GCSE Tutoring Programme
Our chosen students improved 1.19 of a grade on average - 0.45 more than those who didn't have the tutoring.
In order to access this I need to be confident with:
Factors and multiples Highest common factor (HCF) Simplifying fractions Prime factorsThis topic is relevant for:
Here we will learn about simplifying ratios, including how to simplify a ratio, how to write equivalent ratios and using simplifying ratio in real life problems.
There are also ratio worksheets and word problems based on Edexcel, AQA and OCR exam questions, along with further guidance on where to go next if youβre still stuck.
Simplifying ratios is a way of using common factors to divide all the numbers in a ratio until they cannot be divided further.
A simplified ratio is equivalent to an non simplified ratio.
E.g.
Letβs look at katex is not defined red counters and katex is not defined blue counters.
If we organised them into a the two colours, we can get something that looks like this:
Each of the four rows contains katex is not defined red counters and katex is not defined blue counters. This allows us to express the ratio katex is not defined as the ratio in the simplest form katex is not defined.
We can simplify the ratio by finding the highest common factor or HCF (otherwise known as the greatest common factor or GCF) of each of the numbers in the ratio.
For the example above, the highest common factor would be katex is not defined as we can divide katex is not defined and katex is not defined by katex is not defined.
We can use bar modelling to help us simplify ratios.
E.g.
Here are some circles.
Rearranging them into coloured bars, we can show the number of orange circles and the number of purple circles.
The number of orange circles is katex is not defined, and the number of purple circles is katex is not defined. This can be written as the ratio katex is not defined
We can also rearrange these circles into another pattern:
Here, we still have the ratio of katex is not defined but we have arranged them into two rows, each with the same proportion of circles.
Each row contains katex is not defined orange circles, and three purple circles. This means that we can divide each side of the ratio by katex is not defined, and still obtain the same ratio of orange to purple.
So by using bar modelling, we can simplify the ratio katex is not defined to get katex is not defined
The fastest way to simplify a ratio is to use the highest common factor of the numbers in the ratio.
For example
If you are given a ratio of katex is not defined to simplify it: first find the highest common factor which is katex is not defined next divide each part of the ratio by the highest common factor katex is not defined which gives you the answer katex is not defined
We can summarise simplifying ratios using this formula
The simplified ratio katex is not defined
For example.
Simplify katex is not defined
katex is not defined of katex is not defined
katex is not defined
In order to simplify a ratio:
Get your free simplifying ratios worksheet of 20+ questions and answers. Includes reasoning and applied questions.
DOWNLOAD FREEGet your free simplifying ratios worksheet of 20+ questions and answers. Includes reasoning and applied questions.
DOWNLOAD FREESimplifying ratios is part of our series of lessons to support revision on ratio. You may find it helpful to start with the main ratio lesson for a summary of what to expect, or use the step by step guides below for further detail on individual topics. Other lessons in this series include:
Simplify the ratio
katex is not definedListing the factors of katex is not defined and katex is not defined, we have:
katex is not defined katex is not defined katex is not defined2Divide each part of the ratio by the highest common factor.
Solution:
katex is not definedSimplify the following ratio
katex is not definedCalculate the highest common factor of the parts of the ratio.
Listing the factors of all katex is not defined parts of the ratio, we have
katex is not defined katex is not defined katex is not defined katex is not definedDivide each part of the ratio by the highest common factor.
Solution:
katex is not definedSimplify the following ratio
katex is not definedCalculate the highest common factor of the parts of the ratio.
We currently have two decimal values within the ratio. As we only find factors of whole numbers, we need to determine an equivalent ratio first. By multiplying both sides by katex is not defined, we obtain the ratio katex is not defined which we can now find common factors of.
Listing the factors of each part of the ratio, we have
katex is not defined katex is not defined katex is not definedDivide each part of the ratio by the highest common factor.
Solution:
katex is not definedSimplify the ratio
katex is not defined
Calculate the highest common factor of the parts of the ratio.
Here we have two fractions. Instead of looking at finding the highest common factor of fractions, we need to determine an equivalent ratio that contains whole numbers. We can do this by multiplying both fractions by a number (preferably their lowest common multiple) to give a ratio with integers:
katex is not defined katex is not definedWe can now state that the ratio katex is not defined is equivalent to katex is not defined. (Be careful of the order) and katex is not defined
Divide each part of the ratio by the highest common factor.
Solution:
katex is not definedSimplify the ratio
katex is not definedCalculate the highest common factor of the parts of the ratio.
As we have two mixed numbers, we can convert them to improper fractions or decimals. We will look at converting the mixed numbers to decimals and then simplify the ratio.
