Wyatt was hanging out at the mall when a woman walking by dropped her purse, scattering coins all over the ground. Wyatt rushed over to help. He picked up 30 pennies, 15 nickels, 10 dimes, and 13 quarters and returned them to the grateful woman. In order to tell his parents the story at dinner, Wyatt wants to figure out exactly how much money he helped the woman collect. He could go through and add all the coin values together, but is there an easier way for Wyatt to figure out the total value of the coins?

In this concept, you will learn how to use expressions to solve real-world problems.

### Describing Patterns

Sometimes you will need to create your own variable expressions in order to solve problems. Consider this real-world problem that can be written as a variable expression:

*Joanne has a pile of nickels and a pile of dimes. She counts her money and discovers that she has 25 nickels and 36 dimes. How much money does Joanne have in total?*

First, underline all of the important information in the problem.

*Joanne has a pile of nickels and a pile of dimes. She counts her money and discovers that she has 25 nickels and 36 dimes. How much money does Joanne have in total?*

A nickel is 5 cents; use decimal .05 to show that amount in dollars.

A dime is 10 cents; use decimal .10 to show that amount in dollars.

Next, write an expression with variables.

\begin{align*}.05x +.10y\end{align*}

The \begin{align*}x\end{align*} represents the number of nickels, and the \begin{align*}y\end{align*} represents the number of dimes. The expression represents the total amount of money, in dollars, there would be with nickels and dimes.

In this case, you have been given the number of dimes and nickels. Substitute those values into your expression for \begin{align*}x\end{align*} and \begin{align*}y\end{align*}.

\begin{align*}.05(25) + .10(36)\end{align*}

Next, evaluate the expression.

\begin{align*}1.25 + 3.60 = 4.50\end{align*}

The answer is $4.50.

By changing the format of the coin value to decimal values of a dollar, the answer can be seen in dollars. It is much easier to think about the answer as $4.50, than 450 cents!

### Examples

#### Example 1

Earlier, you were given a problem about Wyatt and his good deed.

Wyatt needs to figure out the total value of all the coins he collected: 30 pennies, 15 nickels, 10 dimes and 13 quarters.

First, write an expression.

\begin{align*}.01a + .05b + .10c + .25d\end{align*}

The coin values are written here as decimal values of a dollar: .01 is a penny, .05 is a nickel, .10 is a dime, and .25 is a quarter. This is important as it allows you to consider the different coins on the same scale. It will also means the final answer will be in dollars.

Next, substitute the given values for the variables.

\begin{align*}.01(30)+.05(15)+.10(10)+.25(13)\end{align*}

Then, follow order of operations to multiply and then add.

\begin{align*}.01(30)+.05(15)+.10(10)+.25(13)\\
.30+.75+1.0+3.25\\
5.30\\\end{align*}

The answer is $5.30.

Wyatt can tell his parents that he helped the woman collect 5 dollars and 30 cents.

#### Example 2

Write an expression for this word problem and evaluate.

Imagine that you have found a pile of money in a drawer. There are 10 nickels, 5 dimes and 15 quarters. What is the total sum of money you have found?

First, write an expression. Because you are working with money, it can be helpful to express all different coins as decimal values of a dollar. This means nickels are .05, dimes are .10 and quarters are .25. Write an expression using *nickels, * to represent the number of *dimes*, and to represent the number of *quarters*.

\begin{align*} .05n +.10d + .25q\end{align*}

Next, substitute the numbers of each coin for the variables. There are 10 nickels, 5 dimes and 15 quarters.

\begin{align*} .05(10) +.10(5) + .25(15) \end{align*}

Then, follow order of operations to evaluate the expression.

\begin{align*} .50 + .50 + 3.75 \end{align*}

The answer is 4.75. You found $4.75.

#### Example 3

If you have 6 nickels and five dimes, what is the sum?

First, write an expression using a variable, like

, to represent the number of nickels, and different variable, like , to represent the number of dimes. Multiply the number of nickels by 0.05, and the number of dimes by 0.10.\begin{align*}.05n+.10d\end{align*}

Then, substitute the given number of each coin in place of the correct variable.

\begin{align*}.05n +.10d\\ .05(6)+.10(5)\\\end{align*}

Finally, simplify the expression using the correct order of operations by multiplying and then adding.

\begin{align*}.05(6)+.10(5)\\ .30+.50\\ .80\end{align*}

The solution is $0.80

#### Example 4

If you have 15 nickels and 20 dimes, what is the sum?

First, write an expression to represent the dollar value of some number of nickels added to some number of dimes:

\begin{align*}.05n+.10d\end{align*}

Next, substitute the number of nickels and dimes in place of the variables.

\begin{align*}.05n +.10d\\ .05(15)+.10(20)\\\end{align*}

Finally, simplify the expression using the correct order of operations.

\begin{align*}.05(15)+.10(20)\\ .75+2\\ 2.75\end{align*}The sum is $2.75.

#### Example 5

If you have 35 dimes and 12 quarters, what is the sum?

First, write an expression to represent the dollar value of some number of dimes and quarters

\begin{align*}.10d+.25q\end{align*}

Then, substitute the given number of dimes and quarters into the expression.

\begin{align*}.10d+.25q\\ .10(35)+.25(12)\\\end{align*}

Finally, simplify the expression.

\begin{align*}.10(35)+.25(12)\\ 3.50+3\\ 6.50\\\end{align*}

The sum is $6.50.

### Review

Write an expression for each money amount and evaluate it using the given information.

- 15 quarters
- 10 dimes and 3 quarters
- 30 nickels and 15 dimes
- 6 quarters and 60 nickels
- 21 quarters and 14 dimes
- 6 dimes, 10 nickels and 120 pennies
- 18 quarters and 12 half-dollars
- 32 dimes, 16 nickels and 11 quarters
- 18 nickels, 33 dimes and 39 quarters
- 27 dimes, 87 pennies, 12 quarters
- 10 pennies, 15 nickels, 9 dimes and 27 quarters
- 35 quarters and 98 nickels
- 95 dimes, 27 nickels and 82 quarters
- 77 dimes, 15 nickels and 81 quarters
- 70 nickels, 63 dimes, 82 pennies and 55 quarters
- 12 nickels, 33 dimes, 17 pennies and 80 quarters

### Review (Answers)

To see the Review answers, open this PDF file and look for section 1.15.