Fractions

Section 1.3 Fractions

Fractions play a key role in our daily lives. For example, Onyimbo, a father of four children, bought an orange for Ksh 50 and wanted to share it equally among his children. To ensure fairness, he first cut the orange into two equal halves. Then, he divided each half into two more equal parts, resulting in four equal pieces. Each child received one piece, showing how fractions help in dividing things equally.

🔗

$Circle$

Subsection 1.3.1 Arranging fractions in ascending order

Arranging fractions in ascending order means arranging fractions from smallest to greatest in value.

🔗

To arrange fractions in order, we need to make sure they all have the same numerator or the same denominator. This makes it easier to compare them and put them in the correct sequence.

🔗

When fractions have the same denominator, the numerators are compared. The greater the numerator, the greater the fraction. Also, when fractions have the same numerators, the denominators are compared. The greater the denominators, the smaller the fraction.

🔗

Activity 1.3.1.

Work in groups

🔗

(a)

Make circular paper cutouts as the ones shown below.

🔗

$A circle divided into eight equal parts, with two sections shaded$

$A circle divided into eight equal parts, with one section shaded$

$A circle divided into eight equal parts, with five sections shaded$

$A circle divided into eight equal parts, with four sections shaded$

$A circle divided into eight equal parts, with three sections shaded$

🔗

(b)

Arrange the circles paper in order, starting with the one that has the smallest colored part and ending with the one that has the largest colored part.

🔗

(c)

Identify the fractions that correspond to the shaded parts in the ordered sequence of circles.

🔗

(d)

Share your work with learners in the class.

🔗

Example 1.3.1.

Arrange the following fractions in ascending order.
🔗

🔗

🔗

\begin{equation*} \frac{1}{3},\frac{2}{5},\frac{3}{4},\frac{1}{2} \end{equation*}

🔗

Solution.

We first find the LCM of denominators of the fractions in the given sequence. ${\color{blue}3,5,4} \text{ and } {\color{blue}2}.$
🔗

${\color{red}2}$ ${\color{blue}3}$ ${\color{blue}5}$ ${\color{blue}4}$ ${\color{blue}2}$

${\color{red}2}$ 3 5 2 1

${\color{red}3}$ 3 5 1 1

${\color{red}5}$ 1 5 1 1

1 1 1 1

${\color{red}2} \times {\color{red}2} \times {\color{red}3} \times {\color{red}5} = 60$
🔗

The LCM of ${\color{blue}3,5,4} \text{ and } {\color{blue}2} \text{ is } 60.$
🔗

🔗
Create equivalent fractions with a common denominator of $60\text{.}$ Multiply the numerator and denominator of each fraction by the same value.
🔗

\begin{align*} \frac{1}{3} \amp = \frac{1 \times 20}{3 \times 20} \\ \\ \amp = \frac{20}{60} \end{align*}

🔗

\begin{align*} \frac{2}{5} \amp = \frac{2 \times 12}{5 \times 12} \\ \\ \amp = \frac{24}{60} \end{align*}

🔗

\begin{align*} \frac{3}{4} \amp = \frac{3 \times 15}{ 4 \times 15} \\ \\ \amp = \frac{45}{60} \end{align*}

🔗

\begin{align*} \frac{1}{2} \amp = \frac{1 \times 30}{2 \times 30} \\ \\ \amp = \frac{30}{60} \end{align*}

🔗

Now, $\frac{1}{3}$ is equivalent to $\frac{20}{60}\text{.}$ $\frac{2}{5}$ is equivalent to $\frac{24}{60}\text{.}$ $\frac{3}{4}$ is equivalent to $\frac{45}{60}\text{.}$ $\frac{1}{2}$ is equivalent to $\frac{30}{60}.$
🔗

🔗
Arrange the equivalent fractions in ascending order based on their numerator, since we have a common denominator of $60\text{.}$
🔗

$\frac{20}{60},\frac{24}{60},\frac{30}{60},\frac{45}{60}$
🔗

Now, write the corresponding fractions in their original form as per the arranged order above.
🔗

$\frac{1}{3},\frac{2}{5},\frac{1}{2},\frac{3}{4}$
🔗

🔗

🔗

Example 1.3.2.

Arrange the following fractions in ascending order.

🔗

\begin{equation*} \frac{1}{4}, \frac{3}{5}, \frac{1}{2}, \frac{7}{10} \end{equation*}

🔗

Solution.

