110r. Operations with Fractions and Mixed Numerals

Learning Intentions

• add and subtract fractions by first finding a lowest common multiple of the denominators
• multiply and divide fractions
• To understand that an integer can be written as a fraction with $1$ as the denominator
• perform the four operations on mixed numerals, converting to improper fractions as required

Pre-requisite Summary

• Know that equivalent fractions have the same numerical value
• Be able to find the lowest common multiple of two numbers
• Be able to simplify fractions fully
• Understand the difference between proper fractions, improper fractions and mixed numerals
• Know how to convert between mixed numerals and improper fractions
• Recall that multiplication and division facts can help simplify fraction calculations
• Understand that division by a fraction is related to multiplying by its reciprocal
• Know that any integer can be written as a fraction over $1$

Worked Examples

Worked Example 1

Add or subtract by first finding a lowest common multiple of the denominators:
a) ${\displaystyle \frac{1}{3}}+{\displaystyle \frac{1}{4}}$
b) ${\displaystyle \frac{5}{6}}-{\displaystyle \frac{1}{8}}$

Worked Example 2

Add or subtract by first finding a lowest common multiple of the denominators:
a) ${\displaystyle \frac{3}{10}}+{\displaystyle \frac{2}{15}}$
b) ${\displaystyle \frac{7}{12}}-{\displaystyle \frac{1}{18}}$

Worked Example 3

Multiply or divide fractions:
a) ${\displaystyle \frac{2}{3}}\times {\displaystyle \frac{5}{7}}$
b) ${\displaystyle \frac{4}{9}}÷{\displaystyle \frac{2}{3}}$
c) ${\displaystyle \frac{3}{8}}\times {\displaystyle \frac{4}{5}}$

Worked Example 4

Write each integer as a fraction and evaluate:
a) $3+{\displaystyle \frac{2}{5}}$
b) $7÷{\displaystyle \frac{1}{2}}$
c) $4\times {\displaystyle \frac{3}{8}}$

Worked Example 5

Convert to improper fractions, then evaluate:
a) $1{\displaystyle \frac{1}{2}}+2{\displaystyle \frac{1}{3}}$
b) $3{\displaystyle \frac{1}{4}}-1{\displaystyle \frac{2}{5}}$

Worked Example 6

Convert to improper fractions, then evaluate:
a) $2{\displaystyle \frac{1}{3}}\times 1{\displaystyle \frac{1}{2}}$
b) $4{\displaystyle \frac{1}{5}}÷2{\displaystyle \frac{1}{10}}$

Problems

Problem 1

Add or subtract by first finding a lowest common multiple of the denominators:
a) ${\displaystyle \frac{1}{2}}+{\displaystyle \frac{1}{5}}$
b) ${\displaystyle \frac{7}{8}}-{\displaystyle \frac{1}{6}}$

Problem 2

Add or subtract by first finding a lowest common multiple of the denominators:
a) ${\displaystyle \frac{1}{6}}+{\displaystyle \frac{3}{14}}$
b) ${\displaystyle \frac{11}{15}}-{\displaystyle \frac{1}{10}}$

Problem 3

Multiply or divide fractions:
a) ${\displaystyle \frac{3}{4}}\times {\displaystyle \frac{2}{5}}$
b) ${\displaystyle \frac{5}{6}}÷{\displaystyle \frac{1}{3}}$
c) ${\displaystyle \frac{7}{9}}\times {\displaystyle \frac{3}{7}}$

Problem 4

Write each integer as a fraction and evaluate:
a) $5+{\displaystyle \frac{1}{4}}$
b) $6÷{\displaystyle \frac{3}{4}}$
c) $2\times {\displaystyle \frac{5}{6}}$

Problem 5

Convert to improper fractions, then evaluate:
a) $2{\displaystyle \frac{1}{4}}+1{\displaystyle \frac{2}{3}}$
b) $4{\displaystyle \frac{1}{6}}-2{\displaystyle \frac{3}{8}}$

Problem 6

Convert to improper fractions, then evaluate:
a) $1{\displaystyle \frac{3}{5}}\times 2{\displaystyle \frac{1}{4}}$
b) $3{\displaystyle \frac{1}{2}}÷1{\displaystyle \frac{3}{4}}$

Exercises

Understanding and Fluency

1. Add or subtract the fractions:
   a) ${\displaystyle \frac{1}{4}}+{\displaystyle \frac{1}{6}}$
   b) ${\displaystyle \frac{5}{8}}-{\displaystyle \frac{1}{3}}$
   c) ${\displaystyle \frac{2}{5}}+{\displaystyle \frac{3}{10}}$

