028. Dividing Fractions Using Reciprocals

Learning Intentions

• find 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 $1$
• 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 $1$

Worked Examples

Worked Example 1

Find the reciprocal of each number:
a) ${\displaystyle \frac{3}{5}}$
b) ${\displaystyle \frac{7}{2}}$
c) $2{\displaystyle \frac{1}{4}}$

Worked Example 2

Use reciprocals to divide:
a) ${\displaystyle \frac{1}{2}}÷{\displaystyle \frac{3}{4}}$
b) ${\displaystyle \frac{2}{3}}÷{\displaystyle \frac{5}{6}}$

Worked Example 3

Divide and simplify:
a) ${\displaystyle \frac{3}{4}}÷2$
b) $3÷{\displaystyle \frac{2}{5}}$

Worked Example 4

Convert mixed numerals to improper fractions, then divide:
a) $1{\displaystyle \frac{1}{2}}÷{\displaystyle \frac{3}{4}}$
b) ${\displaystyle \frac{5}{6}}÷2{\displaystyle \frac{1}{3}}$

Worked Example 5

Divide mixed numerals and give the answer in simplest form:
a) $2{\displaystyle \frac{1}{4}}÷1{\displaystyle \frac{1}{2}}$
b) $3{\displaystyle \frac{1}{3}}÷1{\displaystyle \frac{1}{9}}$

Problems

Problem 1

Find the reciprocal of each number:
a) ${\displaystyle \frac{4}{7}}$
b) ${\displaystyle \frac{9}{5}}$
c) $1{\displaystyle \frac{2}{3}}$

Problem 2

Use reciprocals to divide:
a) ${\displaystyle \frac{2}{5}}÷{\displaystyle \frac{3}{10}}$
b) ${\displaystyle \frac{4}{9}}÷{\displaystyle \frac{2}{3}}$

Problem 3

Divide and simplify:
a) ${\displaystyle \frac{5}{6}}÷3$
b) $4÷{\displaystyle \frac{3}{7}}$

Problem 4

Convert mixed numerals to improper fractions, then divide:
a) $2{\displaystyle \frac{1}{2}}÷{\displaystyle \frac{5}{6}}$
b) ${\displaystyle \frac{7}{8}}÷1{\displaystyle \frac{3}{4}}$

Problem 5

Divide mixed numerals and give the answer in simplest form:
a) $3{\displaystyle \frac{1}{5}}÷1{\displaystyle \frac{3}{5}}$
b) $4{\displaystyle \frac{1}{2}}÷2{\displaystyle \frac{1}{4}}$

Exercises

Understanding and Fluency

1. Find the reciprocal of each number:
   a) ${\displaystyle \frac{1}{3}}$
   b) ${\displaystyle \frac{5}{8}}$
   c) ${\displaystyle \frac{9}{4}}$

2. Find the reciprocal of each number:
   a) $2$
   b) $3{\displaystyle \frac{1}{2}}$
   c) $1{\displaystyle \frac{3}{4}}$

3. Divide by multiplying by the reciprocal:
   a) ${\displaystyle \frac{1}{2}}÷{\displaystyle \frac{1}{4}}$
   b) ${\displaystyle \frac{3}{5}}÷{\displaystyle \frac{2}{3}}$
   c) ${\displaystyle \frac{7}{8}}÷{\displaystyle \frac{7}{16}}$

4. Divide by multiplying by the reciprocal:
   a) ${\displaystyle \frac{2}{3}}÷{\displaystyle \frac{4}{5}}$
   b) ${\displaystyle \frac{5}{6}}÷{\displaystyle \frac{1}{2}}$
   c) ${\displaystyle \frac{9}{10}}÷{\displaystyle \frac{3}{4}}$

5. Divide fractions and whole numbers:
   a) ${\displaystyle \frac{3}{4}}÷2$
   b) $5÷{\displaystyle \frac{1}{2}}$
   c) ${\displaystyle \frac{7}{9}}÷3$

6. Divide fractions and whole numbers:
   a) $6÷{\displaystyle \frac{2}{3}}$
   b) ${\displaystyle \frac{5}{8}}÷4$
   c) $2÷{\displaystyle \frac{5}{6}}$

7. Convert mixed numerals, then divide:
   a) $1{\displaystyle \frac{1}{2}}÷{\displaystyle \frac{1}{3}}$
   b) ${\displaystyle \frac{3}{4}}÷1{\displaystyle \frac{1}{2}}$
   c) $2{\displaystyle \frac{1}{3}}÷{\displaystyle \frac{7}{8}}$

8. Convert mixed numerals, then divide:
   a) $3{\displaystyle \frac{1}{4}}÷1{\displaystyle \frac{1}{2}}$
   b) $2{\displaystyle \frac{2}{5}}÷1{\displaystyle \frac{1}{5}}$
   c) $4{\displaystyle \frac{1}{2}}÷{\displaystyle \frac{3}{5}}$

Reasoning

9. Explain why the reciprocal of ${\displaystyle \frac{4}{7}}$ is ${\displaystyle \frac{7}{4}}$.

10. A student says ${\displaystyle \frac{2}{3}}÷{\displaystyle \frac{5}{6}}={\displaystyle \frac{2}{3}}÷{\displaystyle \frac{6}{5}}$. Explain the mistake.

11. Explain why dividing by ${\displaystyle \frac{1}{2}}$ gives a larger answer than the starting number.

12. A student converts $2{\displaystyle \frac{1}{4}}$ to ${\displaystyle \frac{2}{4}}$. Explain why this is incorrect.

Problem-solving

13. A recipe uses ${\displaystyle \frac{3}{4}}$ cup of flour per batch. How many batches can be made from $3$ cups of flour?

14. A rope is $2{\displaystyle \frac{1}{2}}$ m long. Pieces of length ${\displaystyle \frac{5}{8}}$ m are cut from it. How many such pieces fit into the rope?

15. A tank contains $4{\displaystyle \frac{1}{2}}$ L of water. Cups of size ${\displaystyle \frac{3}{4}}$ L are filled from it. How many cups can be filled?

16. A ribbon of length ${\displaystyle \frac{7}{8}}$ m is cut into pieces each of length ${\displaystyle \frac{1}{8}}$ m. How many pieces are made?

17. A container has $3{\displaystyle \frac{1}{3}}$ kg of rice. Each bag holds ${\displaystyle \frac{2}{3}}$ kg. How many bags can be filled?

18. A student walks $2$ km and each lap is ${\displaystyle \frac{2}{5}}$ 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 $1$
• Students may simplify incorrectly before multiplying
• Students may think dividing always makes a number smaller, even when dividing by a fraction less than $1$
• Students may leave answers unsimplified or in improper form when a mixed numeral is preferred

Next: 029. Fractions and Percentages