Relationships between Fractions, Decimals, Ratios and Percentages (2022)

Fractions, decimals and percentages are related and can be used to express the same number, or proportion in different ways.

Understanding 1

Relating Decimals, Fractions and Percents

The learning activities in the previous two modules focussed on numbers represented as fractions , decimals and ratios. This module focuses on percents, another way of representing rational numbers.

Any rational number, whether a fraction or a whole number, can be written as a fraction, decimal or percent.

The term percent is simply another name for hundredths and as such percents are rational numbers with a denominator of 100. For example, 25% (twenty five per cent) is the same asRelationships between Fractions, Decimals, Ratios and Percentages (1)(twenty five hundredths). 25% orRelationships between Fractions, Decimals, Ratios and Percentages (2)can also be written in decimal notation as 0.25 (zero point two five).

By the end of this module you should be able to fill in a chart similar to this one.

Number

Fraction

Decimal

Percent

five

Relationships between Fractions, Decimals, Ratios and Percentages (3)

5.0

500%

two and one eighth

Relationships between Fractions, Decimals, Ratios and Percentages (4)

2.125

212.5%

three quarters

Relationships between Fractions, Decimals, Ratios and Percentages (5)

0.75

75%

Learning Activity 1

Relating Decimals, Fractions and Percent

Please go to the link below and complete the activities suggested below.

Math Is Fun Virtual Manipulative

Activities to demonstrate the relationship between fractions, decimals and percents, and reinforce and extend your understandings of percents being another way to represent fractions:

1. Place your curser on the pizza at the 3 o’clock position, or 90 degrees. At this position one whole pizza is shown, the 100 grid is fully shaded (100 percent or 100%) and the number one is identified on the zero to one number line.

Relationships between Fractions, Decimals, Ratios and Percentages (6)(one whole) = 100% (one hundred out of one hundred equal parts) = 1

2. By rotating the curser in an anti-clockwise direction around the pizza, shading the grid or moving along the number line, you can select a portion of the pizza.

Shade one of the 100 squares on the grid. This is one out of 100 equal parts, therefore, 1% (per cent) of the grid. Notice that one hundredth of the pizza appears, and the pointer is a very small distance past zero on the number line.

Imagine the number line from zero to 1 divided into one hundred equal parts. One of these parts is one hundredth, or 0.01. This part is also one tenth of a tenth, or one tenth of 0.1.

Relationships between Fractions, Decimals, Ratios and Percentages (7)(one one-hundredth) = 1% = 0.01 (zero point zero one)

3. Highlight the top row of the grid, that is, ten of the one hundred squares. You have highlighted one tenth of the square, and will notice that one out of ten equal parts of the pizza Relationships between Fractions, Decimals, Ratios and Percentages (8) has appeared. The pointer shows one tenth, or 0.1 (zero point one) on the number line. This can also be written as 0.10, showing that one tenth is exactly the same as ten hundredths. Relationships between Fractions, Decimals, Ratios and Percentages (9)and 10/100 are equivalent fractions (add link – FDRP LO1).

Relationships between Fractions, Decimals, Ratios and Percentages (10)(one tenth) = 10% = 0.1 (zero point one)

4. Move your curser to show:

Relationships between Fractions, Decimals, Ratios and Percentages (11)(one half) = 50% = 0.5 (zero point five) or 0.50
Relationships between Fractions, Decimals, Ratios and Percentages (12)(one quarter) = 25% = 0.25 (zero point two five)
Relationships between Fractions, Decimals, Ratios and Percentages (13)(seven hundredths) = 7% = 0.07 (zero point zero seven)
Relationships between Fractions, Decimals, Ratios and Percentages (14)(three quarters) = 75% = 0.75 (zero point seven five)
Relationships between Fractions, Decimals, Ratios and Percentages (15)(seven hundredths) = 7% = 0.07 (zero point zero seven)
Relationships between Fractions, Decimals, Ratios and Percentages (16)(nine tenths) = 90% = 0.9 (zero point nine) or 0.90
Relationships between Fractions, Decimals, Ratios and Percentages (17)(ninety nine hundredths) = 99% = 0.99 (zero point nine nine)

Understanding 2

Representing Decimals to Thousandths

A one thousand grid can be used to represent one whole (1), and to demonstrate decimals up to thousandths.

Relationships between Fractions, Decimals, Ratios and Percentages (18)

The entire grid represents one (1), or one whole.

(Video) Converting Between Fractions, Decimals, and Percentages

The grid can be divided into 10 equal parts, or tenths. One of these ten equal parts, or one tenth of the grid (Relationships between Fractions, Decimals, Ratios and Percentages (19)), is shaded in red.

