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Multiplication of Rational Numbers

Multiply fractions: multiply straight across

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Multiplication of Rational Numbers

Suppose you wrote a computer program that multiplies \begin{align*}\frac{1}{5}\end{align*} by a random number. What if the random number were –1? What if it were 0? What if it were 1? In fact, the random number doesn't even have to be an integer. What if it were \begin{align*}\frac{3}{4}\end{align*}? What if it were \begin{align*}- \frac{2}{7}\end{align*}?

Multiplying Rational Numbers

Basic Multiplication Properties and Rules

When you began learning how to multiply whole numbers, you replaced repeated addition with the multiplication sign \begin{align*}(\times)\end{align*}. For example:

\begin{align*}6 + 6 + 6 + 6 + 6 = 5 \times 6 = 30\end{align*}

Multiplying rational numbers is performed the same way. 

There are five different important multiplication properties that you should know for rational numbers:

  • The Multiplication Property of -1: For any real number \begin{align*}a, (-1) \times a = -a\end{align*}. Any number multiplied by -1 is the original numbers opposite.
  • The Multiplicative Identity Property:  For any real number \begin{align*}a, \ (1) \times a = a\end{align*}. Any number multiplied by 1 is the original number. 
  • The Zero Property of MultiplicationFor any real number \begin{align*}a, \ (0) \times a = 0\end{align*}. Any number multiplied by 0 is 0.
  • The Associative Property of Multiplication: For any real numbers \begin{align*}a\end{align*}, \begin{align*}b\end{align*}, and \begin{align*}c\end{align*},

\begin{align*}(a \cdot b) \cdot c = a \cdot (b \cdot c)\end{align*}

 This is similar to the Associative Property of Addition. 

  • The Commutative Property of Multiplication: For any real numbers \begin{align*}a\end{align*} and \begin{align*}b\end{align*},

\begin{align*}a(b) = b(a)\end{align*} 

This is similar to the Commutative Property of Addition.

Rational numbers include positive and negative numbers and when multiplying those, there are two rules that you should follow:

  • The Same Sign Multiplication Rule: The product of two positive or two negative numbers is positive. If the signs are the same then the result is positive.
  • The Different Sign Multiplication Rule: The product of a positive number and a negative number is a negative number. If the signs are different then the result is negative. 

Let's evaluate the following expressions:

  1. \begin{align*}-1 \cdot 9,876\end{align*}

Using the Multiplication Property of \begin{align*}-1\end{align*}: \begin{align*} \ -1 \cdot 9,876 = -9,876\end{align*}.

  1. \begin{align*}0 \cdot -322\end{align*}

Using the Zero Property of Multiplication: \begin{align*} \ 0 \cdot -322 = 0\end{align*}.

  1. \begin{align*}-2\times -5\end{align*} 

Using the Same Sign Multiplication Rule\begin{align*}-2 \times -5=10\end{align*}.

Now, let's show that the following properties are true by showing that the equations are equal:

  1. The Commutative Property: \begin{align*}3 \times 4 = 4 \times 3\end{align*} 

We will check each side separately to see that they equal the same thing. 

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

So we conclude that the equality is satisfied.

  1. The Associative Property: \begin{align*}-6 \times (2 \times 3) = (-6 \times 2)\times 3\end{align*} 

 Like in #1, we will check each side separately to see that they equal the same thing.

\begin{align*}-6 \times (2\times3)=-6\times 6 = -36\\ (-6 \times 2)\times3=-12\times 3 = -36\end{align*} 

So we conclude that the equality is satisfied.

Multiplying Decimals

You can multiply decimals in the same way that you multiply whole numbers with several digits. When you multiply decimals, just follow these two steps:

First, multiply normally, ignoring the decimal points.

Next, put the decimal point in the answer. The answer will have as many decimal places as the original numbers combined.

Let’s multiply the following decimals:

 \begin{align*}34.67 \times 8.2\end{align*}

First, line up the numbers in order to get ready to multiply.

\begin{align*}34.67\\ \underline{ \times \;\; 8.2}\end{align*}

Next, multiply each digit in the top number by each digit in the bottom number, just like whole numbers.

\begin{align*}34.67\\ \underline{ \times \;\; 8.2}\\ 6934\\ \underline{\;\; 27736 \ \ }\\ 284294\end{align*} 

Then, place the decimal point in the product by counting the number of decimal places in each of the numbers that were multiplied. The first number has two decimal places, and the second number has one decimal place. So move the decimal point three places.

\begin{align*}& 2 \ 8 \ 4 . \ \ 2 \quad 9 \quad 4 \\ & \quad \quad \quad {\color{red}\leftarrow} \ {\color{red}\leftarrow} \ \ {\color{red}\leftarrow} \\ & \qquad \quad 3 \quad 2 \quad \ 1\end{align*}

The answer is 284.294. 