Convert each fraction into a percentage.
🔗

$\frac{1}{4} \times 100\% = 25\%$
🔗

$\frac{3}{5} \times 100\% = 60\%$
🔗

$\frac{1}{2} \times 100\% = 50\%$
🔗

$\frac{7}{10} \times 100\% = 70\%$
🔗

🔗
Arrange the percentages from smallest to the largest.
🔗

$25\%, 50\%, 60\%, 70\%$
🔗

Finally, write the corresponding fractions in the order of the arranged percentages above.
🔗

The correct order is $\frac{1}{4},\frac{1}{2},\frac{3}{5},\frac{7}{10}.$
🔗

🔗

🔗

Checkpoint 1.3.3.

Exercise on arranging fractions in ascending order goes here.

🔗

Subsection 1.3.2 Arranging fractions in descending order

Activity 1.3.2.

Work in groups

🔗

1. Make circular paper cutouts as the ones shown below.

🔗

2. Arrange the circles paper in order, starting with the one that has the largest colored part and ending with the one that has the smallest colored part.

🔗

3. Identify the fractions representing the shaded parts in the way you have arranged the colored part from the largest to the smallest colored part.

🔗

Descending order means from the largest to the smallest.

🔗

To arrange fractions in descending order, rewrite the fractions to have the same denominator.

🔗

Example 1.3.4.

Arrange the following fractions in descending order.

🔗

$\frac{1}{12},\frac{2}{3},\frac{5}{6},\frac{9}{24}$

🔗

Solution.

Find the LCM of $3,6,12 \text{ and } 24.$
🔗

2 3 6 12 24

2 3 2 6 12

2 3 1 3 6

3 3 1 3 3

1 1 1 1

$2 \times 2 \times 2 \times 3 = 24$
🔗

The LCM of $3,6,12 \text{ and } 24 \text{ is } 24.$
🔗

🔗
Rewrite the fractions to have a common denominator.
🔗

\begin{align*} \frac{1}{12} \amp = \frac{1 \times 2}{12 \times 2} \\ \\ \amp = \frac{2}{24} \end{align*}

🔗

\begin{align*} \frac{5}{6} \amp = \frac{5 \times 4}{6 \times 4} \\ \\ \amp = \frac{20}{24} \end{align*}

🔗

\begin{align*} \frac{2}{3} \amp = \frac{2 \times 8}{ 3 \times 8} \\ \\ \amp = \frac{16}{24} \end{align*}

🔗

\begin{align*} \frac{9}{24} \amp = \frac{9 \times 1}{24 \times 1} \\ \\ \amp = \frac{9}{24} \end{align*}

🔗

🔗
Arrange the fractions in descending order.
🔗

$\frac{20}{24},\frac{16}{24},\frac{9}{24},\frac{2}{24}$
🔗

$\frac{5}{6},\frac{2}{3},\frac{9}{24},\frac{1}{12}$
🔗

🔗

🔗

Example 1.3.5.

Arrange the following fractions in descending order.

🔗

$\frac{2}{5},\frac{9}{10},\frac{3}{4},\frac{7}{20}$

🔗

Solution.

Convert each fraction into a percentage.
🔗

$\frac{2}{5} \times 100 = 40\%$
🔗

$\frac{9}{10} \times 100 = 90\%$
🔗

$\frac{3}{4} \times 100 = 75\%$
🔗

$\frac{7}{20} \times 100 = 35\%$
🔗

🔗
Arrange the percentages in descending order.
🔗

$90\%, 75\%, 40\%, 35\%$
🔗

Finally, write the corresponding fractions in the order of the arranged percentages above.
🔗

The correct order is $\frac{9}{10},\frac{3}{4},\frac{2}{5},\frac{7}{20}.$
🔗

🔗

🔗

Checkpoint 1.3.6.

Exercise for arranging fractions in descending order.

🔗

Subsection 1.3.3 Adding proper fractions

Activity 1.3.3.

Work in groups

🔗

(a)

Make a circular paper cut-out, divide it into 8 equal parts and shade or color it as shown below.

🔗

(b)

What fraction of the whole circle do the colored parts a and b represent?

🔗

(c)

Combine together the two colored parts. Which fraction of the whole circle do the combined colored parts represent?

🔗

Example 1.3.7.

Add: $\frac{5}{6} + \frac{2}{7}$

🔗

Solution.

To add two proper fractions that have different denominators, we find LCM of denominators first.

🔗

Now, to find the LCM of $6\text{ and }7\text{,}$ list multiples of $6$ and $7\text{.}$

🔗

Multiples of $6$ are $6,12,18,24,30,36,{\color{red}42},48,54,...$

🔗

Multiples of $7$ are $7,14,21,28,35,{\color{red}42},49,56,...$

🔗

We can see that, the common multiple is ${\color{red}42}$ and it is the least common multiple that we have.

🔗

Hence, the LCM of $6\text{ and }7$ is ${\color{red}42}\text{.}$

🔗

Rewrite the fractions as equivalent fractions using the LCM as the common denominator.

