to do a calculation to get an answer to a mathematical question

We need to work out the total cost of the project. 我们需要算出这项工程的总造价。

to understand something or to find the answer to something by thinking about it

[ + question word ] There will be a full investigation to work out what caused the accident. 将进行全面的调查以找出事故发生的原因。

[ + that ] Investigators needed several months to work out that a fraud had been committed. 调查人员需要几个月的时间才能查出欺诈行为。

to exercise in order to improve the strength or appearance of your body

Huw works out in the gym two or three times a week. 休每周在健身房锻炼两三次。

to be the result of a calculation

These figures work out differently each time I add them. 我每次把这些数字相加得出的结果都不一样。

The safe load for a truck of this size works out at nearly 20 tons. 这种大小的卡车的安全载重量算出来是将近20吨。

In fact the trip worked out cheaper than we'd expected. 实际上，这次旅行花的钱比我们预计的少。

to happen or develop in a particular way

How is the new monitoring procedure working out? 新的监督程序运行得如何？

Let's hope this new job works out well for him. 希望这项新工作会非常适合他。

to understand the reasons for someone's behaviour

Why does he behave like that? I can't work him out at all. 他为什么那样做？——我完全搞不清楚。

If a problem or difficult situation works out, it gradually becomes better or satisfactory, and if you work it out, you make it better or satisfactory.

Don't worry about anything - it'll all work out (for the best) in the end, you'll see. 什么也别担心——最后一切都会好转，等着看吧。