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Algebra Expressions with Fraction Bars

Evaluate the numerator, evaluate the denominator, then simplify

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Algebraic Expressions with Fraction Bars
Multi-colored balls

The volume of a sphere can be represented by the formula \begin{align*}V=\frac{4 \pi r^3}{3}\end{align*}, where \begin{align*}r\end{align*} is the radius, and that the radius of a sphere is 4 feet. How would the fraction bar in the formula affect the way that you found the sphere's volume? Which operation do you think the fraction bar represents? 

Algebraic Expressions with Fraction Bars

Fraction bars count as grouping symbols for PEMDAS, and should be treated as a set of parentheses. All numerators and all denominators can be treated as if they have invisible parentheses. When real parentheses are also present, remember that the innermost grouping symbols should be evaluated first. If, for example, parentheses appear in a numerator, they would take precedence over the fraction bar. If the parentheses appear outside of the fraction, then the fraction bar takes precedence.

 

 Let's use PEMDAS to solve the following problems:

  1. Simplify the expression: \begin{align*}\frac{z+3}{4}-1 \text{ when }z=2.\end{align*}

Begin by substituting the appropriate value for the variable:

\begin{align*}\frac{(2+3)}{4} -1 = \frac{5}{4} -1\end{align*}

Rewriting 1 as a fraction, the expression becomes:

\begin{align*}\frac{5}{4} - \frac{4}{4} = \frac{1}{4}\end{align*}

  1. Simplify the expression \begin{align*}\left (\frac{a+2}{b+4} - 1 \right ) + b\end{align*} when \begin{align*}a = 3\end{align*} and \begin{align*}b = 1\end{align*}

Begin by substituting the appropriate value for the variable:

\begin{align*}\frac{(3+2)}{(1+4)} = \frac{5}{5} = 1\end{align*}

\begin{align*}(1 - 1) + b\end{align*} Substituting 1 for b, the expression becomes \begin{align*} 0 + 1 = 1\end{align*} 

  1. Simplify the expression \begin{align*}2 \times \left( \frac{w+(x-2z)}{(y+2)^2} -1 \right)\end{align*} , when  \begin{align*}w=11, x=3, y=1, \text{ and }z= \text{-}2.\end{align*} 

Begin by substituting the appropriate values for the variables:

\begin{align*}2 \left ( \frac{[11+(3-2(-2))]}{[(1+2)^2)]} - 1 \right ) = 2 \left ( \frac{(11+7)}{3^2} -1 \right ) = 2 \left (\frac{18}{9} - 1 \right )\end{align*}

Continue simplifying:

\begin{align*}2\left ( \frac{18}{9} - \frac{9}{9} \right ) = 2 \left ( \frac{9}{9} \right ) = 2(1)= 2\end{align*}

Examples

Example 1

Earlier, you were told that the volume of a sphere can be represented by the formula \begin{align*}V=\frac{4 \pi r^3}{3},\end{align*} where \begin{align*}r\end{align*} is the radius. You were also told that there is a sphere that has a radius of 4 feet. 

To find the volume of the sphere, first plug in 4 for the radius:

\begin{align*}V=\frac{4\pi r^3}{3}=\frac{4\pi (4)^3}{3}\end{align*} 

Now, treat the fraction bar as a grouping symbol and complete the operations above and below the bar independently until the numerator and denominator are simplified. 

Working with the numerator, simplify the exponent first:

\begin{align*}\frac{4\pi (4)^3}{3}=\frac{4\pi (64)}{3}\end{align*}

Now multiply the numerator. You can either leave the symbol \begin{align*}\pi\end{align*} and just multiply 4 and 64 or you can use 3.14 as an estimate for \begin{align*}\pi\end{align*} and end up with a decimal answer. In this case, we will leave the symbol  \begin{align*}\pi\end{align*} so that we have a more exact answer.

The final solution is:

\begin{align*}\frac{256\pi}{3}\end{align*} Both the numerator and denominator are simplified and there is no way to further simplify the fraction without moving to decimals

Example 2

Use the order of operations to evaluate the following expression when \begin{align*}x=6 \text{ and }y=2:\end{align*} 

\begin{align*}\left (\frac{x-1}{y-1} \right )^2 + \frac{2x}{y^2}\end{align*}

Begin by substituting in 6 for \begin{align*}x\end{align*} and 1 for \begin{align*}y\end{align*}.

\begin{align*}\left (\frac{6-1}{2-1} \right )^2 + \frac{2(6)}{2^2}\end{align*}

First, we work with what is inside the parentheses. There, we have a fraction so we have to simplify the fraction first, simplifying the numerator and then the denominator before dividing. We can simply the other fraction at the same time.

\begin{align*}\left (\frac{5}{1} \right )^2 + \frac{12}{4}\end{align*}

Next we simplify the fractions, and finish with exponents and then addition:

\begin{align*} (5 )^2 + 3=25+3=28\end{align*}

Review

Use the order of operations to simplify the following expressions.

  1. \begin{align*}\frac{2 \cdot (3 + (2 - 1))}{4 - (6 + 2)} - (3 - 5)\end{align*}
  2. \begin{align*}\frac{(2+3)^2}{3-8} - \frac{3\cdot(10-4)}{7-4}\end{align*}

Evaluate the following expressions involving variables.

  1. \begin{align*}\frac{jk}{j + k}\end{align*} when \begin{align*}j = 6\end{align*} and \begin{align*}k = 12\end{align*}
  2. \begin{align*}\frac{4x}{9x^2 - 3x + 1}\end{align*} when \begin{align*}x = 2\end{align*}
  3. \begin{align*}\frac{z^2}{x + y} + \frac{x^2}{x - y}\end{align*} when \begin{align*}x = 1, \ y = -2\end{align*}, and \begin{align*}z = 4\end{align*}
  4. \begin{align*}\frac{4xyz}{y^2 - x^2}\end{align*} when \begin{align*}x = 3, \ y = 2\end{align*}, and \begin{align*}z = 5\end{align*}
  5. \begin{align*}\frac{x^2 - z^2}{xz - 2x(z - x)}\end{align*} when \begin{align*}x = -1\end{align*} and \begin{align*}z = 3\end{align*}

The formula to find the volume of a square pyramid is \begin{align*}V=\frac{s^2 (h)}{3}.\end{align*} Evaluate the volume for the given values:

  1. \begin{align*}s=4\ inches,h=18\ inches\end{align*}
  2. \begin{align*}s=10\ feet,h=50\ feet\end{align*}
  3. \begin{align*}h=7\ meters,s=12\ meters\end{align*}
  4. \begin{align*}h=27\ feet,s=13\ feet\end{align*}
  5. \begin{align*}s=16\ cm,h=90\ cm\end{align*}

Review (Answers)

To see the Review answers, open this PDF file and look for section 1.4. 

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Vocabulary

Fraction Bar

A fraction bar is a line used to divide the numerator and the denominator of a fraction. The fraction bar means division.

fraction

A fraction is a part of a whole. A fraction is written mathematically as one value on top of another, separated by a fraction bar. It is also called a rational number.

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.

Parentheses

Parentheses "(" and ")" are used in algebraic expressions as grouping symbols.

PEMDAS

PEMDAS (Please Excuse My Daring Aunt Sally) is a mnemonic device used to help remember the order of operations: Parentheses, Exponents, Multiplication/Division, Addition/Subtraction.

Volume

Volume is the amount of space inside the bounds of a three-dimensional object.

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