🔗

\begin{align*} \frac{5}{6} \amp = \frac{5 \times 7}{6 \times 7} = \frac{35}{42}\\ \\ \frac{2}{7} \amp = \frac{2 \times 6}{7 \times 6} = \frac{12}{42} \end{align*}

🔗

Now, since all the denominators are the same, we add the numerators and write the result as the final numerator on top of the common denominator.

🔗

\begin{align*} \frac{35}{42} + \frac{12}{42} \amp = \frac{47}{42} \\ \\ \amp = \frac{47}{42} \end{align*}

🔗

finally convert the improper fraction to a mixed fraction.

🔗

\begin{align*} \frac{47}{42} \amp = 1 \frac{5}{42}\\ \\ \amp = 1 \frac{5}{42} \end{align*}

🔗

Checkpoint 1.3.8.

Exercise for adding proper fractions

🔗

Subsection 1.3.4 Adding mixed fractions

A mixed fraction is a combination of a whole number and a fraction. To add two mixed fractions, we first change them into improper fractions. Before adding these improper fractions, we need to make sure they have the same denominator. When fractions have the same denominators we add the numerators and put the result over the common denominator.

🔗

Activity 1.3.4.

Work in groups

🔗

Juma’s family was expecting visitors that evening. In preparation, Juma’s mother bought $1 \frac{1}{2}$ kg of meat. Later, his father bought an additional $2 \frac{1}{4}$ kg of meat to ensure there was enough for everyone. How much meat did they have altogether?

🔗

Share your work with other learners in the class.

🔗

Example 1.3.9.

Work out: $7 \frac{2}{5} + 3 \frac{1}{4}$

🔗

Solution.

Convert the mixed fractions into improper fractions.

🔗

$7 \frac{2}{5} = \frac{(5 \times 7)+2}{5} = \frac{37}{5}$

🔗

$3 \frac{1}{4} = \frac{(4 \times 3)+1}{4} = \frac{13}{4}$

🔗

The denominators are different, so we have to find the Least Common Multiple (LCM) of the denominators, i.e., LCM of $5$ and $4 =$ $20\text{.}$

🔗

With the help of the LCM, we will write their respective equivalent fractions so that they become like fractions.

🔗

Multiply the numerator and denominator of $\frac{37}{5}$ by $4$ to get $\frac{148}{20}\text{,}$ and also multiply the numerator and denominator of $\frac{13}{4}$ by $5$ to get $\frac{65}{20}\text{.}$

🔗

Now, we have both the fractions with the same denominators, that is, they have been converted to like fractions. So, we can add them, i.e,

🔗

\begin{align*} \frac{148}{20} + \frac{65}{20} \amp = \frac{148 + 65}{20} \\ \\ \amp = \frac{213}{20} \end{align*}

🔗

Convert the improper fraction $\frac{213}{20}$ to a mixed fraction as,

🔗

\begin{align*} \frac{213}{20} \amp = 10 \frac{13}{20}\\ \\ \amp = 10 \frac{13}{20} \end{align*}

🔗

Checkpoint 1.3.10.

Exercise for adding mixed fractions goes here.

🔗

Subsection 1.3.5 Subtracting proper fractions

Activity 1.3.5.

🔗

(a)

Create a new fraction by clicking the New Fraction Button.

🔗

(b)

On the top left corner, click on the operation Subtract.

🔗

(c)

Click on a checkbox to select a challenge - same, or different denominators.

🔗

(d)

Enter values for A, B, C, and D.

🔗

(e)

Click Check button.

🔗

Figure 1.3.11.

🔗

Example 1.3.12.

Work out: $\frac{3}{4} - \frac{1}{5}$

🔗

Solution.

Check the denominators (bottom numbers) of each fraction.

🔗

In our fractions the denominators have different numbers this means we are dealing with unlike fraction.

🔗

We need to make the denominators the same by finding the LCM of the denominators.

🔗

To find the LCM of $4$ and $5\text{,}$ list out the multiples of the two denominators and find the smallest number that is the same between the two.

🔗

Multiples of $4$ are $4,8,12,16,{\color{blue}20},24,28,32,36,{\color{blue}40},44,48,52,56,{\color{blue}60},64,...$

🔗

Multiples of $5$ are $5,10,15,{\color{blue}20},25,30,35,{\color{blue}40},45,50,55,{\color{blue}60},65,...$

🔗

Common multiples are ${\color{blue}20},{\color{blue}40},{\color{blue}60}$

🔗

The least common multiples is ${\color{blue}20}$

🔗

Multiply the numerator by whatever number you needed to multiply the denominator by to change it into the lowest common denominator.

