137e. Multiplying and Dividing Algebraic Fractions

Learning Intentions

• To understand that the rules of multiplying and dividing fractions extend to algebraic fractions
• multiply algebraic fractions and Simplify the result
• divide algebraic fractions and simplify the result

Pre-requisite Summary

• Know that an algebraic fraction is a fraction containing pronumerals in the numerator, denominator, or both
• Understand that multiplying fractions involves multiplying numerators together and denominators together
• Understand that dividing fractions involves multiplying by the reciprocal
• Be able to simplify numerical fractions
• Be able to simplify algebraic expressions such as
• Know that common factors can be cancelled only when they are factors of the whole numerator and denominator

Worked Examples

Worked Example 1

Use the same rule as numerical fractions to multiply:

a)

b)

Worked Example 2

Multiply the algebraic fractions and simplify:

a)

b)

c)

Worked Example 3

Multiply the algebraic fractions and simplify:

a)

b)

c)

Worked Example 4

Use the reciprocal to divide:

a)

b)

Worked Example 5

Divide the algebraic fractions and simplify:

a)

b)

c)

Worked Example 6

Divide the algebraic fractions and simplify:

a)

b)

c)

Problems

Problem 1

Use the same rule as numerical fractions to multiply:

a)

b)

Problem 2

Multiply the algebraic fractions and simplify:

a)

b)

c)

Problem 3

Multiply the algebraic fractions and simplify:

a)

b)

c)

Problem 4

Use the reciprocal to divide:

a)

b)

Problem 5

Divide the algebraic fractions and simplify:

a)

b)

c)

Problem 6

Divide the algebraic fractions and simplify:

a)

b)

c)

Exercises

Understanding and Fluency

Exercise 1.

Complete each statement:

a) To multiply fractions, multiply the ______ and multiply the ______

b) To divide by a fraction, multiply by its ______

c) The same multiplication and division rules for numerical fractions also Apply to ______ fractions

Exercise 2.

Multiply and simplify:

a)

b)

c)

Exercise 3.

Multiply and simplify:

a)

b)

c)

Exercise 4.

Multiply and simplify:

a)

b)

c)

Exercise 5.

Divide and simplify:

a)

b)

c)

Exercise 6.

Divide and simplify:

a)

b)

c)

Exercise 7.

Simplify each result:

a)

b)

c)

d)

Exercise 8.

Decide whether to multiply directly or use a reciprocal, then simplify:

a)

b)

c)

d)

Reasoning

Exercise 9.

Explain why can be multiplied in the same way as .

Exercise 10.

A student says that to divide algebraic fractions, you divide the numerators and divide the denominators separately. Explain the mistake.

Exercise 11.

Noah says that cannot be simplified because it contains pronumerals. Is he correct? Explain.

Exercise 12.

Explain why dividing by is the same as multiplying by .

Exercise 13.

A student simplifies by cancelling the . Describe the error.

Problem-solving

Exercise 14.

A formula contains the product . Simplify the expression.

Exercise 15.

A scale factor in a design is written as . Simplify the scale factor.

Exercise 16.

A student combines two rates Use . Write the simplified result.

Exercise 17.

A science formula uses . Simplify the expression.

Exercise 18.

A pattern rule is changed by multiplying by . Simplify the result.

Exercise 19.

A quantity is adjusted by dividing by . Simplify the final expression.

Potential Misunderstandings

• Students may think algebraic fractions follow different multiplication and division rules from numerical fractions
• Students may forget to multiply by the reciprocal when dividing
• Students may cancel terms that are not common factors of the whole numerator and denominator
• Students may cancel across addition, for example in expressions like
• Students may multiply the numerators correctly but forget to multiply the denominators
• Students may simplify the numerical part but not the pronumeral part, or vice versa
• Students may think pronumerals cannot be cancelled
• Students may reverse the wrong fraction when dividing
• Students may make sign errors when simplifying

Next: 138. Expanding Brackets and Simplifying Expressions