Determining half life without logs, given only reduction undergone and total time taken

Question

I have a half-life question that I can't solve. There's very limited information given. Even the half-life formula has not been taught yet.

The mass of a radioactive substance in a certain sample has decreased 32 times in 10 years. Determine the half-life of the substance.

The answer is 2.

From the question I understand that 32 half-lives have gone by in 10 years. How can this be solved using the simplest possible methods? Not using graphs and avoiding logarithms too if possible?

Answer 1

The solution $2$ assumes the following interpretation of the problem statement:

Suppose you have a radioactive sample of mass $m_0$ which after $10$ years is reduced to mass $m_{10} = \frac{m_0}{32}$. What is the half-life of the substance?

Recall that the half-life of a radioactive substance is the time after which the mass of a sample is halved by radioactive decay. If $10$ is a multiple of the half-life, then $$ m_{10} = \left(\left(\left(m_0 \cdot \frac{1}{2}\right) \cdot \frac{1}{2}\right) \dotsm \right) \cdot \frac{1}{2} = \frac{m_0}{2^n} $$ Since $2^5 = 32$, we know that in $10$ years the mass got halved $5$ times; in other words, once every $\frac{10}{5} = 2$ years.

Note: Still, I find your problem statement ambiguous, at best.

Answer 2

Just recall that $2^5=32$ and then compute $10/5=2$