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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

School is about to start back up again and Lucy is at the store buying supplies. In her basket she has 5 notebooks for $2 each, 2 packs of pencils for $4 each, and 2 packs of pens for $5 each. She also picked out a big binder for $12. Luckily, she has a coupon for $3 off the binder. How can Lucy write and evaluate a numerical expression to calculate the total amount she will spend on her school supplies?

In this concept, you will learn how to evaluate variable expressions involving the four basic operations.

Evaluating Expressions with Basic Operations

An expression is a mathematical phrase that contains numbers and operations. A numerical expression is an expression with only numbers and operations and no variables. A variable expression is an expression that contains variables.

To evaluate a variable expression means to find the value of the expression for given values of the variables. To evaluate, substitute the given values for the variables in the expression and simplify using the order of operations (PEMDAS).

Remember that the order of operations are:

  1. Parentheses: Start by simplifying any part of the expression in parentheses using the order of operations.
  2. Exponents: Rewrite any terms that contain exponents without exponents.
  3. Multiplication/Division: Do any multiplication and/or division in order from left to right.
  4. Addition/Subtraction: Do any addition and/or subtraction in order from left to right.

The acronym PEMDAS can help you to remember the order of operations. Just remember that multiplication and division go together and addition and subtraction go together.

Here is an example.

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

First, substitute 6 in for the \begin{align*}b\end{align*} in the expression.

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

Next, notice that this expression has multiplication, addition, division, and subtraction. Remember that \begin{align*}4(6)\end{align*} is multiplication. Start by doing the multiplication and division from left to right. First you will multiply \begin{align*}4(6)\end{align*}.

\begin{align*}4(6)+12\div 4-8=24+12\div 4-8\end{align*}

Next, divide \begin{align*}12\div 4\end{align*}.

\begin{align*}24+12\div 4-8=24+3-8\end{align*}

Now, add and subtract from left to right. First, add \begin{align*}24+3\end{align*}.

\begin{align*}24+3-8=27-8\end{align*}

Finally, subtract.

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

The answer is 19.

Examples

Example 1

Earlier, you were given a problem about Lucy, who is shopping for school supplies.

She has 5 notebooks for $2 each, 2 packs of pencils for $4 each, 2 packs of pens for $5 each, and a big binder for $12. She also has a coupon for $3 off the binder. Lucy wants to know how much she will be spending on supplies.

First, write an expression for each item that represents how much she is spending on that item.

  • Notebooks: \begin{align*}5\times \$2\end{align*}
  • Pencils: \begin{align*}2\times \$4\end{align*}
  • Pens: \begin{align*}2\times \$5\end{align*}
  • Binder: \begin{align*}\$ 12-\$ 3\end{align*}

Now, add everything together to create one expression that represents the total amount she is spending.

\begin{align*}5\times 2+2\times 4+2\times 5+12-3\end{align*}

Next, notice that this expression has multiplication, addition, and subtraction. Start by doing the multiplication from left to right. First you will multiply \begin{align*}5\times 2\end{align*}. Then continue multiplying from left to right.

\begin{align*}\begin{array}{rcl} 5\times 2+2\times 4+2\times 5+12-3& = & 10+2\times 4+2\times 5+12-3\\ & = & 10+8+2\times 5+12-3\\ & = & 10+8+10+12-3 \end{array}\end{align*}

Next, add and subtract from left to right.

\begin{align*}\begin{array}{rcl} 10+8+10+12-3 & = & 18+10+12-3\\ & = & 28+12-3\\ & = & 40-3\\ & = & 37 \end{array}\end{align*}

The answer is that Lucy is spending $37 on school supplies.

Example 2

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

First, substitute 4 in for the \begin{align*}x\end{align*} in the expression.

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

Next, notice that this expression has multiplication, addition, division, and subtraction. Start by doing the multiplication and division from left to right. First you will multiply \begin{align*}6(4)\end{align*}.

\begin{align*}6(4)+36\div 4-3+8=24+36\div 4-3+8\end{align*}

Next, divide \begin{align*}36\div 4\end{align*}.

\begin{align*}24+36\div 4-3+8=24+9-3+8\end{align*}

Now, add and subtract from left to right. First, add \begin{align*}24+9\end{align*}.

\begin{align*}24+9-3+8=33-3+8\end{align*}

Then, subtract \begin{align*}33-3\end{align*}.

\begin{align*}33-3+8=30+8\end{align*}

Finally, add.

\begin{align*}30+8=38\end{align*}

The answer is 38.

