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Multiplying and dividing fractions Multiplying and dividing decimals Multiplying and dividing integersMultiplying and dividing rational numbers
Here you will learn how to multiply and divide rational numbers. You will apply previously learned skills such as multiplying and dividing fractions, multiplying and dividing decimals, and multiplying and dividing integers to multiply and divide rational numbers.
Students first learn how to multiply and divide fractions in 4 th grade, multiply and divide decimals in 5 th grade, and multiply and divide integers in 6 th grade. In 7 th grade, students are required to apply those skills in order to multiply and divide rational numbers.
What is multiplying and dividing rational numbers?
Multiplying and dividing rational numbers occurs when one or more numbers being multiplied or divided are classified as rational numbers.
Recall that rational numbers are numbers that can be expressed as a quotient or fraction of two integers where p is the numerator and q is a non-zero denominator.
The same rules for multiplying integers apply to multiplying all rational numbers.
When multiplying and dividing two rational numbers:
- If the signs of the numbers are the same, the product or quotient is positive.
- If the signs of the numbers are different, the product or quotient is negative.
Step-by-step guide: Multiplying and dividing integers
Also, the steps to multiply rational numbers is very similar to the way you multiply fractions.
\text{Product of rational numbers }=\cfrac{\text{Product of the numerators}}{\text{Product of the denominators}}For example, let’s multiply the rational numbers, - \, 3 \cfrac{1}{4} \times 2 \cfrac{2}{3} .
First, rewrite the mixed numbers as improper fractions.
- \, \cfrac{13}{4} \times \cfrac{8}{3}Then, multiply the numerators together and then the denominators together. Since the first mixed number is negative, you can assign the negative sign to the numerator.
- \, \cfrac{13}{4} \times \cfrac{8}{3}=\cfrac{(- \, 13) \times 8}{4 \times 3}=\cfrac{- \, 104}{12}Since the signs of the two rational numbers are different, the product is negative.
After multiplying, just like with any fraction, simplify the product.
\cfrac{- \, 104}{12}=- \, 8 \cfrac{8}{12}=- \, 8 \cfrac{2}{3}The product of - \, 3 \cfrac{1}{4} \times 2 \cfrac{2}{3} is - \, 8 \cfrac{2}{3} .
It is important to remember that the product of any two rational numbers always results in a rational number.
Similarly, to divide rational numbers, use the same process as you would for dividing fractions. Multiply the first fraction by the reciprocal of the second fraction and then do the same as you would when multiplying rational numbers.
For example, find the quotient of - \, 1 \cfrac{2}{7} \div(- \, 3) .
Like with multiplication, first rewrite the mixed number as improper fractions and the integer with a denominator of 1.
- \, \cfrac{9}{7} \div\left(- \, \cfrac{3}{1}\right)Next, take the reciprocal of the second number which in this case is \left(- \, \cfrac{3}{1}\right).
Recall: The reciprocal is the multiplicative inverse of a number, meaning that when you multiply a number by its reciprocal the product is 1, or a\times\cfrac{1}{a}=1.
So in this case, the reciprocal of \left(- \, \cfrac{3}{1}\right) is \left(- \, \cfrac{1}{3}\right) because - \, \cfrac{3}{1} \times- \, \cfrac{1}{3}=1.
After finding the reciprocal of the second number, rewrite the division problem as a multiplication problem.
- \, \cfrac{9}{7} \div\left(-\cfrac{3}{1}\right) becomes, \cfrac{9}{7} \times\left(- \, \cfrac{1}{3}\right), you can assign the negative sign to the numerators before multiplying.
\cfrac{- \, 9}{7} \times \cfrac{- \, 1}{3}=\cfrac{9}{21}Since both of the rational numbers are negative, the quotient is positive.
The fraction cannot be simplified so the quotient of - \, \cfrac{9}{7} \div\left(- \, \cfrac{3}{1}\right) is \cfrac{9}{21}.
Step-by-step guide: Multiplying and dividing fractions
What is multiplying and dividing rational numbers?
Common Core State Standards
How does this relate to 7 th grade math?
- Grade 7 – Number System (7.NS.A.2)
Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers.
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[FREE] Multiplication and Division Quiz (Grade 4, 5 and 7)
Use this quiz to check your grade 4, 5 and 7 students’ understanding of multiplication and division. 10+ questions with answers covering a range of 4th, 5th and 7th grade multiplication and division topics to identify areas of strength and support!
DOWNLOAD FREE NOW [FREE] Multiplication and Division Quiz (Grade 4, 5 and 7)
Use this quiz to check your grade 4, 5 and 7 students’ understanding of multiplication and division. 10+ questions with answers covering a range of 4th, 5th and 7th grade multiplication and division topics to identify areas of strength and support!
