Fractions, decimals and percentages are related and can be used to express the same number, or proportion in different ways.
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 as(twenty five hundredths). 25% orcan 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.
two and one eighth
Learning Activity 1
Relating Decimals, Fractions and Percent
Please go to the link below and complete the activities suggested below.
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.
(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.
(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 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. and 10/100 are equivalent fractions (add link – FDRP LO1).
(one tenth) = 10% = 0.1 (zero point one)
4. Move your curser to show:
(one half) = 50% = 0.5 (zero point five) or 0.50
(one quarter) = 25% = 0.25 (zero point two five)
(seven hundredths) = 7% = 0.07 (zero point zero seven)
(three quarters) = 75% = 0.75 (zero point seven five)
(seven hundredths) = 7% = 0.07 (zero point zero seven)
(nine tenths) = 90% = 0.9 (zero point nine) or 0.90
(ninety nine hundredths) = 99% = 0.99 (zero point nine nine)
Representing Decimals to Thousandths
A one thousand grid can be used to represent one whole (1), and to demonstrate decimals up to thousandths.
The entire grid represents one (1), or one whole.
The grid can be divided into 10 equal parts, or tenths. One of these ten equal parts, or one tenth of the grid (), 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 () as 100 of these make the whole.
A one hundredth (), the yellow portion, can also be divided into ten equal parts (the blue section shows this). The blue portion is represents one thousandths () of the whole, as 1000 of these thousands makes the whole.
The following statements can be made:
The red area is one tenth () or zero point one (0.1) of the whole grid
The yellow area is one hundredth () or zero point zero one (0.01) of the whole grid
The blue area is one thousandth () or zero point zero zero one (0.001) of the whole grid
10 tenths equal one whole (= 1)
100 hundred hundredths equals one whole (= 1)
1000 thousandths equals one whole (= 1)
10 hundredths (yellows) equal one tenth (red)
; 0.10 = 0.1
10 thousandths (blues) equal one hundredth (yellow)
; 0.010 = 0.01
100 thousandths (blues) = one tenth(red)
;0.100 = 0.1
The shaded area of the grid is one hundred and eleven thousands () 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
() or (0.1 + 0.01 + 0.001) or () or ()
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 asor 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 () is equivalent to 500 thousandths ().
Furthermore, the shaded area in the video can be broken up into five tenths. The fraction five tenths () is equivalent to the fraction 50 hundredths () and 500 thousandths ().
All of these fractions have the same value of one half (), and so they are equivalent fractions.
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:
Follow these instructions:
- 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.
- 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), a mixed number () , 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).
- 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.
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:
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.
* andare close approximations. The decimal 0.115 is actuallyand 5 thousandths andis a little more than, because it is 0.3333333.
Examples of how percentages are used in real life
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 to. We can then work outof $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
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 as. Nowof $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 eighthto one fifth
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 thatis larger than.
Five people sharing above cake, soeach
Eight people sharing same sized cake, soeach.
When we are comparing just one of each part, such as one eighthto one fifth (), the bigger the denominator, the smaller each part will be.
The numerator is one ()
When one or both fractions are not unit fractions:
This time, we will compare one fifth(), and three eighths (). 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 thatis a bigger portion than.
Person A ate one fifth () of the cake.
Person B ate three eighths () 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 () is greater than two fifths (), because each of the parts (fifths) is the same size. Four is greater than two, somust be greater than .
What about comparing four fifths () and seven tenths (), 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:
Changing four fifths () to the equivalent fraction eight tenths () makes it much easier to see that four fifths () is greater than seven tenths ().
Practice Task 1
1)Complete the table so that the numbers in each row represented by fractions, decimals and percents are equivalent:
(Video) Ratios, Fraction, Decimals, Percents
2) Order the following numbers from smallest to largest:
3)Write at least four equivalent fractions for each of the following fractions:
Click here (PDF 362.6 KB) to check your answers
Practice Task 2
1) Relating decimals, fractions and percent using a number line
2)Place the following fractions, decimals and percents on a single number line:
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;
- A number can be represented as a fraction or a decimal.
- 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?
Please proceed to Big Idea 4 in the Relationships between Fractions, Decimals, Ratios and Percentages module.
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.
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.
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.
Number one two hundredths expressed as a decimal is zero point zero two number two zero point four
|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.|
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.