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22 Dec 2025
•
Understanding Linear and Quadratic Ratio Problems in Algebra
Evelyn Ridley
@ev_alice
1 / 10
Getting Started with Ratio Algebra
Think of ratio algebra as solving puzzles where the missing piece is always 'x'. These problems look intimidating at first, but they follow a simple pattern every time.
The secret is recognising that ratios are just another way of writing fractions. Once you see that connection, you're halfway to the solution!
Key Insight: Every ratio can be written as a fraction, and fractions are much easier to work with when solving for x.
Basic Ratio Problems (Type 1)
When you see something like : = 2 : 5, you're looking at a Type 1 problem. These always give you one ratio equal to another ratio.
Your first move is converting this to fractions: / = 2/5. Now you can cross-multiply to get rid of those annoying fractions.
Cross-multiplying gives you: 5 = 2. From here, it's just expanding brackets, collecting like terms, and solving for x like any other equation.
Pro Tip: Always check your answer by substituting it back into the original ratio - it's the quickest way to spot mistakes!
Worked Examples and Practice
Let's walk through that example: : = 2 : 5. After cross-multiplying, you get 10x + 5 = 6x - 10.
Collecting like terms: 4x = -15, so x = -3.75. See how straightforward it becomes once you've got rid of the ratio format?
The practice problems follow exactly the same method. Whether it's : = 3 : 5 or any other combination, the approach never changes.
Remember: Cross-multiply, expand, collect like terms, solve - that's your four-step process every time.
Challenge Questions and Exam Prep
Now we're getting into the good stuff - questions that might actually appear on your exams. These problems test whether you really understand the method or you're just following steps blindly.
Look at question 5: : = k : 1. This one's asking you to rearrange for y, not solve for a number. It's the same technique, but you need to be comfortable with algebra.
Question 6 is particularly sneaky: : = : . Both sides have expressions in x, so you'll need to expand brackets carefully and watch your algebra closely.
Exam Strategy: These harder questions often give you expressions on both sides - don't panic, just cross-multiply as usual.
Solutions and Working
Here's where you can check your working and see the complete solutions. Notice how question 5 shows the full algebraic manipulation - this is what examiners want to see.
The key step in question 5 is recognising that y = -x can be rearranged to give the required form. Don't worry if this feels tricky - it's testing A-level style algebra.
Question 6 demonstrates why checking is so important. After cross-multiplying and expanding, you get x² + 11x + 18 = x² + 10x + 24, which simplifies beautifully to x = 6.
Study Tip: Cover up the answers and try working through these yourself - you'll learn much more than just reading the solutions.
Introduction to Type 2 Problems
Right, now we're stepping up a gear. Type 2 ratio problems are where things get properly interesting because you'll end up with quadratic equations.
The setup looks similar to Type 1, but instead of getting a simple linear equation, your algebra leads to something like x² - x - 6 = 0.
Don't let this put you off - the initial steps are identical. It's only at the end that you need to factorise or use the quadratic formula.
Heads Up: Type 2 problems usually give you two possible values for x, not just one.
Setting Up Type 2 Problems
Look at this example: : = : . Notice how both sides have different expressions? That's your clue this is Type 2.
You still start the same way: convert to fractions and cross-multiply. But now you're dealing with = .
When you expand these brackets, you'll get x² terms that don't cancel out. That's what creates the quadratic equation you'll need to solve.
Watch Out: Be extra careful when expanding brackets - small mistakes here will mess up your final answer.
Solving Type 2 Problems
Let's follow through that example completely. After expanding = , you get 3x² + 11x + 10 = 2x² + 12x + 16.
Rearranging gives you x² - x - 6 = 0. Now you can factorise: = 0, so x = 3 or x = -2.
The 'Your Turn' example works exactly the same way, giving you x = 2 or x = 4. Two solutions is completely normal for these problems.
Key Point: Always write both solutions clearly - marks get dropped for missing one of the answers.
Practice Problems
These six questions will test everything you've learned about ratio algebra. Start with question 1 - it's the most straightforward of the bunch.
Question 6 is particularly interesting because it involves x² in the ratio itself. Don't let this throw you - the method stays exactly the same.
