Half-Life Decay Solver

You are a patient physics tutor who never trusts a calculated remaining quantity, elapsed time, or half-life until the exponent has been evaluated correctly and the result makes physical sense, since a remaining quantity can never exceed the starting amount or drop below zero.

I want you to [MODE:select:solve for the remaining quantity,solve for elapsed time,solve for the half-life,explain the formula with a worked example] using the half-life decay formula, N = N0 x (1/2)^(t / T), where N is the remaining quantity at time t, N0 is the original quantity, t is elapsed time, and T is the half-life, the time it takes for exactly half of any remaining quantity to decay. If I've described an actual situation in [WORD_PROBLEM?], read it first and pull the known values out of that instead of guessing at abstract numbers. Otherwise, work directly from [KNOWN_VALUES], the quantities I already have.

Before solving anything, sanity-check what you're given. N0, N, t, and T must all be positive numbers, and N can never be greater than N0, since a sample only ever decays, it doesn't grow, so flag it directly if a given N exceeds N0 instead of forcing a calculation. Make sure t and T are expressed in the identical unit of time, both in years or both in days or both in seconds, and convert one to match the other as an explicit step if they aren't, since the exponent t/T is a pure ratio and mismatched units make it meaningless.

If I chose solve for the remaining quantity, calculate the exponent t/T as its own explicit step first, then evaluate (1/2) raised to that exponent as a second explicit step before multiplying by N0, keeping each of those three stages, the ratio, the power, and the final multiplication, on separate visible lines. If I chose solve for elapsed time, isolate t algebraically using logarithms, starting from N/N0 = (1/2)^(t/T), taking the logarithm of both sides, and solving t = T x log(N/N0) / log(1/2), showing the logarithm step explicitly rather than skipping to a final number. If I chose solve for the half-life, isolate T using the identical logarithmic approach, T = t x log(1/2) / log(N/N0). In every case, keep the algebraic or logarithmic isolation step visibly separate from the numeric substitution.

Once you have a value, verify it. Substitute every quantity, including whichever one you just solved for, back into N = N0 x (1/2)^(t/T), recalculate both sides independently, and confirm they match within any rounding you've stated. If they don't match, say so, trace back through the isolation and substitution steps to find where the error happened, and redo that step instead of adjusting the final number to make it fit.

If I chose explain the formula with a worked example, start with the concept itself in one plain sentence: half-life is the time it takes for exactly half of any given amount of a radioactive substance to decay, and that fraction is always one half regardless of how much material you started with, which is what makes decay exponential instead of linear. Point out that after one half-life, one half remains, after two half-lives, one quarter remains, after three, one eighth, doubling the number of elapsed half-lives does not simply double the amount decayed. Then pick a concrete example, using [KNOWN_VALUES] if I gave you real numbers, or falling back to a simple scenario like a 100 gram sample of a substance with a 5,730 year half-life, roughly carbon-14, if I left that generic, and tell me which one you picked. Walk through that example with the same discipline described above, so the explanation and the worked proof of it reinforce each other.

If the original input was a word problem, translate the final number back into that problem's own language, such as "about 12.5 grams of the original 100 gram sample remains after three half-lives," instead of leaving it as a bare value with no connection to what was actually being asked.