🔗

Multiply both the numerator and denominator of $\frac{3}{4}$ by $5$ to get $\frac{15}{20}\text{.}$ Then, multiply the numerator and denominator of $\frac{1}{5}$ by $4$ to get $\frac{4}{20}\text{.}$

🔗

Now that the denominators are the same, subtract the numerators just like you would subtract regular whole numbers and maintain the common denominator on the bottom.

🔗

\begin{align*} \frac{15}{20} - \frac{4}{20} = \frac{15 - 4}{20} \amp = \frac{11}{20} \\ \\ \amp = \frac{11}{20} \end{align*}

🔗

Checkpoint 1.3.13.

🔗

Subsection 1.3.6 Subtracting mixed fractions

Activity 1.3.6.

Work in groups

🔗

(a)

Make the fraction cards like the one shown below.

🔗

(b)

Pick any fraction card and subtract the fractions.

🔗

(c)

Share your work with other learners in your class.

🔗

Number cards containing operation between twp mixed numbers

🔗

Example 1.3.14.

Work out: $12 \frac{2}{3} - 7 \frac{3}{5} = $

🔗

Solution.

Convert the mixed fractions into improper fractions.
🔗

To convert a mixed number to an improper fraction, the whole number is multiplied by the denominator and the result is added to the numerator of the proper fraction by retaining the denominator.
🔗

\begin{align*} 12 \frac{2}{3} \amp = \frac{(3 \times 12) + 2}{3} \\ \\ \amp = \frac{38}{3} \end{align*}

🔗

\begin{align*} 7 \frac{3}{5} \amp = \frac{(5 \times 7) + 3}{5} \\ \\ \amp = \frac{38}{5} \end{align*}

🔗

Subtract the improper fractions
🔗

\begin{equation*} \frac{38}{3} - \frac{38}{5} \end{equation*}

🔗

Since the denominators are different, we need to make the denominators same by taking the LCM and multiplying the suitables fractions for both.
🔗

The LCM of $3$ and $5$ is $15\text{.}$
🔗

So, $\frac{38}{3} \times \frac{{\color{red}5}}{{\color{red}5}} = \frac{190}{15}$ and $\frac{38}{5} \times \frac{{\color{red}3}}{{\color{red}3}} = \frac{114}{15}$
🔗

Now, we have common denominators and we proceed to subtract numerators.
🔗

\begin{equation*} \frac{190}{15} - \frac{114}{15} = \frac{76}{15} \end{equation*}

🔗

Convert the improper fraction $\frac{76}{15}$ to a mixed number.
🔗

\begin{align*} \frac{76}{15} \amp = 5 \frac{1}{15} \\ \\ \amp = 5 \frac{1}{15} \end{align*}

🔗

🔗

🔗

Checkpoint 1.3.15.

🔗

Subsection 1.3.7 Multiplying a fraction by a whole number

Activity 1.3.7.

Work in groups

🔗

Otuoma, a baker, is excited to surprise his family with three delicious cakes. He eagerly pulls out his recipe book, only to realize he needs to calculate the sugar needed. Each cake, according to the recipe, requires a $\frac{1}{2}$ kg of sugar. Otuoma wonders the total amount of sugar he will need in total?. Can you help Otuoma figure out how much sugar he needs to bake his surprise cakes?

🔗

Share your calculations and the amount of sugar you determined with the rest of the class.

🔗

Example 1.3.16.

Work out: $\frac{5}{7} \times 4 $

🔗

Solution.

To multiply a proper fraction by a whole number, multiply the numerator of the fraction by the whole number and retain the same denominator.
🔗

\begin{align*} \frac{5}{7} \times 4 \amp = \frac{5 \times 4}{7}\\ \\ \amp = \frac{20}{7} \end{align*}

🔗

We now, convert resulting the improper fraction $\frac{20}{7}$ into a mixed fraction.
🔗

\begin{align*} \frac{20}{7} \amp = 2 \frac{6}{7} \\ \\ \amp = 2 \frac{6}{7} \end{align*}

🔗

🔗

🔗

Checkpoint 1.3.17.

Exercise for multiplying a fraction by a whole number goes in here.

🔗

Subsection 1.3.8 Multiplying a proper fraction by a proper fraction

Activity 1.3.8.

🔗

(a)

Drag the slider Slide to explore to the right direction and observe what happens to the squares.

🔗

(b)

Click NEW FRACTIONS button on the bottom right hand side to regenerate new fractions and follow step $\textbf{(a)}$ again.

🔗

(c)

Discuss with your fellow learners on your observation in this activity.

🔗

Figure 1.3.18.

🔗

Example 1.3.19.

