# In the Hundreds

For how many positive values of $n$ are both $\frac n2$ and $2n$ three-digit whole numbers?

## Problem

For how many positive values of $n$ are both $\frac{1}{2}n$ and $2n$ three-digit whole numbers?

## Student Solutions

**Answer**: $150$

**Using the relationship between $\frac12n$ and $2n$**

To get from $\frac12n$ to $2n$, $\times4$

$\frac12n$ and $\frac12n\times4$ are both between $100$ and $999$

$\frac12n$ small, this will be fine e.g. $100$ and $100\times4$ are both between $100$ and $999$

$\frac12n=250\Rightarrow 2n=1000$ is the first number which is too big

$\therefore \frac12n$ is between $100$ and $249$ - which is $150$ numbers (double them to get $n$)

**Using relationships with $n$**

If $\frac{1}{2}n$ is a three-digit whole number, then $n$ is an even number between $200$ and $1998$ inclusive.

Since we also want $2n$ to be a three-digit whole number, $n$ must be an even number between $200$ and $499$ inclusive.

There are $150$ such numbers.