Why does one inch in an official rain gauge actually only one - tenth of an inch of rain?

If the rain gauge has a 10:1 ratio as described, then the funnel on top has an area that is 10 times the cross-sectional area of the glass tube. Suppose you had a can, that has the same area opening as the funnel, then if it rains 0.5 inch that day, there will be 0.5 inches of water in the can, but if you take the water in that can and pour it into the rain gauge (with cross sectional area 1/10 of the can), that same volume of water will fill up the gauge by 5 inches.

They do this so that it is easier to read the amount of rain. If you did not have the funnel on top, it'd be harder to read how much rain there was (was it 0.1 inches or 0.15 inches for example), plus statistically a smaller opening has less of a chance of 'catching' a raindrop than the wider opening.