026. Subtracting Fractions and Mixed Numerals

Learning Intentions

• To understand that subtracting fractions requires a common denominator
• subtract two fractions by considering their lowest common denominator
• subtract two mixed numerals

Pre-requisite Summary

• Understand that the denominator tells the size of the equal parts
• Understand that fractions can only be subtracted directly when the parts are the same size
• Be able to find equivalent fractions
• Be able to find the lowest common multiple of two denominators
• Be able to convert between improper fractions and mixed numerals
• Be able to simplify fractions where appropriate
• Understand regrouping when subtracting whole numbers and mixed numerals

Worked Examples

Worked Example 1

a) Explain why ${\displaystyle \frac{3}{8}}-{\displaystyle \frac{1}{8}}$ can be subtracted directly.
b) Find ${\displaystyle \frac{3}{8}}-{\displaystyle \frac{1}{8}}$.
c) Explain why ${\displaystyle \frac{3}{8}}-{\displaystyle \frac{1}{4}}$ cannot be subtracted directly.

Worked Example 2

Subtract by using the lowest common denominator:
a) ${\displaystyle \frac{1}{2}}-{\displaystyle \frac{1}{3}}$
b) ${\displaystyle \frac{3}{4}}-{\displaystyle \frac{1}{6}}$

Worked Example 3

Subtract by using the lowest common denominator:
a) ${\displaystyle \frac{4}{5}}-{\displaystyle \frac{1}{2}}$
b) ${\displaystyle \frac{7}{8}}-{\displaystyle \frac{1}{4}}$

Worked Example 4

Subtract two mixed numerals:
a) $3{\displaystyle \frac{1}{2}}-1{\displaystyle \frac{1}{3}}$
b) $5{\displaystyle \frac{3}{4}}-2{\displaystyle \frac{1}{8}}$

Worked Example 5

Subtract two mixed numerals:
a) $4{\displaystyle \frac{2}{5}}-1{\displaystyle \frac{3}{10}}$
b) $6{\displaystyle \frac{5}{6}}-2{\displaystyle \frac{1}{3}}$

Worked Example 6

a) Subtract $5{\displaystyle \frac{1}{4}}-2{\displaystyle \frac{5}{8}}$.
b) Write the answer in simplest form.
c) Explain whether converting to improper fractions first would also work.

Problems

Problem 1

a) Explain why ${\displaystyle \frac{5}{7}}-{\displaystyle \frac{2}{7}}$ can be subtracted directly.
b) Find ${\displaystyle \frac{5}{7}}-{\displaystyle \frac{2}{7}}$.
c) Explain why ${\displaystyle \frac{5}{7}}-{\displaystyle \frac{1}{3}}$ cannot be subtracted directly.

Problem 2

Subtract by using the lowest common denominator:
a) ${\displaystyle \frac{2}{3}}-{\displaystyle \frac{1}{4}}$
b) ${\displaystyle \frac{5}{6}}-{\displaystyle \frac{1}{8}}$

Problem 3

Subtract by using the lowest common denominator:
a) ${\displaystyle \frac{9}{10}}-{\displaystyle \frac{1}{2}}$
b) ${\displaystyle \frac{11}{12}}-{\displaystyle \frac{1}{3}}$

Problem 4

Subtract two mixed numerals:
a) $2{\displaystyle \frac{1}{3}}-1{\displaystyle \frac{1}{4}}$
b) $4{\displaystyle \frac{5}{6}}-2{\displaystyle \frac{1}{2}}$

Problem 5

Subtract two mixed numerals:
a) $5{\displaystyle \frac{2}{5}}-2{\displaystyle \frac{1}{10}}$
b) $7{\displaystyle \frac{1}{4}}-3{\displaystyle \frac{3}{8}}$

Problem 6

a) Subtract $4{\displaystyle \frac{7}{8}}-1{\displaystyle \frac{5}{6}}$.
b) Write the answer in simplest form.
c) Explain whether converting to improper fractions first would also work.

