Affiliate 4: Equivalent fractions
iv.one Types of fractions
A
fraction
is part of a whole. At that place are different ways to write fractions. One way is to write the fraction every bit one whole number on elevation of some other whole number with a dividing line between, as .
The bottom number is called the
denominator. It represents all the parts that the whole is divided into. The top number is called the
numerator. Information technology represents only the parts of the whole we are dealing with. Fractions that are written like this are called
common fractions.
fraction
A fraction is a part of a whole.denominator
The denominator is the number written below the line in a fraction; information technology represents all the parts of a whole.numerator
The numerator is the number written above the line in a fraction; information technology represents only the parts of a whole we are dealing with.common fraction
A mutual fraction is a fraction that is written in the form , where the numerator and denominator are both whole numbers.
For example, if you cut a cake into vii pieces, you can say that the whole block is made up of .
Suppose you consume two of the pieces.
 The fraction of the cake you ate is . The denominator shows that the cake was cut into seven pieces. The numerator shows that 2 pieces were eaten.
 The fraction of the cake that is left over is . The denominator all the same shows that the cake was cut into 7 pieces. The numerator at present shows that 5 pieces are left over.
The fractions and are
proper fractions. A proper fraction is smaller than one. Its numerator is smaller than its denominator.
proper fraction
A proper fraction is a fraction that is smaller than 1, so the numerator is smaller than the denominator.
Exercise 4.ane: Write proper fractions from diagrams
For each diagram, express the shaded part as a proper fraction.
A
mixed number
is made upwardly of a whole number and a proper fraction. For instance, apples hateful you accept two whole apples and one onehalf of another apple.
mixed number
A mixed number is a number that is made up of a whole number and a proper fraction.
A mixed number may be expressed as an
improper fraction. An improper fraction is larger than one. Its numerator is larger than its denominator. Suppose we cut the two whole apples shown above into halves also. We will then accept v halves.
These five halves may be written every bit . The denominator tells united states of america that there are ii parts in one whole apple, so we are dealing with halves. The numerator tells us that we accept 5 halves. Therefore, the mixed number may be expressed equally the improper fraction .
improper fraction
An improper fraction is a fraction that is larger than one, then the numerator is larger than the denominator.
Worked example 4.1: Expressing mixed numbers as improper fractions
Write as an improper fraction.

Pace i: Use the denominator to see how many parts there are in one whole.
 The denominator of the proper fraction in this mixed number is 4.
 This means that each whole is divided into 4 parts. We are dealing with quarters.

Step 2: Determine how may parts the whole number represents. Multiply the whole number by the denominator from Step 1.
 The whole number in this mixed number is v.
 If at that place are 4 parts in ane whole, there must be parts in five wholes.

Pace three: Add the additional parts that are given by the numerator of the proper fraction.
 The numerator of the proper fraction is three.
 So, also as the xx parts given by the whole number, we accept some other three parts.
 The total parts are .

Pace 4: Write the improper fraction in the course .
 The numerator gives the total parts we take: 23
 The denominator tells united states how many parts in that location are in i whole: 4
Worked case 4.2: Expressing improper fractions as mixed numbers
Write as a mixed number.

Step 1: Use the denominator to see how many parts there are in i whole.
 The denominator of this improper fraction is 3.
 This means that each whole is divided into 3 parts. Nosotros are dealing with thirds.

Pace two: Determine how many whole numbers can exist formed from the total parts we have. To exercise this, divide the numerator by the denominator from Stride ane.
 The numerator gives the total parts we take: 26
 If there are three parts in i whole, we can form 8 wholes from 26 parts:
The whole number role of the mixed number is: viii

Step 3: Make up one’s mind what fraction is left over afterwards we formed the whole numbers.
 You know that the denominator, which tells united states how many parts there are in one whole, is three.
 To work out the numerator, use the parts that are left over after we formed the whole numbers. If yous look at the long partition, the remainder is 2.
After nosotros formed the whole numbers, the fraction is left over.

