Rational Numbers • Topic 3 of 3

Comparison, Standard Form & Ordering

What is Standard Form of a Rational Number?

A rational number \(\frac{p}{q}\) is in standard form (or simplest form) when:

  • \(q > 0\) (denominator is positive)
  • \(p\) and \(q\) have no common factor other than 1 (they are co-prime)

Steps to convert to standard form:

  • Make denominator positive (multiply numerator and denominator by -1 if needed)
  • Find the HCF (GCD) of numerator and denominator
  • Divide both numerator and denominator by the HCF

How to Compare Rational Numbers?

Method 1 (Same denominator): Compare numerators directly

  • \(\frac{3}{7} > \frac{2}{7}\) because \(3 > 2\)

Method 2 (Different denominators): Use cross-multiplication

  • For \(\frac{a}{b}\) and \(\frac{c}{d}\), compare \(a \times d\) and \(c \times b\)
  • If \(a \times d > c \times b\), then \(\frac{a}{b} > \frac{c}{d}\)

Method 3 (Decimal conversion): Convert to decimals and compare

Ordering of Rational Numbers

Ascending order (smallest to largest): Arrange from left to right on number line Descending order (largest to smallest): Arrange from right to left on number line

Operations on Rational Numbers

OperationMethodExample
AdditionFind LCM of denominators, convert, add numerators\(\frac{1}{2} + \frac{1}{3} = \frac{3}{6} + \frac{2}{6} = \frac{5}{6}\)
SubtractionSame as addition, then subtract numerators\(\frac{3}{4} - \frac{1}{2} = \frac{3}{4} - \frac{2}{4} = \frac{1}{4}\)
MultiplicationMultiply numerators, multiply denominators\(\frac{2}{3} \times \frac{4}{5} = \frac{8}{15}\)
DivisionMultiply by reciprocal of divisor\(\frac{2}{3} \div \frac{4}{5} = \frac{2}{3} \times \frac{5}{4} = \frac{10}{12} = \frac{5}{6}\)
Comparing & Ordering Rational Numbers-2-3/2-1-1/201/213/22Ordering: −2 < −3/2 < −1 < −1/2 < 0 < 1/2 < 1 < 3/2 < 2Standard Form: p/q where q > 0 and gcd(p,q) = 1Example: −6/−4 = 6/4 = 3/2 (positive denominator, fully reduced)
Solved Examples
1
Worked Example
Example 1: Express \(\frac{-24}{36}\) in standard form and compare it with \(\frac{-2}{3}\).
Solution- Step 1: Denominator is positive (36 > 0) ✓ - Step 2: Find HCF of 24 and 36 - Step 3: Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24 - Step 4: Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36 - Step 5: HCF = 12 - Step 6: Divide numerator and denominator by 12: \(\frac{-24 \div 12}{36 \div 12} = \frac{-2}{3}\) - Step 7: To compare, they are equal! \(\frac{-24}{36} = \frac{-2}{3}\)
2
Worked Example
Example 2: Arrange the rational numbers \(\frac{3}{5}, \frac{-1}{2}, \frac{7}{10}, \frac{-3}{4}\) in ascending order.
Solution- Step 1: Convert to decimals for easy comparison (or use cross multiplication) - Step 2: \(\frac{3}{5} = 0.6\) - Step 3: \(\frac{-1}{2} = -0.5\) - Step 4: \(\frac{7}{10} = 0.7\) - Step 5: \(\frac{-3}{4} = -0.75\) - Step 6: Arrange from smallest to largest: \(-0.75, -0.5, 0.6, 0.7\) - Step 7: Write in original form: \(\frac{-3}{4}, \frac{-1}{2}, \frac{3}{5}, \frac{7}{10}\) - Step 8: Verify using number line: \(-3/4\) is leftmost, then \(-1/2\), then \(3/5\), then \(7/10\)
3
Worked Example
Example 3 (Word Problem): Riya ate \(\frac{2}{5}\) of a pizza. Her brother ate \(\frac{1}{3}\) of the remaining pizza. What fraction of the whole pizza is left?
Solution- Step 1: Whole pizza = 1 - Step 2: Riya ate \(\frac{2}{5}\), so remaining after Riya = \(1 - \frac{2}{5}\) - Step 3: \(1 - \frac{2}{5} = \frac{5}{5} - \frac{2}{5} = \frac{3}{5}\) - Step 4: Brother ate \(\frac{1}{3}\) of remaining = \(\frac{1}{3} \times \frac{3}{5} = \frac{3}{15} = \frac{1}{5}\) - Step 5: Total eaten = Riya's share + Brother's share = \(\frac{2}{5} + \frac{1}{5} = \frac{3}{5}\) - Step 6: Leftover = \(1 - \frac{3}{5} = \frac{2}{5}\)
4
Worked Example
Example 4 (Word Problem - Multi-step): A car travels \(\frac{2}{3}\) of its journey on the first day and \(\frac{1}{4}\) of the remaining journey on the second day. If the total journey is 240 km, how much distance is left to cover on the third day?
Solution- Step 1: Total journey = 240 km - Step 2: Distance covered on first day = \(\frac{2}{3} \times 240 = 160\) km - Step 3: Remaining after day 1 = \(240 - 160 = 80\) km - Step 4: Distance covered on second day = \(\frac{1}{4} \times 80 = 20\) km - Step 5: Total covered = \(160 + 20 = 180\) km - Step 6: Distance left = \(240 - 180 = 60\) km

Key Points

  • Standard form: Denominator positive and numerator & denominator have HCF = 1
  • Comparing fractions: Use cross multiplication: \(\frac{a}{b} > \frac{c}{d}\) if \(a \times d > c \times b\)
  • Ascending order: Smallest to largest; Descending order: Largest to smallest
  • Addition/Subtraction: Find LCM of denominators first
  • Multiplication: Multiply numerators and denominators directly
  • Division: Multiply by the reciprocal of the divisor
  • Always simplify your final answer to standard form
  • For word problems, identify what is given and what needs to be found, then break into steps
  • ---
Quick Check — 4 MCQs
Tap an option to check your answer0 / 4
Q1.In standard form, the denominator is:
Explanation: Positive and in lowest terms.
Q2.The standard form of $\tfrac48$ is:
Explanation: Divide by $4$.
Q3.Which is greater: $\tfrac25$ or $\tfrac37$?
Explanation: $\tfrac{14}{35}<\tfrac{15}{35}$.
Q4.Between any two rational numbers there are:
Explanation: Infinitely many.
Practice more MCQs → 📝 Assignment · 35 marks