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Expression Evaluation with Products of Fractions

Evaluating expressions multiplying fractions < 1.

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Expression Evaluation with Products of Fractions
Credit: Oscar
Source: https://www.flickr.com/photos/batrace/358446273
License: CC BY-NC 3.0

Jae is creating a model building for a design competition. Each floor will be \begin{align*}3\frac{1}{2}\end{align*} inches tall. How can Jae find the total height of his building model. How tall will it be if Jae decides to have 10 floors?

In this concept, you will learn how to evaluate expressions and products of fractions

Evaluating Expressions with Products of Fractions

An expression is a numerical phrase that combines numbers and operations but no equal sign. 

There are two kinds of expressions. Numerical expressions include numbers and operations only. Algebraic expressions include numbers, operations, and variables.

Types of Expressions

Includes Examples
numerical expressions numbers, operations


\begin{align*}\frac{3}{4} \times \frac{2}{3}\end{align*}


\begin{align*}4 \left (\frac{3}{4} \right )\end{align*}

algebraic expressions numbers, operations, variables

\begin{align*}3 + x\end{align*}

\begin{align*}\frac{3}{4} \cdot \frac{b}{3}\end{align*}

\begin{align*}15.6 - q\end{align*}

\begin{align*}c \left (\frac{3}{4}\right )\end{align*}

Since a numerical expression includes numbers and operations, perform the operation required to evaluate. To evaluate means to find the value of the expression.

Evaluate the numerical expression.

 \begin{align*}\left ( \frac{1}{4} \right ) \left ( \frac{3}{4} \right )\end{align*}

Notice that there are two sets of parentheses. Remember that the parentheses can identify groups and also indicate multiplication.

Evaluate by multiplying and then simplifying or by simplifying first then multiplying.

\begin{align*}\frac{1}{4} \times \frac{3}{4} = \frac{3}{16}\end{align*}

The answer is in simplest form.

The value of the expression is \begin{align*}\frac{3}{16}\end{align*}.

As you learn about algebra and higher levels of math, you will be working with algebraic expressions.

An algebraic expression is similar to a numeric expressions, except that it uses variables. Variables are letters that represent an unknown number. Sometimes the values of the variables are given.

Evaluate the algebraic expression.

\begin{align*}xy\end{align*} when \begin{align*}x = \frac{3}{4}\end{align*}, and \begin{align*}y = \frac{1}{3}\end{align*}

To evaluate this expression, substitute the given values for \begin{align*}x\end{align*} and \begin{align*}y\end{align*} into the expression. Notice that the expression has \begin{align*}x\end{align*} and \begin{align*}y\end{align*} next to each other. When two variables are next to each other the operation is multiplication. 

\begin{align*}\frac{3}{4} \times \frac{1}{3}\end{align*}

Then, simplify the expression. Cross simplify the 3s with the greatest common factor (GCF) of 3. Each three becomes a one.

\begin{align*}\frac{3}{4} \times \frac{1}{3} = \frac{1}{4} \times \frac{1}{1}\end{align*}

Next, multiply to evaluate the expression.

\begin{align*}\frac{1}{4} \times \frac{1}{1} = \frac{1}{4}\end{align*}

The value of the expression is \begin{align*}\frac{1}{4}\end{align*}.


Example 1

Earlier, you were given a problem about Jae and his design competition.

Jae is making a model building where each floor is \begin{align*}3\frac{1}{2}\end{align*} inches tall. Create an expression that will help determine the total height of the building depending on the number of floors he decides to make. 

For Jae's model building, he can create an algebraic expression to determine the height of his model. The total height will be the the height of each floor times the number of floors.

\begin{align*}3\frac{1}{2} x, \text {where } x \text { is equal to the total number of floors}\end{align*} 

Find the height of the model when x = 10.

First, substitute the variable with the given value.

 \begin{align*}3\frac{1}{2} \times 10\end{align*}

Then, convert the mixed number and whole number into an improper fraction.

\begin{align*}3\frac{1}{2} \times 10 = \frac {7}{2} \times \frac{10}{1}\end{align*}

Next, simplify the fractions.

 \begin{align*}\frac {7}{2} \times \frac{10}{1}=\frac {7}{1} \times \frac{5}{1}\end{align*}

Finally, multiply the fractions.

 \begin{align*}\frac {7}{1} \times \frac{5}{1}=35\end{align*}

Jae's 10 floor model will be 35 inches tall. 

Example 2

Evaluate \begin{align*}xy\end{align*} when \begin{align*}x\end{align*} is \begin{align*}\frac{2}{3}\end{align*} and \begin{align*}y\end{align*} is \begin{align*}\frac{8}{12}\end{align*}.

First, substitute the values into the expression.

\begin{align*} \frac{2}{3} \times \frac{8}{12}\end{align*}

Then, simplify the expression.

\begin{align*} \frac{2}{3} \times \frac{8}{12} = \frac{1}{3} \times \frac{4}{3} \end{align*}

Next, multiply the fractions. 

\begin{align*} \frac{1}{3} \times \frac{4}{3} = \frac{4}{9}\end{align*}

The answer is \begin{align*}\frac{4}{9}\end{align*}.