One tenth, the red portion, can be divided into ten equal parts (the yellow section shows this). The yellow portion is a one hundredth (Relationships between Fractions, Decimals, Ratios and Percentages (20)) as 100 of these make the whole.
A one hundredth (Relationships between Fractions, Decimals, Ratios and Percentages (21)), the yellow portion, can also be divided into ten equal parts (the blue section shows this). The blue portion is represents one thousandths (Relationships between Fractions, Decimals, Ratios and Percentages (22)) of the whole, as 1000 of these thousands makes the whole.

The following statements can be made:

  • The red area is one tenth (Relationships between Fractions, Decimals, Ratios and Percentages (23)) or zero point one (0.1) of the whole grid

  • The yellow area is one hundredth (Relationships between Fractions, Decimals, Ratios and Percentages (24)) or zero point zero one (0.01) of the whole grid

  • The blue area is one thousandth (Relationships between Fractions, Decimals, Ratios and Percentages (25)) or zero point zero zero one (0.001) of the whole grid

  • 10 tenths equal one whole (Relationships between Fractions, Decimals, Ratios and Percentages (26)= 1)

  • 100 hundred hundredths equals one whole (Relationships between Fractions, Decimals, Ratios and Percentages (27)= 1)

  • 1000 thousandths equals one whole (Relationships between Fractions, Decimals, Ratios and Percentages (28)= 1)

  • 10 hundredths (yellows) equal one tenth (red)

  • Relationships between Fractions, Decimals, Ratios and Percentages (29); 0.10 = 0.1

  • 10 thousandths (blues) equal one hundredth (yellow)

  • Relationships between Fractions, Decimals, Ratios and Percentages (30); 0.010 = 0.01

  • 100 thousandths (blues) = one tenth(red)

  • Relationships between Fractions, Decimals, Ratios and Percentages (31);0.100 = 0.1

  • The shaded area of the grid is one hundred and eleven thousands (Relationships between Fractions, Decimals, Ratios and Percentages (32)) of the grid, which can also be expressed as the decimal fraction zero point one one one (0.111)

  • In expanded form or expanded notation this is:

One tenth + one hundredth + one thousandth
(Relationships between Fractions, Decimals, Ratios and Percentages (33)) or (0.1 + 0.01 + 0.001) or (Relationships between Fractions, Decimals, Ratios and Percentages (34)) or (Relationships between Fractions, Decimals, Ratios and Percentages (35))

Learning Activity 2

One Thousand Grid: A visual model for decimal fractions

The following video uses a thousandths grid in a similar way, to demonstrate writing decimal fractions:


The second example in the video focuses on the shaded area being 500 one thousandths of a whole 1000 (comprising one thousandths). It is written asRelationships between Fractions, Decimals, Ratios and Percentages (36)or 0.500.
It is easy to see that this shaded area is one half of the whole grid.

This shaded area can also be broken up into 50 hundredths. The fraction 50 hundredths (Relationships between Fractions, Decimals, Ratios and Percentages (37)) is equivalent to 500 thousandths (Relationships between Fractions, Decimals, Ratios and Percentages (38)).
Furthermore, the shaded area in the video can be broken up into five tenths. The fraction five tenths (Relationships between Fractions, Decimals, Ratios and Percentages (39)) is equivalent to the fraction 50 hundredths (Relationships between Fractions, Decimals, Ratios and Percentages (40)) and 500 thousandths (Relationships between Fractions, Decimals, Ratios and Percentages (41)).
All of these fractions have the same value of one half (Relationships between Fractions, Decimals, Ratios and Percentages (42)), and so they are equivalent fractions.

Relationships between Fractions, Decimals, Ratios and Percentages (43)

Decimal notation

0.5 = 0.50 = 0.500

The decimal notation does not require the zeros after the five. Unlike whole numbers, a zero on the end (right hand side) does not change the value of the decimal. However, the zeros can sometimes assist when adding and subtracting decimals.

