111. Operations with Negative Fractions

Learning Intentions

• To understand that the techniques for adding, subtracting, multiplying and dividing positive fractions also apply to negative fractions
• To understand that the rules for positive and negative integers also apply to fractions
• add, subtract, multiply and divide negative fractions and mixed numerals

Pre-requisite Summary

• Know how to add and subtract positive fractions using a common denominator
• Know how to multiply and divide positive fractions
• Understand that dividing by a fraction means multiplying by its reciprocal
• Know that integers can be positive or negative
• Know the sign rules for multiplication and division of integers
• Be able to compare fractions on a number line
• Be able to convert between mixed numerals and improper fractions
• Be able to simplify fractions fully

Worked Examples

Worked Example 1

Evaluate each expression:

a) $-{\displaystyle \frac{1}{3}}+{\displaystyle \frac{5}{6}}$
b) ${\displaystyle \frac{3}{4}}+(-{\displaystyle \frac{1}{2}})$
c) $-{\displaystyle \frac{2}{5}}+{\displaystyle \frac{1}{10}}$

Worked Example 2

Evaluate each expression:

a) ${\displaystyle \frac{2}{3}}-(-{\displaystyle \frac{1}{6}})$
b) $-{\displaystyle \frac{7}{8}}-{\displaystyle \frac{1}{4}}$
c) $-{\displaystyle \frac{3}{5}}-(-{\displaystyle \frac{1}{10}})$

Worked Example 3

Multiply or divide:

a) $-{\displaystyle \frac{2}{3}}\times {\displaystyle \frac{9}{10}}$
b) $-{\displaystyle \frac{5}{6}}÷{\displaystyle \frac{1}{3}}$
c) $-{\displaystyle \frac{3}{4}}\times (-{\displaystyle \frac{2}{5}})$

Worked Example 4

Multiply or divide:

a) ${\displaystyle \frac{7}{12}}÷(-{\displaystyle \frac{7}{18}})$
b) $-{\displaystyle \frac{4}{9}}\times (-{\displaystyle \frac{3}{8}})$
c) $-{\displaystyle \frac{5}{6}}÷(-{\displaystyle \frac{5}{12}})$

Worked Example 5

Convert to improper fractions, then evaluate:

a) $-1{\displaystyle \frac{1}{2}}+{\displaystyle \frac{3}{4}}$
b) $2{\displaystyle \frac{1}{3}}-3{\displaystyle \frac{1}{6}}$
c) $-2{\displaystyle \frac{1}{4}}+1{\displaystyle \frac{1}{2}}$

Worked Example 6

Convert to improper fractions, then evaluate:

a) $-1{\displaystyle \frac{2}{3}}\times {\displaystyle \frac{3}{5}}$
b) $2{\displaystyle \frac{1}{2}}÷(-1{\displaystyle \frac{1}{4}})$
c) $-3{\displaystyle \frac{1}{2}}÷1{\displaystyle \frac{3}{4}}$

Problems

Problem 1

Evaluate each expression:

a) $-{\displaystyle \frac{1}{4}}+{\displaystyle \frac{3}{8}}$
b) ${\displaystyle \frac{5}{6}}+(-{\displaystyle \frac{1}{3}})$
c) $-{\displaystyle \frac{7}{10}}+{\displaystyle \frac{1}{5}}$

Problem 2

Evaluate each expression:

a) ${\displaystyle \frac{3}{4}}-(-{\displaystyle \frac{1}{8}})$
b) $-{\displaystyle \frac{5}{6}}-{\displaystyle \frac{1}{3}}$
c) $-{\displaystyle \frac{4}{7}}-(-{\displaystyle \frac{2}{7}})$

Problem 3

Multiply or divide:

a) $-{\displaystyle \frac{3}{5}}\times {\displaystyle \frac{10}{9}}$
b) $-{\displaystyle \frac{2}{7}}÷{\displaystyle \frac{1}{7}}$
c) $-{\displaystyle \frac{4}{9}}\times (-{\displaystyle \frac{3}{2}})$

Problem 4

Multiply or divide:

a) ${\displaystyle \frac{5}{8}}÷(-{\displaystyle \frac{5}{12}})$
b) $-{\displaystyle \frac{7}{10}}\times (-{\displaystyle \frac{5}{14}})$
c) $-{\displaystyle \frac{3}{4}}÷(-{\displaystyle \frac{1}{2}})$

Problem 5

Convert to improper fractions, then evaluate:

a) $-1{\displaystyle \frac{1}{4}}+{\displaystyle \frac{1}{2}}$
b) $3{\displaystyle \frac{1}{5}}-4{\displaystyle \frac{1}{10}}$
c) $-2{\displaystyle \frac{1}{3}}+1{\displaystyle \frac{2}{3}}$

Problem 6

Convert to improper fractions, then evaluate:

a) $-2{\displaystyle \frac{1}{2}}\times {\displaystyle \frac{2}{5}}$
b) $1{\displaystyle \frac{1}{2}}÷(-{\displaystyle \frac{3}{4}})$
c) $-2{\displaystyle \frac{3}{4}}÷1{\displaystyle \frac{3}{8}}$