Multiplying Fractions

Unlike adding and subtracting fractions, multiplying fractions does not require a common denominator. When multiplying fractions, first change any mixed numbers to improper fractions. Then use the following rule:

For any real numbers \begin{align*}a, b, c\end{align*} and \begin{align*}d,\end{align*} where \begin{align*}b\neq 0\end{align*} and \begin{align*}d \neq 0\end{align*}:

\begin{align*}\frac {a}{b}\cdot \frac{c}{d} = \frac {ac}{bd}\end{align*}

In other words, multiply the numerators and then multiply the denominators. Then, simplify if you can. 

Let's multiply the following fractions:

  1.  \begin{align*}2 \frac {3}{5} \cdot -\frac {2}{7}\end{align*} 

 First, note that \begin{align*}2 \frac{3}{5}\end{align*} is a mixed number. Change the mixed number to an improper fraction:

\begin{align*}2 \frac {3}{5} = \frac {13}{5}\end{align*} Now, multiply the fractions:

\begin{align*}2 \frac {3}{5} \cdot -\frac{2}{7} = \frac {13}{5} \cdot -\frac{2}{7}=-\frac {13\cdot 2}{5\cdot 7}=-\frac {26}{35}\end{align*}

Note that since the two fractions have different signs, the answer is negative. 

\begin{align*}-\frac{26}{35}\end{align*} cannot be simplified and is the solution to this multiplication problem. 

  1.   \begin{align*}\frac{4}{9}\cdot \frac{1}{6}\end{align*} 

There are no mixed numbers in this problem so we can just multiply the numerators and denominators. 

\begin{align*}\frac {4}{9}\cdot \frac{1}{6}=\frac{4\cdot 1}{9\cdot 6}=\frac{4}{54}=\frac{2}{27}\end{align*}


Example 1

Earlier you were asked what would happen if the number \begin{align*}\frac{1}{5}\end{align*} was multiplied by -1, 0, 1, \begin{align*}\frac{3}{4}\end{align*}, or \begin{align*}- \frac{2}{7}\end{align*}

By the Multiplicative Property of -1, \begin{align*}\frac{1}{5}\times -1 = -\frac{1}{5}\end{align*}.

By the Zero Property of Multiplication, \begin{align*}\frac {1}{5}\times 0 = 0\end{align*}.

By the Multiplicative Identity Property, \begin{align*}\frac {1}{5} \times 1 = \frac {1}{5}\end{align*}.

Using the Same Sign Multiplication Rule, \begin{align*}\frac {1}{5} \cdot \frac {3}{4}=\frac {1\cdot 3}{5 \cdot 4} = \frac {3}{20}\end{align*}.

Using the Different Sign Multiplication Rule, \begin{align*}\frac {1}{5} \cdot -\frac {2}{7}=-\frac {1\cdot 2}{5 \cdot 7} = -\frac {2}{35}\end{align*}.    

Example 2

Anne has a bar of chocolate and she offers Bill a piece, Bill quickly breaks off \begin{align*}\frac {1}{4}\end{align*} of the bar and eats it. Another friend, Cindy, takes \begin{align*}\frac {1}{3}\end{align*} of what was left. Anne splits the remaining candy bar into two equal pieces, which she shares with a third friend, Dora. How much of the candy bar does each person get?

Think of the bar as one whole. 

\begin{align*}1- \frac{1}{4} = \frac{3}{4}\end{align*}   This is the amount remaining after Bill takes his piece.

\begin{align*}\frac{1}{3} \times \frac{3}{4} = \frac{1}{4}\end{align*}   This is the fraction Cindy receives. 

\begin{align*}\frac{3}{4} - \frac{1}{4} = \frac{2}{4} = \frac{1}{2}\end{align*}   This is the amount remaining after Cindy takes her piece.

Anne divides the remaining bar into two equal pieces. Every person receives \begin{align*}\frac {1}{4}\end{align*} of the bar. 

Example 3

Adrian is making a dog house that is \begin{align*}3 \frac {1}{2}\end{align*} feet long and \begin{align*}2 \frac {2}{3}\end{align*} feet wide. How many square feet of area will the dog house be?

Since the formula for area is

\begin{align*} area=length \times width\end{align*},

we plug in the values for length and width:

\begin{align*} area=3\frac{1}{2} \times 2\frac{2}{3} .\end{align*}

We first need to turn the mixed fractions into improper fractions:

\begin{align*} area=3\frac{1}{2} \times 2\frac{2}{3} =\frac{7}{2} \times \frac{8}{3}= \frac{7\times 8}{2\times 3} = \frac{56}{6}. \end{align*}

Now we turn the improper fraction back into a mixed fraction. Since 56 divided by 6 is 9 with a remainder of 2, we get:

\begin{align*} \frac{56}{6}=9\frac{2}{6}=9\frac{1}{3}\end{align*}

The dog house will have an area of \begin{align*}9\frac{1}{3}\end{align*} square feet.

