What are Rational Numbers?
A rational number is any number that can be written in the form p/q, where p and q are integers and q ≠ 0. The numerator p may be positive, negative or zero, while the denominator q can never be zero. Examples include 3/4, −7/5, 8, 0 and −2 because each can be represented as a fraction with an integer numerator and a non-zero integer denominator.
Standard Form
A rational number is in standard form when the numerator and denominator have no common factor other than 1 and the denominator is positive. For example, 18/24 becomes 3/4 after dividing both parts by their greatest common divisor. Keeping fractions in standard form makes comparison and calculation easier.
Addition and Subtraction
For fractions with the same denominator, add or subtract the numerators and keep the denominator unchanged. For different denominators, first find a common denominator, usually using the least common multiple. Then convert the fractions and perform the operation. A useful formula is a/b + c/d = (ad + bc)/bd. Similarly, a/b − c/d = (ad − bc)/bd. Always simplify the final result.
Multiplication
Multiply numerator by numerator and denominator by denominator: a/b × c/d = ac/bd. Before multiplying, look for common factors across a numerator and the other denominator. Cross-cancellation can reduce large numbers and lower the chance of calculation errors.
Division
To divide by a non-zero rational number, multiply by its reciprocal: a/b ÷ c/d = a/b × d/c = ad/bc. The divisor cannot be zero. Remember that the reciprocal of a non-zero fraction a/b is b/a.
Important Properties
- Closure: Rational numbers are closed under addition, subtraction and multiplication. Division is closed when the divisor is non-zero.
- Commutative: a + b = b + a and ab = ba. Subtraction and division are not commutative.
- Associative: (a + b) + c = a + (b + c) and (ab)c = a(bc). Subtraction and division are not associative.
- Distributive: a(b + c) = ab + ac and a(b − c) = ab − ac.
- Additive identity: 0, because a + 0 = a.
- Multiplicative identity: 1, because a × 1 = a.
- Additive inverse: The additive inverse of a is −a.
- Multiplicative inverse: For a non-zero a/b, the inverse is b/a.
Comparing Rational Numbers
To compare two rational numbers, you can use a common denominator or cross multiplication when denominators are positive. For a/b and c/d, compare ad and bc. On a number line, numbers farther to the right are greater. Negative rational numbers are less than zero, and among negative values the number with the larger absolute value is smaller.
How to Score Better
Read the sign carefully, simplify whenever possible, and estimate the size of your answer before submitting. For every addition, subtraction, multiplication and division problem in this tool, enter your answer and press Check Answer. A correct response turns green and an incorrect response turns red. The answer is deliberately hidden until checking, so learners must attempt the problem independently first.
Why Online Practice Helps
Repeated practice helps students develop fluency with signs, fractions and operations. The shuffle system creates a fresh order and the generator produces varied values, allowing the same learner to practise several times. The progress bar gives an immediate view of completed checks, while the final score summarizes performance. The optional timer can be used for focused revision or test-style practice. Teachers and parents can also print or save the generated worksheet using the PDF option.
Quick Revision Formula Box