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# 1.9: Numerical Expression Evaluation with Basic Operations

Difficulty Level: Basic Created by: CK-12
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Practice Numerical Expression Evaluation

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Have you ever been to the White Mountains? Take a look at this dilemma.

The first few days of Teen Adventure were spent at the base of the White Mountains in the Lafayette Place Campground. There the teens worked on survival skills, map skills and basic first aid. The students were kept all together and Kelly made some terrific new friends. Her favorite part was all of the team building and trust building exercises that they learned. She even surprised herself by being able to complete some tasks that she wasn’t sure that she could do. Kelly and the other students learned to rely on themselves and each other over those first few days.

The night before their first hike, the group leaders told them that they would be divided up into hiking groups. Each week they would meet altogether once again to talk about their adventures, but during the actual week each group would be on a different trail. There would be three groups out of the 30 students. The 6 leaders would be divided into the 3 groups. For the first few days, there would be a manager along and two first aid persons. After the first few days, the manager and the two first aid persons would head back to the base office to be checked in with weekly or in case of emergency. So they would start with one number of people and after the first few days the number would change.

Kelly is curious about how many people will start and how many will be together after the first few days.

This is where you come in. You can write an expression to figure this out-using grouping symbols and the order of operations you should be able to calculate two different group numbers. This Concept will teach you all that you need to know. Pay attention because you will see this problem once again.

### Guidance

To begin this Concept, you have to blast back in math. Think back to last year in math, do you remember using the order of operations to evaluate numerical expressions? Remember that a numerical expression is an expression that has numbers and operations in it. Let’s review.

\begin{align*}2 + 3 \times 5\end{align*}

We could evaluate this expression in two different ways.

In the first way, we simply perform the operations in order from left to right.

\begin{align*}2 + 3 = 5 \times 5 = 25\end{align*}

Wait a minute!! That is incorrect because we did not use the order of operations.

Let’s use the order of operations as we evaluate this expression a second way.

\begin{align*}2 + 3 \times 5\end{align*}

Good, now we can apply this information to our example.

\begin{align*}2 + 3 \times 5\end{align*}

We multiply first.

\begin{align*}3 \times 5 & = 15\\ 2 + 15 &= 17\end{align*}

Our answer is 17.

We can apply this information to variable expressions. How can we do that?

Remember that a variable expression has variables, numbers, operations and sometimes exponents in it. We can start by thinking about evaluating variable expressions. When we evaluate a variable expression, we are finding the value of that expression. If given a value for a variable, we substitute the numerical value of the variable into the expression. Then we can find the total value of the expression.

Some variable expressions will have more than one variable in it. We can evaluate the expression if we have been given values for each of the variables.

Evaluate the expression: \begin{align*}4b+12 \div 4-8\end{align*} if \begin{align*}b=6\end{align*}

First, we substitute the given value in for the variable.

\begin{align*}4(6)+ 12 \div 4-8\end{align*}

Next, we perform multiplication and division in order from left to right.

\begin{align*}24 + 3 - 8\end{align*}

Now we add and subtract in order from left to right.

\begin{align*}27 - 8 = 19\end{align*}

Our answer is 19.

Sometimes, you will see a problem with two variables in it.

Evaluate the expression: \begin{align*}\frac{21}{x}-7+8y\end{align*} if \begin{align*}x=3\end{align*} and \begin{align*}y=4\end{align*}

First, we substitute the given values for \begin{align*}x\end{align*} and \begin{align*}y\end{align*} into the expression.

\begin{align*}\frac{21}{3}-7+8(4)\end{align*}

Next, we perform multiplication and division in order from left to right. Notice that this problem has a fraction bar in it which means division.

\begin{align*}7-7+32\end{align*}

Now we add and subtract in order from left to right.

\begin{align*}0 + 32 = 32\end{align*}

Our answer is 32.

Evaluate the following variable expressions by following the order of operations.

#### Example A

\begin{align*}5a+6 \div 2+9\end{align*} if \begin{align*}a\end{align*} is 5

Solution: 37

#### Example B

\begin{align*}7x+14 \div 7-3\end{align*} if \begin{align*}x\end{align*} is 4

Solution: 27

#### Example C

\begin{align*}\frac{48}{x}+ 5y-7\end{align*} if \begin{align*}x\end{align*} is 6 and \begin{align*}y\end{align*} is 9

Solution: 46

Now back to the hiking trip and the groupings.

Kelly is curious about how many people will start and how many will be together after the first few days.

First, let’s work with the numbers involved.

30 students

1 manager

2 first aid persons

Now we can write an expression to show how one group is created out of the whole.

\begin{align*}30 \div 3+6 \div 3\end{align*}

Next, we solve it for the number of students and leaders per group. Remember to follow the order of operations.

\begin{align*}10 + 2 = 12\end{align*} persons

For the first few days, there will also be a manager and two first aid persons.

\begin{align*}12 + 1 + 2\end{align*}

15 people in a group for the first few days.

Then the manager and two first aid persons leave.

\begin{align*}15 - 3 = 12\end{align*}

The core group will consist of 12 people.

### Vocabulary

Here are the vocabulary words in this Concept.

Numerical Expression
an expression that uses numbers and operations.
Variable Expression
an expression that uses numbers, variables and operations.
Order of Operations
the order that you perform each operation when evaluating an expression.

### Guided Practice

Here is one for you to try on your own.

\begin{align*}6x+36 \div 4-3+8\end{align*} if \begin{align*}x\end{align*} is 4

First, remember to follow the order of operations. Then substitute 4 into the expression for x.

\begin{align*}6(4)+36 \div 4-3+8\end{align*}

Now we can complete multiplication and division in order from left to right.

\begin{align*}24 + 9 - 3 + 8\end{align*}

Next, we can complete addition and subtraction in order from left to right.

Our answer is 38.

### Video Review

Here are videos for review.

http://www.youtube.com/watch?v=moUaatNssoQ - This is a James Sousa video on evaluating an expression using the order of operations.

### Practice

Directions: Use the order of operations to evaluate each numerical expression.

1. \begin{align*}5 + 3 \times 4\end{align*}

2. \begin{align*}6 \times 2 + 5 \times 3\end{align*}

3. \begin{align*}4 + 5 \times 2 - 9\end{align*}

4. \begin{align*}4 + 6 \div 2 + 10 - 3\end{align*}

5. \begin{align*}8 - 15 \div 3 + 4 \times 5\end{align*}

Directions: Evaluate the following expressions when \begin{align*}x\end{align*} is 4.

6. \begin{align*}2x+28 \div 4-3\end{align*}

7. \begin{align*}6x+30 \div 2-10\end{align*}

8. \begin{align*}5x+x-3+8\end{align*}

9. \begin{align*}3x+36 \div 9-3+1\end{align*}

10. \begin{align*}5x+9-3+8\end{align*}

11. \begin{align*}4x+30 \div 15-3+8\end{align*}

12. \begin{align*}8x+2x-3+18\end{align*}

13. \begin{align*}2x+x\div 2-3+8\end{align*}

14. \begin{align*}9x+3x \div 2+8\end{align*}

15. \begin{align*}12x+4x \div 4-6+8\end{align*}

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### Vocabulary Language: English

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

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