Note that the units of the area are square feet, or feet squared, because we multiplied two numbers each with units of feet:

\begin{align*}feet \times feet=feet^2\end{align*},

which we call square feet.

Example 4

Doris’s truck gets \begin{align*}10 \frac {2}{3}\end{align*} miles per gallon. Her tank is empty so she puts in \begin{align*}5 \frac {1}{2}\end{align*}gallons of gas. How far can she travel?

Begin by writing each mixed number as an improper fraction.

\begin{align*}10 \frac{2}{3} = \frac{32}{3} && 5 \frac{1}{2} = \frac{11}{2}\end{align*}

Now multiply the two values together.

\begin{align*}\frac{32}{3} \cdot \frac{11}{2} = \frac{352}{6} = 58\frac{4}{6} = 58\frac{2}{3}\end{align*}

Doris can travel \begin{align*}58 \ \frac{2}{3}\end{align*} miles on \begin{align*}5 \frac {1}{2}\end{align*} gallons of gas.


In 1-3, multiply by negative one.

  1. \begin{align*}\pi\end{align*}
  2. \begin{align*}(x + 1)\end{align*}
  3. -105

In 4-13, multiply the rational numbers and simplify.

  1. \begin{align*}\frac{1}{2} \cdot \frac{3}{4}\end{align*}
  2. \begin{align*}\frac{15}{11} \times \frac{9}{7}\end{align*}
  3. \begin{align*}\frac {2}{5} \cdot -\frac{5}{9}\end{align*} 
  4. \begin{align*}\frac {1}{3} \cdot -\frac{2}{7}\cdot\frac{2}{5}\end{align*} 
  5. \begin{align*}4\frac {1}{2} \cdot -3\end{align*} 
  6. \begin{align*}(\frac{1}{2}\cdot\frac{2}{3})\cdot\frac{3}{4}\cdot\frac{4}{5}\end{align*} 
  7. \begin{align*}\frac{5}{12}\times 1 \frac{9}{10}\end{align*} 
  8. \begin{align*}\frac{27}{5} \cdot 0 \end{align*}
  9. \begin{align*}\left (\frac{3}{5} \right )^2 \end{align*}
  10. \begin{align*}\frac{1}{11} \times \frac{22}{21} \times \frac{7}{10} \end{align*}

In 14-16, state the property that applies to each of the following situations.

  1. A gardener is planting vegetables for the coming growing season. He wishes to plant potatoes and has a choice of a single 8 by 7 meter plot, or two smaller plots of 3 by 7 meters and 5 by 7 meters. Which option gives him the largest area for his potatoes?
  2. Andrew is counting his money. He puts all his money into $10 piles. He has one pile. How much money does Andrew have?
  3. Nadia and Peter are raising money by washing cars. Nadia is charging $3 per car, and she washes five cars in the first morning. Peter charges $5 per car (including a wax). In the first morning, he washes and waxes three cars. Who has raised the most money?
  1. Theo is making a flower box that is 5-and-a-half inches by 15-and-a-half inches. How many square inches will he have in which to plant flowers?

Quick Quiz

  1. Order from least to greatest: \begin{align*}\left (\frac{5}{6}, \ \frac{23}{26}, \ \frac{31}{32}, \ \frac{3}{14} \right )\end{align*}.
  2. Simplify \begin{align*}\frac{5}{9} \times \frac{27}{4}.\end{align*}
  3. Simplify \begin{align*}|-5 + 11| - |9 - 37|\end{align*}.
  4. Add \begin{align*}\frac{21}{5}\end{align*} and \begin{align*}\frac{7}{8}.\end{align*}

Review (Answers)

To see the Review answers, open this PDF file and look for section 2.6. 


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Multiplication Property of –1 For any real number a, (-1) \times a = -a.
multiplicative identity property The product of any number and one is the number itself.
Zero Property of Multiplication For any real number a, \ (0) \times a = 0.
Associative Property The associative property states that you can change the groupings of numbers being added or multiplied without changing the sum. For example: (2+3) + 4 = 2 + (3+4), and (2 X 3) X 4 = 2 X (3 X 4).
Commutative Property The commutative property states that the order in which two numbers are added or multiplied does not affect the sum or product. For example a+b=b+a \text{ and\,} (a)(b)=(b)(a).
distributive property The distributive property states that the product of an expression and a sum is equal to the sum of the products of the expression and each term in the sum. For example, a(b + c) = ab + ac.
Integer The integers consist of all natural numbers, their opposites, and zero. Integers are numbers in the list ..., -3, -2, -1, 0, 1, 2, 3...
Mixed Number A mixed number is a number made up of a whole number and a fraction, such as 4\frac{3}{5}.

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