The different pieces of American currency have different monetary values that are used to make purchases in society. The units for American currency are cents and dollars.
Money with a value amount in dollars is marked with the sign $ in front of the number.
Money with a value amount in cents is marked with the sign ??? behind the number.
Also, there are 100 cents in a U.S. dollar.
With that in mind, we can introduce the different coins and bills and tell you how much they are worth:
(add image for each coin/bill)
A penny is worth 1 cent.
A nickel is worth five cents.
A dime is worth ten cents.
A quarter is worth 25 cents.
A half-dollar is worth 50 cents.
A dollar is worth $1.
There are other bills that are worth different numbers of dollars too: a 2-dollar bill, a 5-dollar bill, a 10-dollar bill, a 20-dollar bill, a 50-dollar bill, and a 100-dollar bill.
In this article we will focus on counting up sums of money and applying this practice to everyday situations.
Here is a basic example of counting up the value of a pile of coins.
Example 1: What is the total value of one quarter, one dime, one nickel, and one penny?
Solution: To find the total value of the pile of money, we must add the values of each individual coin together, in cents, adding the values of the coins (in the order the coins are listed) is
$$25 + 10 + 5 + 1 = 35 + 6 = 41$$
Here is another similar example, except some of the coins have the same value.
Example 2: What is the total value of 3 pennies and a nickel?
Solution 1: We add up the individual value of each coin, disregarding whether some of the coins have the same value. In cents, we add from left to right:
$$1 + 1 + 1 + 5 = 2 + 6 = 8$$
Solution 2: The pennies all have the same value, and there are three of them, so we can multiply the value of one penny by three and then add on the value of the nickel (in cents):
$$3(1) + 5 = 3 + 1 = 8$$
When there are a lot of coins, a solution involving multiplication like Solution 2 is usually preferred to avoid excessive adding.
Here is a word problem that uses the same concepts as Examples 1 and 2.
Example 3: A child wants to buy a keychain at a garage sale for fifty cents. He has two quarters, a dime, and a nickel. Does he have enough money?
Solution: We can combine the price value of each individual coin to see if its value exceeds the price of the key chain:
$$25 + 25 + 10 + 5 = 50 + 15 = 65$$
This number is in cents. Since the boy has more money than the key chain is worth, he has enough money to buy it, and he will have some left over.
Often multiplication when you have coin duplicates speeds up the solution. The next example emphasizes this.
Example 4: What is the total value of twelve nickels?
Solution: You could add up the individual value of each nickel, but it is faster to multiply the value of the coin that is duplicated by the quantity of that coin you have. This gives, in cents,
$$5 \times 12 = 60$$
When you have \(n\) copies of the same coin or bill, and the value of one of these coins or bills is \(v\), then the total value of all the coins or bills is \(nV\).
This basically condenses our calculations into a formula. It works for coins and bills, not just coins. The next example demonstrates a basic use of the formula.
Example 5: What is the total value of three quarters?
Solution: The value of the coin involved is \(25\) cents, and there are three coins, so \(n = 3\) and \(V = 25\), so the total value of all the coins is
$$nV = 3 \cdot 25 = 75$$
in cents.
This formula can also be applied if there is more than one type of coin or bill, but then you have multiple values of \(n\) and \(V\). Example 6, shown below, will show how this is possible. In general, we will let \(T\) denote the total value of some pile of money.
Example 6: What is the total value of \(2\) dimes and \(17\) pennies?
Solution: We can find the total value of all the dimes, and find the total value of all the pennies, then add those together. For the dimes, \(n = 2\) and \(v = 10\); for the pennies, \(n = 17\) and \(v = 1\):
$$T = 2(10) + 17(1) = 20 + 17 = 37$$
As always, this answer is in cents.
Example 7: Find the total value of 3 pennies, 6 quarters, and 5 nickels.
Solution: Regardless of how many different types of coins or bills we have to combine, the strategy remains the same--multiply the value and the quantity for each coin and add them together (in cents):
$$3(1) + 6(25) + 5(5) = 3 + 150 + 25 = 178$$
The central idea for counting bills is exactly the same as counting coins, except the values of the bills are different. We can still multiply values and quantities, and the calculations are nearly identical. Here are some examples.
Example 8: What is the total value of a 2-dollar bill and a 10-dollar bill?
Solution: There is only one of each bill, so just add the values together (in dollars):
$$2 + 10 = 12$$
Example 9: You use a 5-dollar bill, a 1-dollar bill, and a 20-dollar bill to buy a gift for your friend. How much money did you spend?
Solution: The bills are used to buy the gift, so the amount spent is the combined value of all the bills used. In dollars, we add up the values of the bills:
$$5 + 1 + 20 = 6 + 20 = 26$$
The next problem involves multiplication and addition.
Example 10: What is the total value of three 10-dollar bills and four 5-dollar bills?
Solution: We find the value of all the 10-dollar bills and all the 5-dollar bills and combine. As usual, calculations are in dollars:
$$3(10) + 4(5) = 30 + 20 = 50$$
Here is a final, more difficult example.
Example 11: What is the total value of three fifty-dollar bills, two twenty-dollar bills, six pennies, three dimes, and a nickel?
Solution: We add everything together. We combine the total value of all the bills with the total value of all the coins.
Bills (in dollars):
$$3(50) + 2(20) = 150 + 40 = 190$$
Coins (in cents):
$$6(1) + 3(10) + 1(5) = 6 + 30 + 5 = 41$$
In the grand total, the dollar amount if 190, and the cent amount is 41 cents, so we write it as 190 dollars and 41 cents, \($190.41\).
Counting money is made easy by the use of basic multiplication and breaking up coins an bills. Watch your calculations carefully, and make sure you know whether they are in dollars or cents.