Solve problems with positive and negative fractions

Ontario Grade 9 Mathematics

Number

Ontario MTH1W — Expectation B3.4

Fractions show up constantly in real life — cooking, money, temperature changes, and building plans all use them. In Grade 8 you worked with positive fractions and with negative integers. In Grade 9 we bring those two ideas together: fractions can now be positive or negative. A fraction like

-\frac{3}{4}

means three-quarters of a unit below zero, or three-quarters of a debt, or three-quarters of a degree below a reference point. Being able to add, subtract, multiply, and divide such fractions — and to use them in realistic problems — is the focus of this lesson.

What you will learn

Add, subtract, multiply, and divide positive and negative fractions.
Apply the correct order of operations when fractions are combined in a multi-step expression.
Solve real-world problems whose answers involve positive and negative fractions.
Identify and avoid common sign and fraction errors.

What You Already Know — A Quick Review

Before working with negative fractions, let's recall two things you already know well.

First, every fraction has a numerator (the top number) and a denominator (the bottom number). The denominator tells you how many equal pieces the whole is cut into; the numerator tells you how many of those pieces you have. For example,

\frac{5}{8}

means 5 out of 8 equal pieces.

Second, negative numbers sit to the left of zero on a number line. The number

-4

is 4 units left of zero. The same idea applies to fractions:

-\frac{3}{4}

sits three-quarters of a unit to the left of zero. A negative sign in front of a fraction applies to the entire fraction, so

-\frac{3}{4} = \frac{-3}{4} = \frac{3}{-4}

. All three forms mean the same thing, but writing the negative sign in front of the fraction — like

-\frac{3}{4}

— is the clearest habit.

The numerator counts pieces; the denominator names the size of each piece.
$-\frac{a}{b}$ means the opposite of $\frac{a}{b}$ ; it sits to the left of zero on the number line.
A negative sign in front of a fraction belongs to the whole fraction, not just the top or bottom.

Adding and Subtracting Positive and Negative Fractions

To add or subtract any two fractions, they must share a common denominator — the same bottom number. Once they do, you add or subtract the numerators and keep the denominator. The sign rules you know from integers apply to the numerators.

Think of a real situation: you are tracking a bank balance that changes by fractional amounts. You start with

\frac{1}{2}

of a dollar. You then spend

\frac{3}{4}

of a dollar. Spending means subtracting, so the change is

-\frac{3}{4}

. Your new balance is

\frac{1}{2} + \left(-\frac{3}{4}\right)

. To add these, find a common denominator. The smallest number that both 2 and 4 divide into evenly is 4 (this is called the lowest common denominator, or LCD). Rewrite

\frac{1}{2}

as

\frac{2}{4}

. Now add:

\frac{2}{4} + \left(-\frac{3}{4}\right) = \frac{2 + (-3)}{4} = \frac{-1}{4} = -\frac{1}{4}

. You owe one-quarter of a dollar.

Subtracting a negative fraction follows the same integer rule: subtracting a negative is the same as adding a positive. For example,

\frac{1}{3} - \left(-\frac{1}{6}\right) = \frac{1}{3} + \frac{1}{6}

. The LCD of 3 and 6 is 6. Rewrite

\frac{1}{3}

as

\frac{2}{6}

. Then

\frac{2}{6} + \frac{1}{6} = \frac{3}{6} = \frac{1}{2}

.

Always simplify your answer. A fraction is in simplest form when the numerator and denominator share no common factor other than 1.

\frac{a}{c} + \frac{b}{c} = \frac{a + b}{c}

Find the LCD before adding or subtracting fractions.
Once denominators match, add or subtract the numerators; keep the sign rules for integers.
Subtracting a negative fraction is the same as adding its positive: $a - (-b) = a + b$ .
Simplify the answer by dividing numerator and denominator by their greatest common factor.

Multiplying and Dividing Positive and Negative Fractions

Multiplying fractions does not require a common denominator. Multiply the numerators together and multiply the denominators together. Then apply the sign rule: two fractions with the same sign give a positive product; two fractions with different signs give a negative product.

For example,

\left(-\frac{2}{3}\right) \times \frac{5}{4}

: multiply numerators to get

2 \times 5 = 10

, multiply denominators to get

3 \times 4 = 12

. The signs differ (one negative, one positive), so the product is negative:

-\frac{10}{12}

. Simplify by dividing by 2:

-\frac{5}{6}

.

