# Calculating amount of Gray of radiation given to a person: divide or multiply?

Hello,

This is what the question says:

--- start of question below ---

Radiation doses are calculated by the energy deposited per kilogram of body tissue, for which the SI unit is the gray (Gy).

1 Gy = 1 J kg−1

The average energy from a single annihilation is approximately 10−13 J and there is an average of 109 annihilations per second over the period of 20 minutes needed for a PET scan.

What is the best estimate of the whole-body dose received by a 60 kg person during a PET scan?

A 2.0 Gy
B 0.20 Gy
C 0.020 Gy
D 0.0020 Gy

--- end of question above ---

I know that the middle paragraph is calculated as
10−13 J per annihilation x 109 annihilations per second x 20 minutes needed for a PET scan x 60 seconds per minute.

This gives us 0.12.

What is bothering me is the last sentence: "What is the best estimate of the whole-body dose received by a 60 kg person during a PET scan?"

I did 0.12 x 60 since you are depositing joules per kilogram and there is 60kg to deal with.

As an analogy, say you are donating money to kids in schools. For a single class with ten kids, each of whom has a bank account, if you deposit into the bank accounts of each child one pound, (ie donation rate = £1/kid) then 10 kids x £1 each = £10, so for a single class of 10 kids (or a single collective group of 60kg of body tissue), they get £10.

So if a PET machine is depositing 0.12 Joules per kilo, shouldn't a 60kg person get 60 x 0.12 Joules?

But the correct answer is actually 0.12 divided by 60.

Am I overthinking this or is the question worded badly? I cannot see how it should be divided by 60kg. If it was "divided by" then in my analogy the entire class of 10 kids would have to share £1 and get one tenth of a pound each.

Thanks.
(edited 11 months ago)
The answer they are looking for isn't in J, it's in Gy.
Original post by black tea
The answer they are looking for isn't in J, it's in Gy.

Yes I know the answer has to be in Gray.

I did this question before, a long time ago and got it right. It's just at the moment I just can't seem to see my obvious mistake.

So here's another example of what I'm thinking. Say it takes 1mg of an anaesthetic per kilo (1mg/kg) to knock out a person for surgery. If a person weighs 60kg, you'd need to multiply 1mg by 60 to get the dose needed to knock them out, 60mg, and not divide. In other words you need to deposit into the person, 60mg of that anaesthetic, not 1/60mg.
Original post by martinthemartian
Yes I know the answer has to be in Gray.

I did this question before, a long time ago and got it right. It's just at the moment I just can't seem to see my obvious mistake.

So here's another example of what I'm thinking. Say it takes 1mg of an anaesthetic per kilo (1mg/kg) to knock out a person for surgery. If a person weighs 60kg, you'd need to multiply 1mg by 60 to get the dose needed to knock them out, 60mg, and not divide. In other words you need to deposit into the person, 60mg of that anaesthetic, not 1/60mg.

The other poster has answered your question, but you have not really considered what it means.
The answer you calculated, 1.2, has units of joule.
If this is the total energy absorbed by a 60 kg person, then how much energy is absorbed by 1 kg? (This is what a Gray is, and the unit for the answer)
It appears the question is giving you the total energy absorbed, and asking you to say what this is in Gray, by dividing it by the mass to find the value per kg.
I agree the question is not very well worded.
Original post by Stonebridge
It appears the question is giving you the total energy absorbed, and asking you to say what this is in Gray, by dividing it by the mass to find the value per kg.
I agree the question is not very well worded.

Ahh okay.

I haven't actually studied radiation, I think last time I guessed, and this question isn't a radiation subject practise question, it is a standalone question in a booklet with various unrelated questions if that makes sense. That booklet is notorious for badly worded questions and in some cases downright incorrect so I want to separate my own factual mistakes from their bad wording-induced errors. So I hope you don't mind if I ask another question.

How would you rephrase this question to make it clearer? I ask because I want to know what part of the wording is confusing me so if they come up with a similar question and I am unfamiliar with the actual topic, which in this case is radiation given to a person (or absorbed by a person, as 'given to' is different to 'absorbed by'), I would know whether I should multiply or divide. (If the question was just said "Gy = J kg−1, J is 0.12, kg is 60, find Gy" I would have divided by 60 and be done.)

To help me answer the question, I tried to replace the word "radiation" in the first sentence to "anaesthetics" and "Joules" to "micrograms" and it makes me want to multiply 0.12 micrograms needed to be "deposited [into the patient] per kilogram of body tissue" by 60kg to get the dose needed (to hypothetically knock out a 60kg person) and that dose is provided by the PET machine. I see deposit as the act of giving the recipient, eg deposit money into a bank, deposit into a person's body etc. I do not see similarity in the words "absorbed" and "deposited" and did not see that the question has given me the total energy absorbed because to me the word dosage per kilo means you need X amount per kilo of body weight, therefore multiply by body weight.

Ahh wait I think I just figured it out, is "Gray" not a unit of dosage and they used the word dosage wrong?
(edited 11 months ago)
@martinthemartian It seems that the question is giving you the information initially to find the total amount of radiation needed for the scan. (In joule)
It then says, effectively, if this is spread over 60kg, what is the dose in Gy.
This is why you then divide by 60.

In the examples you thought up, you are starting with the 'dose' , the amount per kg, and then multiplying it by the number of kg to find the total.
So those calculations are going the opposite way. They are correct, but not what it appears this question is asking for.
Hope this helps.
(edited 11 months ago)
Original post by Stonebridge
@martinthemartian It seems that the question is giving you the information initially to find the total amount of radiation needed for the scan. (In joule)
It then says, effectively, if this is spread over 60kg, what is the dose in Gy.
This is why you then divide by 60.

In the examples you thought up, you are starting with the 'dose' , the amount per kg, and then multiplying it by the number of kg to find the total.
So those calculations are going the opposite way.

Ahhh okay I see now, thank you!