DOWNLOAD FREE NOWHow to multiply and divide rational numbers
In order to multiply and divide rational numbers:
- Apply previously learned skills for multiplication and division of fractions, decimals, and integers.
- Make the calculation.
- Simplify the answer if necessary.
Multiplying and dividing rational numbers examples
Example 1: multiplying rational numbers that are mixed numbers
Find the product of 1\cfrac{1}{2}\times\left(- \, 1\cfrac{2}{3}\right) .
- Apply previously learned skills for multiplication and division of fractions, decimals, and integers.
Since the signs are different on the rational numbers, the product is going to be negative.
Before multiplying, change each mixed number to an improper fraction.
1\cfrac{1}{2}\times\left(- \, 1\cfrac{2}{3}\right)=\cfrac{3}{2}\times\left(- \, \cfrac{5}{3}\right)2Make the calculation.
Assign the negative to the numerator. In order to multiply fractions, multiply numerators together and denominators together.
\cfrac{3}{2}\times\cfrac{- \, 5}{3}=\cfrac{3\times{- \, 5}}{2\times{3}}=\cfrac{- \, 15}{6}3Simplify the answer if necessary.
The product can be simplified:
\cfrac{- \, 15}{6}=\cfrac{- \, 5}{2}=- \, 2\cfrac{1}{2}Example 2: multiplying rational numbers that are decimal numbers
Find the product of (- \, 3)\times(- \, 2.3).
Apply previously learned skills for multiplication and division of fractions, decimals, and integers.
Both of the rational numbers are negative so the product is positive. This means that (- \, 3)\times(- \, 2.3) will give the same result as 3\times{2.3}.
Use the standard algorithm for multiplication to multiply these numbers together.
Make the calculation.
As 3\times{23}=69 and 230 is 10 times larger than 23,
3\times{2.3}=6.9
So, (- \, 3)\times(- \, 2.3)=3\times{2.3}=6.9.
Simplify the answer if necessary.
The product does not need to be simplified.
Example 3: dividing rational numbers that are fractions
Find the quotient of - \, 1\div\cfrac{10}{13}.
Apply previously learned skills for multiplication and division of fractions, decimals, and integers.
The signs of the rational numbers are different so the quotient is negative.
Write the whole number with a denominator of 1 and assign the numerator to be negative.
Take the reciprocal of the second number, since it is a division with fractions. The reciprocal of \cfrac{10}{13} is \cfrac{13}{10}.
Make the calculation.
Simplify the answer if necessary.
The quotient can be rewritten as a mixed number.
\cfrac{- \, 13}{10}=- \, 1\cfrac{3}{10}
Example 4: dividing rational numbers that are decimals
Find the quotient of - \, 3.07\div(- \, 0.1).
Apply previously learned skills for multiplication and division of fractions, decimals, and integers.
The signs of the rational numbers are the same, both negative, so the quotient is positive.
- \, 3.07\div(- \, 0.1)=3.07\div{0.1}
Make the calculation.
Use a division method that you are comfortable using.
Simplify the answer if necessary.
The quotient to - \, 3.07 \div(- \, 0.1)=30.7
Example 5: multiplication of 3 rational numbers
Find the product of 3 \cfrac{2}{7} \times\left(- \, 1 \cfrac{1}{4}\right) \times\left(- \, \cfrac{2}{3}\right).
Apply previously learned skills for multiplication and division of fractions, decimals, and integers.
Moving left to right, the rational numbers are (+) \times(-) \times(-). A positive number times a negative number is negative, which becomes (-) \times(-).
A negative number times a negative number is positive. The final result will therefore be a positive number.
3\cfrac{2}{7}\times\left(- \, 1\cfrac{1}{4}\right)\times\left(- \, \cfrac{2}{3}\right)=3\cfrac{2}{7}\times1\cfrac{1}{4}\times\cfrac{2}{3}
Make the calculation.
Change any mixed numbers to improper fractions.
3\cfrac{2}{7}\times1\cfrac{1}{4}\times\cfrac{2}{3}=\cfrac{23}{7}\times\cfrac{5}{4}\times\cfrac{2}{3}
Carry out the calculation by multiplying the first two fractions together, then the result by the third fraction.
\cfrac{23}{7}\times\cfrac{5}{4}\times\cfrac{2}{3}=\cfrac{115}{28}\times\cfrac{2}{3}=\cfrac{230}{84}
Simplify the answer if necessary.
Example 6: mixed operations of 3 rational numbers
Find the value of \left(- \, 2\cfrac{1}{4}\right)\div\cfrac{3}{8}\times{5}.