Exercises

Understanding and Fluency

1. Evaluate each sum:
   a) $-{\displaystyle \frac{1}{2}}+{\displaystyle \frac{3}{4}}$
   b) ${\displaystyle \frac{5}{6}}+(-{\displaystyle \frac{1}{2}})$
   c) $-{\displaystyle \frac{2}{3}}+{\displaystyle \frac{1}{6}}$

2. Evaluate each difference:
   a) ${\displaystyle \frac{3}{5}}-(-{\displaystyle \frac{1}{5}})$
   b) $-{\displaystyle \frac{7}{8}}-{\displaystyle \frac{1}{4}}$
   c) $-{\displaystyle \frac{4}{9}}-(-{\displaystyle \frac{1}{3}})$

3. Multiply the fractions:
   a) $-{\displaystyle \frac{2}{3}}\times {\displaystyle \frac{3}{8}}$
   b) $-{\displaystyle \frac{5}{7}}\times (-{\displaystyle \frac{7}{10}})$
   c) ${\displaystyle \frac{4}{9}}\times (-{\displaystyle \frac{3}{2}})$

4. Divide the fractions:
   a) $-{\displaystyle \frac{3}{4}}÷{\displaystyle \frac{1}{2}}$
   b) ${\displaystyle \frac{5}{6}}÷(-{\displaystyle \frac{5}{12}})$
   c) $-{\displaystyle \frac{7}{9}}÷(-{\displaystyle \frac{14}{27}})$

5. Determine whether each answer will be positive or negative, then calculate:
   a) $-{\displaystyle \frac{2}{5}}\times {\displaystyle \frac{3}{4}}$
   b) $-{\displaystyle \frac{3}{8}}÷(-{\displaystyle \frac{1}{2}})$
   c) ${\displaystyle \frac{5}{6}}\times (-{\displaystyle \frac{9}{10}})$

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

7. Perform the operation on mixed numerals:
   a) $-1{\displaystyle \frac{1}{2}}+2{\displaystyle \frac{1}{4}}$
   b) $3{\displaystyle \frac{1}{3}}-4{\displaystyle \frac{1}{6}}$
   c) $-2{\displaystyle \frac{1}{5}}+1{\displaystyle \frac{3}{10}}$

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

Reasoning

9. Explain why $-{\displaystyle \frac{2}{3}}+{\displaystyle \frac{1}{6}}$ is found using the same common denominator method as ${\displaystyle \frac{2}{3}}+{\displaystyle \frac{1}{6}}$.

10. A student says that $-{\displaystyle \frac{3}{4}}\times (-{\displaystyle \frac{2}{5}})$ must be negative because both fractions are negative. Explain the mistake.

11. Noah says that ${\displaystyle \frac{1}{2}}-(-{\displaystyle \frac{1}{3}})={\displaystyle \frac{0}{5}}$ because he subtracted both numerators and denominators. Explain why this is incorrect.

12. Explain why dividing by $-{\displaystyle \frac{2}{3}}$ gives a negative answer when the dividend is positive.

13. A student converts $-2{\displaystyle \frac{1}{4}}$ to ${\displaystyle \frac{-2}{4}}+{\displaystyle \frac{1}{4}}$. Describe the error.

Problem-solving

14. A scuba diver is at $-1{\displaystyle \frac{1}{2}}$ m relative to sea level and rises by ${\displaystyle \frac{3}{4}}$ m. What is the diver’s new position?

15. A bank balance changes by $-{\displaystyle \frac{5}{6}}$ of a dollar each day for $3$ days. What is the total change?

16. A temperature of $2{{\displaystyle \frac{1}{4}}}^{\circ }\text{C}$ drops by $3{{\displaystyle \frac{1}{2}}}^{\circ }\text{C}$. What is the new temperature?

17. A length of rope is cut into pieces of length ${\displaystyle \frac{3}{4}}$ m. If the total change in measured length is $-3$ m, how many pieces does this represent?

18. A machine multiplies an input by $-{\displaystyle \frac{2}{5}}$. What is the output when the input is $1{\displaystyle \frac{1}{2}}$?

19. A hiker is $-2{\displaystyle \frac{3}{4}}$ km relative to a checkpoint and then moves forward $1{\displaystyle \frac{1}{4}}$ km. What is the new position?

Potential Misunderstandings

• Students may think the sign rules for integers do not apply to fractions
• Students may forget to find a common denominator when adding or subtracting negative fractions
• Students may add or subtract denominators instead of using equivalent fractions
• Students may ignore the negative sign when simplifying
• Students may think two negative fractions multiplied together give a negative result
• Students may forget that dividing by a fraction means multiplying by its reciprocal
• Students may reverse the wrong fraction when dividing
• Students may convert a negative mixed numeral to an improper fraction incorrectly
• Students may place the negative sign on only one part of a mixed numeral instead of on the whole number
• Students may simplify arithmetic correctly but leave the final sign incorrect
• Students may think subtraction with negative fractions follows different fraction rules from subtraction with positive fractions
• Students may not recognise that the same fraction methods are used first, and then the sign rules are applied