Step iv: Write down the mixed number every bit a whole number followed by a proper fraction.
Exercise 4.2: Catechumen between mixed numbers and improper fractions
 Limited the following mixed numbers as improper fractions.
Parts in ane whole: 7
Parts in half dozen wholes:
Total parts:Parts in i whole: 11
Parts in two wholes:
Total parts:Parts in one whole: 3
Parts in xiii wholes:
Total parts:  Express the following improper fractions equally mixed numbers.
Parts in one whole: 7
Full parts: thirtyWhole number: 4
Fraction:Parts in 1 whole: 9
Total parts: 112Whole number: 12
Fraction:Parts in one whole: eleven
Total parts: 85Whole number: vii
Fraction:
In one case you understand how to express a mixed number every bit an improper fraction, you lot may employ this brusk method.
Example:
Denominator:
Numerator:
Answer:
iv.two Equivalent fractions
Fractions that represent the aforementioned part of a whole in different means are called
equivalent fractions. Look at the 3 diagrams beneath. The fraction of the rectangle that is shaded is given side by side to each diagram.
All three fractions represent half of the rectangle. The fractions take unlike numerators and denominators, but they all represent the value .
equivalent fractions
Equivalent fractions are fractions that have different numerators and denominators, but they represent the aforementioned part of a whole, and they accept the same value.
Find equivalent fractions
We notice equivalent fractions past multiplying or dividing the numerator and the denominator by the same whole number.
Worked case 4.three: Working out equivalent fractions
Find the missing numbers in these equivalent fractions:

Step one: Start with the first pair. Place what is given, which in this example is the two denominators. Was the first denominator multiplied or divided to become the second denominator, and by what whole number?
The denominator was multiplied by 4 to become sixteen:

Step 2: Echo the action identified in Step 1 for the remaining part of the fraction.
The numerator must also be multiplied past iv:

Step 3: Repeat Steps i and 2 to discover the next missing number.
The numerator was divided by 2:
The denominator must also be divided by 2:

Footstep 4: Echo Steps one and 2 for the last fraction. If it is not possible to perform Step 1 with the fraction to the left of the missing number, apply any other fraction that is complete.
It is not possible to multiply the numerator 6 or the numerator 12 by a whole number and so get 15 as an answer. You have to use the fraction .
The numerator was multiplied by five:
The denominator must also exist multiplied past five:

Step 5: Write downwardly the total answer.
Improper fractions can also have equivalent fractions. Earlier on in this affiliate, we showed that apples is equal to apples. The improper fraction tells u.s.a. that we are dealing with apple halves and that nosotros accept 5 apple halves.
Suppose we cut each apple onehalf into two pieces. We volition and then take ten apple quarters.
This may be expressed equally . The denominator tells us that in that location are iv parts in ane whole apple, and so we are dealing with quarters. The numerator tells us that nosotros have 10 quarters. Therefore, the fractions and are equivalent fractions. In this instance, the numerator and denominator were multiplied by two:
Exercise four.3: Make up one’s mind equivalent fractions
Discover the missing numbers in each list of equivalent fractions.
Simplify equivalent fractions
A fraction is in its
simplest form
when the numerator and the denominator have no common factor. For example, is the simplest form of the fractions and . These fractions are not in their simplest form, considering in , 2 and 4 accept 2 as a common factor, and in , 4 and eight accept 2 and 4 every bit common factors.
simplest form
A fraction is in its simplest form when the numerator and denominator have no mutual gene.
Worked example four.4: Finding the simplest form of a fraction
Find the simplest grade of the fraction .

Method i (Prime number numbers)
1.
Discover the smallest prime number that is a factor of both the numerator and the denominator. In this case the factor is 2.
2.
Split both the numerator and denominator by this prime. Write down the equivalent fraction.
3.
Repeat until the prime number tin can no longer dissever into both the numerator and the denominator.
4.
Move on to the side by side prime number that is a factor of both the numerator and the denominator. In this case the cistron is 5. Split both the numerator and the denominator by 5.
5.
Repeat this process until the numerator and the denominator have no more than mutual factors.The simplest form of is .

