Rational Numbers

• What a rational number is and why every fraction, integer, and terminating/repeating decimal belongs to this set.
• How to write rational numbers in standard form (lowest terms, positive denominator).
• The four operations — addition, subtraction, multiplication, and division — on rational numbers.
• How to compare, order, and locate rational numbers on the number line.
• Real-life uses: sharing quantities, recipe scaling, currency conversion, and percentage calculations.

1. Definition — A rational number is any number that can be written as p/q, where p and q are integers and q ≠ 0. Examples: 3/4, −7/2, 5 (= 5/1), 0 (= 0/1).

2. Standard form — Divide numerator and denominator by their HCF so they have no common factor except 1, and keep the denominator positive. Example: −6/8 → −3/4.

3. Addition & subtraction — Find the LCM of denominators, convert to equivalent fractions, then add/subtract numerators. Example: 1/2 + 1/3 = 3/6 + 2/6 = 5/6.

4. Multiplication — Multiply numerators and denominators: (a/b) × (c/d) = ac/bd. Simplify at the end.

5. Division — Multiply by the reciprocal: (a/b) ÷ (c/d) = (a/b) × (d/c).

6. Decimal forms — Terminating decimals (0.25 = 1/4) and repeating decimals (0.333… = 1/3) are rational. Only non-repeating, non-terminating decimals are irrational.

7. Where it shows up — Splitting a pizza among friends, mixing paint in given ratios, calculating discounts, and converting measurements in science labs.

Add 2/3 and 5/6, then express the answer in standard form.

Step 1 — LCM of 3 and 6 is 6
Step 2 — 2/3 = 4/6  and  5/6 stays as 5/6
Step 3 — 4/6 + 5/6 = 9/6
Step 4 — Simplify: 9/6 = 3/2  (divide by HCF 3)
Answer: 3/2  or  1½

Application: A tailor uses 2/3 m of cloth for one shirt and 5/6 m for another. Total cloth needed = 3/2 m = 1.5 m.

• Forgetting that 0 is rational (0 = 0/1) but division by 0 is undefined.
• Adding fractions by adding numerators and denominators separately (1/2 + 1/3 ≠ 2/5).
• Leaving answers without simplifying to lowest terms.
• Changing the sign of only the numerator when the denominator is negative (−3/−4 = 3/4, not −3/4).

• Write −12/18 in standard form.
• Compute 3/5 × 10/9 and simplify.
• Which is greater: 7/8 or 5/6?
• Express 0.375 as a fraction in lowest terms.

Open the Practice tab for graded questions on Rational Numbers.