Mathematics › 🔄 Grade 7 › Rational Numbers: Adding and Subtracting Signed Fractions

Rational Numbers: Adding and Subtracting Signed Fractions

Fractions can be negative too — and once you find a common denominator, the same sign rules you already know for integers take over.

Grade 7Middle school

Rational Numbers: Adding and Subtracting Signed Fractions — illustration

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The big idea: A rational number is any fraction that can be positive or negative — think of it as a point some distance to the left or right of zero. Adding and subtracting signed fractions works exactly like adding and subtracting integers, as long as the fractions share a common denominator first. Once the denominators match, you just combine the numerators using the sign rules you already know.

🎯 By the end, you'll be able to

Define rational numbers as fractions that may be positive or negative
Rewrite fractions with unlike denominators using a common denominator
Apply integer sign rules to add and subtract signed fractions
Simplify the result of a fraction operation to lowest terms

📎 You should already know

Adding and subtracting integers
Equivalent fractions and common denominators

Fractions get a sign

A rational number is just a fraction — positive or negative — that names a point on the number line. −½ sits half a unit to the left of zero, exactly as far from zero as +½ sits to the right. Money owed, a temperature below a reference point, a golf score under par: all of these are naturally negative fractions.

The good news is that adding and subtracting them is not a new skill. It is two skills you already have — finding a common denominator, and combining signed numbers — used together.

🔑 Same denominator, then integer rules

To add or subtract signed fractions: first rewrite them so they share a common denominator. Once the denominators match, add or subtract the numerators using the same sign rules you use for integers, and keep the shared denominator underneath.

\[ \dfrac{a}{d} + \dfrac{c}{d} = \dfrac{a + c}{d} \]

With a common denominator d, you only need to combine the numerators — the denominator does not change.

Building a common denominator

If two fractions do not already share a denominator, multiply each fraction by a clever form of 1 — the same number over itself — so both fractions land on the same-sized pieces without changing their value.

\[ \dfrac{a}{b} = \dfrac{a \times k}{b \times k} \]

Multiplying top and bottom by the same number k gives an equivalent fraction, cut into smaller, matching pieces.

🎮 Rational Number Operator LIVE

Add and subtract signed fractions on a common scale.

Negative fractions, three ways

A negative fraction can be written with the minus sign in front, on the top, or on the bottom — all three mean exactly the same value. Once the fractions share a denominator, treat the numerators as signed integers: combine like signs by adding their sizes, combine unlike signs by subtracting the smaller size from the larger and keeping the sign of the bigger one.

✨ Three faces of the same negative fraction

−3/4 (sign out front), −3 over 4 (the minus on the numerator), and 3 over −4 (the minus on the denominator) are all the same number. Rewriting a negative fraction with the sign out front, next to the fraction bar, usually makes it easiest to work with.

📝 Worked example: Find 3/4 + (−1/2).

Rewrite −1/2 with denominator 4: −1/2 = −2/4.
Now both fractions share denominator 4: 3/4 + (−2/4).
Combine the numerators like integers: 3 + (−2) = 1, over the shared denominator 4.

✓ 3/4 + (−1/2) = <strong>1/4</strong>.

📝 Worked example: Find −2/3 − 5/6.

Rewrite the subtraction as adding the opposite: −2/3 + (−5/6).
Convert −2/3 to sixths: −2/3 = −4/6.
Add the numerators: −4 + (−5) = −9, over the shared denominator 6, giving −9/6.
Simplify −9/6 by dividing top and bottom by 3.

✓ −2/3 − 5/6 = <strong>−3/2</strong> (that is, −1½).

⚠️ Never add the denominators

A common mistake is combining denominators along with numerators, turning 3/4 + (−2/4) into something like 1/8. The denominator only tells you the size of each piece — once the pieces match, it stays the same; only the numerators combine.

Check your understanding

1. Find 1/4 + (−3/4).

The denominators already match: 1 + (−3) = −2, over 4, which simplifies to −1/2.

2. What common denominator would you use to add 1/3 and 1/4?

The least common multiple of 3 and 4 is 12, so both fractions can be rewritten as twelfths.

3. Find −2/5 + 3/5.

The denominators match, so combine numerators: −2 + 3 = 1, over 5, giving 1/5.

4. Which expression is equal to −3/7?

Moving the negative sign to the denominator gives 3/−7, which equals −3/7. The other options simplify to positive values.

5. Find −1/2 − 1/4.

Rewrite −1/2 as −2/4, then −2/4 − 1/4 = −3/4.

✅ Key takeaways

A rational number is a fraction that can be positive or negative, sitting to either side of zero on the number line.
To add or subtract signed fractions, first rewrite them with a common denominator.
Once denominators match, combine the numerators using the same sign rules used for integers.
A negative fraction can be written with the sign in front, on the numerator, or on the denominator — all equal.
Always simplify your final answer to lowest terms, and never add the denominators themselves.