## Making sense of sunscreen protection percents. Where does “SPF 30 absorbs 97% of the UV” come from?

You may have heard that: “SPF 30 absorbs 97% of the UV” or “SPF 50 absorbs 98% of the UV”. These numbers are from a math model and it’s quite simple!

The math model is:

1 – (1 ÷ SPF)

1 divided by the sunscreen’s SPF, subtracted from 1.

With an SPF 30:

1 – (1 ÷ 30) = 1 – (1/30 or 0.0333…) = 1 – 0.0333… = 0.9666…

The ellipses (…) means repeating, the 666 in the decimal number 0.9666 repeats forever.

For simplicity, we can round up 0.9666… to 0.97. We can then convert a decimal number to a percent by multiplying it by 100.

0.97 x 100 = 97%

What’s the basis of this math model? The SPF of our sunscreens are tested experimentally on real people. SPF is the ratio between the amount of UV the participants’ skin can be exposed to before sunburn with and without the sunscreen.

SPF can be affected by things that aren’t absorbing or reflecting UV – like antioxidants, anti-inflammatories, protection boosters, and an individual’s skin. We also know that not every wavelength of UV causes sunburn equally. The math model only accounts for the amount of UV the sunscreen passes through to the skin and the amount of UV it doesn’t.

That’s why these percentage protection numbers are a model, they’re a simplified representation. But models can be useful in understanding complicated things.

So let’s break down this model

1 – (1 ÷ SPF)

1 ÷ SPF represents the fraction of UV that the sunscreen lets through.

So in the model, an SPF 25 exposes the skin to 1 ÷ 25 or 1/25 or 0.04

To convert a decimal number into a percent we multiply by 100
0.04 x 100 gives us 4%.

If we want to know the fraction of UV that the sunscreen prevents from reaching the skin in this model, we subtract it from the total, which is 100%. 100% can be written as 1/1 or 1 or 25/25.

1 – (1 ÷ SPF)

With an SPF 25, we can write 1 as 25/25

1 – (1 ÷ 25) = 1 – (1/25) = 25/25 – (1/25) = 24/25 or 0.96

To convert a decimal number into a percent we multiply 0.96 by 100, which gives us 96%.

The model doesn’t account for how, or really what form of UV. Just the UV that causes sunburn – which SPF is a ratio of, and what is being allowed through and not let through.

1 ÷ SPF gives us the fraction of UV the sunscreen lets through.

1 – (1 ÷ SPF) gives us the fraction of UV that the sunscreen doesn’t let through.

The fraction of UV that is being let through and not being let through add up to all of the UV, 1 or 100%.

On the previous slides, we showed that an SPF 25 in the model lets through 4% and doesn’t let through 96% of the UV.

4% and 96% add up to 100%.

Let’s run through this for an SPF 60. Working it out with your calculator can make it easier to understand!

1 ÷ SPF gives us the fraction of UV the sunscreen lets through.

1 – (1 ÷ SPF) gives us the fraction of UV that the sunscreen doesn’t let through.

Since the SPF is 60, we can put that in

1 ÷ SPF gives us the fraction of UV the sunscreen lets through. We can write 1 ÷ 60

1 – (1 ÷ SPF) gives us the fraction of UV that the sunscreen doesn’t let through.

We can write 1 – (1 ÷ 60)

What fraction or percent of the UV does this model show an SPF 60 letting through and not letting through?

So the amount of UV that an SPF 60 lets through in this model is:

1 ÷ SPF, since SPF is 60, we write 1 ÷ 60

1 ÷ 60 can be written as 1/60. Enter that into a calculator and you get the decimal number, which is 0.01666… for simplicity, we can round that up to 0.0167. We multiply that by 100 to get a percent, 1.67%

The amount of UV that the SPF doesn’t let through is 1 – (1 ÷ SPF). We know 1 ÷ SPF is 1.67%, so 100% minus 1.67% gives us 98.33%

1 – (1 ÷ SPF) = 1 – (1/60) = 60/60 – (1/60) = 59/60 = 0.98333… = 98.333% rounded to 98.33%

We can check our work by seeing that 1.67% and 98.33% add up to 100%.

Sometimes the percentages don’t add up to exactly 100% – that’s usually because of how the decimal numbers were rounded.

The math here might look complicated, but it is just fractions.

If you know a quarter is 1/4 and can be written as 0.25 or 25%

That 4 quarters is equal to 1 and can be written as 4/4 or 100%

Then you can do this!