21 Nickels, Dimes, and Pennies in a Dollar
From: Mary C. DuChateau Subject: Coin question Date: Tue, 8 Nov 94 20:15:23 EST Hello Dr. math- I am showing my eleven year-old electronic mail on the Internet, and he also has a math question for school homework. We are going to "kill two birds with one stone." His question is: How can you have 21 coins of nickels, dimes, and pennies, that equal one dollar using no quarters? Please help us out. Thank you so much - Marie and John DuChateau
Date: Tue, 8 Nov 1994 20:55:15 -0500 From: Doctor Melissa Subject: Re: Coin question Hello! I'm not sure how much math John has had, but I'll try to show this using algebra. If he needs more of an explanation, please write back! We're trying to find what number of coins of pennies nickels and dimes add up to $1.00. We can rewrite this equation as: 0.10D + 0.05N + 0.01P = 1.00 Multiply everything by 100 to make the numbers easier to read on a computer: 10D + 5N + P = 100 where D is the number of dimes, N is the number of nickels, and P is the number of pennies. But that isn't enough information to solve the problem - we were also told that there are 21 coins. So we have another equation: D + N + P = 21 We can then rewrite it as: P = 21 - D - N and substitute this into the original equation to get: 10D + 5N + (21 - D - N) = 100 Which is equal to: 9D + 4N + 21 = 100 So: 9D + 4N = 79 Since there are two variables here, we know there are many different answers. To find an answer with only integers (no fractions) in it, you can either graph it or just try values for D (Hint: there is a value less than 5 that gives a counting number answer for N) and solve for N. Good luck! Dr. Melissa
Date: 6/25/96 at 13:27:30 From: Doctor Gary Subject: Re: Coin question Here's another answer. Since 20 nickels are worth a dollar, at least one of the coins must be worth less than five cents, so there has to be at least one penny. If there's at least one penny, there have to be at least 5 pennies, because the value of all the coins (like the value of each nickel and dime) is a multiple of five cents. So, the possible numbers of pennies are five, ten, fifteen, and twenty. We can rule out fifteen and twenty, because all the other coins (even if they were all dimes) wouldn't be worth enough to make up the difference between the value of the pennies and one dollar. So we've only to consider two possible scenarios: (1) 5 pennies, 16 nickels and dimes 16 nickels are worth 80 cents. "Trade" three nickels for dimes and you've got 13 nickels (worth 65 cents) and 3 dimes (worth 30 cents), to go along with your five pennies. (2) 10 pennies, 11 nickels and dimes 11 nickels are worth 55 cents. "Trade" 7 nickels for dimes and you've got 4 nickels (worth 20 cents) and 7 dimes (worth 70 cents) to go along with your ten pennies. So the two solutions to the problem are: 5 pennies, 13 nickels, 3 dimes, or 10 pennies, 4 nickels, 7 dimes. Math is just common sense with a pencil. Sometimes, common sense is even faster without the pencil. -Doctor Gary, The Math Forum Check out our web site! http://mathforum.org/dr.math/
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