How to multiply and divide rational numbers

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Use this seventh-grade math worksheet to provide students with practice multiplying and dividing rational numbers, including positive and negative fractions, decimals, and mixed numbers. Students will need to be able to multiply and divide positive and negative integers and convert between decimals, fractions, and mixed numbers before completing this worksheet. This practice worksheet challenges learners by progressing from multiplying or dividing rational numbers that have the same form, to multiplying or dividing rational numbers that have different forms. Check out the Adding and Subtracting Rational Numbers worksheet for more practice on operations with rational numbers!

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Multiplication :

To multiply two rational numbers 'a/b' and 'c/d', multiply the numerators and denominators separately as shown below. 

(a/b) ⋅ (c/d)  =  (a ⋅ c) / (b ⋅ d)

Simplify the product to its lowest form.

Division : 

To divide a rational number 'a/b' by another rational number 'c/d', multiply the first rational number 'a/b' by  the multiplicative inverse of the second rational number 'c/d'.

Multiplicative inverse of 'c/d' is 'd/c'. 

(a/b) ÷ (c/d)  =  (a/b) ⋅ (d/c)  =  (a ⋅ d) / (b ⋅ c)

Note :

In case we have mixed fractions, first we have to convert them to improper fractions and do the multiplication and division as explained above.

Example 1 :

Multiply 2/3 and 5/7.

Solution :

=  (2/3) ⋅ (5/7)

Multiply the numerators and denominators. 

=  (2 ⋅ 5) / (3 ⋅ 7)

=  10/21

Example 2 :

Evaluate : 

(4/-11) ⋅ (-22/8)

Solution :

=  (4/-11) ⋅ (-22/8)

Before we multiply numerators and denominators, we can simplify as shown below.

  =  (-2) / (-2)

=  1

Example 3 :

Simplify :

(2/5) ÷ (7/9)

Solution :

  =  (2/5) ÷ (7/9)

To divide by 7/9, multiply by 9/7.

=  (2/5) ⋅ (9/7)

Multiply the numerators and denominators. 

=  (2 ⋅ 9) / (5 ⋅ 7)

=  18/35

Example 4 :

Simplify : 

(-4/9) ÷ (9/-4)

Solution :

  =  (-4/9) ÷ (9/-4)

To divide by 9/-4, multiply by -4/9.

=  (-4/9) ⋅ (-4/9)

Multiply the numerators and denominators. 

=  [(-4)(-4)] / (9 ⋅ 9)

=  16/81

Example 5 :

Multiply 2¼ and 3½.

Solution :

=  (2¼) ⋅ (3½)

Converting the mixed fractions to improper fractions.

=  (9/4) ⋅ (7/2)

=  (9 ⋅ 7) / (4 ⋅ 2)

=  63/8

=  7⅞

Example 6 :

Simplify : 

-9¾ ÷  1⅞

Solution :

=  -9¾ ÷  1⅞

Converting the mixed numbers to improper fractions

=  (-39/4) ÷ (15/8)

To divide by 15/8, multiply by 8/15.

=  (-39/4) ⋅ (8/15)

Simplify 39 and 15 using 3 times table, 4 and 8 using 4 times table. 

=  (-13/1) ⋅ (2/5)

=  (-13 ⋅ 2) / (1 ⋅ 5)

=  -26/5

=  -5⅕

Example 7 :

Find the product of -11/2 and 4/5.

Solution :

  =  (-11/2) ⋅ (4/5)

Multiplying the numerators and denominators.

=  (-11 ⋅ 4) / (2 ⋅ 5)

=  -44/10

=  -22/5

Converting the improper fraction into mixed fraction.

  =  -4⅖

Example 8 :

What is two-fifth of 7/18 ?

Solution :

Two-fifth of 5/18 :

=  (2/5) ⋅ (7/18)

Simplify 2 and 18 using 2 times table.

=  (1/5) ⋅ (7/9)

Multiply numerators and denominators. 

=  (1 ⋅ 7) / (5 ⋅ 9)

=  7/45

Example 9 :

Lily wants to share three-fourth of a pizza equally to 4 of her friends. How much pizza will each friend get ?

Solution :

Amount of pizza each friend will get :

=  (3/4) ÷ 4

=  (3/4) ÷ (4/1)

To divide by 4/1, multiply by 1/4.

=  (3/4) ⋅ (1/4)

Multiply numerators and denominators. 

=  (3 ⋅ 1) / (4 ⋅ 4)

=  3/16

Each friend will get 3/16 of a pizza.  

Example 10 :

Last month, John spent of three-fifth of his salary for food. If John's salary is $5000, how much money did he spend for food ?

Solution :

Money spent for food :

=  (3/5) ⋅ 5000

=  (3/5) ⋅ (5000/1)

Simplify 5 and 5000 using 5 times table.

=  (3/1) ⋅ (1000/1)

Multiply numerators and denominators. 

=  (3 ⋅ 1000) / (1 ⋅ 1)

=  3000/1

=  3000

John spent $3000 for food.  

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How will you multiply and divide the rational expression?