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# Words that Describe Patterns

## Use variable expressions to solve real-world problems.

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Words that Describe Patterns
License: CC BY-NC 3.0

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 \begin{align*}.05x+.10y \ \\\end{align*}represents the total amount of money, in dollars, there would be with \begin{align*}x\end{align*} nickels and \begin{align*}y\end{align*} 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*}

#### Example 3

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

First, write an expression using a variable, like \begin{align*}n\end{align*}, to represent the number of nickels, and different variable, like \begin{align*}d\end{align*}, 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*}

#### 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.

1. 15 quarters
2. 10 dimes and 3 quarters
3. 30 nickels and 15 dimes
4. 6 quarters and 60 nickels
5. 21 quarters and 14 dimes
6. 6 dimes, 10 nickels and 120 pennies
7. 18 quarters and 12 half-dollars
8. 32 dimes, 16 nickels and 11 quarters
9. 18 nickels, 33 dimes and 39 quarters
10. 27 dimes, 87 pennies, 12 quarters
11. 10 pennies, 15 nickels, 9 dimes and 27 quarters
12. 35 quarters and 98 nickels
13. 95 dimes, 27 nickels and 82 quarters
14. 77 dimes, 15 nickels and 81 quarters
15. 70 nickels, 63 dimes, 82 pennies and 55 quarters
16. 12 nickels, 33 dimes, 17 pennies and 80 quarters

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

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### Vocabulary Language: English

$\ge$

The greater-than-or-equal-to symbol "$\ge$" indicates that the value on the left side of the symbol is greater than or equal to the value on the right.

$\le$

The less-than-or-equal-to symbol "$\le$" indicates that the value on the left side of the symbol is lesser than or equal to the value on the right.

$\ne$

The not-equal-to symbol "$\ne$" indicates that the value on the left side of the symbol is not equal to the value on the right.

constant

A constant is a value that does not change. In Algebra, this is a number such as 3, 12, 342, etc., as opposed to a variable such as x, y or a.

Equation

An equation is a mathematical sentence that describes two equal quantities. Equations contain equals signs.

Evaluate

To evaluate an expression or equation means to perform the included operations, commonly in order to find a specific value.

Expression

An expression is a mathematical phrase containing variables, operations and/or numbers. Expressions do not include comparative operators such as equal signs or inequality symbols.

greater than

The greater than symbol, $>$, indicates that the value on the left side of the symbol is greater than the value on the right.

greater than or equal to

The greater than or equal to symbol, $\ge$, indicates that the value on the left side of the symbol is greater than or equal to the value on the right.

inequality

An inequality is a mathematical statement that relates expressions that are not necessarily equal by using an inequality symbol. The inequality symbols are $<$, $>$, $\le$, $\ge$ and $\ne$.

less than

The less-than symbol "<" indicates that the value on the left side of the symbol is lesser than the value on the right.

less than or equal to

The less-than-or-equal-to symbol "$\le$" indicates that the value on the left side of the symbol is lesser than or equal to the value on the right.

not equal to

The "not equal to" symbol, $\ne$, indicates that the value on the left side of the symbol is not equal to the value on the right.

Variable

A variable is a symbol used to represent an unknown or changing quantity. The most common variables are a, b, x, y, m, and n.

Variable Expression

A variable expression is a mathematical phrase that contains at least one variable or unknown quantity.