Work out: $\frac{5}{9} \times \frac{2}{3}$

🔗

Solution.

To multiply two fractions, multiply numerators and denominators separately and simplify the resulting fraction to its simplest form.

🔗

\begin{align*} \frac{5}{9} \times \frac{2}{3} \amp = \frac{5 \times 2}{9 \times 3}\\ \\ \amp = \frac{10}{27} \end{align*}

🔗

Checkpoint 1.3.20.

Multiplying a fraction by a fraction goes in here.

🔗

Subsection 1.3.9 Multiplying a fraction by a mixed number

Activity 1.3.9.

Work in groups

🔗

A delivery lorry set off from Kitale town to Kisumu city, the lorry is transporting fresh farm produce, and the distance between Kiatle and Kisumu is $80 \frac{1}{5}$ km. The lorry uses $\frac{4}{5}$ litres of fuel per kilometre.

🔗

How many litres of fuel does the lorry need for the whole journey?

🔗

Share your work with learners in your class.

🔗

Example 1.3.21.

Work out: $\frac{2}{5} \times 1 \frac{3}{7} $

🔗

Solution.

First, we convert $1 \frac{{\color{blue}3}}{{\color{red}7}}$ to improper fraction.
🔗

\begin{align*} 1 \frac{{\color{blue}3}}{{\color{red}7}} \amp = \frac{({\color{red}7} \times 1) + {\color{blue}3}}{{\color{red}7}} \\ \\ \amp = \frac{10}{7} \end{align*}

🔗

🔗
Multiply the fractions
🔗

\begin{align*} \frac{2}{5} \times \frac{10}{7} \amp = \frac{2 \times 10}{5 \times 7}\\ \\ \amp = \frac{20}{35} \end{align*}

🔗

🔗
Simplify the numerator and denominator by dividing through by $5.$
🔗

\begin{align*} \frac{20}{35} \amp = \frac{4}{7}\\ \\ \amp = \frac{4}{7} \end{align*}

🔗

🔗

🔗

Checkpoint 1.3.22.

Exercise for Multiplying a fraction by a mixed number goes in here

🔗

Subsection 1.3.10 Reciprocal of fractions

The reciprocal of a fraction is obtained by interchanging the numerator and the denominator. The product of a fraction and its reciprocal is $1.$

🔗

In general, the reciprocal of a fraction $\frac{a}{b}$ is written as $\frac{b}{a}$ where $(a,b \text{ are whole numbers})\text{.}$ For example, the reciprocal of $\frac{2}{3}$ is $\frac{3}{2}.$

🔗

We can also find the reciprocal of fraction by dividing 1 by the fraction.

🔗

Activity 1.3.10.

Work in groups

🔗

(a)

Form groups of at least five learners.

🔗

(b)

Cut out $10$ small pieces of paper, each labeled with a unique number from $1$ to $10\text{.}$

🔗

(c)

Fold the papers so the numbers are hidden. One learner in the group should shuffle them.

🔗

(d)

This is a two-player game where players take turns picking a folded paper until all are taken.

🔗

(e)

The first player randomly picks a paper, reveals the number written on it, and keep it aside.

🔗

(f)

The second player does the same, picking a paper, revealing the number, and keep it aside.

🔗

(g)

Write a fraction using the first player’s number as the numerator and the second player’s number as the denominator.

🔗

(h)

Now, write another fraction by swapping the positions use the second player’s number as the numerator and the first player’s number as the denominator.

🔗

(i)

Multiply the two fractions and observe the result.

🔗

(j)

Discuss with your classmates what you notice about the fractions and their product.

🔗

Example 1.3.23.

What is the reciprocal of $\frac{17}{39}\text{?}$

🔗

Solution.

To get the reciprocal of $\frac{17}{39}\text{,}$ we swap the numerator and the denominator.

🔗

$\frac{{\color{red}17}}{{\color{blue}39}} \text{ to get } \frac{{\color{blue}39}}{{\color{red}17}}$

🔗

Therefore, the reciprocal of $\frac{{\color{red}17}}{{\color{blue}39}} \text{ is } \frac{{\color{blue}39}}{{\color{red}17}}.$

🔗

Example 1.3.24.

What is the reciprocal of $3 \frac{2}{7}\text{?}$

🔗

Solution.

Convert $3 \frac{2}{7}$ into an improper fraction.
🔗

${\color{red}3} \frac{{\color{blue}2}}{{\color{green}7}} = \frac{({\color{green}7} \times {\color{red}3})+{\color{blue}2}}{{\color{green}7}} = \frac{23}{7}$
🔗

🔗
Swap the numerator and the denominator.
🔗

$\frac{{\color{red}23}}{{\color{blue}7}} \text{ to get } \frac{{\color{blue}7}}{{\color{red}23}}$
🔗

Therefore, the reciprocal of $3 \frac{2}{7} \text{ is } \frac{7}{23}.$
🔗

🔗

🔗

Checkpoint 1.3.25.

