Solving Problems Involving Coins | Prealgebra

Learning Outcomes

Determine the value of a given number of coins
Apply the problem-solving method to solve word problems involving coins

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.

To determine the total value of a stack of nickels, multiply the number of nickels times the value of one nickel.

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

$17$ 17

dimes; how much are they worth? Each dime is worth

$\text{\$0.10}$ $0.10

—that is the value of one dime. To find the total value of the pile of

$17$ 17

dimes, multiply

$17$ 17

$\text{\$0.10}$ $0.10

to get

$\text{\$1.70}$ $1.70

. This is the total value of all

$17$ 17

dimes.

$\begin{array}{c}\qquad 17\cdot \text{\$0.10}=\text{\$1.70}\qquad \\ \qquad \text{number}\cdot \text{value}=\text{total value}\qquad \end{array}$ 17⋅$0.10=$1.70number⋅value=totalvalue

Finding the Total Value for Coins of the Same Type

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

$\text{number}\cdot \text{value}=\text{total value}$ number⋅value=totalvalue

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

$14$ 14

quarters,

$17$ 17

dimes,

$21$ 21

nickels, and

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

PROBLEM-SOLVING STRATEGY

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

Step 2. Identify what you are looking for.

Step 3. Name what you are looking for.

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.

Step 6. Check.

Step 7. Answer the question.

Let's see how this method is used to solve a coin word problem.

Example

Adalberto has

$\text{\$2.25}$ $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

$\text{\$0.10}$ $0.10

and the value of a nickel is

$\text{\$0.05}$ $0.05

. The total value of all the coins is

$\text{\$2.25}$ $2.25

Type	$\text{Value (\$)}$ Value($)	$\text{Total Value (\$)}$ TotalValue($)
Dimes	$0.10$ 0.10
Nickels	$0.05$ 0.05
		$2.25$ 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.
$\text{Let }d=\text{number of dimes.}$ Letd=numberofdimes.

$\text{Let }d+9=\text{number of nickels.}$ Letd+9=numberofnickels.
See Also
Mercury Dime Values | Discover Their Worth
Fill in the "number" column to help get everything organized.

Type	$\text{Number}$ Number	$\text{Value (\$)}$ Value($)	$\text{Total Value (\$)}$ TotalValue($)
Dimes	$d$ d	$0.10$ 0.10
Nickels	$d+9$ d+9	$0.05$ 0.05
			$2.25$ 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

${\text{number}}\cdot {\text{value}}$ number⋅value

and write the results in the Total Value column.

Type	$\text{Number}$ Number	$\text{Value (\$)}$ Value($)	$\text{Total Value (\$)}$ TotalValue($)
Dimes	$d$ d	$0.10$ 0.10	$0.10d$ 0.10d
Nickels	$d+9$ d+9	$0.05$ 0.05	$0.05\left(d+9\right)$ 0.05(d+9)
			$2.25$ 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.	$0.10d+0.05(d+9)=2.25$ 0.10d+0.05(d+9)=2.25
Distribute.	$0.10d+0.05d+0.45=2.25$ 0.10d+0.05d+0.45=2.25
Combine like terms.	$0.15d+0.45=2.25$ 0.15d+0.45=2.25
Subtract 0.45 from each side.	$0.15d=1.80$ 0.15d=1.80
Divide to find the number of dimes.	$d=12$ d=12
The number of nickels is d + 9	$d+9$ d+9 $\color{red}{12}+9$ 12+9 $21$ 21

Step 6. Check.

$\begin{array}{ccc}12 dimes: 12\left(0.10\right)\qquad & =\qquad & 1.20\qquad \\ 21\text{ nickels: }21\left(0.05\right)\qquad & =\qquad & \underset{\text{------}}{1.05}\qquad \\ & & \text{\$2.25}\quad\checkmark \qquad \end{array}$ 12dimes:12(0.10)21nickels:21(0.05)==1.20——1.05$2.25✓

Step 7. Answer the question.

$\mathit{\text{Adalberto has twelve dimes and twenty-one nickels.}}$ Adalbertohastwelvedimesandtwenty-onenickels.

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

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

$\text{\$2.43}$ $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?

Show Solution

Solution:

Step 1.Read the problem carefully.

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	$\text{Value (\$)}$ Value($)	$\text{Total Value (\$)}$ TotalValue($)
Quarters	$0.25$ 0.25
Pennies	$0.01$ 0.01
		$2.43$ 2.43

Step 2. Identify what you are looking for.

$\text{We are looking for the number of quarters and pennies.}$ Wearelookingforthenumberofquartersandpennies.

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

$\text{We know Maria has twice as many pennies as quarters. The number of pennies is defined in terms of quarters.}$ WeknowMariahastwiceasmanypenniesasquarters.Thenumberofpenniesisdefinedintermsofquarters.

$\text{Let }q\text{represent the number of quarters.}$ Letqrepresentthenumberofquarters.