Converting both of the mixed numbers to decimals we have katex is not defined s the same as katex is not defined
Multiplying both of these parts of the ratio by katex is not defined will quickly remove the decimal so we can then simplify the ratio.
katex is not definedCalculating the highest common factor of katex is not defined and katex is not defined quickly by using prime factor trees, we get:
From this, we can show that the highest common factor of katex is not defined and katex is not defined is katex is not defined
katex is not definedDivide each part of the ratio by the highest common factor.
Solution:
katex is not definedGiven that katex is not defined and katex is not defined are all positive integers, simplify the ratio
katex is not definedCalculate the highest common factor of the parts of the ratio.
Divide each part of the ratio by the highest common factor.
Solution:
katex is not definedUse a bar model to simplify the ratio katex is not defined
Calculate the highest common factor of the parts of the ratio.
Representing the ratio katex is not defined as a bar model, we have
Where red represents katex is not defined shares, and purple represents katex is not defined shares.
katex is not definedDivide each part of the ratio by the highest common factor.
As the highest common factor is katex is not defined, we divide the bar above into two bars containing the same amount of red shares, and purple shares.
Each bar is the ratio katex is not defined
Solution:
katex is not definedA common error is that the ratio is not cancelled to its simplest form due to not using the highest common factor,
E.g.
Simplify the ratio katex is not defined
We can simplify by dividing parts of the ratio by dividing both numbers by katex is not defined, leaving the ratio katex is not defined.
This is not the final answer as it can be simplified further by dividing by katex is not defined to get the ratio katex is not defined
A common error is to change the order of the ratio. It is important that the order of the numbers in the ratio remains the same while simplifying.
When simplifying a ratio with more than two parts, a common misconception is to not simplify all of the parts of the ratio.
E.g.
βSimplify the ratio katex is not definedβ could be incorrectly written as katex is not defined as the middle part is not cancelled down.
Some numbers are more tricky to divide than others, especially when calculating it mentally.
Common incorrect calculations include katex is not defined and katex is not defined
We cannot use addition and subtraction to simplify a ratio, we can only use multiplication and division.
Letβs look at why using a visual representation:
Below are katex is not defined rectangles katex is not defined and katex is not defined stars katex is not defined.
If we were to express the number of stars and rectangles as a ratio, we would write the ratio katex is not defined. Remember that the order is important and so we have written the number of stars first and the number of rectangles second katex is not defined.
Dividing the number of rectangles by the number of stars we get katex is not defined.
Here the number katex is not defined is known as the constant of proportionality (Step-by-step guide: Direct proportion).
Now if we were to subtract katex is not defined from each of the stars and the rectangles above and divide the quantities again the answer should be katex is not defined if the ratio was equivalent.
Subtracting katex is not defined from each we now have katex is not defined stars katex is not defined and katex is not defined rectangles katex is not defined. I katex is not defined.
As this is not katex is not defined the ratios are not equivalent.
Simplified ratios should not include decimals.
E.g.
If we want to simplify the ratio katex is not defined, we need to divide each quantity in the ratio by a common factor.
Here we have simplified by a common factor of katex is not defined, and then simplified again by dividing these new numbers by katex is not defined.
When simplifying a ratio, your solution must contain whole numbers (integers)
So the correct simplified ratio is:
1. Simplify
katex is not defined
2. Simplify
katex is not defined
3. Simplify
katex is not defined
4. Simplify
katex is not defined
5. Simplify
katex is not defined
6. Given that katex is not defined and katex is not defined are positive integers, simplify
katex is not defined
1. The number of visitors to a zoo is written as the ratio of adults to children as katex is not defined respectively. Write this ratio in its simplest form.
(1 mark)
(1)
2. (a) A taxi firm is researching the distances travelled for each journey. The results are shown below.
State the ratio of journeys between katex is not defined and katex is not defined miles and the total number of journeys. Write your answer in its simplest form.
(b) Another katex is not defined journeys are added to the table, katex is not defined of which are within the category katex is not defined. What is the new ratio of the number of journeys between katex is not defined and katex is not defined miles and the total number of journeys. Write your answer in its simplest form.
(3 marks)
(a)
katex is not defined(1)
(b)
New ratio katex is not defined
(1)
katex is not defined(1)
3. Below is a recipe to make custard for katex is not defined people.
(a) Write the ratio of caster sugar to double cream to milk in its simplest form.
(b) I only have katex is not defined egg yolks. How much double cream should I use?
(4 marks)
(a)
katex is not defined(1)
katex is not defined(1)
(b)
katex is not defined(1)
katex is not defined double cream
(1)
You have now learned how to:
Prepare your KS4 students for maths GCSEs success with Third Space Learning. Weekly online one to one GCSE maths revision lessons delivered by expert maths tutors.
Find out more about our GCSE maths tuition programme.
Notifications