Example 3

Evaluate \begin{align*}\left ( \frac{4}{7} \right ) \left ( \frac{21}{28} \right )\end{align*}. Answer in simplest form.

First, simplify the fractions.

\begin{align*}\frac{4}{7} \times \frac{21}{28} = \frac{1}{1} \times \frac{3}{7} \end{align*}

Then, multiply.

\begin{align*}1 \times \frac{3}{7} = \frac{3}{7} \end{align*}

The value of the expression is \begin{align*}\frac{3}{7}\end{align*}.

Example 4

Evaluate \begin{align*}(xy)\end{align*} when \begin{align*}x\end{align*} is \begin{align*}\frac{3}{5}\end{align*} and \begin{align*}y\end{align*} is \begin{align*}\frac{10}{11}\end{align*}. Answer in simplest form.

First, substitute the values into the expression.

 \begin{align*}\frac{3}{5}\times \frac{10}{11}\end{align*}

Then, simplify the fractions. 

\begin{align*}\frac{3}{5}\times \frac{10}{11}= \frac{3}{1}\times \frac{2}{11}\end{align*}

Next, multiply.

\begin{align*}\frac{3}{1}\times \frac{2}{11} = \frac{6}{11}\end{align*}

The value of the expression is \begin{align*}\frac{6}{11}\end{align*}.

Example 5

Evaluate \begin{align*}\left ( \frac{5}{9} \right ) \left ( \frac{45}{60} \right )\end{align*}. Answer in simplest form.

First, simplify the fractions. 

\begin{align*} \frac{5}{9} \times \frac{45}{60} = \frac {5}{3} \times \frac {1}{4}\end{align*}

Next, multiply. 

\begin{align*} \frac {5}{3} \times \frac {1}{4}= \frac{5}{12}\end{align*}

The value of the expression is \begin{align*}\frac{5}{12}\end{align*}.


Evaluate each expression. Answer in simplest form. 

  1. Evaluate \begin{align*}(xy)\end{align*} when \begin{align*}x = \frac{2}{3}\end{align*} and \begin{align*}y = \frac{6}{10}\end{align*}
  2. Evaluate \begin{align*}(xy)\end{align*} when \begin{align*}x = \frac{1}{3}\end{align*} and \begin{align*}y = \frac{4}{10}\end{align*}
  3. Evaluate \begin{align*}(xy)\end{align*} when \begin{align*}x = \frac{12}{13}\end{align*} and \begin{align*}y = \frac{2}{6}\end{align*}
  4. Evaluate \begin{align*}(xy)\end{align*} when \begin{align*}x = \frac{1}{3}\end{align*} and \begin{align*}y = \frac{4}{5}\end{align*}
  5. Evaluate \begin{align*}(xy)\end{align*} when \begin{align*}x = \frac{7}{9}\end{align*} and \begin{align*}y = \frac{3}{21}\end{align*}
  6. Evaluate \begin{align*}(xy)\end{align*} when \begin{align*}x = \frac{4}{5}\end{align*} and \begin{align*}y = \frac{16}{20}\end{align*}
  7. Evaluate \begin{align*}\left ( \frac{4}{6} \right ) \left ( \frac{1}{2} \right )\end{align*}
  8. Evaluate \begin{align*}\left ( \frac{1}{9} \right ) \left ( \frac{6}{18} \right )\end{align*}
  9. Evaluate \begin{align*}\left ( \frac{4}{9} \right ) \left ( \frac{1}{4} \right )\end{align*}
  10. Evaluate \begin{align*}\left ( \frac{4}{11} \right ) \left ( \frac{11}{12} \right )\end{align*}
  11. Evaluate \begin{align*}\left ( \frac{9}{10} \right ) \left ( \frac{5}{6} \right )\end{align*}
  12. Evaluate \begin{align*}\left ( \frac{8}{9} \right ) \left ( \frac{3}{6} \right )\end{align*}
  13. Evaluate \begin{align*}\left ( \frac{18}{19} \right ) \left ( \frac{3}{6} \right )\end{align*}
  14. Evaluate \begin{align*}\left ( \frac{4}{9} \right ) \left ( \frac{36}{40} \right )\end{align*}
  15. Evaluate \begin{align*}\left ( \frac{12}{14} \right ) \left ( \frac{7}{8} \right )\end{align*}

Review (Answers)

To see the Review answers, open this PDF file and look for section 7.5. 


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Algebraic Expression

An expression that has numbers, operations and variables, but no equals sign.

Numerical expression

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


The product is the result after two amounts have been multiplied.

Image Attributions

  1. [1]^ Credit: Oscar; Source: https://www.flickr.com/photos/batrace/358446273; License: CC BY-NC 3.0

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