Half-life of substance is 5 years — what fraction remains after 15 years

Question

The half-life of a radioactive substance is 5 years. What fraction of the original amount remains after 15 years?

Solution — Step by Step

Half-life $T_{1/2} = 5$ years. Total time $t = 15$ years.

Number of half-lives $= \frac{t}{T_{1/2}} = \frac{15}{5} = 3$

Three complete half-lives have passed.

After each half-life, the amount remaining is halved:

After 1 half-life: $\frac{1}{2}$ remains

After 2 half-lives: $\frac{1}{2} \times \frac{1}{2} = \frac{1}{4}$ remains

After 3 half-lives: $\frac{1}{2} \times \frac{1}{2} \times \frac{1}{2} = \frac{1}{8}$ remains

After 15 years (3 half-lives), the fraction remaining is:

N/N_0 = \left(\frac{1}{2}\right)^3 = \frac{1}{8}

One-eighth (1/8) of the original amount remains.

If you started with, say, 80 g, after 15 years only 10 g remains.

Why This Works

Radioactive decay is a random, spontaneous process — each nucleus independently has a fixed probability of decaying per unit time. The half-life is the time for exactly half of any given sample to decay, regardless of how much you start with.

The general formula is:

N(t) = N_0 \left(\frac{1}{2}\right)^{t/T_{1/2}} = N_0 e^{-\lambda t}

where $\lambda = \ln 2 / T_{1/2}$ is the decay constant. The exponential form and the fraction form are equivalent — use whichever is cleaner for the numbers given.

Alternative Method

Using the exponential decay formula:

N = N_0 e^{-\lambda t}

\lambda = \frac{\ln 2}{T_{1/2}} = \frac{0.693}{5} = 0.1386\text{ yr}^{-1}

\frac{N}{N_0} = e^{-0.1386 \times 15} = e^{-2.079} = e^{-\ln 8} = \frac{1}{8}

Same result. For integer numbers of half-lives, the $(1/2)^n$ method is faster; for non-integer times, the exponential formula is necessary.

When the total time is an exact multiple of the half-life, use the shortcut $(1/2)^n$ where $n = t/T_{1/2}$ . This avoids calculator work entirely. Look for this pattern in JEE MCQs — the numbers are almost always chosen to give integer half-lives.

Common Mistake

Students sometimes subtract rather than divide: they think “after 3 half-lives, 3/2 has decayed, so 1/2 remains” — which confuses fraction decayed with total fraction remaining. The correct logic: after each half-life, you MULTIPLY the remaining amount by 1/2. After 3 half-lives: $1 \times \frac{1}{2} \times \frac{1}{2} \times \frac{1}{2} = \frac{1}{8}$ . Don’t add or subtract the 1/2 fractions — multiply them.

Question

Solution — Step by Step

Why This Works

Alternative Method

Common Mistake

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