Before you ride your horse you need to ensure he's clean and ready to go - but this quick groom, or quartering, should not take long at all. Here's what you need to do.

Quartering is a quick groom before exercise and should take less than 10 minutes.  It is a quick 'tidy up'  to ensure your horse is clean and tidy.  Old horseman would be horrified to see a horse being ridden with straw, shavings or any other form of bedding in his tail as it was seen as a poor reflection on standards and the owner.
To Quarter Your Horse
1. Tie your horse up safely.
2. Pick out feet.
3. If wearing a rug undo the breast buckles and surcingles, remembering to tie up the surcingles.
4. Turn the front of the rug back over the quarters and quickly brush off, removing any stable stains.

HA0769Bob Atkins
5. Replace the front of the rug, then turn the back of the rug forwards, resting it over the withers and shoulders.  Brush quickly, removing any stable stains, then replacethe rug.
6. Brush out the mane and tail.
7. Sponge eyes, nose and dock.

HA2072Bob Atkins 

Assuming the horse was being ridden after quartering, it would be left tied up and the rug over his back (not fastened up) ready for the tack to be put on.