Exercises

Understanding and Fluency

1. Subtract fractions with the same denominator:
   a) ${\displaystyle \frac{4}{9}}-{\displaystyle \frac{1}{9}}$
   b) ${\displaystyle \frac{7}{10}}-{\displaystyle \frac{3}{10}}$
   c) ${\displaystyle \frac{11}{12}}-{\displaystyle \frac{5}{12}}$

2. State the lowest common denominator and then subtract:
   a) ${\displaystyle \frac{3}{4}}-{\displaystyle \frac{1}{2}}$
   b) ${\displaystyle \frac{5}{6}}-{\displaystyle \frac{1}{3}}$
   c) ${\displaystyle \frac{7}{10}}-{\displaystyle \frac{1}{5}}$

3. Subtract by using the lowest common denominator:
   a) ${\displaystyle \frac{2}{3}}-{\displaystyle \frac{1}{6}}$
   b) ${\displaystyle \frac{5}{8}}-{\displaystyle \frac{1}{4}}$
   c) ${\displaystyle \frac{4}{5}}-{\displaystyle \frac{1}{10}}$

4. Subtract by using the lowest common denominator:
   a) ${\displaystyle \frac{7}{12}}-{\displaystyle \frac{1}{3}}$
   b) ${\displaystyle \frac{5}{6}}-{\displaystyle \frac{1}{4}}$
   c) ${\displaystyle \frac{9}{10}}-{\displaystyle \frac{2}{5}}$

5. Subtract two mixed numerals:
   a) $3{\displaystyle \frac{1}{2}}-1{\displaystyle \frac{1}{4}}$
   b) $4{\displaystyle \frac{1}{3}}-2{\displaystyle \frac{1}{6}}$
   c) $5{\displaystyle \frac{3}{5}}-2{\displaystyle \frac{1}{5}}$

6. Subtract two mixed numerals:
   a) $6{\displaystyle \frac{1}{8}}-2{\displaystyle \frac{3}{4}}$
   b) $5{\displaystyle \frac{5}{6}}-1{\displaystyle \frac{1}{3}}$
   c) $4{\displaystyle \frac{7}{10}}-2{\displaystyle \frac{2}{5}}$

7. Subtract and simplify where needed:
   a) ${\displaystyle \frac{8}{9}}-{\displaystyle \frac{2}{9}}$
   b) ${\displaystyle \frac{3}{4}}-{\displaystyle \frac{1}{6}}$
   c) $3{\displaystyle \frac{1}{2}}-1{\displaystyle \frac{1}{2}}$

8. Subtract and write the answer in simplest form:
   a) ${\displaystyle \frac{7}{8}}-{\displaystyle \frac{1}{3}}$
   b) ${\displaystyle \frac{11}{12}}-{\displaystyle \frac{3}{4}}$
   c) $6{\displaystyle \frac{2}{3}}-2{\displaystyle \frac{5}{6}}$

Reasoning

9. Explain why fractions must have a common denominator before they can be subtracted.

10. A student says ${\displaystyle \frac{3}{4}}-{\displaystyle \frac{1}{2}}={\displaystyle \frac{2}{2}}$. Explain the mistake.

11. Explain why the denominator usually stays the same after subtracting fractions with a common denominator.

12. A student finds ${\displaystyle \frac{5}{6}}-{\displaystyle \frac{1}{3}}={\displaystyle \frac{4}{3}}$. Explain why this is incorrect.

Problem-solving

13. Mia used ${\displaystyle \frac{3}{4}}$ m of ribbon from a piece that was $1$ m long. How much ribbon is left?

14. A tank contained ${\displaystyle \frac{5}{6}}$ L of water. ${\displaystyle \frac{1}{3}}$ L was poured out. How much water remains?

15. A rope is $4{\displaystyle \frac{1}{2}}$ m long. A piece of $1{\displaystyle \frac{3}{4}}$ m is cut off. What length remains?

16. A container held $6{\displaystyle \frac{1}{5}}$ kg of rice. $2{\displaystyle \frac{7}{10}}$ kg was used. How much rice remains?

17. A student completed ${\displaystyle \frac{7}{8}}$ of a task and still had ${\displaystyle \frac{1}{4}}$ left to revise. How much more had been completed than remained?

18. A ribbon piece of $5{\displaystyle \frac{2}{3}}$ m is shortened by $1{\displaystyle \frac{5}{6}}$ m. What is the new length?

Potential Misunderstandings

• Students may think numerators and denominators can both be subtracted directly
• Students may not recognise that denominators represent the size of the parts
• Students may choose a common denominator that is not a common multiple
• Students may find a common denominator but forget to rename the numerators
• Students may subtract the denominator after converting to equivalent fractions
• Students may use a common denominator correctly but fail to simplify the final answer
• Students may subtract the whole-number parts and fractional parts of mixed numerals without first finding a common denominator for the fractions
• Students may convert mixed numerals to improper fractions incorrectly
• Students may forget to regroup when the fractional part being subtracted is larger than the fractional part in the starting mixed numeral

Next: 027. Multiplying Fractions, Mixed Numerals and Whole Numbers