My bestie, Detan was once laughed at in a Maths class for interchanging decimals and percentages (I too was secretly guilty). Not to worry, herein, you would understand the relationship between fractions, decimals, and percentages - very similar concepts.
In this article, we will explore the concepts of fractions, decimals, and percentages, examples of each, and how to convert from one form to another.
What are fractions, decimals, and percentages? To start with, let us recall the definitions of these three concepts - fractions, decimals and percentages.
What is a fraction?
A fraction is as a portion or part of a whole number, expression, or anything other than zero.
Visually, a fraction is represented as two numbers or expressions separated by a slash or the over sign.
The slash or over symbol means "divided by".
Fig. 1. An illustration of fractions with a pie chart.
The number or expression above the slash is called the numerator while the number or expression below the slash is called the denominator:
\[\text{fraction} = \frac{\text{numerator}}{\text{denominator}}.\]
Proper and Improper Fractions
A fraction can amount to more than a whole (improper fraction), or just to a portion of the whole (proper fraction).
A proper fraction is a fraction in which the numerator is smaller than the denominator.
Let's have a look at some examples:
\(\frac{1}{2}, \frac{2}{3}, \frac{3}{8}\) are all proper fractions.
Proper fractions amount always to less than a whole.
An improper fraction is a fraction in which the numerator is greater than or equal to the denominator.
Let's have a look at some examples:
\(\frac{3}{2}, \frac{3}{3}, \frac{8}{3}\) are all improper fractions.
Proper fractions amount always to a whole or more.
Also, they are often converted to mixed numbers. For example,
\[\frac{3}{2} = 1\frac{1}{2}\]
Observe that proper fractions are less than 1 while improper fractions are more than 1.
Converting improper to mixed fractions
In other to convert an improper fraction to a mixed fraction, you should follow these few steps.
Step 1. Divide the numerator by the denominator.
Step 2. Place your whole number from the division towards the left, then towards your right, and put the remainder as the new numerator while retaining the former denominator.
Step 3. Verify that no factor is common between the new numerator and the denominator. If they share a common factor, then simplify by that factor until there is no common factor between both.
You should see the example below to get a clearer picture of it.
Convert the following to mixed fractions.
a. \(\frac{5}{3}\)
b. \(\frac{27}{6}\)
Solution
a. Converting \(\frac{5}{3}\) to mixed fraction.
Step 1: Divide the numerator by the denominator.
\[5\div 3=1\, r\, 2\]
where \(r\) signifies 'remainder'.
Step 2: since your whole number is \(1\) and your remainder is \(2\), you can place \(1\) at the left side. Then \(2\) becomes the new numerator while \(3\) is retained as the denominator, so that you have
\[\frac{5}{3}=1\frac{2}{3}\]
Step 3: Verify that no factor is common between the new numerator and the denominator. \(2\) and \(3\) are both prime numbers hence without a common factor.
Therefore, the improper fraction, \(\frac{5}{3}\), has the mixed fraction, \(1\frac{2}{3}\).
b. Converting \(\frac{27}{6}\) to mixed fraction.
Step 1: Divide the numerator by the denominator.
\[27\div 6=4\, r\, 3\]
where \(r\) signifies 'remainder'.
Step 2: since your whole number is \(4\) and your remainder is \(3\), you can place \(4\) at the left side. Then \(3\) becomes the new numerator while \(6\) is retained as the denominator, so that you have
\[\frac{27}{6}=4\frac{3}{6}\]
Step 3: Verify that no factor is common between the new numerator and the denominator. Between the numerator, \(3\), and the denominator, \(6\), there is common factor which is \(3\). Hence, simplify \(\frac{3}{6}\) by dividing by \(3\) to arrive at
\[4+\frac{3\div 3}{6\div 3}=4\frac{1}{2}\]
Since the current numerator and denominator cannot be further divided, therefore, the improper fraction, \(\frac{27}{6}\), has the mixed fraction, \(4\frac{1}{2}\).
What is a decimal?
