When you find an explanation for or a solution to a problem or mystery, you solve it. To solve a math problem, you must find the answer to a problem.
Ex: The famous fictional detective, Sherlock Holmes, was able to solve mysteries using deductive reasoning. When he announced the solution, he always said to his companion, "Elementary, my dear Watson!"
Ex: Solving a math problem may require several steps. You may have to know how to solve for x in an equation by using addition, subtraction, and multiplication in the same problem.
You can solve a puzzle like the contestants do on the television show, Wheel of Fortune. The successful contestant is the puzzle solver, the person who finds the solution of the puzzle. When you solve a code, a mystery or a difficult problem, you can use the verb crack as well. If you crack a code, you have solved it.
Ex: The New York Times Sunday crossword puzzle is very challenging today, and its solver has many reasons to be proud.
Work out can mean to solve a problem. If something works out, like a relationship, it can be said to be successful.
Ex: I had to use all my previous knowledge of the game Cluedo to work out who had committed the murder and win the game. Colonel Mustard did it in the library with the rope.
Ex: John's boss is very demanding. Their relationship at work is getting worse. After meeting this afternoon, John and his boss have decided to take steps to work it out.
A workout is an exercise routine. When you perform that routine, you are working out. Your workout might leave you with some stiff muscles.
Ex: My mother works out every morning. She goes for a run before doing some reps with free weights. Thanks to her workout, she is in great shape.
Ex: Last weekend, Sharon painted her dining room. The muscles in her arms and shoulders became stiff and painful by the end of the day. A long, hot bath helped her work out the stiffness, so she felt great on Monday morning.