Half-life of a radioactive substance — how much remains after 5 half-lives

Question

A radioactive substance has a half-life of 20 minutes. If you start with 1000 g of the substance, how much of it remains after 5 half-lives?

Solution — Step by Step

5 half-lives have passed. Since each half-life is 20 minutes, the total time elapsed is $5 \times 20 = 100$ minutes. We don’t actually need this number for the calculation, but it’s good to keep track.

After every half-life, exactly half the remaining substance decays. So after 1 half-life, you have $\frac{1}{2}$ of the original. After 2 half-lives, you have $\frac{1}{2} \times \frac{1}{2} = \frac{1}{4}$ of the original. See the pattern?

After $n$ half-lives, the remaining amount is:

N = N_0 \left(\frac{1}{2}\right)^n

Here, $N_0 = 1000$ g and $n = 5$ .

N = 1000 \times \left(\frac{1}{2}\right)^5 = 1000 \times \frac{1}{32} = \frac{1000}{32} = 31.25 \text{ g}

After 5 half-lives, 31.25 g of the substance remains.

Why This Works

Radioactive decay is a purely probabilistic process. Each nucleus has a fixed probability of decaying per unit time — it has no memory of how long it’s been sitting there. This is why the fraction that decays in each half-life is always exactly $\frac{1}{2}$ , regardless of how much is left.

The formula $N = N_0 \left(\frac{1}{2}\right)^n$ captures this beautifully. Every time $n$ increases by 1, we multiply by another $\frac{1}{2}$ . This is exponential decay — the same mathematics that governs compound interest, just in reverse.

This topic has high weightage in both NEET and CBSE Class 12 boards. The half-life formula alone fetches direct 1-mark questions in NEET almost every year.

Alternative Method

Instead of the formula, we can track it step by step in a table. This is slower but makes the concept crystal clear — and helps when the examiner asks you to show working.

Half-life No.	Amount Remaining
Start (0)	1000 g
After 1st	500 g
After 2nd	250 g
After 3rd	125 g
After 4th	62.5 g
After 5th	31.25 g

Each row is just the previous row divided by 2. Same answer, different route.

In MCQs, if $n$ is a whole number, compute $2^n$ mentally and divide $N_0$ by it. $2^5 = 32$ , so the answer is $1000 \div 32 = 31.25$ g. Takes under 10 seconds.

Common Mistake

Students often confuse “5 half-lives have passed” with “the substance is 50% gone” — that’s only true after 1 half-life. After 5 half-lives, only $\frac{1}{32}$ ≈ 3.125% remains. The decay is exponential, not linear. Drawing the step table (shown above) for 30 seconds in the exam prevents this slip every time.

Question

Solution — Step by Step

Why This Works

Alternative Method

Common Mistake

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