Example

Adalberto has [latex]\text{\$2.25}[/latex] 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 [latex]\text{\$0.10}[/latex] and the value of a nickel is [latex]\text{\$0.05}[/latex]. The total value of all the coins is [latex]\text{\$2.25}[/latex].

Type	[latex]\text{Number}[/latex]	[latex]\text{Value (\$)}[/latex]	[latex]\text{Total Value (\$)}[/latex]
Dimes		[latex]0.10[/latex]
Nickels		[latex]0.05[/latex]
			[latex]2.25[/latex]

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. [latex]\text{Let }d=\text{number of dimes.}[/latex] [latex]\text{Let }d+9=\text{number of nickels.}[/latex] Fill in the "number" column to help get everything organized.

Type	[latex]\text{Number}[/latex]	[latex]\text{Value (\$)}[/latex]	[latex]\text{Total Value (\$)}[/latex]
Dimes	[latex]d[/latex]	[latex]0.10[/latex]
Nickels	[latex]d+9[/latex]	[latex]0.05[/latex]
			[latex]2.25[/latex]

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 [latex]{\text{number}}\cdot {\text{value}}[/latex] and write the results in the Total Value column.

Type	[latex]\text{Number}[/latex]	[latex]\text{Value (\$)}[/latex]	[latex]\text{Total Value (\$)}[/latex]
Dimes	[latex]d[/latex]	[latex]0.10[/latex]	[latex]0.10d[/latex]
Nickels	[latex]d+9[/latex]	[latex]0.05[/latex]	[latex]0.05\left(d+9\right)[/latex]
			[latex]2.25[/latex]

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.	[latex]0.10d+0.05(d+9)=2.25[/latex]
Distribute.	[latex]0.10d+0.05d+0.45=2.25[/latex]
Combine like terms.	[latex]0.15d+0.45=2.25[/latex]
Subtract 0.45 from each side.	[latex]0.15d=1.80[/latex]
Divide to find the number of dimes.	[latex]d=12[/latex]
The number of nickels is d + 9	[latex]d+9[/latex] [latex-display]\color{red}{12}+9[/latex-display] [latex]21[/latex]

Step 6. Check. [latex-display]\begin{array}{ccc}12\text{ dimes: }12\left(0.10\right)\hfill & =\hfill & 1.20\hfill \\ 21\text{ nickels: }21\left(0.05\right)\hfill & =\hfill & \underset{\text{_____}}{1.05}\hfill \\ & & \text{\$2.25}\quad\checkmark \hfill \end{array}[/latex-display] Step 7. Answer the question. [latex-display]\mathit{\text{Adalberto has twelve dimes and twenty-one nickels.}}[/latex-display] If this were a homework exercise, our work might look like this:

Example

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

Answer:

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	[latex]\text{Number}[/latex]	[latex]\text{Value (\$)}[/latex]	[latex]\text{Total Value (\$)}[/latex]
Quarters		[latex]0.25[/latex]
Pennies		[latex]0.01[/latex]
			[latex]2.43[/latex]

Step 2. Identify what you are looking for. [latex-display]\text{We are looking for the number of quarters and pennies.}[/latex-display] Step 3. Name: Represent the number of quarters and pennies using variables. [latex-display]\text{We know Maria has twice as many pennies as quarters. The number of pennies is defined in terms of quarters.}[/latex-display] [latex-display]\text{Let }q\text{represent the number of quarters.}[/latex-display] [latex]\text{Then the number of pennies is }2q[/latex].

Type	[latex]\text{Number}[/latex]	[latex]\text{Value (\$)}[/latex]	[latex]\text{Total Value (\$)}[/latex]
Quarters	[latex]q[/latex]	[latex]0.25[/latex]
Pennies	[latex]2q[/latex]	[latex]0.01[/latex]
			[latex]2.43[/latex]

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

Type	[latex]\text{Number}[/latex]	[latex]\text{Value (\$)}[/latex]	[latex]\text{Total Value (\$)}[/latex]
Quarters	[latex]q[/latex]	[latex]0.25[/latex]	[latex]0.25q[/latex]
Pennies	[latex]2q[/latex]	[latex]0.01[/latex]	[latex]0.01\left(2q\right)[/latex]
			[latex]2.43[/latex]

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.	[latex]0.25q+0.01(2q)=2.43[/latex]
Multiply.	[latex]0.25q+0.02q=2.43[/latex]
Combine like terms.	[latex]0.27q=2.43[/latex]
Divide by [latex]0.27[/latex].	[latex]q=9[/latex] quarters
The number of pennies is [latex]2q[/latex].	[latex]2q[/latex] [latex-display]2\cdot\color{red}{9}[/latex-display] [latex]18[/latex] pennies

Step 6. Check the answer in the problem. Maria has [latex]9[/latex] quarters and [latex]18[/latex] pennies. Does this make [latex]\text{\$2.43}?[/latex] [latex-display]\begin{array}{cccccc}\text{9 quarters}\hfill & & & 9\left(0.25\right)\hfill & =\hfill & 2.25\hfill \\ \text{18 pennies}\hfill & & & 18\left(0.01\right)& =\hfill & \underset{\text{_____}}{0.18}\hfill \\ \text{Total}\hfill & & & & & \hfill \text{\$2.43}\quad\checkmark \end{array}[/latex-display] Step 7. Answer the question. Maria has nine quarters and eighteen pennies.

Example

Danny has [latex]\text{\$2.14}[/latex] 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?

Answer:

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 [latex]$0.01[/latex]. Nickels are worth [latex]$0.05[/latex].
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 [latex]p=\text{number of pennies}[/latex]
The number of nickels is two more than ten times the number of pennies.	[latex]10p+2=\text{number of nickels}[/latex]

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

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

Step 4. Translate: Write the equation by adding the total value of all the types of coins. Step 5. Solve the equation.

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

Step 6. Check. Is the total value of [latex]4[/latex] pennies and [latex]42[/latex] nickels equal to [latex]\text{\$2.14}?[/latex] [latex-display]\begin{array}{}\\ 4\left(0.01\right)+42\left(0.05\right)\stackrel{?}{=}2.14\\ 2.14=2.14\quad\checkmark \end{array}[/latex-display] Step 7. Answer the question. Danny has [latex]4[/latex] pennies and [latex]42[/latex] nickels.