Solving Ratio Problems

If you don’t yet know how to work out the 4th number in a ratio grid, watch this video first!

The videos on this page use the worksheet “Ratio Problems” which you can download:

Simplify a Ratio using a calculator
Split Money using a ratio
Wordy Ratio problem about socks
Wordy ratio problem given one of the values and not the total
Ratio problem sharing sweets


Introduction to Ratio Grids

Ratio Grids can be used whenever you have a problem that starts with 3 numbers, and you have to times and divide. They are a clear way of laying out your working, and knowing which order to do the calculation.

You will need a calculator to do the practice questions, but first, here are 2 videos. The first shows you how to solve a small ratio grid, the second deals with the larger ones:

Using Ratio Grids
Larger Ratio Grids

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Now you can use Ratio Grids to solve a Maths Problems, here are some types of problem you can try:

Working with angles in Polygons

There are loads of different ways to approach the problem of “How do I remember that the total of the angles in a hexagon is 720?” (and all the other polygons for that matter!). Which one suits YOU is pretty dependent on your learning style, and on how much Maths you do week on week….

Memorisation (and a cheat!)

This may work for you, to some extent. You need to KNOW that the tota180 degree protractorl of angles in a triangle is 180 degrees. If you really struggle with this, perhaps it can help you to dig out your protractor from your pencil case. The biggest number on it is 180 and that’s the number you need to remember!

Again most people can remember that a RIGHT ANGLE is 90 degrees so the TOTAL of the angles in a rectangle is 360. (4×90).

If you are really good at memorising, then it may be helpful for you to remember that, every time you add one more side to your polygon, you add another 180 degrees. That means that a 5 sided shape (pentagon) has 540, 6 sided has 720, 7 sided has 900, 8 sided has 1080, etc etc…..

One single angle in a regular polygon

For example, an 8 sided regular octagon has a total of 1080 degrees. Each internal angle must be 1080/8 = 135 degrees.

Another approach…. less to memorise.regular octagon

This may suit you if you really have a flaky memory…. it IS important that you know The total of the angles in a triangle is 180 degrees. Then you can sketch the shape…. split it into triangles….. see the video below.

I have taught this the “traditional” way where you break the octagon up into 4 triangles. It’s cool. Mathematically very clever. But VERY hard for some of my students to actually remember… so this (less elegant) way is what I tend to teach more often.

Working Backwards is almost as easy….

Sometimes you are given the interior angle and told “This is from a regular polygon. How many sides has it got? Here’s how to do that kind of question….






Mental Maths – how to square any number between 11 and 19

Stage 1 – practice squaring with pen and paper, until you are confident (see video below)

Stage 2 – just jot the answers into the grid

Stage 3 – jot the grid and imagine the numbers

Stage 4 – do it all in your head

The video for that would be a bit boring!

Chinese Multiplication, or how to multiply together VERY large numbers!

Chinese multiplication has been explained many times in many places on the Internet. This is a quick recap of the way I do it….

The kids I’ve taught, especially the more able ones, really like this way of multiplying numbers because it’s SOOOO easy to build up to very large numbers.

Within 20 minutes, a group of ambitious mathematicians has commandeered the class whiteboard and tried to do an ENORMOUS sum like 185936296722 x 15436796 and got an answer. This gives the teacher a problem. How can the sum be checked?  Calculators and EXCEL will round the answer to only 10 or so significant figures, which is pretty hopeless for checking the work.

Here is a link to an EXCEL spreadsheet  that will do these HUGE sums so you can check pupils’ (or your own) work.

The extra challenge that these interpid mathematicians give themselves, of course, is how to add together huge long lists of numbers. Here’s an example of one of the additions in the sum mentioned above:

One of the additions needed to do the sum...
One of the additions needed to do the sum…

The student has to add 4,1,7,5,2,1,2,5,7,1,4,2,5,1,2 and 0. It’s tough to add all that without errors, so encourage them to look for TENS, and cross them out, “carrying” them into the next column…

They could make ten from the 4,1 and 5, then another from 7,2 and 1, and anotherfrom 1,4 and 5. Cross them out neatly and there’s not really much more to add! The nice thing is you can tackly any column you like, in any order, which is great for mathematicians who don’t know their right from their left! (except of course the TENS have to move left!).

Most of the numbers made TENS! Each ten has been done in a distinct colour for clarity.
Most of the numbers made TENS! Each ten has been done in a distinct colour for clarity.

About half the adding done now...
About half the adding done now…

Once this TEN-hunting is complete, the final pass is to add up any digits that are left.

Most of the adding done now...
Most of the adding done now…

completed sum
Completed! The answer!

Finally, some thoughts about the process of learning Chinese Multiplication:

  • It’s great practice USING TABLES
  • It’s great practice at ADDING long lists of numbers
  • Pupils will normally self-differentiate and settle with the size of sum that suits them. For GCSE only a 2-digit by 3-digit sum is normally required (which seems a shame really!)
  • They take time to learn how to draw the grids, and need to practice regularly. Sadly this pus some schools off teaching the method as “THE” method of multiplication. It is the most powerful, and handles decimals really easily too:

Decimal points in Chinese Multiplication
Decimal points in Chinese Multiplication