Work through these systematically. If you get stuck, go back to the examples and check you're following the same steps.
Study Method: Try to do these without looking at the answers first - you'll learn much more from making mistakes and fixing them.
We thought you’d never ask...
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4.8/5
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The app is very easy to use and well designed. I have found everything I was looking for so far and have been able to learn a lot from the presentations! I will definitely use the app for a class assignment! And of course it also helps a lot as an inspiration.
Stefan S
iOS user
This app is really great. There are so many study notes and help [...]. My problem subject is French, for example, and the app has so many options for help. Thanks to this app, I have improved my French. I would recommend it to anyone.
Samantha Klich
Android user
Wow, I am really amazed. I just tried the app because I've seen it advertised many times and was absolutely stunned. This app is THE HELP you want for school and above all, it offers so many things, such as workouts and fact sheets, which have been VERY helpful to me personally.
Anna
iOS user
Best app on earth! no words because it’s too good
Thomas R
iOS user
Just amazing. Let's me revise 10x better, this app is a quick 10/10. I highly recommend it to anyone. I can watch and search for notes. I can save them in the subject folder. I can revise it any time when I come back. If you haven't tried this app, you're really missing out.
Basil
Android user
This app has made me feel so much more confident in my exam prep, not only through boosting my own self confidence through the features that allow you to connect with others and feel less alone, but also through the way the app itself is centred around making you feel better. It is easy to navigate, fun to use, and helpful to anyone struggling in absolutely any way.
David K
iOS user
The app's just great! All I have to do is enter the topic in the search bar and I get the response real fast. I don't have to watch 10 YouTube videos to understand something, so I'm saving my time. Highly recommended!
Sudenaz Ocak
Android user
In school I was really bad at maths but thanks to the app, I am doing better now. I am so grateful that you made the app.
Greenlight Bonnie
Android user
very reliable app to help and grow your ideas of Maths, English and other related topics in your works. please use this app if your struggling in areas, this app is key for that. wish I'd of done a review before. and it's also free so don't worry about that.
Rohan U
Android user
I know a lot of apps use fake accounts to boost their reviews but this app deserves it all. Originally I was getting 4 in my English exams and this time I got a grade 7. I didn’t even know about this app three days until the exam and it has helped A LOT. Please actually trust me and use it as I’m sure you too will see developments.
Xander S
iOS user
THE QUIZES AND FLASHCARDS ARE SO USEFUL AND I LOVE THE SCHOOLGPT. IT ALSO IS LITREALLY LIKE CHATGPT BUT SMARTER!! HELPED ME WITH MY MASCARA PROBLEMS TOO!! AS WELL AS MY REAL SUBJECTS ! DUHHH 😍😁😲🤑💗✨🎀😮
Elisha
iOS user
This apps acc the goat. I find revision so boring but this app makes it so easy to organize it all and then you can ask the freeeee ai to test yourself so good and you can easily upload your own stuff. highly recommend as someone taking mocks now
Paul T
iOS user
The app is very easy to use and well designed. I have found everything I was looking for so far and have been able to learn a lot from the presentations! I will definitely use the app for a class assignment! And of course it also helps a lot as an inspiration.
Stefan S
iOS user
This app is really great. There are so many study notes and help [...]. My problem subject is French, for example, and the app has so many options for help. Thanks to this app, I have improved my French. I would recommend it to anyone.
Samantha Klich
Android user
Wow, I am really amazed. I just tried the app because I've seen it advertised many times and was absolutely stunned. This app is THE HELP you want for school and above all, it offers so many things, such as workouts and fact sheets, which have been VERY helpful to me personally.
Anna
iOS user
Best app on earth! no words because it’s too good
Thomas R
iOS user
Just amazing. Let's me revise 10x better, this app is a quick 10/10. I highly recommend it to anyone. I can watch and search for notes. I can save them in the subject folder. I can revise it any time when I come back. If you haven't tried this app, you're really missing out.
Basil
Android user
This app has made me feel so much more confident in my exam prep, not only through boosting my own self confidence through the features that allow you to connect with others and feel less alone, but also through the way the app itself is centred around making you feel better. It is easy to navigate, fun to use, and helpful to anyone struggling in absolutely any way.
