Quadrupling Leads to Halving

Task

Give an explanation, in terms of the structure of the expression below, why it halves in value when $n$ is quadrupled: $$\frac{s}{\sqrt n}.$$

IM Commentary

This question provides students with an opportunity to see expressions as constructed out of a sequence of operations: first taking the square root of $n$, then dividing the result of that operation into $s$. This way of looking at the expression helps the student see that $$\frac{s}{\sqrt{4n}}$$ can be rewritten as $$\frac12 \frac{s}{\sqrt{n}}.$$

Students studying statistics encounter the expression in this question as the standard deviation of a sampling distribution with samples of size $n$ when the distribution from which the sample is taken has standard deviation $s$.

Solution

The expression is a fraction in which the denominator is $\sqrt{n}$. The square root of $4n$ is twice the square root of $n$, because $\sqrt{4n} = \sqrt{4}\times\sqrt{n} = 2\sqrt{n}$. So quadrupling $n$ multiplies the denominator of the expression by $2$: $$\frac{s}{\sqrt {4n}}=\frac{s}{\sqrt 4 \times \sqrt n} = \frac{s}{2 \times \sqrt n}.$$ Multiplying the denominator of a fraction by $2$ halves the value of the fraction: $$\frac{s}{2 \times \sqrt n} = \frac {1}{2} \times \frac {s}{\sqrt n}.$$ So multiplying $n$ by $4$ multiplies the value of the expression by $1/2$.