025. Adding Fractions and Mixed Numerals

Learning Intentions

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

Pre-requisite Summary

• Understand that the denominator tells the number of equal parts in the whole
• Understand that fractions can only be combined 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

Worked Examples

Worked Example 1

a) Explain why ${\displaystyle \frac{1}{4}}+{\displaystyle \frac{2}{4}}$ can be added directly.
b) Find ${\displaystyle \frac{1}{4}}+{\displaystyle \frac{2}{4}}$.
c) Explain why ${\displaystyle \frac{1}{4}}+{\displaystyle \frac{1}{3}}$ cannot be added directly.

Worked Example 2

Add 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

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

Worked Example 4

Add two mixed numerals:
a) $1{\displaystyle \frac{1}{2}}+2{\displaystyle \frac{1}{3}}$
b) $3{\displaystyle \frac{3}{4}}+1{\displaystyle \frac{1}{8}}$

Worked Example 5

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

Worked Example 6

a) Add $2{\displaystyle \frac{3}{4}}+3{\displaystyle \frac{5}{8}}$.
b) Write the answer in simplest form.
c) Explain whether converting to improper fractions first would also work.

Problems

Problem 1

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

Problem 2

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

Problem 3

Add by using the lowest common denominator:
a) ${\displaystyle \frac{3}{10}}+{\displaystyle \frac{1}{2}}$
b) ${\displaystyle \frac{5}{12}}+{\displaystyle \frac{1}{3}}$

Problem 4

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

Problem 5

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

Problem 6

a) Add $1{\displaystyle \frac{7}{8}}+2{\displaystyle \frac{5}{6}}$.
b) Write the answer in simplest form.
c) Explain whether converting to improper fractions first would also work.

Exercises

Understanding and Fluency

1. Add fractions with the same denominator:
   a) ${\displaystyle \frac{1}{5}}+{\displaystyle \frac{2}{5}}$
   b) ${\displaystyle \frac{3}{8}}+{\displaystyle \frac{4}{8}}$
   c) ${\displaystyle \frac{5}{12}}+{\displaystyle \frac{1}{12}}$

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

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

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

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

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

7. Add and simplify where needed:
   a) ${\displaystyle \frac{2}{9}}+{\displaystyle \frac{5}{9}}$
   b) ${\displaystyle \frac{1}{6}}+{\displaystyle \frac{1}{2}}$
   c) $2{\displaystyle \frac{1}{4}}+1{\displaystyle \frac{3}{4}}$

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

Reasoning

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

10. A student says ${\displaystyle \frac{1}{2}}+{\displaystyle \frac{1}{3}}={\displaystyle \frac{2}{5}}$. Explain the mistake.

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

12. A student finds ${\displaystyle \frac{3}{4}}+{\displaystyle \frac{1}{8}}={\displaystyle \frac{4}{12}}$. Explain why this is incorrect.

Problem-solving

13. Mia walked ${\displaystyle \frac{1}{2}}$ km in the morning and ${\displaystyle \frac{1}{4}}$ km in the afternoon. How far did she walk altogether?

14. A recipe uses ${\displaystyle \frac{2}{3}}$ cup of milk and ${\displaystyle \frac{1}{6}}$ cup of cream. How much liquid is used altogether?

15. A rope is $1{\displaystyle \frac{1}{2}}$ m long and another rope is $2{\displaystyle \frac{3}{4}}$ m long. What is their total length?

16. A tank is filled by $3{\displaystyle \frac{1}{5}}$ L from one jug and $2{\displaystyle \frac{7}{10}}$ L from another jug. How much water is added altogether?

17. A student completed ${\displaystyle \frac{5}{8}}$ of a task on Monday and ${\displaystyle \frac{1}{4}}$ on Tuesday. What fraction of the task was completed altogether?

18. A ribbon piece of $2{\displaystyle \frac{2}{3}}$ m is joined to a piece of $1{\displaystyle \frac{5}{6}}$ m. What is the total length?

Potential Misunderstandings

• Students may think numerators and denominators can both be added 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 use a common denominator correctly but fail to simplify the final answer
• Students may add 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 of a mixed numeral sum is greater than $1$

Next: 026. Subtracting Fractions and Mixed Numerals