Often, when answering questions for GCSE mathematics, we are required to express our answers as either a ratio or fraction. Therefore, it is important that we are able to convert fractions to ratios and vice versa. Suppose, for example, that a piece of string is cut in the ratio of 2:3, we need to be able to work out the lengths of the two pieces of string as a fraction of the original piece of string. Luckily, we will cover everything there is to know about fractional ratios in this article. We will start by defining ratios and fractions, and establish the differences between them.
Difference between Ratio and Fraction
A ratio is a way of comparing two quantities. The ratio of two quantities highlights how much bigger one number is in comparison to another. A fraction tells us how much something is as a proportion of something else.
Really, fractions and ratios mean the same thing. However, we express them slightly differently. In the below example, you can see the same things expressed as both a fraction and ratio.
- In a class of students, the ratio of girls to boys is 2:3. This means that for every five students, 2 are girls and 3 are boys. We could say that the fraction of girls is \(\dfrac{2}{5}\), and the fraction of boys is \(\dfrac{3}{5}\).
- A piece of string is cut in the ratio of \(1:4\). Thus, if the string was 5 cm, the shorter piece would be 1 cm and the longer piece would be 4 cm. We could say that the shorter length is \(\dfrac{1}{5}\) of the original piece of string and the longer length is \(\dfrac{4}{5}\) of the original piece.
- The ratio of blue sweets to orange sweets in a bag is 3:7. Thus, for every 10 sweets, we could say that 3 are blue and 7 are orange. We could therefore say that \(\dfrac{3}{10}\) of the sweets are blue and that \(\dfrac{7}{10}\) of the sweets are orange.
Can you spot what we are doing here? In the below section we will discuss in more detail how to convert fractions to ratios and ratios to fractions. However, it is important to note that ratios and fractions can both be used to represent the same thing and so you may see the words "fraction" and "ratio" used interchangeably.
When we see a fraction, we are comparing a single part to a whole. For example, for the fraction \(\dfrac{2}{3}\), we are looking at two parts out of three.
When we see a ratio, we are looking at two or more of the components that make up the whole. For example, for the ratio \(2:3\), two and three are both separate parts of the same whole.
Fraction to Ratio
How to Convert a Fraction to a Ratio
If we have a fraction, we can convert it to a ratio quite simply. The method is as follows:
Step 1: Determine the fraction that makes up each quantity. For example, if we have a bag of red, blue, and orange counters, we need to work out the fraction of red, blue, and orange counters.
Step 2: Write each of the fractions in the order of the ratio specified with colons separating them.
Step 3: Multiply each component of the ratio by a number so that each part of the ratio is an integer (ie, a whole number, not a fraction).
In the below examples we will discuss how to convert fractions to ratios.
Fraction to Ratio Examples
On a school trip, \(\dfrac{1}{3}\) of the students go to a museum and the rest of the students go to an art gallery. What is the ratio of students who go to the museum to the art gallery?
Solution:
Since \(\dfrac{1}{3}\) of the students go to a museum, we can deduce that \(\dfrac{2}{3}\) of the students go to the art gallery since fractions add up to one.
We next write the fractions as a ratio, like this: \(\dfrac{1}{3}:\dfrac{2}{3}\).
Now, to simplify this ratio, we can multiply both sides by three. Thus, we end up with \(1:2\). Therefore, we can say that for every single student who goes to the museum, we have two students who go to the art gallery.
In a cinema, \(\dfrac{5}{6}\) of people are adults and the rest are children. What is the ratio of adults to children?
Solution:
Since \(\dfrac{5}{6}\) of the people are adults, \(\dfrac{1}{6}\) must be children since fractions add to one. Writing this as a ratio is \(\dfrac{5}{6}:\dfrac{1}{6}\).
Now, by multiplying both sides of the ratio by 6, we obtain \(5:1\). Thus, the ratio of adults to children is \(5:1\).
In a bag of sweets, \(\dfrac{1}{4}\) are red, \(\dfrac{1}{3}\) are green and the rest are orange. What is the ratio of red to green to orange sweets?
Solution:
The sum of red sweets and green sweets is \(\dfrac{1}{4}+\dfrac{1}{3}=\dfrac{3}{12}+\dfrac{4}{12}=\dfrac{7}{12}\).
Since the rest are orange, we can say that \(1-\dfrac{7}{12}=\dfrac{12}{12}-\dfrac{7}{12}=\dfrac{5}{12}\) are orange.
Putting our fractions into ratios, we have the ratio of red to green to orange sweets is \(\dfrac{1}{4}:\dfrac{1}{3}:\dfrac{5}{12}\). Now, to get integer values for our ratio, we need to multiply each element of the ratio by 12. We obtain \(3:4:5\). Thus, the ratio of red sweets to green sweets to orange sweets is \(3:4:5\).