$\text{Then the number of pennies is }2q$ Thenthenumberofpenniesis2q

Type	$\text{Number}$ Number	$\text{Value (\$)}$ Value($)	$\text{Total Value (\$)}$ TotalValue($)
Quarters	$q$ q	$0.25$ 0.25
Pennies	$2q$ 2q	$0.01$ 0.01
			$2.43$ 2.43

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

Type	$\text{Number}$ Number	$\text{Value (\$)}$ Value($)	$\text{Total Value (\$)}$ TotalValue($)
Quarters	$q$ q	$0.25$ 0.25	$0.25q$ 0.25q
Pennies	$2q$ 2q	$0.01$ 0.01	$0.01\left(2q\right)$ 0.01(2q)
			$2.43$ 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.	$0.25q+0.01(2q)=2.43$ 0.25q+0.01(2q)=2.43
Multiply.	$0.25q+0.02q=2.43$ 0.25q+0.02q=2.43
Combine like terms.	$0.27q=2.43$ 0.27q=2.43
Divide by $0.27$ 0.27 .	$q=9$ q=9 quarters
The number of pennies is $2q$ 2q .	$2q$ 2q $2\cdot\color{red}{9}$ 2⋅9 $18$ 18 pennies

Step 6. Check the answer in the problem.

Maria has

$9$ 9

quarters and

$18$ 18

pennies. Does this make

$\text{\$2.43}?$ $2.43?

$\begin{array}{cccccc}9 quarters\qquad & & & 9\left(0.25\right)\qquad & =\qquad & 2.25\qquad \\ \text{18 pennies}\qquad & & & 18\left(0.01\right)& =\qquad & \underset{\text{------}}{0.18}\qquad \\ \text{Total}\qquad & & & & & \qquad \text{\$2.43}\quad\checkmark \end{array}$ 9quarters18penniesTotal9(0.25)18(0.01)==2.25——0.18$2.43✓

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

In the next example, we'll show only the completed table—make sure you understand how to fill it in step by step.

Example

Danny has

$\text{\$2.14}$ $2.14

worth of pennies and nickels in his piggy bank. The number of nickels is two more than ten times the number of pennies. How many nickels and how many pennies does Danny have?

Show Solution

Solution:

Step 1: Read the problem.
Determine the types of coins involved. Create a table.	Pennies and nickels
Write in the value of each type of coin.	Pennies are worth $\$0.01$ $0.01 . Nickels are worth $\$0.05$ $0.05 .
Step 2: Identify what you are looking for.	the number of pennies and nickels
Step 3: Name. Represent the number of each type of coin using variables. The number of nickels is defined in terms of the number of pennies, so start with pennies.	Let $p=\text{number of pennies}$ p=numberofpennies
The number of nickels is two more than ten times the number of pennies.	$10p+2=\text{number of nickels}$ 10p+2=numberofnickels

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

Type	$\text{Number}$ Number	$\text{Value (\$)}$ Value($)	$\text{Total Value (\$)}$ TotalValue($)
pennies	$p$ p	$0.01$ 0.01	$0.01p$ 0.01p
nickels	$10p+2$ 10p+2	$0.05$ 0.05	$0.05\left(10p+2\right)$ 0.05(10p+2)
			$\text{\$2.14}$ $2.14

Step 4. Translate: Write the equation by adding the total value of all the types of coins.

Step 5. Solve the equation.

	$0.01p+0.50p+0.10=2.14$ 0.01p+0.50p+0.10=2.14
	$0.51p+0.10=2.14$ 0.51p+0.10=2.14
	$0.51p=2.04$ 0.51p=2.04
	$p=4$ p=4 pennies
How many nickels?	$10p+2$ 10p+2
	$10(\color{red}{4})+2$ 10(4)+2
	$42$ 42 nickels

Step 6. Check. Is the total value of

$4$ 4

pennies and

$42$ 42

nickels equal to

$\text{\$2.14}?$ $2.14?

$\begin{array}{}\\ 4\left(0.01\right)+42\left(0.05\right)\stackrel{?}{=}2.14\\ 2.14=2.14\quad\checkmark \end{array}$ 4(0.01)+42(0.05)=?2.142.14=2.14✓

Step 7. Answer the question. Danny has

$4$ 4

pennies and

$42$ 42

nickels.

In the following video we show another example of how to solve a word problem that involves findingan amount of coins.

Licenses and Attributions

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Coins. Authored by: KMR Photography. Located at: https://www.flickr.com/photos/morbokat/16441865195/. License: CC BY: Attribution

CC licensed content, Specific attribution

Prealgebra. Provided by: OpenStax. License: CC BY: Attribution. License terms: Download for free at http://cnx.org/contents/[emailprotected]

Solving Problems Involving Coins | Prealgebra | (2024)

Learning Outcomes

Finding the Total Value for Coins of the Same Type

PROBLEM-SOLVING STRATEGY

Example

Solve a coin word problem

Example

Example

Licenses and Attributions

CC licensed content, Shared previously

CC licensed content, Specific attribution

References