Dividing by a fraction means multiplying by its reciprocal. The reciprocal of a fraction flips the numerator and denominator: the reciprocal of

\frac{b}{c}

is

\frac{c}{b}

. The sign stays with the number when you flip it. So

\left(-\frac{3}{5}\right) \div \frac{2}{7}

becomes

\left(-\frac{3}{5}\right) \times \frac{7}{2} = -\frac{21}{10}

. As a mixed number this is

-2\frac{1}{10}

.

Sign summary: same signs → positive result; different signs → negative result. This rule applies to both multiplication and division.

\frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d} \qquad \frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}

To multiply fractions: multiply numerators, multiply denominators, then apply the sign rule.
Same signs give a positive result; different signs give a negative result.
To divide by a fraction, multiply by its reciprocal (flip the second fraction, then multiply).
Always simplify the final answer.

Order of Operations with Positive and Negative Fractions

When an expression has more than one operation, follow the order of operations: Brackets first, then multiplication and division (left to right), then addition and subtraction (left to right). This is the same rule you used with integers — it still applies when the numbers are fractions.

Consider the expression

\frac{1}{2} - \frac{3}{4} \times \left(-\frac{2}{3}\right)

. First, handle the multiplication:

\frac{3}{4} \times \left(-\frac{2}{3}\right) = -\frac{6}{12} = -\frac{1}{2}

. Now substitute back:

\frac{1}{2} - \left(-\frac{1}{2}\right) = \frac{1}{2} + \frac{1}{2} = 1

.

Notice how a subtraction followed by a negative fraction turned into addition. Always write out each step clearly so you do not lose track of signs.

\text{Order: } (\;) \rightarrow \times \div \rightarrow + -

Order of operations: Brackets → Multiply/Divide → Add/Subtract.
Do not add or subtract before finishing all multiplications and divisions.
Write each step on its own line to keep track of signs.

Solving Problems in Context

Real problems give you a situation in words. Your job is to translate the words into a mathematical expression, compute carefully, and then interpret the answer in the context of the problem.

Key words to watch for: 'lost', 'dropped', 'below', 'spent', and 'decreased by' usually signal a negative value. 'Gained', 'above', 'earned', and 'increased by' usually signal a positive value.

After you calculate an answer, always ask: does this answer make sense in the situation? If a temperature starts below zero and drops further, your answer should be more negative. If you earn more than you owe, your answer should be positive. This check takes only a few seconds and catches many errors.

Mixed numbers (like

-2\frac{3}{4}

) appear in real problems. Convert a mixed number to an improper fraction before computing:

-2\frac{3}{4} = -\frac{11}{4}

. Do this by multiplying the whole number by the denominator and adding the numerator:

2 \times 4 + 3 = 11

, then place that over the denominator and keep the negative sign.

-n\frac{a}{b} = -\frac{n \times b + a}{b}

Translate problem keywords into positive or negative values before computing.
Convert mixed numbers to improper fractions before any calculation.
After computing, check that the sign and size of the answer fit the real situation.
Show all steps — it helps you find sign errors quickly.

Sign Rules for Multiplying and Dividing Fractions

First fraction	Second fraction	Result sign	Example
Positive $+$	Positive $+$	Positive $+$	$\frac{2}{3} \times \frac{3}{4} = \frac{1}{2}$
Positive $+$	Negative $-$	Negative $-$	$\frac{2}{3} \times \left(-\frac{3}{4}\right) = -\frac{1}{2}$
Negative $-$	Positive $+$	Negative $-$	$\left(-\frac{2}{3}\right) \times \frac{3}{4} = -\frac{1}{2}$
Negative $-$	Negative $-$	Positive $+$	$\left(-\frac{2}{3}\right) \times \left(-\frac{3}{4}\right) = \frac{1}{2}$

Worked example

A Multi-Step Temperature Problem

At sunrise, the temperature outside was

-1\frac{1}{2}

°C. During the morning it rose by

\frac{7}{4}

°C. In the afternoon a cold front caused the temperature to drop by

2\frac{3}{4}

°C. What was the final temperature?

