 What is the Sum of Fractions Rule?
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# What is the Sum of Fractions Rule?

Dear Dr. Zoomie – I was reading about posting areas that hold radioactive materials and they said something about the “sum of fractions” that I didn’t quite understand. There was some mathematical equation, but I didn’t exactly understand that either. Can you tell me what’s up with the “sum of fractions?” Thanks!

This one is actually sort of important, but it’s not as hard as the regs make it seem. And you can use the same thinking for a couple of things, so it’s worth trying to understand. Here’s how it works.

Say you’ve got a room where you’re storing radioactive materials and you’re trying to figure out whether or not you need to put up a sign indicating it’s being used for radioactive materials storage. If you look in the regulations you’ll find a table telling you what level of activity requires licensing (anything less than this limit is exempt from licensing) and possibly another table telling you what level requires labeling (if you can’t find the second table then the labeling level is 10 times as high as the exemption limit). So it seems pretty simple – if you have more than the limit for labeling then you have to post the room and if you have less than the limit, you don’t. In reality it’s a little more complicated.

Let’s try an easy one – according to 10 CFR 30 Appendix B (which is titled Quantities of Licensed Materials Requiring Labeling) you will find that if you are storing tritium (H-3) in a room then you have to post the room if you have more than 1000 µCi (or 1 mCi). So if you have, say, 900 µCi of tritium in the room you don’t have to worry about posting it. Similarly, the level for P-32 is 10 µCi and the limit for I-125 (both of these are used in research) is 1 µCi. So if you have less than 10 µCi of P-32 or less than 1 µCi of I-125 you’re also exempt from having to post the room for radioactive materials storage. But what if you’ve got a bit of all of these nuclides – or others?

Say, for example, you have a room with 400 µCi of H-3, 4 µCi of P-32, and 0.4 µCi of I-125? None of these are high enough to require posting in and of themselves. But using the “sum of fractions” you need to put a sign on the room anyhow. The table here explains why.

 Nuclide Labeling limit (µCi) Amount on-hand (µCi) Fraction of limit H-3 1000 400 0.4 P-32 10 4 0.4 I-125 1 0.4 0.4 Sum of fractions 1.2

So – the first thing you need to do is to figure out how much activity you’ve got compared to the labeling limit (half, a third, three quarters, etc.). To do this you divide the amount of activity you’ve got by the labeling limit (for example, 400 µCi is 40% of the limit of 1000 µCi for tritium). After you’ve done this for each of the nuclides you’re storing you just add up all of the fractions – if the sum is greater than 1 you have to post the room.

So let’s take the table above. For each of these nuclides you have 40% of the allowable limit (40% is the same as 0.4). Since you have three nuclides, each of them with 40% of the allowable limit, the sum of the fractions comes out to 0.4+0.4+0.4+1.2. Since this is greater than 1.0 the room has to be posted. Easy, right?

You see the Sum of Fractions show up in a number of places. For example, 10 CFR 37 talks about when you have to take increased controls over the security of radioactive materials – and you use the sum of fractions if you have multiple nuclides in storage. If you’re discharging radionuclides into the sanitary sewer system (or letting them escape into the air) you’ll have limits for each radionuclide being discharged – the sum of fractions is used here as well. There’s more, but you get the idea. Once you understand how the idea works then you can use the same technique – the same calculations (or the same spreadsheet) across the board.