef}{30}\\ &=\frac{2}{3}\times \textcolor{colordef}{30}\\ &=\frac{2\times 30}{3}\\ &=(2\times 30)\div 3\\ &=\textcolor{colorprop}{20}.\\ \end{aligned}$$

Check. $\dfrac{20}{30}=\dfrac{2}{3}$.

Skills to practice

Calculating a Fraction Times a Whole Number

VII Going further

Skills to practice

Solving Problems with Fractions
Inventing Problems and Explaining Calculations
Investigating Patterns in Fractions
Quiz

\(+\)	\(=\)	\(+\)
	\(=\)
\(\dfrac{1}{2}+\dfrac{1}{4}\)	\(=\)	\(\dfrac{2}{4}+\dfrac{1}{4}\)
	\(=\)	\(\dfrac{3}{4}\)