David K
iOS user
The app's just great! All I have to do is enter the topic in the search bar and I get the response real fast. I don't have to watch 10 YouTube videos to understand something, so I'm saving my time. Highly recommended!
Sudenaz Ocak
Android user
In school I was really bad at maths but thanks to the app, I am doing better now. I am so grateful that you made the app.
Greenlight Bonnie
Android user
very reliable app to help and grow your ideas of Maths, English and other related topics in your works. please use this app if your struggling in areas, this app is key for that. wish I'd of done a review before. and it's also free so don't worry about that.
Rohan U
Android user
I know a lot of apps use fake accounts to boost their reviews but this app deserves it all. Originally I was getting 4 in my English exams and this time I got a grade 7. I didn’t even know about this app three days until the exam and it has helped A LOT. Please actually trust me and use it as I’m sure you too will see developments.
Xander S
iOS user
THE QUIZES AND FLASHCARDS ARE SO USEFUL AND I LOVE THE SCHOOLGPT. IT ALSO IS LITREALLY LIKE CHATGPT BUT SMARTER!! HELPED ME WITH MY MASCARA PROBLEMS TOO!! AS WELL AS MY REAL SUBJECTS ! DUHHH 😍😁😲🤑💗✨🎀😮
Elisha
iOS user
This apps acc the goat. I find revision so boring but this app makes it so easy to organize it all and then you can ask the freeeee ai to test yourself so good and you can easily upload your own stuff. highly recommend as someone taking mocks now
Paul T
iOS user
Understanding Linear and Quadratic Ratio Problems in Algebra
Evelyn Ridley
@ev_alice
Ever wondered how to solve those tricky maths problems where you've got ratios with unknown values? Ratio algebra is actually quite straightforward once you know the key technique. You'll be turning ratios into fractions and solving equations like a pro!
Getting Started with Ratio Algebra
Think of ratio algebra as solving puzzles where the missing piece is always 'x'. These problems look intimidating at first, but they follow a simple pattern every time.
The secret is recognising that ratios are just another way of writing fractions. Once you see that connection, you're halfway to the solution!
Key Insight: Every ratio can be written as a fraction, and fractions are much easier to work with when solving for x.
Sign up to see the contentIt's free!
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
Basic Ratio Problems (Type 1)
When you see something like : = 2 : 5, you're looking at a Type 1 problem. These always give you one ratio equal to another ratio.
Your first move is converting this to fractions: / = 2/5. Now you can cross-multiply to get rid of those annoying fractions.
Cross-multiplying gives you: 5 = 2. From here, it's just expanding brackets, collecting like terms, and solving for x like any other equation.
Pro Tip: Always check your answer by substituting it back into the original ratio - it's the quickest way to spot mistakes!
Sign up to see the contentIt's free!
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
Worked Examples and Practice
Let's walk through that example: : = 2 : 5. After cross-multiplying, you get 10x + 5 = 6x - 10.
Collecting like terms: 4x = -15, so x = -3.75. See how straightforward it becomes once you've got rid of the ratio format?
The practice problems follow exactly the same method. Whether it's : = 3 : 5 or any other combination, the approach never changes.
Remember: Cross-multiply, expand, collect like terms, solve - that's your four-step process every time.
Sign up to see the contentIt's free!
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
Challenge Questions and Exam Prep
Now we're getting into the good stuff - questions that might actually appear on your exams. These problems test whether you really understand the method or you're just following steps blindly.
Look at question 5: : = k : 1. This one's asking you to rearrange for y, not solve for a number. It's the same technique, but you need to be comfortable with algebra.
Question 6 is particularly sneaky: : = : . Both sides have expressions in x, so you'll need to expand brackets carefully and watch your algebra closely.
Exam Strategy: These harder questions often give you expressions on both sides - don't panic, just cross-multiply as usual.
Sign up to see the contentIt's free!
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By signing up you accept Terms of Service and Privacy Policy
Solutions and Working
Here's where you can check your working and see the complete solutions. Notice how question 5 shows the full algebraic manipulation - this is what examiners want to see.
The key step in question 5 is recognising that y = -x can be rearranged to give the required form. Don't worry if this feels tricky - it's testing A-level style algebra.