Method 2 (HCF)
i.
Express both the numerator and the denominator as a product of their prime numbers.
two.
Find the HCF of the numerator and the denominator.
three.
Split the numerator and the denominator by their HCF.\begin{array}{r  r} ii & 40 \newline \hline 2 & twenty \newline \hline 2 & 10 \newline \hline 5 & 5 \newline \hline & 1 \finish{array} \begin{assortment}{r  r} ii & sixty \newline \hline 2 & 30 \newline \hline three & 15 \newline \hline 5 & 5 \newline \hline & 1 \end{array}
The simplest class of is .
If necessary, go back to Chapter 2 to see how to detect the highest common gene (HCF) of two or more numbers.

Method 3 (Inspection)
1.
Find any common factor of the numerator and the denominator.
2.
Divide both the numerator and denominator past this factor. Write downwardly the equivalent fraction.
3.
Repeat this procedure until the numerator and the denominator have no more common factors.The simplest form of is .
Do iv.four: Simplify fractions
Observe the simplest form of each of the postobit fractions, using the method of your choice.

Using Method 1 (Prime numbers):
The simplest form of is .

Using Method 1 (Prime numbers):
The simplest grade of is .

Using Method 2 (HCF):
The simplest form of is .

Using Method two (HCF):
The simplest form of is .

Using Method 3 (Inspection):
The simplest form of is .
Identify equivalent fractions
Fractions that are equivalent have the aforementioned simplest grade. Await at the two diagrams beneath. The fraction of the rectangle that is shaded is given next to each diagram.
It is easy to run across that the fractions and are not equivalent. They do non correspond the same office of the whole rectangle. Nosotros can cheque this by finding the simplest form of each fraction.
This shows that the fractions and practise not accept the same value. They are therefore not equivalent.
Remember that equivalent fractions have the same simplest form.
Exercise 4.5: Determine whether fractions are equivalent or not
For each of the following pairs of fractions, decide whether they are equivalent or non.

and
The fractions and are equivalent.

and
The fractions and are equivalent.

and
The fractions and are Non equivalent.

and
The fractions and are equivalent.

and
The fractions and are Not equivalent.
4.three Comparing fractions
It may exist difficult to compare fractions that practise not have the same denominator. Expect at the ii diagrams beneath. The fraction of the circle that is shaded is given below each diagram.
Which fraction is larger, or ? It is not and so easy to decide this just by looking at the diagrams or the fractions themselves. However, if we express both fractions with the aforementioned denominator, it becomes easier to compare them.
Now it is articulate that is the larger fraction. We may write this in two ways:

\frac{2}{iii}\frac{2}{3} is smaller than

, which means is larger than
To exist able to compare fractions, nosotros first need to find the
lowest common denominator (LCD)
for the fractions. This is the lowest mutual multiple (LCM) of the denominators. Then we need to limited the two fractions as equivalent fractions using this same denominator.
everyman common denominator (LCD)
The LCD is the smallest whole number into which the denominators of two or more fractions can divide without a residual.
If necessary, get back to Chapter two to see how to observe the LCM of 2 or more numbers.
If we express more than than two fractions using the same denominator, this allows us to club the fractions in
ascending club
or
descending order. Fractions that are arranged in ascending lodge outset with the smallest fraction, then accept the next smallest, and so on, and the largest fraction is final. For case . Fractions that are arranged in descending order kickoff with the largest fraction, then have the next largest, so on, and the smallest fraction is last. For instance .
ascending order
When numbers of any kind are arranged in ascending order, they are arranged from the smallest to the largest.descending order
When numbers of any kind are bundled in descending social club, they are arranged from the largest to the smallest.
Worked example iv.5: Ordering fractions
Adapt the following fractions in ascending guild:

Step ane: Before we tin compare fractions, nosotros demand to give equivalent fractions using the same denominator. To do this, first discover the LCD of the fractions.
The denominators expressed as products of their prime factors are:

Step two: For each given fraction, find an equivalent fraction using the denominator calculated in Stride one.

