Word Lesson: Coin Problems

By S Taylor

In order to solve coin problems, or closely related problems, you should be able to:

translate word problem language into equations
work with literal equations
solve one variable equations
solve a system of equations by substitution

Coin problems usually involve knowing how many coins and how much money someone has and trying to find out how many of each coin the person has. The same strategy used for solving coin problems can also be applied to other types of money or anything where there are several monetary values involved.

Suppose Ken has 25 coins in nickels and dimes only and has a total of $1.65. How many of each coin does he have?

Usually coin problems involve two equations: one that describes how many coins there are and one that describes the amount of money.

The equation that describes the number of coins is

since there are only nickels (N) and dimes (D) and there are 25 coins total.

The equation the describes the amount of money is

since each nickel is worth 5 cents (0.05) and each dime is worth 10 cents (0.10) and the total amount of money is $1.65. To determine value we always multiply the worth of the item by how many of an item we have. Therefore, we are multiplying the value times the number of coins we have to come up with this equation.

To solve the equation that describes the amount of money might look impossible because there are two variables. But this is where we will make use of the equation that describes the number of coins.

The equation that describes the number of coins can be re-written so that it is now

By subtracting N from both sides, we have a different version of the same equation.

Now that we have an expression for D, we can substitute that back into the equation that describes the amount of money.

Now we just need to solve the new equation.

This tells us that the number of nickels is equal to 17. Since we know the total number of coins is 25, that means the number of dimes is 8. Ken has 17 nickels and 8 dimes.

Terry has 2 more quarters than dimes and has a total of $6.80. The number of quarters and dimes is 38. How many quarters and dimes does Terry have?

In my wallet, I have one-dollar bills, five-dollar bills, and ten-dollar bills. The total amount in my wallet is $43. I have four times as many one-dollar bills as ten-dollar bills. All together, there are 13 bills in my wallet. How many of each bill do I have?

Marsha has three times as many one-dollar bills as she does five dollar bills. She has a total of $32. How many of each bill does she have?

2 ones and 6 fives
12 ones and 4 fives
4 ones and 12 fives

A vending machine has $41.25 in it. There are 255 coins total and the machine only accepts nickels, dimes and quarters. There are twice as many dimes as nickels. How many of each coin are in the machine.

45 nickels, 90 dimes, 120 quarters
82 nickels, 41 dimes, 132 quarters
8 nickels, 16 dimes, 231 quarters