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ICSE • Class 7 • Mathematics • Ch 3
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Fractions

In ICSE Class 7 Mathematics, "Fractions" provides an authoritative, pedagogical master study guide analyzing the arithmetic, properties, and applications of fractional quantities. This comprehensive chapter explores Classification of Fractions (Proper fractions [numerator < denominator], Improper fractions [numerator >= denominator], Mixed fractions, Like vs Unlike fractions, Unit fractions, and Equivalent fractions), Reduction to Lowest Terms (Dividing numerator and denominator by their HCF), Comparison and Ordering of Unlike Fractions (Cross-multiplication and LCM of denominators methods; Ascending and Descending order), Operations on Fractions (Addition and Subtraction via LCM of denominators; Multiplication of fractions: product of numerators divided by product of denominators, interpretation of "of" as multiplication; Division of fractions: multiplying by reciprocal / multiplicative inverse), Reciprocal of a Fraction and Division by Zero, Complex Fractions and Simplification (Continued fractions, multi-tiered brackets), Order of Operations using the BODMAS Rule, and Real-World Applied Word Problems (Fractional division of shared quantities, rates of work, recipe scaling, area calculations) aligned with the 2026–27 CISCE curriculum.

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How Did Egyptian Scribes Build the Great Pyramids Using Only Fractions with a Numerator of One?

Nearly 4,000 years ago, scribes in ancient Egypt sat along the banks of the Nile River writing the famous Rhind Mathematical Papyrus. The Pharaohs commanded them to calculate exact bread rations, beer allocations, and stone block dimensions for the construction of the Great Pyramids of Giza. But the ancient Egyptians had a peculiar mathematical rule: they refused to write any fraction with a numerator greater than 1! Every fraction had to be expressed exclusively as a sum of distinct Unit Fractions (fractions with a numerator of $1$, like $\frac{1}{2}, \frac{1}{3}, \frac{1}{7}$). If an Egyptian surveyor wanted to write $\frac{2}{5}$, he could not simply write $\frac{2}{5}$; he had to decompose it into $\frac{1}{3} + \frac{1}{15}$! What happens when you divide a fraction by another fraction? Why does dividing by a tiny fraction like $\frac{1}{100}$ make a number explode one hundred times larger? How do you simplify multi-tiered Complex Fractions? Let's master the arithmetic of fractions.

Why This Chapter Matters

Fractions are the primary bridge between elementary whole-number counting and advanced rational algebra. Mastery of fractional operations is essential in physics (focal lengths, kinetic energy), chemistry (chemical stoichiometry), culinary arts, carpentry, medicine dosages, and scoring top marks in ICSE school examinations.

Before You Begin (Prerequisites)

  • Multiplication tables and basic whole-number division.
  • Finding the Least Common Multiple (LCM) and Highest Common Factor (HCF).
  • Basic concept of a fraction as a part of a whole.

What You Will Learn (Core Objectives)

  • Classify fractions into proper, improper, mixed, like, unlike, and unit fractions.
  • Convert improper fractions to mixed fractions and vice versa.
  • Compare and order unlike fractions using cross-multiplication and LCM methods.
  • Perform accurate addition, subtraction, multiplication, and division of mixed fractions.
  • Calculate the reciprocal of proper, improper, and mixed fractions.
  • Simplify complex, multi-tiered fractional expressions using the BODMAS convention.
  • Solve multi-step real-world word problems involving fractional parts and rates.

Chapter Roadmap & Progression

1 1. Classification & Conversion of F...
2 2. Comparison & Ordering of Unlike...
3 3. Multiplication & Division of Fra...
4 4. Complex Fractions & The BODMAS H...

Complete Concept Guide (100% Curriculum Coverage)

1. Classification & Conversion of Fractions

Understand
A. Types of Fractions:
  • Proper Fraction: Numerator is strictly less than denominator ($N < D$). Represents a quantity strictly less than 1 (e.g., $\frac{3}{7}, \frac{5}{11}$).
  • Improper Fraction: Numerator is greater than or equal to denominator ($N \ge D$). Represents a quantity greater than or equal to 1 (e.g., $\frac{11}{4}, \frac{7}{7}$).
  • Mixed Fraction: A combination of a whole number and a proper fraction (e.g., $2\frac{3}{4}$).
  • Like Fractions: Fractions having the exact same denominator (e.g., $\frac{2}{9}, \frac{5}{9}, \frac{7}{9}$).
  • Unlike Fractions: Fractions having different denominators (e.g., $\frac{2}{3}, \frac{4}{5}, \frac{7}{8}$).
  • Unit Fraction: A fraction with a numerator of $1$ (e.g., $\frac{1}{2}, \frac{1}{8}, \frac{1}{25}$).
B. Inter-conversion of Mixed & Improper Fractions:
$$\text{Improper Fraction} = \frac{(\text{Whole Number} \times \text{Denominator}) + \text{Numerator}}{\text{Denominator}}$$ $$3\frac{5}{8} = \frac{(3 \times 8) + 5}{8} = \frac{24 + 5}{8} = \frac{29}{8}$$ $$\frac{37}{7} = 37 \div 7 = 5 \text{ (Quotient)} \text{ with remainder } 2 \implies 5\frac{2}{7}$$

