Thread by @MrYoungMaths

Have you ever wondered what the percentages mean in those road signs about an upcoming gradient change? I know I have! Maths teachers, here is what I discovered... #mathschat @standupmaths

You might have some guesses about what the percentages mean. Does it mean the percentage change of the gradient? Is it the percentage of an angle? Is it to do with how you should adjust your speed? Alas, none of the above.

In order to understand their origin, we must first travel to the mid-1970s where the gradient signs showed not percentages but ratios! How fascinating! Apparently some of these old signs still exist and I would love to find them.

The signs would say 1:3, for example. This meant that for every 3 units travelled horizontally, there would be a 1 unit vertical increase/decrease. It’s irrelevant what unit you’re using (metres, feet, etc) because it’s the ratio that’s important.

So far, so easy. The problem is this: the ratio 1:5 (1 vertical for every 5 horizontal) is a shallower gradient than the ratio 1:10 (1 vertical for every 10 horizontal) and that’s actually quite confusing for some people. Surely bigger number = more steep?

Add that most people are viewing these signs in a split-second while driving a vehicle at speed and you’ve got an even bigger problem. A road sign needs to clearly and instantly communicate a warning and the ratio form of the sign simply doesn’t do that. It’s confusing!

And so, the powers that be converted the ratio signs into percentage signs. A bigger percentage means a bigger steepness and that’s straightforward for people to grasp. 20% uphill? I’d better change down a gear or two! 5% downhill? Time to ready my brakes!

But I haven’t actually answered the question I posed...what do the percentages actually mean? Well, they are derived from the original ratios. 1:5 means that the vertical change is one fifth of the horizontal change...or 20%! 1:10 would become 10%, 1:4 would be 25%...

Put generally, if the sign said x% it means that the vertical change is x% of the horizontal change. Therefore a bigger % translates as a steeper hill. If a sign said 50%, it means the vertical change is only half the horizontal change - that’s steep! Not for the faint-hearted!

While the percentages seem a bit nonsensical at first, they are actually rooted in proper maths. It’s just a way of turning something potentially alienating into something that can be easily understood by all. And a road sign absolutely must be easy to understand!

As a bonus, have you ever wondered what angle the percentages correspond to? Well now you know what they mean, you can use some simple trigonometry (tangent ratio) to calculate this! A good answer for when a student asks when they might need trig in real life?

There are calculated by doing inverse tan of the vertical change divided by the horizontal change.

All angles in degrees and to 3sf.

5% is 2.86
10% is 5.71
20% is 11.3
25% is 14.0

And so on...

(100% would be 45 degrees because tan45 = 1)

I hope this has enlightened you in some way! Perhaps it’ll make you a safer driver or maybe cause some accidents because you’ll be distracted while driving. I take no responsibility, I’m just happy to spread knowledge!

Of course, I would love for as many maths teachers and beyond to see this so please share the start of this thread far and wide! @mathsjem @mrbartonmaths @MrDraperMaths @tkendalluk @robeastaway @jamesgrime @DrFrostMaths

EDIT: apologies, this should say steeper not shallower!