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# Patterns and Expressions

## Write variable expressions to represent word problems.

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Practice Patterns and Expressions
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Estimated11 minsto complete
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Patterns and Expressions

What if you knew the Booster Club sold 855 spaghetti dinners and collected $6840? How could you write an equation to find the amount each diner paid? After completing this Concept, you'll be able to write equations like this one. ### Guidance In mathematics, and especially in algebra, we look for patterns in the numbers we see. The tools of algebra help us describe these patterns with words and with equations (formulas or functions). An equation is a mathematical recipe that gives the value of one variable in terms of another. For example, if a theme park charges$12 admission, then the number of people who enter the park every day and the amount of money taken in by the ticket office are related mathematically, and we can write a rule to find the amount of money taken in by the ticket office.

In words, we might say “The amount of money taken in is equal to twelve times the number of people who enter the park.”

We could also make a table. The following table relates the number of people who visit the park and the total money taken in by the ticket office.

#### Example B

The following table shows the relationship between two quantities. First, write an equation that describes the relationship. Then, find out the value of b\begin{align*}b\end{align*} when a\begin{align*}a\end{align*} is 750.

a0 10203040  50b20406080100120\begin{align*}& a \qquad 0 \qquad \ 10 \qquad 20 \qquad 30 \qquad 40 \ \ \qquad 50\\ & b \qquad 20 \qquad 40 \qquad 60 \qquad 80 \qquad 100 \qquad 120 \end{align*}

Solve Problems Using Equations

Let’s solve the following real-world problem by using the given information to write a mathematical equation that can be solved for a solution.

#### Example C

A group of students are in a room. After 25 students leave, it is found that 23\begin{align*}\frac{2}{3}\end{align*} of the original group is left in the room. How many students were in the room at the start?

Write a Verbal Equation

In the examples above, we had a rule, written in words, and from that developed an algebraic equation. In the following example, we will develop a verbal equation based on a table, and use that to write an algebraic equation.

#### Example D

The following table shows the values of two related quantities. Write an equation that describes the relationship mathematically.

x\begin{align*}x-\end{align*}value y\begin{align*}y-\end{align*}value
-2 10
0 0
2 -10
4 -20
6 -30

### Vocabulary

An equation is a term used to describe a collection of numbers and variables related through mathematical operators. An algebraic equation will contain letters that represent real quantities.

### Guided Practice

Zarina has a $100 gift card, and she has been spending money on the card in small regular amounts. She checks the balance on the card weekly and records it in the following table. Week Number Balance ($)
1 100
2 78
3 56
4 34

Write an equation for the money remaining on the card in any given week.

### Practice

Day Profit
1 20
2 40
3 60
4 80
5 100

For 1-3, use the above table, which depicts the profit in dollars taken in by a store each day.

1. Write a mathematical equation that describes the relationship between the variables in the table.
2. What is the profit on day 10?
3. If the profit on a certain day is 200, what is the profit on the next day? For 4-6, Write a mathematical equation that describes each situation below, assuming the cookie jar starts with 24 cookies. 1. How many cookies are left in the jar after you have eaten some? 2. How many cookies are in the jar after you have eaten 9 cookies? 3. How many cookies are in the jar after you have eaten 9 cookies and then eaten 3 more? For 7-12, write a mathematical equation for the following situations and solve. 1. Seven times a number is 35. What is the number? 2. Three times a number, plus 15, is 24. What is the number? 3. Twice a number is three less than five times another number. Three times the second number is 15. What are the numbers? 4. One number is 25 more than 2 times another number. If each number were multiplied by five, their sum would be 350. What are the numbers? 5. The sum of two consecutive integers is 35. What are the numbers? 6. Peter is three times as old as he was six years ago. How old is Peter? For 13-16, Jae just took a math test with 20 questions, each worth an equal number of points. The test is worth 100 points total. 1. Write an equation relating the number of questions Jae got right to the total score he will get on the test. 2. If a score of 70 points earns a grade of C\begin{align*}C-\end{align*}, how many questions would Jae need to get right to get a C\begin{align*}C-\end{align*} on the test? 3. If a score of 83 points earns a grade of B\begin{align*}B\end{align*}, how many questions would Jae need to get right to get a B\begin{align*}B\end{align*} on the test? 4. Suppose Jae got a score of 60% and then was allowed to retake the test. On the retake, he got all the questions right that he got right the first time, and also got half the questions right that he got wrong the first time. What is his new score? For 17-22, solve the problem by writing an equation. 1. How much water should be added to one liter of pure alcohol to make a mixture of 25% alcohol? 2. A mixture of 50% alcohol and 50% water has 4 liters of water added to it. It is now 25% alcohol. What was the total volume of the original mixture? 3. In Crystal’s silverware drawer there are twice as many spoons as forks. If Crystal adds nine forks to the drawer, there will be twice as many forks as spoons. How many forks and how many spoons are in the drawer right now? 4. Mia is exploring different routes to drive to Javier's house. 1. Mia drove to Javier’s house at 40 miles per hour. Javier’s house is 20 miles away. Mia arrived at Javier’s house at 2:00 pm. What time did she leave? 2. Mia left Javier’s house at 6:00 pm to drive home. This time she drove 25% faster. What time did she arrive home? 3. The next day, Mia took the expressway to Javier’s house. This route was 24 miles long, but she was able to drive at 60 miles per hour. How long did the trip take? 4. When Mia took the same route back, traffic on the expressway was 20% slower. How long did the return trip take? 5. The price of an mp3 player decreased by 20% from last year to this year. This year the price of the player is120. What was the price last year?
6. SmartCo sells deluxe widgets for \$60 each, which includes the cost of manufacture plus a 20% markup. What does it cost SmartCo to manufacture each widget?

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