Footstep iii: Write the original fractions in the required order.
Ascending order means we have to beginning with the smallest fraction and end with the largest fraction: .
The original fractions, arranged in ascending order, are:
Do 4.6: Order and compare fractions

Conform the following fractions in ascending order:
Ascending lodge:

Conform the postobit fractions in descending guild:
Descending order:

Look at this listing of fractions: .
From the list, identify:
 two fractions that are equivalent
 the fraction with the highest value
The proper fractions are all smaller than 1. The 2 improper fractions are larger than 1, so work but with them.
 the fraction with the everyman value
Both improper fractions are larger than 1. The three proper fractions are smaller than i, so work only with them.
From a) above:
Just:
 an improper fraction that is smaller than 3
From b) above:
\therefore\frac{17}{half dozen}<3
 a fraction that is equivalent to
From c) to a higher place:
iv.iv Decimal fractions
Instead of writing a fraction as a mutual fraction, we tin also write it as a
decimal fraction. We apply
decimal notation
to write decimal fractions. In this notation, the numerator is written afterward a decimal betoken and the denominator is not written down. Earlier y’all tin can write a common fraction as a decimal fraction, the denominator always has to be a power of ten.
In the kickoff section of this chapter, you learnt that a mixed number is a whole number and a proper fraction. Mixed numbers may as well be written in decimal note. The decimal signal separates the whole numbers from the fractions. For example, 2.five means that we accept 2 wholes and a fraction of .
decimal fraction
A decimal fraction is a fraction that is written in decimal notation, with the whole number earlier the decimal signal and the fraction after it.decimal annotation
Decimal notation is a manner of writing a fraction by putting the numerator later a decimal point. The denominator is not written down.
In decimal notation, the number of digits after the decimal point shows u.s.a. which power of ten is the fraction’s denominator. You learnt about these decimal fractions in Chapter 1:
\begin{array}{50lll} \hline \textbf{Name} & \textbf{Common fraction} & \textbf{Powers of ten} & \textbf{Decimal fraction} \newline \hline \text{I 10th} & \frac{1}{10} & \frac{1}{10}=10^{one} & 0.1 \newline \hline \text{1 hundredth} & \frac{1}{100} & \frac{1}{ten\times10}=ten^{2} & 0.01 \newline \hline \text{I thousandth} & \frac{one}{ane,000} & \frac{1}{10\times10\times10}=10^{3} & 0.001 \newline \hline \text{One 10 thousandth} & \frac{1}{10,000} & \frac{1}{10\times10\times10\times10}=10^{4} & 0.0001 \newline \hline \end{array}
You do not need to write zeros at the finish of decimal fractions. For instance, we write 0.5 rather than 0.50 or 0.500. This is because .
Converting decimal fractions to common fractions
The fraction 0.5 and the fraction represent exactly the aforementioned function of a whole, namely a half. The fractions and are equivalent fractions. A decimal fraction and a common fraction are merely different ways of expressing the same value.
Worked case 4.vi: Converting decimal fractions to common fractions
Write 4.08 as a common fraction.

Pace i: Employ the number of digits after the decimal point to determine which power of ten is the denominator.
At that place are ii digits after the decimal point. Therefore, the denominator is:

Step two: Starting from the first notzippo digit of the decimal fraction, write the digits without the decimal point. This becomes the numerator of the common fraction.

Step 3: Find the simplest form of the fraction obtained in Step 2.
Do iv.7: Convert decimal fractions to common fractions
Catechumen the following decimal fractions to common fractions.