Question 6 demonstrates why checking is so important. After cross-multiplying and expanding, you get x² + 11x + 18 = x² + 10x + 24, which simplifies beautifully to x = 6.
Study Tip: Cover up the answers and try working through these yourself - you'll learn much more than just reading the solutions.
Sign up to see the contentIt's free!
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Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
Introduction to Type 2 Problems
Right, now we're stepping up a gear. Type 2 ratio problems are where things get properly interesting because you'll end up with quadratic equations.
The setup looks similar to Type 1, but instead of getting a simple linear equation, your algebra leads to something like x² - x - 6 = 0.
Don't let this put you off - the initial steps are identical. It's only at the end that you need to factorise or use the quadratic formula.
Heads Up: Type 2 problems usually give you two possible values for x, not just one.
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Improve your grades
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By signing up you accept Terms of Service and Privacy Policy
Setting Up Type 2 Problems
Look at this example: : = : . Notice how both sides have different expressions? That's your clue this is Type 2.
You still start the same way: convert to fractions and cross-multiply. But now you're dealing with = .
When you expand these brackets, you'll get x² terms that don't cancel out. That's what creates the quadratic equation you'll need to solve.
Watch Out: Be extra careful when expanding brackets - small mistakes here will mess up your final answer.
Sign up to see the contentIt's free!
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Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
Solving Type 2 Problems
Let's follow through that example completely. After expanding = , you get 3x² + 11x + 10 = 2x² + 12x + 16.
Rearranging gives you x² - x - 6 = 0. Now you can factorise: = 0, so x = 3 or x = -2.
The 'Your Turn' example works exactly the same way, giving you x = 2 or x = 4. Two solutions is completely normal for these problems.
Key Point: Always write both solutions clearly - marks get dropped for missing one of the answers.
Sign up to see the contentIt's free!
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
Practice Problems
These six questions will test everything you've learned about ratio algebra. Start with question 1 - it's the most straightforward of the bunch.
Question 6 is particularly interesting because it involves x² in the ratio itself. Don't let this throw you - the method stays exactly the same.
Work through these systematically. If you get stuck, go back to the examples and check you're following the same steps.
Study Method: Try to do these without looking at the answers first - you'll learn much more from making mistakes and fixing them.
Sign up to see the contentIt's free!
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
We thought you’d never ask...
What is the Knowunity AI companion?
Our AI Companion is a student-focused AI tool that offers more than just answers. Built on millions of Knowunity resources, it provides relevant information, personalised study plans, quizzes, and content directly in the chat, adapting to your individual learning journey.
Where can I download the Knowunity app?
You can download the app from Google Play Store and Apple App Store.
Is Knowunity really free of charge?
That's right! Enjoy free access to study content, connect with fellow students, and get instant help – all at your fingertips.
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4.9/5
App Store
4.8/5
Google Play
The app is very easy to use and well designed. I have found everything I was looking for so far and have been able to learn a lot from the presentations! I will definitely use the app for a class assignment! And of course it also helps a lot as an inspiration.
Stefan S
iOS user
This app is really great. There are so many study notes and help [...]. My problem subject is French, for example, and the app has so many options for help. Thanks to this app, I have improved my French. I would recommend it to anyone.
Samantha Klich
Android user
Wow, I am really amazed. I just tried the app because I've seen it advertised many times and was absolutely stunned. This app is THE HELP you want for school and above all, it offers so many things, such as workouts and fact sheets, which have been VERY helpful to me personally.
Anna
iOS user
Best app on earth! no words because it’s too good
Thomas R
iOS user
Just amazing. Let's me revise 10x better, this app is a quick 10/10. I highly recommend it to anyone. I can watch and search for notes. I can save them in the subject folder. I can revise it any time when I come back. If you haven't tried this app, you're really missing out.
Basil
Android user
This app has made me feel so much more confident in my exam prep, not only through boosting my own self confidence through the features that allow you to connect with others and feel less alone, but also through the way the app itself is centred around making you feel better. It is easy to navigate, fun to use, and helpful to anyone struggling in absolutely any way.