Apply previously learned skills for multiplication and division of fractions, decimals, and integers.
Moving left to right, the rational numbers are (-) \div(+) \times(+). A negative number divided by a positive number is negative which becomes (-) \times(+) and a negative number times a positive number is negative.
The result will be negative, so leave the rational numbers that contain a negative sign. Assign the negative to the numerator.
Make the calculation.
Change the fractions to improper fractions.
\left(- \, 2\cfrac{1}{4}\right)\div\cfrac{3}{8}\times{5}=\left(\cfrac{- \, 9}{4}\right) \div\cfrac{3}{8}\times{5}
Change the second fraction to its reciprocal and then multiply the first two fractions:
\left(\cfrac{- \, 9}{4}\right)\times\cfrac{8}{3}\times{5}=\cfrac{- \, 9\times{8}}{4\times{3}}\times{5}=\cfrac{- \, 72}{12}\times{5}=- \, 6\times{5}=- \, 30
Simplify the answer if necessary.
Teaching tips for multiplying and dividing rational numbers
- Infuse digital platforms with embedded tutorial videos to help struggling students review strategies.
- Instead of using worksheets to practice skills, incorporate game playing such as collaborative scavenger hunts, learning centers with manipulatives, and digital math games.
Easy mistakes to make
- Mixing up integer rules
For example, applying the integer rules for addition and subtraction to multiplication and division.
\left(- \, \cfrac{2}{3}\right)\times\left(- \, \cfrac{1}{8}\right)≠ \left(- \, \cfrac{1}{12}\right)
When multiplying rational numbers, if the signs are different the product is positive.
\left(- \, \cfrac{2}{3}\right)\times\left(- \, \cfrac{1}{8}\right)=\left(\cfrac{1}{12}\right)
- Confusing the division strategy of fractions
For example, when dividing rational numbers, not remembering to multiply by the reciprocal.
\left(\cfrac{11}{9}\right)\div\left(- \, \cfrac{1}{5}\right)≠ \left(- \, \cfrac{11}{45}\right)
When dividing, take the reciprocal of the second number and multiply it to the first number.
\begin{aligned}\left(\cfrac{11}{9}\right)\div\left(- \, \cfrac{1}{5}\right)&= \\\\ \left(\cfrac{11}{9}\right)\times\left(- \, \cfrac{5}{1}\right)&=\left(- \, \cfrac{55}{9}\right)=\left(- \, 6\cfrac{1}{9}\right) \end{aligned}
Related multiplication and division lesson
Multiplying and dividing rational numbers practice questions
1. Find the product of \left(- \, 3\cfrac{1}{3}\right)\times\left(- \, 7\cfrac{1}{9}\right).
When multiplying rational numbers, apply integer rules and rules for multiplying fractions.
Both rational numbers are negative so the product is negative.
Also, change the mixed numbers to improper fractions before multiplying.
\begin{aligned}&\left(- \, 3\cfrac{1}{3}\right)\times\left(- \, 7\cfrac{1}{9}\right) \\\\ &\left(- \, \cfrac{10}{3}\right)\times\left(- \, \cfrac{64}{9}\right)=\cfrac{640}{27}=24\cfrac{2}{27} \end{aligned}
2. Find the quotient of \left(- \, 8 \cfrac{2}{3}\right) \div\left(2 \cfrac{2}{3}\right).
When dividing rational numbers, apply integer rules and rules for dividing fractions.
The signs of the rational numbers are different so the quotient is negative.
Also, change the mixed numbers to improper fractions and find the reciprocal of the second number.
Then rewrite the problem to be a multiplication.
\begin{aligned}&\left(- \, 8\cfrac{2}{3}\right)\div\left(2\cfrac{2}{3}\right) \\\\ &\left(\cfrac{- \, 26}{3}\right)\div\left(\cfrac{8}{3}\right) \\\\ &\left(\cfrac{- \, 26}{3}\right)\times\left(\cfrac{3}{8}\right)=\cfrac{- \, 78}{24}=\cfrac{- \, 13}{4}=- \, 3\cfrac{1}{4} \end{aligned}
3. Find the product of \left(- \, \cfrac{5}{9}\right)\left(1\cfrac{1}{6}\right)\left(- \, 2\cfrac{2}{3}\right).
Reading the problem from left to right, the signs of the numbers are (-) \times(+) \times(-).
Since the first two numbers have different signs, the product is negative which means the problem becomes, (-) \times(-).
Now, since both numbers are negative, the final product is positive.
