The ratio of boys to girls at a school is 5:7
There are 600 children at the school.
How many boys are there at the school?
I've seen other teachers use totally different algebraic methods for questions like this (there's a nice example here).
I don't always use algebra for ratio problems - I also teach my students the 'scaling' method. This involves taking the initial ratio and finding a series of equivalent ratios until you find one that works (essentially finding x by trial and error). In this example, we start with 7:3 and multiply by two to get 14:6. Now if Alice started with 14 sweets and Olivia started with 6 sweets, then Alice giving three to Olivia would result in a ratio of 11:9, which is not what we want. So we try again. Our next equivalent ratio is 21:9, and again if we take three from Alice and give them to Olivia we'd get 18:12, which again is not equivalent to 5:3. So we keep going. Our next try is 28:12 which works.
Scaling is sometimes really time consuming. In this example we could have skipped straight to the correct answer if we'd realised that the total number of sweets has to split into 10 parts (ie 7:3) and also into 8 parts (ie 5:3). If we try the lowest common multiple of 10 and 8 then we get the answer really quickly.
We'd use the lowest common multiple if we were going to use bar modelling too.
If you're not familiar with bar modelling, Kris Boulton's TES article and William Emeny's blog post are a good place to start for the basics. Your local Maths Hub may run a bar modelling course. If you know the basics but struggle with the harder questions, check out this Twitter thread to see a bar model in action for a trickier ratio problem. It's also worth watching this excellent video on solving harder ratio problems using bar models from Colin Hegarty.
Bar modelling certainly is a clear and accessible approach for simple ratio questions. I would say though that for some trickier ratio questions it's not always quite as intuitive and obvious as expert bar modellers suggest it is.
Let's have a look at methods for another 'ratio change' question - this one is from Don Steward:
Scaling works too:
5:6 gives 7:6 when Jan gains 2 marbles ✗
10:12 gives 12:12 when Jan gains 2 marbles ✗
15:18 gives 17:18 when Jan gains 2 marbles ✗
20: 24 gives 22:24 when Jan gains 2 marbles ✗
25: 30 gives 27:30 when Jan gains 2 marbles ✗
30: 36 gives 32:36 when Jan gains 2 marbles ✗
35: 42 gives 37:42 when Jan gains 2 marbles ✗
40: 48 gives 42:48 when Jan gains 2 marbles - this simplifies to 7:8 ✓
Again, we can get there more quickly if we think about multiples. We start with a multiple of 11 and end with a multiple of 15, but we gained two marbles along the way. So we can list all multiples of 11 and all multiples of 15 and find a pair which are two apart.
11, 22, 33, 44, 55, 66, 77, 88
15, 30, 45, 60, 75, 90
So we started with 88 marbles.
Not all tricky ratio questions are in the form of these 'ratio change' problems. We also now get GCSE questions like this:
If the ratio a:b is 4:7, write a in terms of b
Though this may seem obvious to many of us (a is the smaller of the two, so a is four sevenths of b), writing the ratios as equivalent fractions can help students get their numbers the right way round (providing they are confident in rearranging equations).
Another type of question is this:
If the ratio a:b is 2:5 and the ratio b:c is 3:10, what is the ratio a:c?
Again, fractions might help.
A quicker alternative is scaling here. We can write a:b:c as one ratio if we get the b parts to match.
a: b can be written as 6:15
b:c can be written as 15:50
So a:b:c is 6:15:50
This shows that the ratio a:c is 6:50, which simplifies to 3:25
And here's another type of question:
Punch is made my mixing orange juice and cranberry juice in the ratio 7:2. Mark has 30 litres of orange juice and 8 litres of cranberry juice. What is the maximum amount of punch that Mark can make?Again, I think that scaling is probably the quickest approach here. Multiplying the punch ratio by 4 gives us 28:8. This is the most punch we can make because we're using all the cranberry juice. So in total we're making 36 litres of punch.
In summary, there are a variety of approaches for solving trickier ratio problems - most can be solved efficiently using algebra, scaling or bar modelling. If you're teaching this topic for the first time at GCSE it's worth spending some time looking at the various methods. I think our students will need a lot of practice of numerous different types of ratio question to prepare for their GCSE.
I've created a lesson to accompany this post - download it from TES here.
Here are some other resources that you might find helpful:
- Mel from JustMaths collated ratio Higher GCSE questions from sample and specimen papers here, and has written up her solutions here.
- If you subscribe to MathsPad then you'll be pleased to hear that they have lovely resources for ratio including a set of questions for Higher GCSE with loads of examples like the problems I've featured in this post.
- Don Steward has plenty of ratio tasks including his set of 'Harder Ratio Questions' and a really helpful collection of GCSE ratio and proportion questions.
- On MathsBot you can generate ratio questions, revision grids and practice papers. Select 'ratio, proportion and rates of change' at the top.
- There are exam style questions in this collection from Lucy Kilgariff on TES.
- OCR has a 'Calculations with Ratio' Topic Check In and AQA has a Ratio and Proportion Topic Test.
- David Morse of Maths4Everyone has shared a set of revision exercises and ratio exam style questions.
I hope this post will be helpful when you teach ratio at GCSE. Do tweet me to let me know what methods you use if I haven't mentioned them here.