Learning Activity 3

Fractions Greater than One

Click on the following link from Illuminations Resources for Teaching Math:

FRACTION MODELS

Follow these instructions:

  1. Select the tab ‘wide range’ at the top of the display screen. This sets the numerator range at the bottom of the screen as 0 – 100, and the denominator range at 1 – 25. The fractions will therefore be improper, or greater than 1, because the numerator will be greater than the denominator.
  2. Select the ‘area’ model option, located on the right hand side under the table. Use the plus and minus tabs either side of the numerator and denominator settings to select a numerator of 5 and a denominator of 3. You will see five thirds represented on the area model on the screen. Above this you will see how this number is expressed as a fraction (or improper fraction)Relationships between Fractions, Decimals, Ratios and Percentages (44), a mixed number (Relationships between Fractions, Decimals, Ratios and Percentages (45)) , a decimal (1.6667), and a percent (166.67%). Note that the decimal and percent have been rounded up; otherwise they would go on forever.
    Look at the different models (length, area, region, set).
  3. Try other numbers greater than one, looking at the different visual representations. Note how they are expressed in improper fractions, mixed numbers, decimals and percents.

Understanding 3

Relating Decimals, Fractions and Percent using a Number Line

The number line below is marked in increments of one hundredths from zero to 0.36. Notice where the following decimal numbers, all containing similar digits but in different places, are placed on the number line:

0.257

0.05

0.023

0.307

0.175

0.12

Relationships between Fractions, Decimals, Ratios and Percentages (46)

(Video) Relationship of Percent, Fractions, Ratios and Decimals

The decimal 0.023 has a zero in the tenths place, so it is less than one tenth (0.1). It decimal 0.023 has two hundredths. It also has 3 thousandths, so it is just past the 2 hundredths mark (three tenths past the mark).

The decimal 0.05 has a zero in the tenths place, so it is less than one tenth (0.1). is bigger than 0.023, as it has more hundredths.

The decimal is 0.12 has 1 tenth and 2 hundredths, so is two hundredths past the one tenth (0.1) mark.

The decimal 0.175 is also between 0.1 and 0.2, but it is closer to 0.2 because it has seven tenths. It is half way between the seven and eight tenths marks because it also has 5 thousandths.

The decimal 0.257 is between 0.2 and 0.3. It has five hundredths, so it is about half way between 0.2 and 0.3. It also has 7 thousandths, so it is just past half way between 0.2 and 0.3.

The decimal 0.302 is only 2 thousandths more than 0.3, so it is only slightly past the 0.3 mark.

This time the three different representations of rational numbers, fractions, decimals and percents, have been placed on a blank number line.

15%

0.28

Relationships between Fractions, Decimals, Ratios and Percentages (47)

70%

0.115

1

Relationships between Fractions, Decimals, Ratios and Percentages (48)

Relationships between Fractions, Decimals, Ratios and Percentages (49)

0.3

Relationships between Fractions, Decimals, Ratios and Percentages (50)

15%

0.28

Relationships between Fractions, Decimals, Ratios and Percentages (51)

70%

0.115

1

Relationships between Fractions, Decimals, Ratios and Percentages (52)

Relationships between Fractions, Decimals, Ratios and Percentages (53)

0.3

Relationships between Fractions, Decimals, Ratios and Percentages (54)

Relationships between Fractions, Decimals, Ratios and Percentages (55)

Relationships between Fractions, Decimals, Ratios and Percentages (56)

Relationships between Fractions, Decimals, Ratios and Percentages (57)

Relationships between Fractions, Decimals, Ratios and Percentages (58)*

Relationships between Fractions, Decimals, Ratios and Percentages (59)

Relationships between Fractions, Decimals, Ratios and Percentages (60)*

Relationships between Fractions, Decimals, Ratios and Percentages (61)

Relationships between Fractions, Decimals, Ratios and Percentages (62)

* Relationships between Fractions, Decimals, Ratios and Percentages (63)andRelationships between Fractions, Decimals, Ratios and Percentages (64)are close approximations. The decimal 0.115 is actuallyRelationships between Fractions, Decimals, Ratios and Percentages (65)and 5 thousandths andRelationships between Fractions, Decimals, Ratios and Percentages (66)is a little more thanRelationships between Fractions, Decimals, Ratios and Percentages (67), because it is 0.3333333.

Examples of how percentages are used in real life

Example 1

(Video) Fractions, Decimals and Percentages

There is a sale on homewares at a department store with 25% of selected items. A dinner set before the sale cost $130.
What will it cost you now?Solution: We recognise that 25% is equal toRelationships between Fractions, Decimals, Ratios and Percentages (68). We can then work outRelationships between Fractions, Decimals, Ratios and Percentages (69)of $130 which is $32.50.
(we know this because half of 130 is 65 and half of 65 is 32.5. This is the same as dividing 130 by 4).
Therefore you can purchase the dinner set for $130 - $32.5 = $97.50

Example 2

A property that was on the market for $450,000 last year has decreased in value by 10%. How much will you save by buying it now?Solution: We recognise that 10% is the same asRelationships between Fractions, Decimals, Ratios and Percentages (70). NowRelationships between Fractions, Decimals, Ratios and Percentages (71)of $450,000 is $45,000. Therefore you would save $45,000 by buying the property now.

