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By the end of this section, you will be able to:
  • Solve coin word problems
  • Solve ticket and stamp word problems

Before you get started, take this readiness quiz.

  1. Multiply: 14 ( 0.25 ) .
    If you missed this problem, review Decimal Operations .
  2. Simplify: 100 ( 0.2 + 0.05 n ) .
    If you missed this problem, review Distributive Property .
  3. Solve: 0.25 x + 0.10 ( x + 4 ) = 2.5
    If you missed this problem, review Solve Equations with Fraction or Decimal Coefficients .

Solve coin word problems

Imagine taking a handful of coins from your pocket or purse and placing them on your desk. How would you determine the value of that pile of coins?

If you can form a step-by-step plan for finding the total value of the coins, it will help you as you begin solving coin word problems.

One way to bring some order to the mess of coins would be to separate the coins into stacks according to their value. Quarters would go with quarters, dimes with dimes, nickels with nickels, and so on. To get the total value of all the coins, you would add the total value of each pile.

An image of a large stack of pennies, a large stack of nickels, a shorter stack of dimes, and a stack of quarters is shown. There are several coins in the background.
(Credit: Darren Hester via ppdigital)

How would you determine the value of each pile? Think about the dime pile—how much is it worth? If you count the number of dimes, you'll know how many you have—the number of dimes.

But this does not tell you the value of all the dimes. Say you counted 17 dimes, how much are they worth? Each dime is worth $0.10 —that is the value of one dime. To find the total value of the pile of 17 dimes, multiply 17 by $0.10 to get $1.70 . This is the total value of all 17 dimes.

17 · $0.10 = $1.70 number · value = total value

Finding the total value for coins of the same type

For coins of the same type, the total value can be found as follows:

number · value = total value

where number is the number of coins, value is the value of each coin, and total value is the total value of all the coins.

You could continue this process for each type of coin, and then you would know the total value of each type of coin. To get the total value of all the coins, add the total value of each type of coin.

Let's look at a specific case. Suppose there are 14 quarters, 17 dimes, 21 nickels, and 39 pennies. We'll make a table to organize the information – the type of coin, the number of each, and the value.

Type Number Value ($) Total Value ($)
Quarters 14 0.25 3.50
Dimes 17 0.10 1.70
Nickels 21 0.05 1.05
Pennies 39 0.01 0.39
6.64

The total value of all the coins is $6.64 . Notice how [link] helped us organize all the information. Let's see how this method is used to solve a coin word problem.

Adalberto has $2.25 in dimes and nickels in his pocket. He has nine more nickels than dimes. How many of each type of coin does he have?

Solution

Step 1. Read the problem. Make sure you understand all the words and ideas.

  • Determine the types of coins involved.

Think about the strategy we used to find the value of the handful of coins. The first thing you need is to notice what types of coins are involved. Adalberto has dimes and nickels.

  • Create a table to organize the information.
    • Label the columns ‘type’, ‘number’, ‘value’, ‘total value’.
    • List the types of coins.
    • Write in the value of each type of coin.
    • Write in the total value of all the coins.

We can work this problem all in cents or in dollars. Here we will do it in dollars and put in the dollar sign ($) in the table as a reminder.

The value of a dime is $0.10 and the value of a nickel is $0.05 . The total value of all the coins is $2.25 .

Type Number Value ($) Total Value ($)
Dimes 0.10
Nickels 0.05
2.25

Step 2. Identify what you are looking for.

  • We are asked to find the number of dimes and nickels Adalberto has.

Step 3. Name what you are looking for.

  • Use variable expressions to represent the number of each type of coin.
  • Multiply the number times the value to get the total value of each type of coin.
    In this problem you cannot count each type of coin—that is what you are looking for—but you have a clue. There are nine more nickels than dimes. The number of nickels is nine more than the number of dimes.
    Let d = number of dimes.
    d + 9 = number of nickels
    Fill in the “number” column to help get everything organized.
Type Number Value ($) Total Value ($)
Dimes d 0.10
Nickels d + 9 0.05
2.25

Now we have all the information we need from the problem!

You multiply the number times the value to get the total value of each type of coin. While you do not know the actual number, you do have an expression to represent it.

And so now multiply number · value and write the results in the Total Value column.

Type Number Value ($) Total Value ($)
Dimes d 0.10 0.10 d
Nickels d + 9 0.05 0.05 ( d + 9 )
2.25

Step 4. Translate into an equation    . Restate the problem in one sentence. Then translate into an equation.
The sentence “Sum of the value of the dimes and value of the nickels is total value of the coins,” is written. Below “value of the dimes” is 0.10d. Below “and” is a plus sign. Below “value of the nickels” is 0.05(d plus 9). Below “is” is an equal sign. Below “total value of the coins” is 2.25.

Step 5. Solve the equation using good algebra techniques.

Write the equation. .
Distribute. .
Combine like terms. .
Subtract 0.45 from each side. .
Divide to find the number of dimes. .
The number of nickels is d + 9 .
.
.

Step 6. Check.

12 dimes: 12 ( 0.10 ) = 1.20 21 nickels: 21 ( 0.05 ) = 1.05 _____ $2.25

Step 7. Answer the question.

Adalberto has twelve dimes and twenty-one nickels.

If this were a homework exercise, our work might look like this:

How many of each type does he have?” Below this is a table with 4 rows and 4 columns. The first row is a header row. The headings are, “Type”, “Number”, “Value ($)”, and “Total Value ($)” Under the “Type” column are the entries dimes and nickels. Under the “Number”,column are d and d plus 9. Under the “Value”,column are the values 0.10 and 0.05. Under the “Total Value”,column are 0.10d and 0.05(d plus 9) followed by 2.25. Below the table is the word “nickels”,in bold. The equation 0.10d plus 0.05(d plus 9) equals 2.25 is shown. Below that are 2 columns. The left column says 0.10d plus 0.05d plus 0.45 equals 2.25, then 0.15d plus 0.45 equals 2.25, then 0.15d equals 1.80, then d equals 12 dimes. There is a red arrow pointing to the right column. The right column says d plus 9, then a red 12 plus 9, then 21 nickels.

Check:

12 dimes 12 ( 0.10 ) = 1.20 21 nickels 21 ( 0.05 ) = 1.05 _____ $2.25

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Source:  OpenStax, Prealgebra. OpenStax CNX. Jul 15, 2016 Download for free at http://legacy.cnx.org/content/col11756/1.9
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