Ratio to Fraction
We can also quite easily convert ratios to fractions. In questions, it is often easier to work with fractions than it is to work with ratios and so this is particularly useful. To do this, we simply add together the parts of the ratio. This is the denominator of all of the fractions. Then, each part of the ratio will be the numerator of the different fractions. It is really simple; in the below examples we will be converting ratios to fractions.
Ratio to Fraction Examples
In a reception class, the ratio of students to teachers is \(5:1\). What fraction of the class are students and what fraction are teachers?
Solution:
First, add together the different parts of the ratio. In this case, we add five and one to get six. This is the denominator of the fractions. Then, we can say that \(\dfrac{5}{6}\) are students and \(\dfrac{1}{6}\) of the class are teachers.
In a company, the ratio of female employees to male employees is \(2:3\). What fraction of the employees are male?
Solution:
First, add two and three to get five. Thus, \(\dfrac{2}{5}\) are female and \(\dfrac{3}{5}\) are male.
A piece of string is cut into three pieces in the ratio \(1:2:3\). Work out the fraction of the original piece that each of the three pieces makes up.
Solution:
\(1+2+3=5\). Thus, the smallest piece is \(\dfrac{1}{5}\), the middle piece is \(\dfrac{2}{5}\) and the largest piece is \(\dfrac{3}{5}\).
Fractions, Percentages and Ratios
We can also convert fractions and ratios to percentages.
The ratio of boys to girls taking A-Level English in a school is \(3:7\). Work out the percentage of boys who take A-Level English.
Solution:
First, we can say that the fraction of boys taking A-level English is\(\dfrac{3}{10}\). Converting this to a percentage, we get 30%. Thus, 30% of students taking A-Level English are boys.
There are 300 people at the fair. The ratio of adults to children is \(1:2\). 20% of the children are under the age of 6 and get free entry. Work out the fraction of children who get free entry.
Solution:
First, we could say that the fraction of children is \(\(\dfrac{2}{3}\). Since there are 300 people at the fair, we can say that \(\dfrac{2}{3}\) of 300 are children. Since \(\dfrac{1}{3}\) of 300 is 100 since \(300\div 3=100\), we know that \(\dfrac{2}{3}\) of 300 is 200. Thus, 200 of the people at the fair are children. 20% of the children are under the age of 6, so we need to work out 20% of 200. Since 10% of 200 is 20, 20% of 200 is 40.
Thus, 40 attendees of the fair got free entry.
Fractional Ratio Methods
Fractional ratio often comes up in other topics in GCSE mathematics. One where it is particularly useful is vectors. Here, we will look at two vectors questions where a fractional ratio is incorporated. However, if you are not overly familiar with vectors, it may be useful to recap this topic before reading any further.
In the below triangle DEF, vector \(\overrightarrow{DE}=a\) and \(\overrightarrow{EF}=b\). Point \(A\) cuts the line \(DF\) such that \(\overrightarrow{DA}:\overrightarrow{AF}=2:3\). Work out \(\overrightarrow{DA}\).
Fig. 1. Triangle DEF with point A.
Solution:
First, we can find the vector \(\overrightarrow{DF}\).
\[\overrightarrow{DF}=\overrightarrow{DE}+\overrightarrow{EF}=a+b\]
Now, since \(\overrightarrow{DA}:\overrightarrow{AF}=2:3\), we can say that,
\[\overrightarrow{DA}=\dfrac{2}{5}\overrightarrow{DF}=\dfrac{2}{5}(a+b)\]
Thus,
\[\overrightarrow{DA}=\dfrac{2}{5}(a+b)\]
In the below quadrilateral GHIJ, the point K cuts the vector \(\overrightarrow{IJ}\) in the ratio \(1:2\).
Given: \(\overrightarrow{GH}=a\), \(\overrightarrow{HI}=b\) and \(\overrightarrow{IJ}=c\), find an expression for the vector \(\overrightarrow{GK}\).
Fig. 2. Quadrilateral GHIJ with point K.
Solution:
First, note that the vector \(\overrightarrow{GI}=\overrightarrow{GH}+\overrightarrow{HI}=a+b\).
Now, K splits the vector \(\overrightarrow{IJ}\) in the ratio \(1:2\) and so we can say that,
\[\overrightarrow{IK}=\dfrac{1}{3}\overrightarrow{IJ}=\dfrac{1}{3}c\]
Thus,
\[\overrightarrow{GK}=a+b+\dfrac{1}{3}c\]
Fractional ratio - Key takeaways
- A ratio is a way of comparing two quantities.
- A fraction tells us how much something is as a proportion of something else.
- Really, fractions and ratios mean the same thing. However, we express them slightly differently.
- If we have a ratio, we can convert it to a fraction and if we have fractions that make up a whole, we can convert it to a ratio.
- We can also convert ratios to percentages.
- Being able to deal with fractional ratios is particularly useful when studying vectors.
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