(Note that a 10% discount off a small item such as a $20 t-shirt amounts to just a few dollars, in this case $2. Whereas a 10% discount of a $450,000 property is a very worthwhile $45,000. So the significance of what a 10% discount might mean to us depends on the whole we started with).

Common Misconceptions for Decimals and Fractions

Decimals stop at hundredths - NO

Examples of decimals beyond hundredths:

A millimetre (mm) is one thousandth of a metre

1mm = 0.001m

2.44 micrograms is equal to 0.00244 milligrams.

Common Misconceptions for Ordering Fractions

1. The larger the denominator, the bigger the fraction

This is true for unit fractions (fractions with a numerator of one). There is an inverse relationship between the number of parts and the size of each part: The larger the number of parts (the denominator), the smaller the size of each part (the numerator). Unless the problem context indicates that two fractions relate to different wholes, we assume both relate to the same whole. With this in mind, it makes sense that the more parts into which the whole is divided, the smaller they will be.

Example: Compare one eighthRelationships between Fractions, Decimals, Ratios and Percentages (72)to one fifthRelationships between Fractions, Decimals, Ratios and Percentages (73)
If we are referring to the same whole, such as a portion of a cake (modelled below), we can see that the more parts into which it is divided, the smaller each part will be.

In the visual representation, we can clearly see thatRelationships between Fractions, Decimals, Ratios and Percentages (74)is larger thanRelationships between Fractions, Decimals, Ratios and Percentages (75).

Relationships between Fractions, Decimals, Ratios and Percentages (76)


Five people sharing above cake, soRelationships between Fractions, Decimals, Ratios and Percentages (77)each


Relationships between Fractions, Decimals, Ratios and Percentages (78)

Eight people sharing same sized cake, soRelationships between Fractions, Decimals, Ratios and Percentages (79)each.

When we are comparing just one of each part, such as one eighthRelationships between Fractions, Decimals, Ratios and Percentages (80)to one fifth (Relationships between Fractions, Decimals, Ratios and Percentages (81)), the bigger the denominator, the smaller each part will be.

The numerator is one (Relationships between Fractions, Decimals, Ratios and Percentages (82))

When one or both fractions are not unit fractions:

This time, we will compare one fifth(Relationships between Fractions, Decimals, Ratios and Percentages (83)), and three eighths (Relationships between Fractions, Decimals, Ratios and Percentages (84)). We know that eighths are smaller than fifths, but we must note that this time there are three eighths, not just one.
In the diagram below, we can see thatRelationships between Fractions, Decimals, Ratios and Percentages (85)is a bigger portion thanRelationships between Fractions, Decimals, Ratios and Percentages (86).

Relationships between Fractions, Decimals, Ratios and Percentages (87)

Person A ate one fifth (Relationships between Fractions, Decimals, Ratios and Percentages (88)) of the cake.


Relationships between Fractions, Decimals, Ratios and Percentages (89)

Person B ate three eighths (Relationships between Fractions, Decimals, Ratios and Percentages (90)) of the cake.

If we cannot reliably compare the fractions with different denominators visually, as in the diagram above, we need to change one or both of the fractions into equivalent fractions for a common denominator.
It is easy to recognise that four fifths (Relationships between Fractions, Decimals, Ratios and Percentages (91)) is greater than two fifths (Relationships between Fractions, Decimals, Ratios and Percentages (92)), because each of the parts (fifths) is the same size. Four is greater than two, soRelationships between Fractions, Decimals, Ratios and Percentages (93)must be greater than Relationships between Fractions, Decimals, Ratios and Percentages (94).
What about comparing four fifths (Relationships between Fractions, Decimals, Ratios and Percentages (95)) and seven tenths (Relationships between Fractions, Decimals, Ratios and Percentages (96)), which have different denominators?
As was seen on the fraction wall, each fifth is equivalent to two tenths. This is demonstrated in the model below:

Relationships between Fractions, Decimals, Ratios and Percentages (97)

Changing four fifths (Relationships between Fractions, Decimals, Ratios and Percentages (98)) to the equivalent fraction eight tenths (Relationships between Fractions, Decimals, Ratios and Percentages (99)) makes it much easier to see that four fifths (Relationships between Fractions, Decimals, Ratios and Percentages (100)) is greater than seven tenths (Relationships between Fractions, Decimals, Ratios and Percentages (101)).

