What Do the Percentages Mean on Steep Road Warning Signs?

Maths teacher Simon Young tweeted a thread about percentages on road signs, and he's given me permission to write it up as a blog post. I've stayed faithful to Simon's original wording, with a little editing here and there, mostly based on his own corrections and some of the conversation resulting from his thread (the original thread is linked at the end).

Road steepness is expressed as a percentage on UK road signs: what does it mean, and where do the numbers come from?

Have you ever wondered what the percentages mean in those road signs about an upcoming gradient change? I know I have! Here is what I discovered...

20% gradient sign By the B836 road at the top of the hill west of Loch Striven. © Copyright Thomas Nugent and licensed for reuse under this Creative Commons Licence.

You might have some guesses about what the percentages mean. Do they mean...

a) the percentage change of the gradient?

b) the percentage of an angle?

c) something to do with how you should adjust your speed?

or 

d) none of the above.

Did you pick d? Of course you did! Correct!

In order to understand their origin, we must first travel back to the mid- 1970s when the gradient signs showed not percentages but ratios! 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.

An old-style gradient sign on the Bishop's Road near Downhill, Northern Ireland. Albert Bridge / CC BY-SA

So far, so easy. The problem is this: the ratio 1:5 (a change of 1 unit vertically for every 5 units travelled horizontally) is a steeper 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%! (so the two examples in the photographs above show the same gradients). 1:10 would become 10%, 1:4 would be 25%

Put generally, if the sign says "𝓍%" it means that the vertical change is 𝓍% of the horizontal change: a bigger percentage 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 confusing to some into a system that's a bit more intuitive (a road sign absolutely must be easy to understand!)

Bonus maths!

Have you ever wondered what angle the percentages correspond to? Well now you know what they mean, you can use some simple trigonometry to calculate this!

Trig Gradient TOA by T.Briggs is licensed under CC BY-SA 4.0

Compare this to calculating the gradient of a straight line, as would be taught in a maths class ("rise over run" or "change in 𝓎 divided by change in 𝓍"). A good answer for when a student asks when they might need trig in real life, maybe? The angle can be calculated by doing inverse tan of the vertical change divided by the horizontal change.

Some examples, with all angles correct 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 tan(45°) = 1)

Some people responding to the original thread pointed out that in the actual maths the gradient would be most likely calculated by comparing the change in elevation with the actual length of the road travelled, in which case we get:

Trig Gradient SOH by T.Briggs is licensed under CC BY-SA 4.0

Simon signed off with:

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!

It certainly enlightened me! Here's the tweet that kicked off the thread (click through to see the whole thing):

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 pic.twitter.com/Jkq2EkJNf1

— Simon Young (@MrYoungMaths) July 11, 2020