"Coin" Word Problems

What is an example of a coin word problem?

The following is a typical example of a coin problem:

Your uncle walks in, jingling the coins in his pocket. He grins at you and tells you that you can have all the coins if you can figure out how many of each kind of coin he is carrying. You're not too interested until he tells you that he's been collecting those gold-tone one-dollar coins. The twenty-six coins in his pocket are all dollars and quarters, and they add up to seventeen dollars in value.

How many of each coin does he have?

To solve your uncle's puzzle, you need to use the total number of coins, the values of the two types of coins, and the total value of those coins.

There are twenty-six coins in total. Some of them are quarter coins; let "q" stand for the number of quarters. The rest of the coins are dollar coins. Since there are 26 coins in total and q of them are quarter coins, then there are 26 − q coins left to be dollar coins. (This "how much is left over" construction is important. Make sure you understand what it is and how to use it.)

If your uncle has only one quarter, then 25×1 = 25 cents comes from quarters. If he has two quarters, then 25×2 = 50 cents comes from quarters. Since he has q quarters, then 25×q = 25q cents comes from quarters.

For the dollar coins, we need first to convert their value to cents; one dollar is one hundred cents. Since he has 26 − q dollars, then he has 100(26 − q) cents from the dollar coins.

He has seventeen dollars in total, or 1700 cents, part of which is from quarters and part of which is from dollars. To help keep things straight, we can set up a table:

The total value comes from adding the value of the quarters and the value of the dollar coins. So we add the "total cents" expressions from the right-hand column above, and set this sum equal to the given total:

Then solve:

In other words, 12 of the coins are quarters. Since the remainder of the twenty-six coins are dollar coins, then there are 26 − 12 = 14 dollar coins. I can check to make sure this works:

Since the answer works in the original exercise, it must be right.

(Remember: You can *always* check your answers to "solving" problems by plugging your answers back into the original contexts. If everything checks out, then you know that your answer is right.)

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• A collection of 33 coins, consisting of nickels, dimes, and quarters, has a value of $3.30. If there are three times as many nickels as quarters, and one-half as many dimes as nickels, how many coins of each kind are there?

"But," I hear you say, "we never used the fact that the coins add up to $3.30. Shouldn't we have?" Well, we can use that information to check our answer. But it turns out that this information was not actually necessary for finding the solution.

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• A wallet contains the same number of pennies, nickels, and dimes. The coins total $1.44. How many of each type of coin does the wallet contain?

Remember that you can always check your answers to "solving" problems by plugging them back in to the original question. In this case, I would check whether nine pennies plus nine nickels plus nine dimes totalled $1.44:

So the solution is correct.

When you have the time, this type of checking is a good idea on tests. That way, you can make sure your answer is correct before you hand in your work.

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How do you solve coin word problems?

The take-aways for solving coin word problems are these:

1. If the quantity of one type of coin is given in terms of the quantity of another coin, then pick a variable for the second type of coin, and use that variable to create an expression to stand for the quantity of the first type of coin.
2. Convert the relationships between the numbers of coins (if given, as in the second examle) into an equation.
3. Convert the statements about the values of the coins (if given, as in the first example) into equations that state the values all in the same unit (for instance, in cents).
4. Solve the equation for the value of the variable.
5. Check your definitions to confirm the type of coin.
6. Back-solve to find the other types of coins.
7. (Optional) Use the information from the original exercise statement to confirm your answer.

And, as always, label everything!