Practice Task 1

1)Complete the table so that the numbers in each row represented by fractions, decimals and percents are equivalent:

Fraction

Decimal

Percent

Relationships between Fractions, Decimals, Ratios and Percentages (102)

1.1

110%

Relationships between Fractions, Decimals, Ratios and Percentages (103)

0.04

(Video) Ratios, Fraction, Decimals, Percents

25%

Relationships between Fractions, Decimals, Ratios and Percentages (104)

350%

0.125

2) Order the following numbers from smallest to largest:

0.125

Relationships between Fractions, Decimals, Ratios and Percentages (105)

1.5

1.45

0.25

Relationships between Fractions, Decimals, Ratios and Percentages (106)

0.81

Relationships between Fractions, Decimals, Ratios and Percentages (107)

0.09

1.1065

3)Write at least four equivalent fractions for each of the following fractions:

Relationships between Fractions, Decimals, Ratios and Percentages (108)

Relationships between Fractions, Decimals, Ratios and Percentages (109)

Relationships between Fractions, Decimals, Ratios and Percentages (110)

Click here (PDF 362.6 KB) to check your answers

Practice Task 2

1) Relating decimals, fractions and percent using a number line

Click on the link below and complete the activity by placing all of the fractions, decimals and percents on the number lines from ICT games.

Equivalence of Fractions, Decimals and Percents

2)Place the following fractions, decimals and percents on a single number line:

10%

0.375

Relationships between Fractions, Decimals, Ratios and Percentages (111)

Relationships between Fractions, Decimals, Ratios and Percentages (112)

50%

1.3

128%

0.002

Relationships between Fractions, Decimals, Ratios and Percentages (113)

Relationships between Fractions, Decimals, Ratios and Percentages (114)


3)Take a look in the day's newspaper and highlight every time percentages are referred to or used. This is an activity that students could do as well.

Click here (PDF 275.9 KB) to check your answers

Check your understanding of Big Idea 3

The purpose of this big idea was to demonstrate the following understandings;

  1. A number can be represented as a fraction or a decimal.
  2. A percent is a fraction out of one hundred and are a very commonly used in everyday life. Percents can also be understood as hundredths

Does this make sense to you now?

(Video) FRACTIONS, RATIOS, DECIMALS, PERCENTAGES

Please proceed to Big Idea 4 in the Relationships between Fractions, Decimals, Ratios and Percentages module.

FAQs

What is the relationship between fractions decimals and percentages? ›

To a fraction: Read the decimal and reduce the resulting fraction. To a decimal: Move the decimal point 2 places to the left and drop the % symbol. To a percent: Convert the fraction first to a decimal, then move the decimal point 2 places to the right and add the % symbol.

What is the relationship between ratios and percentages? ›

Percent means hundredths or per hundred and is written with the symbol, %. Percent is a ratio were we compare numbers to 100 which means that 1% is 1/100.

What is the importance of having a good understanding of fractions decimals and percent? ›

Being able to convert between fractions, decimals, and percentages is an essential skill at Key Stage 3 as it greatly develops a student's concept of what the quantities mean. It also makes fractions and percentages of amounts much simpler to visualise and compare.

How do you teach relationships between fractions and decimals? ›

Number one two hundredths expressed as a decimal is zero point zero two number two zero point four

What is the relationship between a ratio and a fraction? ›

FractionRatio
A fraction is defined as a part of a whole.Ratio compares the size of two or more quantity in relation to each other.
A fraction is used to show a part of or compare to something.It shows the relation among items.
1 more row

What is the relationship between the numerator and denominator? ›

In a fraction, the denominator represents the number of equal parts in a whole, and the numerator represents how many parts are being considered. You can think of a fraction as p/q is as p parts, which is the numerator of a whole object, which is divided into q parts of equal size, which is the denominator.

Videos

1. Ratios, Fractions and Percentage Problems! Common Exam Questions!! | Grade 5+ | GCSE Maths Tutor
(The GCSE Maths Tutor)
2. 4.5 Relationships between ratios, rates and proportions with fractions, decimals and percentages
(Muhammad Syafiq)
3. Relationship of Percent, Fractions, Ratios and Decimals
(Teacher Lenny Nicanor)
4. Recognising the relationship between Fractions, Decimals and Percentages - Pictorial
(HG Walker)
5. Relationships Between Fractions, Decimals and Percent
(TrueMaths)
6. Relationship between Fractions, Decimals and Percentages
(Mr Greene)

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