Exercise for Reciprocal of fractions goes here.

🔗

Subsection 1.3.11 Division of a fraction by a whole number

Activity 1.3.11.

🔗

(a)

Use the blue slider labeled Numerator to set the top number of the fraction.

🔗

(b)

Use the blue slider labeled Denominator to set the top number of the fraction.

🔗

(c)

Use the red slider labeled Divisor to select the whole number by which the fraction will be divided.

🔗

(d)

Click the Calculate button to see the visualization result of the fraction division.

🔗

(e)

Observe how dividing a fraction by a whole number affects the fraction.

🔗

(f)

Click the Reset button to clear the settings and start a new calculation.

🔗

Try different numerators, denominators, and divisors to explore different cases.

🔗

Figure 1.3.26.

🔗

When dividing a fraction by a whole number, multiply the fraction by the reciprocal of the whole.

🔗

Example 1.3.27.

Work out: $\frac{4}{15} \div 10 $

🔗

Solution.

First, we have to find the reciprocal of the divisor.
🔗

In this case, our divisor is $10\text{,}$ which is a whole number and we can express $10$ as $\frac{10}{1}.$
🔗

Then, find the reciprocal of $\frac{{\color{red}10}}{{\color{blue}1}}$ by swapping the numerator and denominator to get $\frac{{\color{blue}1}}{{\color{red}10}}.$
🔗

🔗
Multiply $\frac{4}{15}$ by $\frac{1}{10}$ and simplify the result.
🔗

\begin{align*} \frac{4}{15} \times \frac{1}{10} \amp = \frac{4 \times 1}{15 \times 10}\\ \\ \amp = \frac{4}{150} \\ \\ \amp = \frac{2}{75} \end{align*}

🔗

🔗

🔗

Checkpoint 1.3.28.

Exercise for Division of a fraction by a whole number

🔗

Subsection 1.3.12 Division of a proper fraction by a proper fraction

Division of proper fractions is similar to multiplication. We change division sign to multiplication sign and we multiply the first proper fraction by the reciprocal of the second proper fraction.

🔗

Activity 1.3.12.

Work in groups

🔗

1. Cut out a circular shape and divide the circle into $8$ equal parts using a ruler and pencil. Shade the parts to match the coloring in the provided example.

🔗

2. What fraction of the whole does the shaded part a and b represent?

🔗

3. How many times does the smaller part a fit into the larger part b?

🔗

4. Share your work with other learners in class.

🔗

Example 1.3.29.

Work out: $\frac{1}{7} \div \frac{1}{4}$

🔗

Solution.

Find the reciprocal of the divisor.
🔗

The reciprocal of $\frac{{\color{red}1}}{{\color{blue}4}} $ is $\frac{{\color{blue}4}}{{\color{red}1}}.$
🔗

🔗
Multiply $\frac{1}{7} \text { by } \frac{4}{1}.$
🔗

\begin{align*} \frac{{\color{red}1}}{{\color{blue}7}} \times \frac{{\color{red}4}}{{\color{blue}1}} \amp = \frac{{\color{red}1} \times {\color{red}4}}{{\color{blue}7} \times {\color{blue}1}}\\ \\ \amp = \frac{4}{7} \end{align*}

🔗

🔗

🔗

Checkpoint 1.3.30.

🔗

Subsection 1.3.13 Division of a fraction by a mixed number

Activity 1.3.13.

🔗

Amina is a baker in Umoja Market. She has $7 \frac{1}{2}$ kilograms of flour to bake muffins.

🔗

(a)

How many muffins can she bake if each muffin requires $1 \frac{1}{4}$ kilograms of flour?

🔗

(b)

Share your work with other learners in your class.

🔗

Example 1.3.31.

Work out: $\frac{4}{7} \div 9 \frac{1}{5} = $

🔗

Solution.