2. Add or subtract the fractions:
   a) ${\displaystyle \frac{3}{7}}+{\displaystyle \frac{1}{14}}$
   b) ${\displaystyle \frac{11}{12}}-{\displaystyle \frac{1}{8}}$
   c) ${\displaystyle \frac{5}{9}}-{\displaystyle \frac{1}{6}}$

3. Multiply the fractions:
   a) ${\displaystyle \frac{2}{5}}\times {\displaystyle \frac{3}{4}}$
   b) ${\displaystyle \frac{7}{8}}\times {\displaystyle \frac{2}{7}}$
   c) ${\displaystyle \frac{5}{6}}\times {\displaystyle \frac{3}{10}}$

4. Divide the fractions:
   a) ${\displaystyle \frac{3}{4}}÷{\displaystyle \frac{1}{2}}$
   b) ${\displaystyle \frac{5}{9}}÷{\displaystyle \frac{5}{12}}$
   c) ${\displaystyle \frac{7}{10}}÷{\displaystyle \frac{7}{20}}$

5. Write each integer as a fraction with denominator $1$, then evaluate:
   a) $2+{\displaystyle \frac{3}{7}}$
   b) $5\times {\displaystyle \frac{2}{3}}$
   c) $8÷{\displaystyle \frac{4}{5}}$

6. Convert each mixed numeral to an improper fraction:
   a) $1{\displaystyle \frac{2}{3}}$
   b) $3{\displaystyle \frac{1}{4}}$
   c) $5{\displaystyle \frac{5}{6}}$

7. Perform the operation on mixed numerals:
   a) $1{\displaystyle \frac{1}{2}}+2{\displaystyle \frac{1}{4}}$
   b) $3{\displaystyle \frac{2}{3}}-1{\displaystyle \frac{1}{6}}$
   c) $2{\displaystyle \frac{1}{5}}\times 1{\displaystyle \frac{1}{2}}$

8. Perform the operation on mixed numerals:
   a) $4{\displaystyle \frac{1}{2}}÷1{\displaystyle \frac{1}{2}}$
   b) $2{\displaystyle \frac{3}{4}}+1{\displaystyle \frac{5}{8}}$
   c) $5{\displaystyle \frac{1}{3}}-2{\displaystyle \frac{1}{2}}$

Reasoning

9. Explain why ${\displaystyle \frac{1}{3}}+{\displaystyle \frac{1}{4}}$ cannot be found by adding the denominators.

10. A student says that ${\displaystyle \frac{2}{5}}÷{\displaystyle \frac{3}{4}}={\displaystyle \frac{6}{20}}$. Explain the mistake.

11. Explain why the integer $6$ can also be written as ${\displaystyle \frac{6}{1}}$.

12. A student converts $2{\displaystyle \frac{1}{3}}$ to ${\displaystyle \frac{3}{3}}+{\displaystyle \frac{1}{3}}={\displaystyle \frac{4}{3}}$. Describe the error.

13. Noah says that to divide by a fraction, you divide both numerators and both denominators. Is he correct? Explain.

Problem-solving

14. A recipe uses ${\displaystyle \frac{1}{2}}$ cup of milk in one step and ${\displaystyle \frac{3}{4}}$ cup in another step. How much milk is used altogether?

15. A ribbon of length ${\displaystyle \frac{7}{8}}$ m is cut and ${\displaystyle \frac{1}{3}}$ m is used. How much ribbon is left?

16. A container holds $2{\displaystyle \frac{1}{2}}$ L of juice. It is shared equally among $5$ people. How much juice does each person get?

17. A student walks $1{\displaystyle \frac{3}{4}}$ km in the morning and $2{\displaystyle \frac{1}{2}}$ km in the afternoon. How far does the student walk altogether?

18. A baker makes $3{\displaystyle \frac{1}{3}}$ trays of muffins, and each tray holds ${\displaystyle \frac{3}{4}}$ of a dozen muffins. How many dozens of muffins are made altogether?

19. A rope $4{\displaystyle \frac{1}{2}}$ m long is cut into pieces of length ${\displaystyle \frac{3}{4}}$ m. How many pieces can be cut?

Potential Misunderstandings

• Students may add or subtract denominators when working with fractions
• Students may forget to find a common denominator before adding or subtracting unlike fractions
• Students may use a common denominator that is not a common multiple
• Students may forget to simplify the final answer
• Students may multiply fractions correctly but not simplify before or after multiplying
• Students may confuse dividing by a fraction with dividing the numerators and denominators separately
• Students may forget that dividing by a fraction means multiplying by its reciprocal
• Students may not recognise that an integer such as $4$ can be written as ${\displaystyle \frac{4}{1}}$
• Students may convert mixed numerals to improper fractions incorrectly
• Students may forget to convert mixed numerals before multiplying or dividing
• Students may convert correctly but then make errors with the fraction operation
• Students may leave answers as improper fractions when a mixed numeral is expected, or vice versa