A decimal number is a non-integer number with digits after a decimal point. These digits amount to part of a whole quantity.
Usually, we work with numbers in the "Base Ten Number System", in which numbers are expressed as combinations of the digits between 0 and 9.
Other number systems exist: for example, computers work with the "Binary System", in which every number is expressed as a combination of 0s and 1s.
Fig. 2. Using a number line to illustrate decimals.
A decimal is marked with a decimal point. Numbers before the decimal point (left-hand side) are whole numbers while those after the decimal point (right-hand side) are parts of a whole:
0.5, 2.46, and 0.0057 are decimal numbers.
In the decimal 2.46, 2 is a whole number and is an integer while 0.4 (4 tenths) and 0.06 (6 hundredths) are non-integers because they are just parts of a whole number.
What is a percentage?
A percentage can be defined as the ratio or a part of a number in a hundred.
Fig. 3. An illustration of
percentage using a pie chart.
Percentage responds to how much of a quantity can be found in a 100 of another quantity.
If you think of the word 'percent' literally, PER-CENT. CENT in numbers means 100, PER means "in every..." Thus, combine the meaning of both and it just means "in every 100".
Hence, a number in percentage means that the number is divided by 100.
Percentage is represented with the symbol % written in front of numbers.
The % sign can be translated to a 1 represented by a slash and two 0s beside the upper left and lower right of the slash; 1 and two 0s - 100 . Don't forget the slash is a division sign .
\(50\%\), \(40\%\), \(13\%\) are examples of percentages.
Actually \(50\%\) means
\[50\%=\frac{50}{100}=\frac{1}{2}\]
Likewise, \(40\%\) means
\[40\%=\frac{40}{100}=\frac{2}{5}\]
and \(13%\) means
\[13\%=\frac{13}{100}\]
What is the relationship between fractions, decimals, and percentages?
Fraction, decimals, and percentages are all used to represent how much a portion is, compared to a whole. However, they are represented differently. Fractions and decimals define the part of a whole to the lowest term; on the other hand, percentages define the part of a whole with respect to 100.
Furthermore, when represented, fractions and decimals are not mixed up with other forms (such as fraction, decimal or percentage). However, percentages can be represented in the form of fractions or decimals where they could have a fraction part like \(53\frac{2}{3}\%\); or percentages may have a decimal part such as \(53.67\%\).
Fractions and percentages
Fractions and percentages are very related. All fractions can be converted to percentages as well as all percentages being able to be converted to fractions. These two are just ways of expressing parts out of a whole.
How do we convert fractions to percentages?
If you wish to change fractions to percentages, multiply the fraction by 100%. Recall that any number with the percentage symbol means the number is divided by 100; so 100% is actually equal to 1 because:
\[100\% \frac{100}{100} = 1\]
Therefore, note that multiplying by 100% does not change the real value of the fraction: it only changes the way it is being expressed.
For a matter of organisation, follow these steps:
Step 1: Multiply by 100%;
Step 2: Simplify the result obtained in step 1, until you go no further.
See the below examples for proper guidance.
Convert the following fractions to percentages:
a) \(\frac{1}{4}\)
b) \(2 \cdot \frac{1}{5}\)
Solution
a) Step 1: Multiply the fraction by 100%.
\[\frac{1}{4} \cdot 100\%\]
b) Step 2: Simplify the previous result, by dividing it by 4
\[\frac{1}{4} \cdot 100\% = \frac{100}{4} \% = \frac{\cancel{100}}{\cancel{4}} \% = 25\%\]
b) When you are to convert a mixed fraction to a percentage, you have an extra step. Firstly, convert the mixed fraction to an improper fraction:
\[2 \cdot \frac{1}{5} = \frac{(2 \cdot 5) +1}{5} = \frac{11}{5}\]
Now to the steps.
Step 1: Multiply the improper fraction by 100%; \(\frac{11}{5} \cdot 100\%\)
Step 2: Simplify by dividing by 5;
\(\frac{11}{\cancel{5}}\cdot \cancel{100}\% = 11 \cdot 20\%= 220\%\)
How do you convert percentages to fractions?