Example 3

Evaluate \begin{align*}5a+6\div 2+9\end{align*} if \begin{align*}a=5\end{align*}.

First, substitute 5 in for the \begin{align*}a\end{align*} in the expression.

\begin{align*}5(5)+6\div 2+9\end{align*}

Next, notice that this expression has multiplication, addition, and division. Start by doing the multiplication and division from left to right. First you will multiply \begin{align*}5(5)\end{align*}. Then divide.

\begin{align*}\begin{array}{rcl} 5(5)+6\div 2+9& = & 25+6\div 2+9\\ & = & 25+3+9 \end{array}\end{align*}

Next, add from left to right.

\begin{align*}\begin{array}{rcl} 25+3+9 & = & 28+9\\ &= &37 \end{array}\end{align*}

The answer is 37.

Example 4

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

First, substitute 4 in for the \begin{align*}x\end{align*} in the expression.

\begin{align*}7(4)+14\div 7-3\end{align*}

Next, notice that this expression has multiplication, addition, division, and subtraction. Start by doing the multiplication and division from left to right. First you will multiply \begin{align*}7(4)\end{align*}. Then divide.

\begin{align*}\begin{array}{rcl} 7(4)+14\div 7-3 &= & 28+14\div 7-3\\ & = & 28+2-3 \end{array}\end{align*}

Next, add and subtract from left to right. First you will add \begin{align*}28+2\end{align*}. Then subtract.

\begin{align*}\begin{array}{rcl} 28+2-3 & = & 30-3\\ & = & 27 \end{array}\end{align*}

The answer is 27.

Example 5

Evaluate \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.

First, substitute 6 in for the \begin{align*}x\end{align*} and 9 in for the \begin{align*}y\end{align*} in the expression.

\begin{align*}\frac{48}{6} + 5(9)-7\end{align*}

Next, notice that this expression has division, addition, multiplication, and subtraction. Remember that a fraction bar means division so \begin{align*}\frac{48}{6}\end{align*} is the same as \begin{align*}48\div 6\end{align*}.

Start by doing the multiplication and division from left to right. First you will divide \begin{align*}\frac{48}{6}\end{align*}. Then multiply.

\begin{align*}\begin{array}{rcl} \frac{48}{6}+5(9)-7& = & 8+5(9)-7\\ & = & 8+45-7 \end{array}\end{align*}

Next, add and subtract from left to right. First you will add \begin{align*}8+45\end{align*}. Then subtract.

\begin{align*}\begin{array}{rcl} 8+45-7 & = & 53-7\\ & = & 46 \end{array}\end{align*}

The answer is 46.

Review

Use the order of operations to evaluate each numerical expression.

  1. \begin{align*}5+3\times4\end{align*} 
  2. \begin{align*}6\times2+5\times3\end{align*} 
  3. \begin{align*}4+5\times2-9\end{align*} 
  4. \begin{align*}4+6\div2+10-3\end{align*} 
  5. \begin{align*}8-15\div3+4\times5\end{align*} 

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

  1. \begin{align*}2x+28\div4-3\end{align*} 
  2. \begin{align*}6x+30\div2-10\end{align*} 
  3. \begin{align*}5x+x-3+8\end{align*} 
  4. \begin{align*}3x+36\div9-3+1\end{align*} 
  5. \begin{align*}5x+9-3+8\end{align*} 
  6. \begin{align*}4x+30\div15-3+8\end{align*} 
  7. \begin{align*}8x+2x-3+18\end{align*} 
  8. \begin{align*}2x+x\div2-3+8\end{align*} 
  9. \begin{align*}9x+3x\div2+8\end{align*}
  10. \begin{align*}12x+4x\div4-6+8\end{align*} 

Review (Answers)

To see the Review answers, open this PDF file and look for section 1.9.

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Vocabulary

Numerical expression

A numerical expression is a group of numbers and operations used to represent a quantity.

Order of Operations

The order of operations specifies the order in which to perform each of multiple operations in an expression or equation. The order of operations is: P - parentheses, E - exponents, M/D - multiplication and division in order from left to right, A/S - addition and subtraction in order from left to right.

Variable Expression

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

Image Attributions

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

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