028. Dividing Fractions Using Reciprocals

Learning Intentions

• Solve the reciprocal of a fraction or a mixed numeral
• To understand that dividing fractions can be done by multiplying by a reciprocal
• divide fractions, mixed numerals and/or whole numbers, giving an answer in simplest form

Pre-requisite Summary

• Understand that a fraction can represent part of a whole or division
• Know that a whole number can be written as a fraction with denominator
• Be able to convert a mixed numeral to an improper fraction
• Be able to multiply fractions and Simplify answers
• Understand factors and common factors for simplifying
• Know that the reciprocal of a number is the multiplicative inverse
• Understand that multiplying a number by its reciprocal gives

Worked Examples

Worked Example 1

Find the reciprocal of each number:

a)

b)

c)

Worked Example 2

Use reciprocals to divide:

a)

b)

Worked Example 3

Divide and simplify:

a)

b)

Worked Example 4

Convert mixed numerals to improper fractions, then divide:

a)

b)

Worked Example 5

Divide mixed numerals and give the answer in simplest form:

a)

b)

Problems

Problem 1

Find the reciprocal of each number:

a)

b)

c)

Problem 2

Use reciprocals to divide:

a)

b)

Problem 3

Divide and simplify:

a)

b)

Problem 4

Convert mixed numerals to improper fractions, then divide:

a)

b)

Problem 5

Divide mixed numerals and give the answer in simplest form:

a)

b)

Exercises

Understanding and Fluency

Exercise 1.

Find the reciprocal of each number:

a)

b)

c)

Exercise 2.

Find the reciprocal of each number:

a)

b)

c)

Exercise 3.

Divide by multiplying by the reciprocal:

a)

b)

c)

Exercise 4.

Divide by multiplying by the reciprocal:

a)

b)

c)

Exercise 5.

Divide fractions and whole numbers:

a)

b)

c)

Exercise 6.

Divide fractions and whole numbers:

a)

b)

c)

Exercise 7.

Convert mixed numerals, then divide:

a)

b)

c)

Exercise 8.

Convert mixed numerals, then divide:

a)

b)

c)

Reasoning

Exercise 9.

Explain why the reciprocal of is .

Exercise 10.

A student says . Explain the mistake.

Exercise 11.

Explain why dividing by gives a larger answer than the starting number.

Exercise 12.

A student converts to . Explain why this is incorrect.

Problem-solving

Exercise 13.

A recipe uses cup of flour per batch. How many batches can be made from cups of flour?

Exercise 14.

A rope is m long. Pieces of length m are cut from it. How many such pieces fit into the rope?

Exercise 15.

A tank contains L of water. Cups of size L are filled from it. How many cups can be filled?

Exercise 16.

A ribbon of length m is cut into pieces each of length m. How many pieces are made?

Exercise 17.

A container has kg of rice. Each bag holds kg. How many bags can be filled?

Exercise 18.

A student walks km and each lap is km. How many laps does the student walk?

Potential Misunderstandings

• Students may think the reciprocal is found by subtracting or inverting only one part of the fraction
• Students may forget to convert mixed numerals to improper fractions before finding the reciprocal
• Students may think dividing fractions means dividing numerators and denominators separately
• Students may multiply by the original divisor instead of its reciprocal
• Students may forget that a whole number can be written as a fraction with denominator
• Students may simplify incorrectly before multiplying
• Students may think dividing always makes a number smaller, even when dividing by a fraction less than
• Students may leave answers unsimplified or in improper form when a mixed numeral is preferred

Next: 029. Fractions and Percentages