When you are to convert percentages to fractions, you simply do these two steps:
Step 1: divide the percentage by 100%, and
Step 2: simplify.
Convert the following percentages to fractions:
a) 40%
b) 120%
Solution
a) Step 1: Divide the percentage by 100%.
\(\frac{40\%}{100\%}\)
Step 2: Simplify until you can divide no further.
\(\frac{\cancel{40\%}}{\cancel{100\%}} = \frac{4}{10} = \frac{\cancel{4}}{\cancel{10}} = \frac{2}{5}\)
b) Step 1: Divide the percentage by 100%.
\(\frac{120\%}{100\%}\)
Step 2: Simplify until you can divide no further.
\(\frac{\cancel{120\%}}{\cancel{100\%}} = \frac{12}{10} = \frac{\cancel{12}} {\cancel{10}} = \frac{6}{5} = 1 \cdot \frac{1}{5}\)
Fractions and decimals
Fractions and decimals are similar. They just differ in the mode of expression of numbers although they both show parts out of a whole.
How can you convert fractions to decimals?
Fractions are easily converted to decimals by dividing directly with the appropriate placement of decimal points or through the use of the long division method. But we shall apply only the direct division approach.
Direct division
In this approach, when the numerator is less than the denominator and divided, a 0 is written, and a decimal point is placed after it. After that, the division continues in that manner.
However, once the decimal point is placed, you cannot have another decimal point placed again peradventure you divide another number less than the denominator. All you should do is add a 0 in front each time this happens.
Convert the following to decimal:
a) \(\frac{1}{5}\)
b) \(\frac{1}{8}\)
Solution
a)\(\frac{1}{5}\)
The numerator 1 is less than the denominator 5. So you add a 0 in front of 1 making it 10 but place a 0 and decimal point after it above the fraction as seen below,
\[0.\frac{10}{5}\]
Now the numerator is large enough to be divided by the denominator, hence you can divide;
\[0.\frac{10}{5} = \frac{\cancel{10}}{\cancel{5}}\]
Place your answer after the decimal point. Continue dividing if there is a remainder; but in this case, there is no remainder. Therefore:
\[0.2 \frac{1}{5}\]
So our answer is 0.2.
b) \(\frac{1}{8}\)
The numerator 1 is less than the denominator 5. So you add a 0 in front of 1 making it 10 but place a 0 and decimal point after it above the fraction as seen below;
\[0.\frac{10}{8}\]
10 divided by 8 is 1 remainder 2; write the 1 after the decimal point and leave the remainder on top of 10.
\[0.1 \frac{\cancel{10^2}}{\cancel{8}}\]
Next, you divide the remainder 2 by 8, 2 is less than 8 so you add another 0 beside it making it 20 and divide by 8;
\[0.1 \frac{\cancel{20^4}}{\cancel{8}}\]
Next, you divide the remainder 4 by 8, 4 is less than 8, so you add another 0 beside it making it 40 and divide by 8;
\(0.12 \frac{\cancel{40}}{\cancel{8}}\)
There is no remainder anymore, thus:
\[\frac{1}{8} = 0.125\]
So our answer is 0.125.
How are decimals converted to fractions?
Decimals are converted to fractions by following the following steps:
Determine how many decimal places (d.p.) - this is done by counting the numbers after the decimal point.
The number of decimal places will determine how many 0s, 1 d.p would be 10, 2 d.p is 100, 3 is 1000 and so on.
Remove the decimal point and divide the number by 10, 100, 1000 etc. depending on the d.p.
This would become very interesting when you try the steps accordingly... Easy-peasy .
Convert the following to fraction:
a) 0.2
b) 0.125
Solution
a) Step 1: Determine how many decimal places; the d.p. of 0.2 is 1 because there is only one number that comes after the decimal point.
Step 2: Since it has 1 d.p, it means 10 is what you divide with.
