How can I estimate the number of dimes (or jelly beans, or buttons, etc.) that are in a container without counting them?

You will need a scale. You need to perform an expirement by weighing several of the items (for example, weight 100 dimes). Next, weigh the container and the items together, then weigh the container empty. Subtract the weight of the empty container from the weight of the container with the items to get the weight of just the items alone. Divide the weight of the items alone by the sample weight (for example, the weight of 100 dimes), and multiply that by the number of sample items you weighed (for example, 100) to find the approximate number of items in the bottle. Generally, the more sample items you weigh, the more accurate your estimate will be. It is important to use a large enough sample if your items do not all weigh the same.

For example, you have a jar full of jelly beans. You count out 100 jelly beans and weight them; assume they weigh 4.6 ounces. You weigh the jar full or jelly beans and together they weigh 98.5 ounces. You empty the jelly beans into another container temporarily and weigh the empty jar; it weighs 12.5 ounces. Subtract 12.5 ounces from 98.5 ounces to get the weight of the jelly beans alone; they weigh 86 ounces. Divide 86 ounces by 4.6 ounces and multiply by 100 (since you weighed 100 jelly beans) and the result is 1870 jelly beans (approximately).