Also, change the rational numbers to improper fractions.
\begin{aligned}&\left(- \, \cfrac{5}{9}\right)\left(1\cfrac{1}{6}\right)\left(- \, 2\cfrac{2}{3}\right)=\left(- \, \cfrac{5}{9}\right)\left(\cfrac{7}{6}\right)\left(- \, \cfrac{8}{3}\right) \\\\ &=\cfrac{- \, 5\times{7}}{9\times{6}}\times\left(\cfrac{- \, 8}{3}\right) \\\\ &=\left(\cfrac{- \, 35}{54}\right)\times\left(\cfrac{- \, 8}{3}\right) \\\\ &=\cfrac{- \, 35\times{- \, 8}}{54\times{3}} \\&=\cfrac{280}{162} \\\\ &=\cfrac{140}{81}=1\cfrac{59}{81} \end{aligned}
4. Solve the problem \left(- \, 2\cfrac{2}{11}\right)\div\left(- \, 4\cfrac{1}{5}\right)\times\left(- \, 1\cfrac{3}{4}\right).
Reading the problem from left to right, the signs of the numbers are (-) \div(-) \times(-).
Since the first two numbers have the same signs, the quotient is positive which means the problem becomes, (+) \times(-).
Since the numbers now have different signs, the final answer is negative.
Also, change the rational numbers to improper fractions and find the reciprocal of the second number.
\begin{aligned}&\left(- \, 2\cfrac{2}{11}\right)\div\left(- \, 4\cfrac{1}{5}\right)\times\left(- \, 1\cfrac{3}{4}\right) \\\\ &\left(\cfrac{- \, 24}{11}\right)\div\left(\cfrac{- \, 21}{5}\right)\times\left(\cfrac{- \, 7}{4}\right) \\\\ &\left(\cfrac{- \, 24}{11}\right)\times\left(\cfrac{- \, 5}{21}\right)\times\left(\cfrac{- \, 7}{4}\right)=- \, \cfrac{10}{11}\end{aligned}
5. Find the quotient \left(- \, \cfrac{7}{4}\right)\div\left(- \, \cfrac{3}{8}\right)\div\left(\cfrac{7}{30}\right).
Reading the problem from left to right, the signs of the numbers are (-) \div(-) \div(+).
Since the first two numbers have the same signs, the quotient is positive which means the problem becomes (+) \div(+).
Since the numbers still have the same signs, the final quotient is positive.
Also, be sure to rewrite as multiplication after finding the reciprocals.
\begin{aligned}&\left(- \, \cfrac{7}{4}\right)\div\left(- \, \cfrac{3}{8}\right)\div\left(\cfrac{7}{30}\right) \\\\ &=\cfrac{7}{4}\div\cfrac{3}{8}\div\cfrac{7}{30} \\\\ &=- \, \cfrac{7}{4}\times- \, \cfrac{8}{3}\div\cfrac{7}{30} \\\\ &=\cfrac{56}{12}\div\cfrac{7}{30} \\\\ &=\cfrac{14}{3}\div\cfrac{7}{30} \\\\ &=\cfrac{14}{3}\times\cfrac{30}{7}=\cfrac{20}{1}=20 \end{aligned}
6. Find the product of (- \, 0.9)\times\left(- \, 3\cfrac{2}{9}\right).
In order to multiply, either change both rational numbers to fractions or to decimals.
In this case, it might be easier to change them both to fractions.
\begin{aligned}&- \, 0.9=- \, \cfrac{9}{10} \\\\ &\left(- \, \cfrac{9}{10}\right)\times\left(- \, 3\cfrac{2}{9}\right) \end{aligned}
Also, change the mixed number to an improper fraction.
\left(- \, \cfrac{9}{10}\right) \times\left(- \, \cfrac{29}{9}\right)
Since both rational numbers have the same sign, the product is positive.
\begin{aligned}&\left(- \, \cfrac{9}{10}\right)\times\left(- \, \cfrac{29}{9}\right)=\cfrac{29}{10} \\\\ &\cfrac{29}{10}=2.9 \end{aligned}
Multiplying and dividing rational numbers FAQs
Yes, convert the rational numbers to be either fractions or decimals before multiplying or dividing. This will make the problem easier to solve.
Yes, the rules for integers apply to rational numbers.
The rules for multiplying and dividing rational numbers are the same rules for multiplying and dividing rational expressions. So, you will apply those rules in an algebra class when multiplying and dividing rational expressions.
A complex fraction is simply a fraction divided by another fraction.
For example, \cfrac{\frac{- \, 2}{7}}{\frac{3}{5}}=- \, \cfrac{2}{7} \div \cfrac{3}{5}, you would solve this like any other rational number division problem.
The next lessons are
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