Negative Fractions KS3: Operations and the Number Line

What do negative fractions look like on a number line?

Negative fractions sit between negative integers on the number line, getting closer to zero as their absolute value decreases.

Order from smallest to largest: −2½, −1¾, −1, −½, 0, ½, 1

Key insight: −¼ is greater than −¾ because it is closer to zero. When comparing negative fractions, the one with the smaller absolute value is the greater number.

How do you add and subtract negative fractions?

Use the same rules as adding and subtracting negative integers.

Rule recap:

• Adding a negative is the same as subtracting: a + (−b) = a − b
• Subtracting a negative is the same as adding: a − (−b) = a + b

Method at a glance:

1. Rewrite using the rule above if needed.
2. Find a common denominator if the fractions have different denominators.
3. Add or subtract the numerators as directed.
4. Simplify the answer.

Worked example 1: Calculate ½ + (−¼).

½ + (−¼) = ½ − ¼ = 2/4 − 1/4 = ¼

Worked example 2: Calculate −⅔ − (−⅙).

−⅔ − (−⅙) = −⅔ + ⅙ = −4/6 + 1/6 = −3/6 = −½

Worked example 3: Calculate −¾ + (−⅝).

Common denominator is 8: −6/8 + (−5/8) = −11/8 = −1⅜

When both fractions are negative and you add them, the result is further from zero (more negative).

How do you multiply negative fractions?

Apply the sign rules first, then multiply numerators and denominators.

Worked example 1: Calculate (−⅔) × (¾).

Sign: negative × positive = negative. Magnitude: ⅔ × ¾ = 6/12 = ½. Answer: −½

Worked example 2: Calculate (−⅖) × (−⅗).

Sign: negative × negative = positive. Magnitude: ⅖ × ⅗ = 6/25. Answer: 6/25

How do you divide negative fractions?

Dividing fractions uses KCF: Keep the first fraction, Change ÷ to ×, Flip the second fraction. Then apply the sign rules.

Worked example: Calculate (−¾) ÷ (½).

• Keep: −¾; Change: ×; Flip: 2/1
• (−¾) × 2 = −6/4 = −3/2 = −1½

Sign: negative ÷ positive = negative. ✓

What about negative mixed numbers?

Convert to an improper fraction first, keeping the negative sign.

Worked example: Calculate −1¾ + ⅝.

• −1¾ = −7/4 = −14/8
• ⅝ = 5/8
• −14/8 + 5/8 = −9/8 = −1⅛

Take care: −1¾ is NOT the same as −1 + ¾ = −¼. The mixed number −1¾ means negative one-and-three-quarters, so as an improper fraction it is −7/4, not −1/4.

Frequently asked questions

Is −½ greater or less than −¾?

−½ is greater than −¾. On the number line, −½ is closer to zero (it is only half a unit away, whereas −¾ is three quarters of a unit away). "Greater" means further to the right on the number line.

Does the minus sign go with the numerator or the denominator?

The minus sign applies to the whole fraction. The three equivalent forms −(a/b), (−a)/b, and a/(−b) all have the same value. It is conventional to write the minus sign in front of the fraction: −a/b.

Can you have a negative fraction as a mixed number?

Yes: −1½ means negative one and a half, equal to −3/2. Be careful when converting: −2¼ = −9/4, not −7/4. The whole-number part and the fraction part are both "below zero."

What happens when you multiply two negative fractions?

Two negatives multiply to give a positive, so (−½) × (−⅔) = +⅓. This follows the same rule as integer multiplication: (−2) × (−3) = +6.

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