1.05
There are 2 digits after the decimal signal:
In its simplest course:

0.128
In that location are 3 digits later on the decimal betoken:
In its simplest course:

0.0275
There are 4 digits afterwards the decimal betoken:
In its simplest class:
Converting common fractions to decimal fractions
When a common fraction has a denominator that is a ability of 10, it is easy to convert it to a decimal fraction. You dealt with the identify values of decimal fractions in Chapter 1.
\brainstorm{array}{cccccc} \hline \text{units} & \text{.} & \text{tenths} & \text{hundredths} & \text{thousandths} & \text{ten thousandths} \newline (1) & \text{.} & \left( \frac{one}{ten} \right) & \left( \frac{i}{100} \correct) & \left( \frac{1}{i,000} \correct) & \left( \frac{1}{ten,000} \right) \newline \text{u} & \text{.} & \text{t} & \text{h} & \text{th} & \text{tth} \newline \hline \end{array}
Consider the fraction . If the denominator is i,000, the corresponding decimal fraction must accept 3 digits later on the decimal point. We may write information technology in a identify value table.
\brainstorm{array}{cccccc} \hline (1) & \text{.} & \left( \frac{1}{10} \right) & \left( \frac{1}{100} \right) & \left( \frac{one}{1,000} \right) & \left( \frac{1}{x,000} \correct) \newline \text{u} & \text{.} & \text{t} & \text{h} & \text{th} & \text{tth} \newline \hline 0 & \text{.} & 0 & three & 0 & \newline \hline \end{assortment}
The identify value table shows that we have zero units, nothing tenths, three hundredths and null thousandths. Remember that nosotros do not write zeros at the cease of decimal fraction. This fraction is therefore written as 0.03.
Similarly, the common fraction may be expressed equally 0.30, which is written as 0.3. The common fraction may be expressed as iii.0, which is written equally 3.
Exercise 4.viii: Catechumen common fractions with denominators that are powers of ten to decimal fractions
Catechumen the following common fractions to decimal fractions.

At that place must exist three digits afterwards the decimal indicate: 0.506

There must exist 4 digits later the decimal indicate: 0.0029

The must be 2 digits after the decimal point: 7.20
This is written every bit: 7.2
When the denominator of a common fraction is not a power of ten, we apply division to convert the common fraction to a decimal fraction.
Worked example four.7: Converting mutual fractions to decimal fractions
Write as a decimal fraction.

Step 1: Split up the denominator into the numerator. If it cannot divide, write a zero.

Footstep ii: Put a decimal signal after the zero and after the numerator.

Step three: Add zeros later on the decimal point in the numerator.
The numbers v; 5.0; v.00; 5.000; v.0000 and and then on all correspond the value 5.

Step 4: Behave on with the sectionalization.

Step five: Write down the reply.
The sectionalization volition carry on forever, because we keep on getting the same remainder. This is called a recurring decimal. We write it with a dot above the digit that repeats.
Practice 4.ix: Convert common fractions to decimal fractions
Catechumen the following mutual fractions to decimal fractions.
4.5 Percentages
As well equally writing a fraction as a common fraction or a decimal fraction, we may also express a fraction as a
percentage. A percentage is a fraction in which the denominator is always 100. Only the numerator is written down, followed by a percentage symbol: %. The symbol shows that the number nosotros wrote down gives the parts of a whole that is divided into 100.
percentage
A percent is a fraction in which the denominator is 100 and where merely the numerator is written downwardly, followed past a pct symbol.
This figure is divided into 100 parts. There are 22 blocks that are shaded. The mutual fraction tells you lot what role is shaded. To express this as a percentage, we write only the numerator, followed by a percentage symbol: 22%.
If we shade all the blocks in the effigy, the shaded fraction will be . One full whole is therefore 100%.
Converting percentages to common fractions
Information technology is easy to convert percentages to common fractions. We simply employ the form . We know that the denominator is ever 100, then nosotros take . The numerator is the number before the percentage symbol.
For example, . The simplest form of this fraction is .
Exercise iv.10: Convert percentages to mutual fractions
Convert the following percentages to common fractions.

55%
In its simplest form:

lxxx%
In its simplest grade:

4%
In its simplest form:
Converting common fractions to percentages
To express a common fraction as a percentage, we need to know how many parts out of 100 we are dealing with. Consider the fraction . It shows that we are dealing with 2 parts of a whole that is divided into 5. This square has 100 small blocks in the background, only it is divided into 5 equal rectangles.
We may stand for the fraction by shading 2 of the five rectangles.
If you count the pocketsized blocks that are shaded, you will see that 40 out of the 100 pocketsized blocks are shaded. This means that . The fraction has a denominator of 100, so it may be written every bit a percentage: xl%.
Worked case 4.eight: Converting common fractions to percentages
Write as a percent.

Stride 1: Notice the LCM of 100 and the denominator.

Step 2: Detect the equivalent fraction in which the denominator is equal to the number obtained in Pace i.

Pace iii: Simplify the fraction from Footstep ii until its denominator is 100.
A common fraction cannot have a decimal fraction as its numerator or denominator. The “fraction” obtained in Pace iii is just used to assist us get to the answer.

