Hints and solutions to Guess and Check #1

Barbara has exactly $2.00 in nickels and dimes. She has twice as many dimes as nickels. How many of each does she have?

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• There are (at least!) three approaches to this one. Two are guess-and-check. Probably the most straightforward is to guess a number of nickels, and then see how much money that many nickels plus twice as many dimes makes. For instance, if you guess 5 nickels, then 5 nickels and 10 dimes makes $1.25, which is less than $2.00, so you need to guess more nickels, etc.
• A similar approach would be to guess a number of dimes and nickels that adds up to $2.00, and then adjust by trading nickels for dimes until there are twice as many nickels as dimes. For instance, I could start with 10 dimes and 20 nickels, which makes $2.00, but I clearly have too many nickels. So trade 4 of the nickels for two dimes and check again (makes 12 dimes and 16 nickels, still not right, etc).
• A more sophisticated approach uses algebra - for instance, let x be the number of nickels, then there are 2x dimes, and we have 5x + 20x = 25x cents altogether, and we need 25x = 200. So x is 8 and there should be 8 nickels and 16 dimes.