David K
iOS user
The app's just great! All I have to do is enter the topic in the search bar and I get the response real fast. I don't have to watch 10 YouTube videos to understand something, so I'm saving my time. Highly recommended!
Sudenaz Ocak
Android user
In school I was really bad at maths but thanks to the app, I am doing better now. I am so grateful that you made the app.
Greenlight Bonnie
Android user
very reliable app to help and grow your ideas of Maths, English and other related topics in your works. please use this app if your struggling in areas, this app is key for that. wish I'd of done a review before. and it's also free so don't worry about that.
Rohan U
Android user
I know a lot of apps use fake accounts to boost their reviews but this app deserves it all. Originally I was getting 4 in my English exams and this time I got a grade 7. I didn’t even know about this app three days until the exam and it has helped A LOT. Please actually trust me and use it as I’m sure you too will see developments.
Xander S
iOS user
THE QUIZES AND FLASHCARDS ARE SO USEFUL AND I LOVE THE SCHOOLGPT. IT ALSO IS LITREALLY LIKE CHATGPT BUT SMARTER!! HELPED ME WITH MY MASCARA PROBLEMS TOO!! AS WELL AS MY REAL SUBJECTS ! DUHHH 😍😁😲🤑💗✨🎀😮
Elisha
iOS user
This apps acc the goat. I find revision so boring but this app makes it so easy to organize it all and then you can ask the freeeee ai to test yourself so good and you can easily upload your own stuff. highly recommend as someone taking mocks now
Paul T
iOS user
The app is very easy to use and well designed. I have found everything I was looking for so far and have been able to learn a lot from the presentations! I will definitely use the app for a class assignment! And of course it also helps a lot as an inspiration.
Stefan S
iOS user
This app is really great. There are so many study notes and help [...]. My problem subject is French, for example, and the app has so many options for help. Thanks to this app, I have improved my French. I would recommend it to anyone.
Samantha Klich
Android user
Wow, I am really amazed. I just tried the app because I've seen it advertised many times and was absolutely stunned. This app is THE HELP you want for school and above all, it offers so many things, such as workouts and fact sheets, which have been VERY helpful to me personally.
Anna
iOS user
Best app on earth! no words because it’s too good
Thomas R
iOS user
Just amazing. Let's me revise 10x better, this app is a quick 10/10. I highly recommend it to anyone. I can watch and search for notes. I can save them in the subject folder. I can revise it any time when I come back. If you haven't tried this app, you're really missing out.
Basil
Android user
This app has made me feel so much more confident in my exam prep, not only through boosting my own self confidence through the features that allow you to connect with others and feel less alone, but also through the way the app itself is centred around making you feel better. It is easy to navigate, fun to use, and helpful to anyone struggling in absolutely any way.
David K
iOS user
The app's just great! All I have to do is enter the topic in the search bar and I get the response real fast. I don't have to watch 10 YouTube videos to understand something, so I'm saving my time. Highly recommended!
Sudenaz Ocak
Android user
In school I was really bad at maths but thanks to the app, I am doing better now. I am so grateful that you made the app.
Greenlight Bonnie
Android user
very reliable app to help and grow your ideas of Maths, English and other related topics in your works. please use this app if your struggling in areas, this app is key for that. wish I'd of done a review before. and it's also free so don't worry about that.
Rohan U
Android user
I know a lot of apps use fake accounts to boost their reviews but this app deserves it all. Originally I was getting 4 in my English exams and this time I got a grade 7. I didn’t even know about this app three days until the exam and it has helped A LOT. Please actually trust me and use it as I’m sure you too will see developments.
Xander S
iOS user
THE QUIZES AND FLASHCARDS ARE SO USEFUL AND I LOVE THE SCHOOLGPT. IT ALSO IS LITREALLY LIKE CHATGPT BUT SMARTER!! HELPED ME WITH MY MASCARA PROBLEMS TOO!! AS WELL AS MY REAL SUBJECTS ! DUHHH 😍😁😲🤑💗✨🎀😮
Elisha
iOS user
This apps acc the goat. I find revision so boring but this app makes it so easy to organize it all and then you can ask the freeeee ai to test yourself so good and you can easily upload your own stuff. highly recommend as someone taking mocks now
Paul T
iOS user