Convert $9 \frac{1}{5}$ into an improper fraction.
🔗

\begin{align*} {\color{red} 9} \frac{{\color{green} 1}}{{\color{blue} 5}} \amp = \frac{({\color{red} 9} \times {\color{blue} 5})+{\color{green} 1}}{{\color{blue} 5}} \\ \\ \amp = \frac{45 + 1}{5} \\ \\ \amp = \frac{46}{5} \end{align*}

🔗

🔗
Find the reciprocal of $\frac{46}{5}$ by interchanging the numerator and denominator.
🔗

The reciprocal of $\frac{{\color{red} 46}}{{\color{blue}5}}$ is $\frac{{\color{blue}5}}{{\color{red} 46}}\text{.}$
🔗

🔗
Multiply $\frac{4}{7}$ by $\frac{5}{46}.$
🔗

$\frac{{\color{blue}4}}{{\color{red}7}} \times \frac{{\color{blue}5}}{{\color{red}46}} = \frac{{\color{blue}4} \times {\color{blue}5}}{{\color{red}7} \times {\color{red}46}}$
🔗

$\qquad \qquad= \frac{20}{322}$
🔗

Simplify the fraction:
🔗

$\qquad \qquad= \frac{10}{161}$
🔗

🔗

🔗

Checkpoint 1.3.32.

Exercise for Division of a fraction by a mixed number goes in here.

🔗

Subsection 1.3.14 Division of a whole number by a fraction

When dividing a whole number by a fraction, we are finding how many groups of the fraction can fit in the whole number. To make this easier, instead of dividing, we multiply the whole number by the reciprocal of the fraction. For example, if we divide $4$ by $\frac{1}{2}\text{,}$ we multiply $4$ by $\frac{2}{1}$ (the reciprocal of $\frac{1}{2}$), which is $4 \times \frac{2}{1} = 8\text{.}$

🔗

Activity 1.3.14.

🔗

(a)

Drag the Whole Number slider to select the whole number being divided.

🔗

(b)

Drag the Denominator slider to set the denominator of the fraction divisor.

🔗

(c)

Discuss your observaation with learners in class.

🔗

(d)

Repeat the process using a different whole number and a new denominator for the fraction.

🔗

Figure 1.3.33.

🔗

Example 1.3.34.

Work out: $7 \div \frac{2}{3} $

🔗

Solution.

Convert the whole number to a fraction. To do this, make the whole number the numerator of a fraction. Make the denominator $1\text{.}$
🔗

$7 = \frac{7}{1}$
🔗

Find the reciprocal of the divisor, that is $\frac{2}{3}\text{.}$
🔗

To find the reciprocal of a fraction, reverse the numerator and denominator. The reciprocal of $\frac{{\color{red}2}}{{\color{blue}3}}$ is $\frac{{\color{blue}3}}{{\color{red}2}}.$
🔗

🔗
Multiply $\frac{7}{1}$ by $\frac{3}{2}\text{.}$
🔗

To multiply fractions, first multiply the numerators together. Then, multiply the denominators together.
🔗

\begin{align*} \frac{{\color{green}7}}{1} \times \frac{{\color{green}3}}{2} = \frac{{\color{green}7} \times {\color{green}3}}{1 \times 2} \amp = \frac{21}{2} \\ \amp \\ \amp = \frac{21}{2} \end{align*}

🔗

🔗

🔗

Example 1.3.35.

Evaluate: $48 \div 4 \frac{1}{3}$

🔗

Solution.

Convert the mixed number $4 \frac{1}{3}$ to an improper fraction.
🔗

\begin{align*} {\color{red}4} \frac{{\color{blue}1}}{{\color{green}3}} \amp = \frac{({\color{red}4} \times {\color{green}3}) + {\color{blue}1}}{{\color{green}3}} \\ \\ \amp = \frac{12 + 1}{3}\\ \\ \amp = \frac{13}{3} \end{align*}

🔗

🔗
Find the reciprocal of $\frac{13}{3}$ by interchanging the numerator and denominator.
🔗

The reciprocal of $\frac{{\color{red}13}}{{\color{blue}3}}$ is $\frac{{\color{blue}3}}{{\color{red}13}}$
🔗

Convert the whole number to a fraction. $48 = \frac{48}{1}$
🔗

🔗
Now, multiply $\frac{48}{1}$ by $\frac{3}{13}.$
🔗

$\frac{48}{1} \times \frac{3}{13}$
🔗

To multiply fractions, first multiply the numerators together. Then, multiply the denominators together.
🔗

\begin{align*} \frac{{\color{blue}48}}{1} \times \frac{{\color{blue}3}}{13} = \frac{{\color{blue}48} \times {\color{blue}3}}{1 \times 13} \amp = \frac{144}{13} \\ \amp \\ \amp = \frac{144}{13} \\ \\ \amp = 11 \frac{1}{13} \end{align*}

🔗

🔗

🔗

Checkpoint 1.3.36.

Exercise for Division of a whole number by a fraction goes in here.

🔗

Subsection 1.3.15 Number sequence involving fractions

Activity 1.3.15.

Work in groups

🔗

(a)

Look at the sticks below. Each stick has blue circles and green circle.

🔗

(b)

Count the number of green circles in each stick.

🔗

(c)

Count the number of blue circles in each stick.

