To measure a short distance using the miles scale...
(1) Set divider tips to span the distance.
(2) Move the dividers to the miles scale with one tip on 0 miles to get the approximate distance.
(3) Then slide the other tip to the nearest integer on the scale, and read the fractional part from the tip that was on 0.

General Notes
Miles scales will appear on most charts with scales of 1:80,000 or larger. When no miles scale is given, use the Latitude scale for the miles measure. Each minute of Latitude equals 1 nautical mile.

When using electronic charts, the display software will typically include an option on the menu bar for measuring ranges and bearings. With these functions, you just drag the mouse cursor from one point to the next and the distance between points and the bearing between them show numerically on the screen.

Video Notes
We will measure the distance between the two points marked by these fingers. The dividers used are a speed bow, which we adjust with the knob between the two points. We are measuring the distance from a channel marker buoy to a daymark at the entrance to a channel.

Once the dividers are accurately set to these points, we move the dividers — with out changing their separation — up to the miles scale, which in this case was at the top of the chart. It could be anywhere on the chart.
Notice that we first put left tip on the 0 and look to the right tip to see that we have here a distance of just over 2 and one half miles (nautical miles). Next slide the dividers to the left until the right tip is exactly on the 2, and read the fractional part from where the left tip hits the tenths scale. Count out the number of tick marks from the 0 back to the left to see that the fractional part is about 0.64. The total distance is then 2.64 nautical miles.
Whether or not the 0.04 part has any significance at all depends on how carefully we have done the other parts. Normally, such a measurement would only be done to 0.1 mile precision, so this answer would be stated as 2.6 miles.