Step 1 — Write the expression
Rising means adding a positive value; dropping means adding a negative value. Write one expression that shows all three values combined.
$-1\frac{1}{2} + \frac{7}{4} + \left(-2\frac{3}{4}\right)$
Step 2 — Convert mixed numbers to improper fractions
Convert $-1\frac{1}{2}$ : multiply $1 \times 2 + 1 = 3$ , so $-1\frac{1}{2} = -\frac{3}{2}$ . Convert $-2\frac{3}{4}$ : multiply $2 \times 4 + 3 = 11$ , so $-2\frac{3}{4} = -\frac{11}{4}$ . The middle term $\frac{7}{4}$ is already an improper fraction.
$-\frac{3}{2} + \frac{7}{4} + \left(-\frac{11}{4}\right)$
Step 3 — Find a common denominator
The denominators are 2 and 4. The LCD is 4. Rewrite $-\frac{3}{2}$ with denominator 4 by multiplying numerator and denominator by 2.
$-\frac{3}{2} = -\frac{6}{4}$
Step 4 — Rewrite the expression with the common denominator
All three fractions now share the denominator 4.
$-\frac{6}{4} + \frac{7}{4} + \left(-\frac{11}{4}\right)$
Step 5 — Add the numerators left to right
Combine the numerators: $-6 + 7 = 1$ , then $1 + (-11) = -10$ . Keep the denominator 4.
$\frac{-6 + 7 + (-11)}{4} = \frac{-10}{4}$
Step 6 — Simplify
Divide numerator and denominator by their greatest common factor, which is 2.
$\frac{-10}{4} = -\frac{5}{2} = -2\frac{1}{2}$

Answer: The final temperature was

-2\frac{1}{2}

°C.

Check: Start at

-1.5

°C, add

1.75

°C to get

0.25

°C, then subtract

2.75

°C to get

0.25 - 2.75 = -2.5

°C

= -2\frac{1}{2}

°C. ✓

Common mistakes and how to avoid them

Applying the negative sign to only the numerator when converting a mixed number: writing

-2\frac{3}{4} = \frac{-8+3}{4} = -\frac{5}{4}

instead of

-\frac{11}{4}

.

Correction: The negative sign belongs to the entire mixed number. Convert first —

2 \times 4 + 3 = 11

— then attach the negative:

-\frac{11}{4}

.

Forgetting to find a common denominator before adding or subtracting, for example writing

\frac{1}{2} + \frac{1}{3} = \frac{2}{5}

.

Correction: Always find the LCD first. Here the LCD is 6:

\frac{3}{6} + \frac{2}{6} = \frac{5}{6}

.

Forgetting to flip the second fraction when dividing, for example computing

\frac{3}{4} \div \frac{2}{5}

as

\frac{3}{4} \times \frac{2}{5}

.

Correction: Division means multiply by the reciprocal:

\frac{3}{4} \div \frac{2}{5} = \frac{3}{4} \times \frac{5}{2} = \frac{15}{8}

.

Adding or subtracting before multiplying because the multiplication appears later in the written expression.

Correction: Always follow the order of operations. Multiplication and division must be completed before addition and subtraction, regardless of where they appear in the expression.

Thinking that subtracting a negative fraction makes the answer more negative, for example assuming

\frac{1}{4} - \left(-\frac{1}{4}\right) = 0

.

Correction: Subtracting a negative is the same as adding a positive:

\frac{1}{4} - \left(-\frac{1}{4}\right) = \frac{1}{4} + \frac{1}{4} = \frac{1}{2}

.

Lesson summary

A negative fraction sits to the left of zero on the number line; its negative sign applies to the entire fraction.
To add or subtract fractions, find the LCD, rewrite each fraction with that denominator, then add or subtract the numerators using integer sign rules.
To multiply fractions, multiply numerators together and denominators together, then apply the sign rule: same signs give a positive result, different signs give a negative result.
To divide by a fraction, multiply by its reciprocal; keep the sign with the number when flipping.
Always follow the order of operations: brackets first, then multiply and divide, then add and subtract.
Convert mixed numbers to improper fractions before calculating, and check that your final answer makes sense in the context of the problem.

Check your understanding

Question 1

What is

-\frac{3}{4} + \frac{1}{2}

?

