The basic quick-pickle ratio for mildly tart and sweet pickles is three parts vinegar, three parts water, one part sugar, 1/4 part Kosher salt. Pickles will take about a day to ripen and will last for about a month in the refrigerator. These are not as long lasting as canned pickles. Cut the salt in half if using table salt.
I'm told there's a lot of play in the recipe. I've made pickled cucumbers with no water, two parts vinegar and one part sugar. These come out intensely sweet and sour, very mouth puckering, not sure how long they last.
For carrots, it's important to peel them so the brine will penetrate easily.
Sample recipe for green beans:
Boil the green beans for 2 minutes, then chill in cold water. Prepare the brine by mixing ingredients and bringing to a boil until the sugar and salt are dissolved. Pack the beans in glass mason jars and add any additions. Pour the hot brine over the beans. Refrigerate overnight.
Salt the cucumbers for half an hour. Rinse, drain, and squeeze out some liquid using cheesecloth. This step is optional but will result in a more flavorful cucumber. (The salting step removes some of the water in the cucumbers, so when you add the cucumbers to the pickling liquid they soak more of it up.)
Heat the pickling liquid (water, sugar, vinegar.) Then add the cucumber slices and chill.