Operations on Rational Numbers

• Add and subtract rational numbers with same and different denominators.
• Multiply and divide rational numbers using standard rules.
• Simplify results to standard form after every operation.
• Apply integer sign rules when operating with negative rationals.

Level 1 — Core idea

1. Addition and subtraction — same denominator
   p/q ± r/q = (p ± r)/q
   Example: 5/7 + 2/7 = 7/7 = 1.

2. Addition and subtraction — different denominators
   Find the LCM of denominators, convert to equivalent fractions, then add/subtract numerators.
   Example: 1/3 + 1/4 = 4/12 + 3/12 = 7/12.

3. Additive inverse — The additive inverse of p/q is −p/q, since p/q + (−p/q) = 0.

Level 2 — Process and representation

1. Multiplication
   (p/q) × (r/s) = (p × r)/(q × s)
   Simplify before or after multiplying.
   Example: (−2/3) × (9/4) = −18/12 = −3/2.

2. Division
   (p/q) ÷ (r/s) = (p/q) × (s/r) — multiply by the reciprocal.
   Example: (3/5) ÷ (2/7) = (3/5) × (7/2) = 21/10.

3. Sign rules — Same as for integers: same signs → positive product/quotient; different signs → negative.

Evaluate 2/3 − 5/6 + 1/2.

Step 1 — LCM of 3, 6, 2 is 6
Step 2 — convert: 2/3 = 4/6;  5/6 = 5/6;  1/2 = 3/6
Step 3 — compute: 4/6 − 5/6 + 3/6 = (4 − 5 + 3)/6 = 2/6
Step 4 — standard form: 2/6 = 1/3
Answer: 1/3

Evaluate (−4/5) × (15/8) ÷ (−3/2).

Step 1 — multiply first two: (−4/5) × (15/8) = −60/40 = −3/2
Step 2 — divide by (−3/2): (−3/2) ÷ (−3/2) = (−3/2) × (−2/3) = 6/6 = 1
Answer: 1
Shortcut check: dividing a number by itself gives 1 ✓

• Compute 3/4 + 1/6. (11/12)
• Compute (−2/3) × (9/4). (−3/2)
• Compute 5/6 ÷ 2/3. (5/4)
• What is the additive inverse of 7/9? (−7/9)

Open the Practice tab for graded questions on rational number operations.