$-\frac{1}{4}$
$\frac{1}{4}$
$-\frac{5}{4}$
$\frac{5}{4}$

Show answer and explanation

-\frac{1}{4}

The LCD of 4 and 2 is 4. Rewrite

\frac{1}{2}

as

\frac{2}{4}

. Then

-\frac{3}{4} + \frac{2}{4} = \frac{-3+2}{4} = -\frac{1}{4}

.

Question 2

What is

\left(-\frac{2}{5}\right) \times \left(-\frac{5}{6}\right)

?

$-\frac{1}{3}$
$\frac{1}{3}$
$-\frac{10}{30}$
$\frac{2}{3}$

Show answer and explanation

\frac{1}{3}

Multiply numerators:

2 \times 5 = 10

. Multiply denominators:

5 \times 6 = 30

. Both signs are negative, so the result is positive:

\frac{10}{30} = \frac{1}{3}

.

Question 3

A diver is at

-\frac{3}{2}

m relative to the surface. She dives down another

\frac{5}{4}

m. What is her new position?

$-\frac{11}{4}$ m
$-\frac{1}{4}$ m
$\frac{11}{4}$ m
$-\frac{7}{4}$ m

Show answer and explanation

-\frac{11}{4}

m

Diving down means subtracting (going more negative):

-\frac{3}{2} - \frac{5}{4}

. LCD is 4. Rewrite:

-\frac{6}{4} - \frac{5}{4} = -\frac{11}{4}

m.

Question 4

Evaluate

\frac{1}{3} - \frac{1}{2} \times \left(-\frac{2}{3}\right)

.

$-\frac{1}{9}$
$\frac{2}{3}$
$\frac{1}{9}$
$0$

Show answer and explanation

\frac{2}{3}

Follow order of operations — multiply first:

\frac{1}{2} \times \left(-\frac{2}{3}\right) = -\frac{2}{6} = -\frac{1}{3}

. Then subtract:

\frac{1}{3} - \left(-\frac{1}{3}\right) = \frac{1}{3} + \frac{1}{3} = \frac{2}{3}

.

Key terms

Fraction: A number written as $\frac{a}{b}$ , where $a$ is the numerator (how many pieces you have) and $b$ is the denominator (how many equal pieces the whole is divided into).
Negative fraction: A fraction less than zero, written with a negative sign in front: for example, $-\frac{3}{4}$ . It represents a quantity to the left of zero on the number line.
Lowest common denominator (LCD): The smallest number that two or more denominators both divide into evenly. Required before adding or subtracting fractions.
Reciprocal: The reciprocal of a fraction $\frac{a}{b}$ is $\frac{b}{a}$ — the numerator and denominator are swapped. Used when dividing by a fraction.
Improper fraction: A fraction where the numerator is greater than or equal to the denominator, such as $\frac{11}{4}$ . Easier to use in calculations than a mixed number.
Mixed number: A number written as a whole number and a fraction together, such as $2\frac{3}{4}$ . Convert to an improper fraction before calculating.
Simplest form: A fraction whose numerator and denominator share no common factor other than 1, for example $\frac{3}{4}$ is simpler than $\frac{6}{8}$ .
Order of operations: The agreed sequence for evaluating expressions: brackets first, then multiplication and division (left to right), then addition and subtraction (left to right).

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About this lesson and its review

Published by DoAssignment. This reviewed lesson follows Ontario Grade 9 Mathematics (MTH1W), expectation B3.4. It is a study resource, not an official curriculum publication.

Before publication, content is checked for structure, mathematical or chemical notation, calculations, course boundaries, and readability. Errors can still occur, so corrections are welcomed.

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B3.4 · Solve problems with positive and negative fractions

Ontario Grade 9 Mathematics

Ontario MTH1W — Expectation B3.4

What you will learn

What You Already Know — A Quick Review

Adding and Subtracting Positive and Negative Fractions

Multiplying and Dividing Positive and Negative Fractions

Order of Operations with Positive and Negative Fractions

Solving Problems in Context

Sign Rules for Multiplying and Dividing Fractions

A Multi-Step Temperature Problem

Common mistakes and how to avoid them

Lesson summary

Check your understanding

Question 1

Question 2

Question 3

Question 4

Key terms

Continue through MTH1W

About this lesson and its review