Step 3: Remove the decimal point and divide the number by 10, 100, 1000 etc. depending on the d.p; when you remove the decimal point, the number you have is 02, but the 0 is actually omitted so you have 2. Now divide 2 by 10:
\(\frac{2}{10}= \frac{\cancel{2}}{\cancel{10}} = \frac{1}{5}\)
b) Step 1: Determine how many decimal places; the d.p of 0.125 is 3 because there are three numbers that come after the decimal point.
Step 2: Since it has 3 d.p, it means 1000 is the divisor.
Step 3: Remove the decimal point and divide the number by 10, 100, 1000, etc. depending on the d.p; when you remove the decimal point, the number you have is 0125, but the 0 is actually omitted so you have 125. Now divide 125 by 1000;
\(\frac{125}{1000}\)
Divide by 5:
\(\frac{\cancel{125}}{\cancel{1000}}=\frac{25}{200}\)
Continue dividing until you can no longer divide:
\(\frac{\cancel{125}}{\cancel{1000}}=\frac{25}{200} = \frac{5}{40} = \frac{1}{8}\)
Decimals and percentages
Decimals and percentages are quite linked. All decimals can be converted to percentages, and vice-versa.
How do you convert decimals to percentages?
Decimals are converted to percentages by multiplying the decimal by 100%.
Convert the following to percentage:
a) 0.7
b) 1.6
Solution
a) Multiply the decimal by 100%:
\(0.7 \cdot 100\% = 70\%\)
b) Multiply the decimal by 100%:
\(1.6 \cdot 100\% = 160\%\)
How do you convert percentages to decimals?
Percentages can be converted to decimals by dividing the percentage by 100%. Note that you would arrive at a fraction initially; afterward, you are to convert the fraction to decimal following the steps earlier explained herein.
Convert the following to decimals:
a) 70%
b) 160%
Solution
a) Divide by 100%:
\(\frac{70\%}{100\%} = \frac{\cancel{70\%}}{\cancel{100\%}} = \frac{7}{10}\)
Convert the fraction to decimal using the steps explained earlier;
\(\frac{7}{10} = 0.70.7\)
b) Divide by 100%:
\(\frac{160\%}{100\%} = \frac{\cancel{160\%}}{\cancel{100\%}} = \frac{16}{10} = \frac{8}{5}\)
Convert the fraction to decimal using the steps explained earlier:
\(\frac{8}{5} = 1.6\)
Further examples on fractions, decimals, and percentages
There are several instances in real life where you could be required to present data in all 3 forms.
A man earns £1000, and he spends £400 on accommodation.
a) Determine the fraction he spends on accommodation.
b) Express in decimal the portion spent on groceries if he spends £100 on that.
c) Determine the percentage of his income donates if he gives alms worth £50.
Solution
a) The fraction he spends on accommodation is calculated as
\(\frac{400}{1000} = \frac{\cancel{400}}{\cancel{1000}} = \frac{4}{10}=\frac{2}{5}\)
b) The portion spent on groceries in decimal is
\(\frac{100}{1000} = \frac{\cancel{100}}{\cancel{1000}} = \frac{1}{10}\)
Using the steps explained earlier, the fraction is converted to
\(\frac{1}{10} = 0.10.1\)
c) the percentage he gives to the poor is
\(\frac{50}{1000} \cdot 100\% = \frac{50}{10\cancel{00}}\cdot \cancel{100}\% = \frac{5}{10} \% = \frac{1}{2} \% = \frac{1}{5} \%\)
or this can be expressed as a decimal percentage by converting the fraction part to a decimal
\(\frac{1}{5} \% = 0.2\%\)
Fractions, Decimals and Percentages - Key takeaways
- A fraction is a portion or part of a whole number, expression, or anything other than zero.
- A decimal is a number which expresses numbers in fractions of 10 or multiples of 10.
- A percentage can be defined as the ratio or a part of a number in a hundred.
- All fractions can be converted to percentages, and vice-versa.
- All fractions can be converted to decimals, and vice-versa.
- All decimals can be converted to percentages, and vice-versa.
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