## Solving Coin Word Problems

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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.

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 \(\text{\$0.10}\)—that is the *value* of one dime. To find the total value of the pile of \(17\) dimes, multiply \(17\) by \(\text{\$0.10}\) to get \(\text{\$1.70}.\) This is the total value of all \(17\) dimes.

### Definition: Finding the Total Value for Coins of the Same Type

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

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 | \(\text{Number}\) | \(\text{Value (\$)}\) | \(\text{Total Value (\$)}\) |
---|---|---|---|

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 \(\text{\$6.64}.\) Notice how the table above helped us organize all the information. Let’s see how this method is used to solve a coin word problem.

## Example

Adalberto has \(\text{\$2.25}\) 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 \(\text{\$0.10}\) and the value of a nickel is \(\text{\$0.05}.\) The total value of all the coins is \(\text{\$2.25}.\)

Type | \(\text{Number}\) | \(\text{Value (\$)}\) | \(\text{Total Value (\$)}\) |
---|---|---|---|

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.

\(\phantom{\rule{2.0em}{0ex}}\text{Let}\phantom{\rule{0.2em}{0ex}}d=\text{number of dimes.}\)

\(\phantom{\rule{2.0em}{0ex}}d+9=\text{number of nickels}\)

Fill in the “number” column to help get everything organized.

Type | \(\text{Number}\) | \(\text{Value (\$)}\) | \(\text{Total Value (\$)}\) |
---|---|---|---|

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 \({\text{number}}·{\text{value}}\) and write the results in the Total Value column.

Type | \(\text{Number}\) | \(\text{Value (\$)}\) | \(\text{Total Value (\$)}\) |
---|---|---|---|

Dimes | \(d\) | \(0.10\) | \(0.10d\) |

Nickels | \(d+9\) | \(0.05\) | \(0.05\left(d+9\right)\) |

\(2.25\) |

Step 4. **Translate** into an **equation**. Restate the problem in one sentence. Then translate into an equation.

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.**

\(\begin{array}{ccc}12\phantom{\rule{0.2em}{0ex}}\text{dimes:}\phantom{\rule{0.2em}{0ex}}12\left(0.10\right)\hfill & =\hfill & \phantom{\rule{0.6em}{0ex}}1.20\hfill \\ 21\phantom{\rule{0.2em}{0ex}}\text{nickels:}\phantom{\rule{0.2em}{0ex}}21\left(0.05\right)\hfill & =\hfill & \phantom{\rule{0.3em}{0ex}}\underset{\text{_____}}{1.05}\hfill \\ & & \phantom{\rule{0.2em}{0ex}}\text{\$2.25}✓\hfill \end{array}\)

Step 7. **Answer** the question.

\(\phantom{\rule{2em}{0ex}}\mathit{\text{Adalberto has twelve dimes and twenty-one nickels.}}\)

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

Check:

\(\begin{array}{cccccc}\text{12 dimes}\hfill & & & \hfill 12\left(0.10\right)& =\hfill & \phantom{\rule{0.4em}{0ex}}1.20\hfill \\ \text{21 nickels}\hfill & & & \hfill 21\left(0.05\right)& =\hfill & \phantom{\rule{0.1em}{0ex}}\underset{\text{_____}}{1.05}\hfill \\ & & & & & \text{\$2.25}\hfill \end{array}\)

### How to Solve a coin word problem.

**Read**the problem. Make sure you understand all the words and ideas, and create a table to organize the information.**Identify**what you are looking for.**Name**what you are looking for. Choose a variable to represent that quantity.- Use variable expressions to represent the number of each type of coin and write them in the table.
- Multiply the number times the value to get the total value of each type of coin.

**Translate**into an equation. Write the equation by adding the total values of all the types of coins.**Solve**the equation using good algebra techniques.**Check**the answer in the problem and make sure it makes sense.**Answer**the question with a complete sentence.

You may find it helpful to put all the numbers into the table to make sure they check.

Type | Number | Value ($) | Total Value |
---|---|---|---|

## Example

Maria has \(\text{\$2.43}\) in quarters and pennies in her wallet. She has twice as many pennies as quarters. How many coins of each type does she have?

### Solution

Step 1. **Read** the problem.

- Determine the types of coins involved.
We know that Maria has quarters and pennies.

- 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.

Type | \(\text{Number}\) | \(\text{Value (\$)}\) | \(\text{Total Value (\$)}\) |
---|---|---|---|

Quarters | \(0.25\) | ||

Pennies | \(0.01\) | ||

\(2.43\) |

Step 2. **Identify** what you are looking for.

\(\phantom{\rule{2em}{0ex}}\text{We are looking for the number of quarters and pennies.}\)

Step 3. **Name:** Represent the number of quarters and pennies using variables.

\(\phantom{\rule{2em}{0ex}}\text{We know Maria has twice as many pennies as quarters. The number of pennies is defined in terms of quarters.}\)

\(\phantom{\rule{2em}{0ex}}\text{Let}\phantom{\rule{0.2em}{0ex}}q\phantom{\rule{0.2em}{0ex}}\text{represent the number of quarters.}\)

\(\phantom{\rule{2em}{0ex}}\text{Then the number of pennies is}\phantom{\rule{0.2em}{0ex}}2q.\)

Type | \(\text{Number}\) | \(\text{Value (\$)}\) | \(\text{Total Value (\$)}\) |
---|---|---|---|

Quarters | \(q\) | \(0.25\) | |

Pennies | \(2q\) | \(0.01\) | |

\(2.43\) |

Multiply the ‘number’ and the ‘value’ to get the ‘total value’ of each type of coin.

Type | \(\text{Number}\) | \(\text{Value (\$)}\) | \(\text{Total Value (\$)}\) |
---|---|---|---|

Quarters | \(q\) | \(0.25\) | \(0.25q\) |

Pennies | \(2q\) | \(0.01\) | \(0.01\left(2q\right)\) |

\(2.43\) |

Step 4. **Translate.** Write the equation by adding the ‘total value’ of all the types of coins.

Step 5. **Solve** the equation.

Write the equation. | |

Multiply. | |

Combine like terms. | |

Divide by 0.27. | |

The number of pennies is 2q. |

Step 6. **Check** the answer in the problem.

Maria has \(9\) quarters and \(18\) pennies. Does this make \(\text{\$2.43}?\)

\(\begin{array}{cccccc}\text{9 quarters}\hfill & & & 9\left(0.25\right)\hfill & =\hfill & \phantom{\rule{0.4em}{0ex}}2.25\hfill \\ \text{18 pennies}\hfill & & & 18\left(0.01\right)& =\hfill & \phantom{\rule{0.1em}{0ex}}\underset{\text{_____}}{0.18}\hfill \\ \text{Total}\hfill & & & & & \hfill \text{\$2.43}✓\end{array}\)

Step 7. **Answer** the question. Maria has nine quarters and eighteen pennies.

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