The question you opened
Seven coins that make fifty cents
You have 7 coins worth 50 cents in total, and every coin is 5 or 10 cents.
How many 10 cent coins do you have?
3 coins
Seven 5 cent coins give 35. Each swap to a 10 adds 5, and you need 15 more, so three coins get swapped.
Seven coins add up to 50 cents, and every coin is either 5 or 10 cents. Three of them are 10 cent coins. The neatest way in is to start with a wrong answer on purpose. Imagine all seven coins are 5 cent coins. That gives 35 cents, which is 15 short of 50. Now swap one 5 for a 10. The count of coins stays at seven and the total goes up by 5. Each swap adds another 5.
You need 15 more, and 15 divided by 5 is 3, so three swaps. Three 10 cent coins and four 5 cent coins: 30 plus 20 makes 50, and that is seven coins. Checking is easy and worth doing. Three tens is 30. Four fives is 20. Thirty plus twenty is fifty. Three plus four is seven. Both facts hold. The tempting wrong answer is five, because five 10 cent coins do make 50 cents. It fails the other half of the question, which says there are seven coins.
Puzzles like this always carry two conditions, a total and a count, and an answer has to satisfy both. The start all the same, then swap method is genuinely useful and works well before anyone has met algebra. Cows and chickens with a known number of heads and legs. Tickets at two prices with a known total. Boxes of two sizes filling a shelf. Assume everything is the cheaper or smaller kind, see how far short you are, then work out what each swap is worth.
One swap here is worth 5 cents, which is the difference between the two coin values. That difference is the number to find first in any version of this puzzle, because the shortfall divided by the difference gives you the number of swaps directly. Try building your own with 12 coins making 130 cents and see whether it has an answer at all. Some versions do not, and finding that out is half the fun.
Technique: Start all the same, then swap