Simplify a ratio from a tape diagram

We're told that the following diagram describes the volume of yellow and red paint in an orange mixture. So we can see that for every 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 parts of yellow, we have one, two, three, four, five, six, seven, eight parts of red.

They ask us what is the ratio of red paint to total paint in the mixture. So pause this video and see if you can have a go at that before we do it together.

All right, so if we want to think about red paint, it looks like we have eight parts red paint. And if we think about the ratio of that towards total paint, you might be tempted to put a 12 there, but that's not the total paint; that's just the yellow paint. The total paint for every eight parts red would be the eight parts red plus the twelve parts yellow. So it'd be eight plus twelve, which is twenty.

And this is true: the ratio of red paint to the total paint in the mixture is eight to twenty. Now you might not always see it expressed this way because there's other ways of writing an equivalent ratio, some that people would argue are actually simpler.

Another way to think about it is this diagram where we see the red paint. It has eight parts right over here, but we could also describe it as four groups of two, just like that, because eight is divisible by four. And if you divide the twelve parts of yellow into four groups, it's four groups of three.

And you'll see in a second why this is really interesting. Because we were able to break down both of these into four groups, and when you do that, you see very clearly hopefully that for every two parts of red, you have three parts of yellow. For every two parts of red, you have three parts of yellow: two parts of red, three parts of yellow.

And then last but not least, these two parts of red and then those three parts of yellow. So for every two parts of red, you have three parts yellow. Now once again, the ratio that they're asking isn't the ratio of red to yellow, which is two to three, but you could just take one of these groups and say, all right, for every two parts of red, I have one, two, three, four, five parts of total paint. So you could say that the ratio is also for every two parts of red, I have five parts of total paint.

And hopefully this makes intuitive sense right here why these two things are equivalent. If for every eight you have 20 total, well if you have a—if you have—if you divide that by four, for every two of red, you're going to have five.