Step 4: Write down the numerator from Pace three, followed by a percent symbol.
There are other ways to convert mutual fractions to percentages. You lot will learn more about this in Chapter 6.
Do four.11: Convert common fractions to percentages
Convert the postobit common fractions to percentages.



Use long division to determine the numerator:
We go along on getting the aforementioned remainder, so this partitioning produces a recurring decimal. We write a dot above the digit that repeats.
4.6 Practical applications
Fractions are very useful in everyday life. We utilize equivalent fractions to compare dissimilar fractions and to convert one type of fraction to another.
Exercise 4.12: Use equivalent fractions to solve issues

A instructor has 25 learners in her class. At break time, she decides to give each of them one quarter of an orange to eat. How many oranges does she need?
Parts in one whole: 4
Full parts: 25
She needs oranges.Whole number: 6
Fraction:
She needs oranges. 
Afterwards a altogether party, cakes are left over. If the leftover block is cut into equal pieces, how many people could get one slice each?
Parts in one whole: 5
Parts in ii cakes:
Boosted pieces: 2
Total pieces available: 
The floor of a room is tiled with 64 identical tiles. A carpet is put in the center of the room. The carpet covers of the tiled surface. How many tiles are under the carpet?
There are 24 tiles under the carpet.

A purse with 72 sweets is divided between Musa, Umar and Isa. Musa gets 27 sweets, Umar gets 12 sweets, and Isa gets the rest of the sweets. Determine which fraction of the bag each boy received.
Musa:
Umar:
Isa received sweets:

Atinuke and Eniola buy a bag of rice. Atinuke gets of the bag, Eniola gets of the bag and a little chip is left over. Who received the larger portion of rice, Atinuke or Eniola?
Eniola received the larger portion of the rice.

A wall with a surface surface area of 1 chiliad is painted. The painter knocks over the can of paint and some of the paint cannot be recovered. She is only able to paint 0.85 m of the wall. Express the fraction of the wall that was painted every bit a common fraction.
In that location are ii digits afterwards the decimal point:

Amadi has to walk one km to school. Later walking of the way, he realises that he forgot a volume at abode and he turns around. Determine the distance Amadi walked before he turned effectually. Give the reply in kilometres.
Amadi walked 0.625 km before he turned around.

Jummai got 88% for a class test. The test was for 25 marks. Calculate Jummai’s mark out of 25.
Jummai’s mark was 22 out of 25.

There are 30 students in a class. If 21 students are girls, what percentage of the class is female?
Fraction of the class that is female person:
70% of the class is female.

A children’south article of clothing store advertises a 15% disbelieve on a clothes of ₦3,000. Calculate the value of the disbelieve.
The disbelieve is ₦450.
4.vii Summary
 A common fraction is a fraction that is written in the grade , where the numerator and denominator are both whole numbers.
 A proper fraction is smaller than one. The numerator is smaller than the denominator, for example .
 A mixed number consists of a whole number and a proper fraction, for example .
 An improper fraction is larger than 1. The numerator is larger than the denominator, for example .
 Equivalent fractions have unlike numerators and denominators, only they represent the same role of a whole and therefore the aforementioned value. For example, .
 A fraction is in its simplest class when the numerator and denominator take no common factor.
 To be able to compare fractions, we must express them with the aforementioned denominators. The everyman mutual denominator (LCD) of two or more fractions is the smallest whole number into which all the denominators tin can divide without a remainder.
 A decimal fraction is a fraction expressed in decimal notation, where the numerator is written after a decimal betoken and the denominator is non written downwards. The denominator in a decimal fraction must be a power of x.
 In decimal note, the decimal point separates whole numbers from fractions.
 In decimal notation, the number of digits after the decimal betoken shows united states which power of ten is the fraction’southward denominator. For case, if there are three digits after the decimal point, the denominator is .
 A per centum is a fraction of which the denominator is 100 and where only the numerator is written down, followed by a percentage symbol.
Which Diagram Represents a Fraction Equivalent to 40
Source: https://www.siyavula.com/read/maths/jss1/equivalentfractions/04equivalentfractions?id=45percentages