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Numerical Expression Evaluation with Basic Operations

Evaluate expressions for any basic operation.

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Numerical Expression Evaluation with Basic Operations
License: CC BY-NC 3.0

Margot volunteers at a local bird sanctuary. She just spent two weeks on vacation, so on her first day back she is talking to her supervisor about what she has missed. When Margot left, there were 256 birds. In her absence, 3 birds have each given birth to 5 baby birds, 2 birds were released into the wild and 3 new injured birds were brought in. How can Margot write an expression to figure out the new sanctuary population?

In this concept, you will learn how to evaluate numerical expressions using the four operations.

Order of Operations

An equation is a number sentence that describes two values that are equal to each other. The values are separated by the "equals" sign, e.g. 3 + 4 = 7. In this case, 3 + 4 and 7 are equal entities. Sometimes you will be given an incomplete equation, which you have to "solve" in order to make both sides equal.

An expression is a number sentence without an equals sign. It can be simplified and/or evaluated, but it cannot be "solved".

Expressions can involve more than one operation, so it is important to know which operation to do first. The order of operations tells you the correct order for completing operations within an expression.

 Order of Operations

P - parentheses

E - exponents

MD - multiplication or division, in order from left to right

AS - addition or subtraction, in order from left to right

Let's look at an example.

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

This is an expression with both addition and multiplication. The order of operations tells you that multiplication comes before addition. So, first multiply and then add.

\begin{align*}& 4 + 3 \times 5\\ & 4 + 15\\ & 19\\\end{align*}

When you evaluate this expression correctly, using order of operations, the answer is 19.

If you had NOT followed the order of operations, and instead evaluated from left to right, it would look like this.

\begin{align*}& 4 + 3 \times 5\\ & 7 \times 5\\ & 35 \end{align*}

You would have gotten an incorrect answer. That is why it is important to always follow the order of operations.

Examples

Example 1

Earlier, you were given a problem about Margot and her bountiful birds.

Margot needs to write and evaluate an expression to see how the original population of 256 has changed since 3 birds each gave birth to 5 baby birds, 2 birds were released, and 3 injured birds were brought in.

First, identify the important information in the problem.

256 birds to start

3 birds have 5 babies each

2 birds released

3 birds added

Next, write this as an expression.

\begin{align*}256+3\times 5-2+3\end{align*}

Then, use order of operations to evaluate the expression.

\begin{align*}256 + 3 \times 5 - 2 + 3 & \quad \text{Multiply }3 \times 5\\ 256 + 15 - 2 + 3 & \quad \text{Add }256 + 15\\ 271 - 2 + 3 & \quad \text{Subtract }271 - 2\\ 269 + 3 & \quad \text{Add }269 + 3\\ 272 &\\\end{align*}

The answer is 272.

Margot’s sanctuary now has 272 birds.

Example 2

Evaluate the expression using order of operations.

\begin{align*}6 + 8 \times 4 - 11 + 6 =\underline{\;\;\;\;\;\;\;}\end{align*}

First, complete the multiplication from left to right.

\begin{align*}6+32-11+6\end{align*}

Then, complete the addition and subtraction from left to right.

\begin{align*}6+32-11+6\\ 38-11+6\\ 27+6\\ 33\end{align*}

The answer is 33.

Example 3

Evaluate the following expression.

\begin{align*} 8 - 1 \times 4 + 3 = \underline{\;\;\;\;\;\;\;}\end{align*} 

First, perform the multiplication:

\begin{align*}8 - 1 \times 4 + 3 & \quad \text{ Multiply }1 \times 4 = 4\\ 8 - 4 + 3 & \\\end{align*}

Next, add and subtract from left to right:

\begin{align*}8 - 4 + 3 & \quad \text{Subtract }8 - 4 = 4\\ 4 + 3 & \quad \text{Add }4 + 3 = 7\\ 7 & \\\end{align*}

The solution is 7.

Example 4

Evaluate the following expression.

\begin{align*}5 + 9 \times 3 - 6 + 2 =\underline{\;\;\;\;\;\;\;}\end{align*} 

First, perform multiplication and division from left to right:

\begin{align*}2 \times 6 + 8 \div 2 & \quad \text{Multiply }2 \times 6\\ 12 + 8 \div 2 & \quad \text{Divide }8 \div 2\\ 12 + 4 &\\\end{align*}

Next, add:

\begin{align*}12 + 4 & \quad \text{Add }12 + 4\\ 16 &\\\end{align*}

The answer is 16.

Example 5

Evaluate the following expression.

\begin{align*}5 + 9 \times 3 - 6 + 2 =\underline{\;\;\;\;\;\;\;}\end{align*}

First, perform multiplication and division from left to right:

\begin{align*}5 + 9 \times 3 - 6 + 2 & \quad \text{Multiply }9 \times 3\\ 5 + 27 - 6 + 2 & \\\end{align*}

Next, add and subtract from left to right:

\begin{align*}5 + 27 - 6 + 2 & \quad \text{Add }5 + 27\\ 32 - 6 + 2 & \quad \text{Subtract }32 - 6\\ 26 + 2 & \quad \text{Add }26 + 2\\ 28 & \\\end{align*}

The answer is 28.

Review

Evaluate each expression according to the correct order of operations.

  1. \begin{align*}2+3\times4+7 = \underline{\;\;\;\;\;\;\;}\end{align*} 
  2. \begin{align*}4+5\times2+9-1 = \underline{\;\;\;\;\;\;\;}\end{align*} 
  3. \begin{align*}6\times7+2\times3 = \underline{\;\;\;\;\;\;\;}\end{align*} 
  4. \begin{align*}4\times5+3\times1-9 = \underline{\;\;\;\;\;\;\;}\end{align*} 
  5. \begin{align*}5\times3\times2+5-1 = \underline{\;\;\;\;\;\;\;}\end{align*} 
  6. \begin{align*}4+7\times3+8\times2 = \underline{\;\;\;\;\;\;\;}\end{align*} 
  7. \begin{align*}9-3\times1+4-7 = \underline{\;\;\;\;\;\;\;}\end{align*} 
  8. \begin{align*}10+3\times4+2-8 = \underline{\;\;\;\;\;\;\;}\end{align*} 
  9. \begin{align*}11\times3+2\times4-3 = \underline{\;\;\;\;\;\;\;}\end{align*} 
  10. \begin{align*}6+7\times8-9\times2\end{align*} 
  11. \begin{align*}3+4^2-5\times2+9 = \underline{\;\;\;\;\;\;\;}\end{align*} 
  12. \begin{align*}2^2+5\times2+6^2-11 = \underline{\;\;\;\;\;\;\;}\end{align*} 
  13. \begin{align*}3^2\times2+4-9 = \underline{\;\;\;\;\;\;\;}\end{align*} 
  14. \begin{align*}6+3\times2^2+7-1 = \underline{\;\;\;\;\;\;\;}\end{align*} 
  15. \begin{align*}7+2\times4+3^2-5 = \underline{\;\;\;\;\;\;\;}\end{align*}  

Review (Answers)

To see the Review answers, open this PDF file and look for section 1.10. 

Resources

 

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Vocabulary

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.

Image Attributions

  1. [1]^ License: CC BY-NC 3.0

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