🔗

(d)

Express the number of green circles as numerator of the fraction and the number of blue circles as the denominator of the fraction in each stick.

🔗

(e)

Analyze the pattern in the fractions you have formed and discuss any relationships or trends you observe.

🔗

Example 1.3.37.

Find the next fraction in the sequence.

🔗

$\frac{4}{36},\frac{9}{49},\frac{16}{64},\frac{25}{81},\frac{36}{100},$—

🔗

Solution.

Identify the pattern
🔗

Observing the denominators, we see that they are squares of consecutive numbers:
🔗

$36 = {\color{blue} 6}^{\color{red}2}, \quad 49 = {\color{blue} 7}^{\color{red}2}, \quad 64 = {\color{blue} 8}^{\color{red}2}, \quad 81 = {\color{blue} 9}^{\color{red}2}, \quad 100 = {\color{blue} 10}^{\color{red}2} $
🔗

Thus, the denominators follow the sequence: $6^{\color{red}2},\,\, 7^{\color{red}2},\,\, 8^{\color{red}2},\,\, 9^{\color{red}2},\,\,10^{\color{red}2}$
🔗

Observing the numerators, we see that they are squares of consecutive numbers:
🔗

$4= {\color{blue} 2}^{\color{red}2}, \quad 9 = {\color{blue} 3}^{\color{red}2}, \quad 16 = {\color{blue} 4}^{\color{red}2}, \quad 25 = {\color{blue} 5}^{\color{red}2}, \quad 36 = {\color{blue} 6}^{\color{red}2} $
🔗

Thus, the numerators follow the sequence: $2^{\color{red}2},\,\, 3^{\color{red}2},\,\, 4^{\color{red}2},\,\, 5^{\color{red}2},\,\,6^{\color{red}2}$
🔗

🔗
We find the next numerator
🔗

Since the numerators follow the squares of consecutive natural numbers, the next numerator will be $7$ and $7^2 = {\color{red}7} \times {\color{red}7} = 49$
🔗

So, the next numerator will be $49.$
🔗

We find the next denominator
🔗

Since the denominators follow the squares of consecutive natural numbers, the next number will be $11$ and $11^2 = {\color{red}11} \times {\color{red}11} = 121$
🔗

So, the next denominator will be $121.$
🔗

🔗
Therefore, the next fraction will have a numerator of $49$ and a denominator of $121\text{:}$ $\frac{49}{121}$
🔗

The sequence is $\frac{4}{36},\, \frac{9}{49},\, \frac{16}{64},\,\frac{25}{81},\,\frac{36}{100},\, \frac{49}{121}.$
🔗

🔗

🔗

Checkpoint 1.3.38.

🔗

Subsection 1.3.16 Creating number sequences involving fractions

Activity 1.3.16.

Work in groups

🔗

(a)

Starting at $\frac{3}{5}\text{,}$ create an increasing sequence consisting of 5 terms.

🔗

(b)

Write down each term in your sequence as a fraction and explain how you determined the next term.

🔗

(c)

Compare your sequence with those of other groups. What similarities or differences do you notice?

🔗

Example 1.3.39.

Create an addition sequence of four fractions starting from $1 \frac{1}{3}.$

🔗

Solution.

Since the question allows the freedom to choose any fraction to add to the starting fraction, we have chosen $\frac{1}{3}$ for this sequence.

🔗

Here, we begin with $1 \frac{1}{3}$ and repeatedly add $\frac{1}{3}$ to determine the next term in the sequence.

🔗

\begin{align*} \text{ First term in the sequence } \amp = 1 \frac{1}{3}\\ \\ \text{Second term in the sequence: } 1 \frac{1}{3} + {\color{blue}\frac{1}{3}} \amp = 1 \frac{2}{3} \\ \\ \text{Third term in the sequence: } 1 \frac{2}{3} + {\color{blue}\frac{1}{3}} \amp = 2 \\ \\ \text{Fourth term in the sequence: } \quad 2 + {\color{blue}\frac{1}{3}} \amp = 2 \frac{1}{3} \\ \end{align*}

🔗

The sequence is $1 \frac{1}{3}, \, 1 \frac{2}{3}, \, 2,\, 2 \frac{1}{3}.$

🔗

Checkpoint 1.3.40.

Exercise on Creating number sequences involving fractions

🔗

Prev Top Next

\({\color{red}2}\)	\({\color{blue}3}\)	\({\color{blue}5}\)	\({\color{blue}4}\)	\({\color{blue}2}\)
\({\color{red}2}\)	3	5	2	1
\({\color{red}3}\)	3	5	1	1
\({\color{red}5}\)	1	5	1	1
	1	1	1	1