2. Comparison & Ordering of Unlike Fractions

Comparison
A. The LCM Method for Ordering:

To arrange unlike fractions in Ascending or Descending order:

  1. Find the LCM of all denominators.
  2. Convert each fraction into an equivalent like fraction with the common LCM denominator.
  3. Compare and arrange their numerators in the required order.
B. Cross-Multiplication Method (For Two Fractions):

To quickly compare $\frac{a}{b}$ and $\frac{c}{d}$:

  • Compute $a \times d$ and $b \times c$.
  • If $a \times d > b \times c \implies \frac{a}{b} > \frac{c}{d}$.

3. Multiplication & Division of Fractions (Reciprocals)

Operations
A. Multiplication of Fractions:
$$\frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d} = \frac{\text{Product of Numerators}}{\text{Product of Denominators}}$$
  • "Of" Operator: In mathematics, "of" indicates multiplication: $\frac{3}{4} \text{ of } 28 = \frac{3}{4} \times 28 = 21$.
  • Always convert mixed fractions into improper fractions before multiplying!
B. Reciprocal & Division of Fractions:
  • Reciprocal: Two non-zero fractions are reciprocals of each other if their product is 1. The reciprocal of $\frac{a}{b}$ is $\frac{b}{a}$. (Reciprocal of $2\frac{1}{3} = \frac{7}{3}$ is $\frac{3}{7}$).
  • Division Rule: Dividing by a fraction is identical to multiplying by its reciprocal: $$\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} = \frac{a \cdot d}{b \cdot c}$$

4. Complex Fractions & The BODMAS Hierarchy

Complex Simplifications
A. Complex (Multi-Tiered) Fractions:

A complex fraction is a fraction whose numerator, denominator, or both contain fractions:

$$\frac{\frac{a}{b}}{\frac{c}{d}} = \frac{a}{b} \div \frac{c}{d} = \frac{a \times d}{b \times c}$$
B. Order of Operations (BODMAS):
  1. B (Brackets): Simplify expressions inside parentheses $( )$, braces $\{ \}$, and brackets $[ ]$ starting from the innermost.
  2. O (Of / Orders): Evaluate "of" operations before standard multiplication and division!
  3. D / M: Division and multiplication from left to right.
  4. A / S: Addition and subtraction from left to right.

Key Formulas, Identities & Theorems

Fraction Multiplication Rule
$$\frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d}$$
Reduce common factors between numerators and denominators first.
Fraction Division (Invert and Multiply)
$$\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} = \frac{a \cdot d}{b \cdot c}$$
Multiply by the reciprocal of the divisor.

Classification of Fractions & The Division Reciprocal Flip

Fractions: Typology, Invert-and-Multiply & BODMAS FRACTION TAXONOMY • Proper: N < D • e.g., 3/7 (< 1) • Improper: N ≥ D • e.g., 11/4 (≥ 1) • Mixed: Whole + Proper (2 ¾) • Like: Same denominator (3/8, 7/8) • Unlike: Diff denominators (2/3, 4/5) • Unit Fraction: Numerator = 1 (1/6) • Conversion Formula:   a b/c = (a×c + b) / c INVERT & MULTIPLY Dividing by a Fraction: (a/b) ÷ (c/d) ↓ Flip Divisor ↓ (a/b) × (d/c) • "Of" Operator means MULTIPLY:   ¾ of 28 = ¾ × 28 = 21 • "Of" evaluated BEFORE ÷ or ×! BODMAS HIERARCHY • 1. B: Innermost brackets first   ( ) → { } → [ ] • 2. O: "Of" operations • 3. D / M: Division & Multi (L to R) • 4. A / S: Add & Subtract (L to R) • Complex Fraction Rule:   (a/b) / (c/d) = (a×d) / (b×c) • Extremes × Extremes / Means × Means ALWAYS CONVERT MIXED FRACTIONS TO IMPROPER FRACTIONS BEFORE MULTIPLYING OR DIVIDING

Chapter Summary & 10 Key Takeaways

Takeaway 1
A proper fraction represents a quantity less than 1; an improper fraction represents a quantity >= 1.
Takeaway 2
Mixed fractions combine a whole number and a proper fraction: $a\frac{b}{c} = \frac{ac + b}{c}$.
Takeaway 3
Fractions are reduced to lowest terms by dividing numerator and denominator by their HCF.
Takeaway 4
Unlike fractions are compared by converting to equivalent fractions via the LCM of denominators.
Takeaway 5
The mathematical word "of" signifies multiplication: $\frac{a}{b} \text{ of } c = \frac{a}{b} \times c$.
Takeaway 6
The product of two fractions is the product of numerators divided by the product of denominators.
Takeaway 7
The reciprocal of a non-zero fraction $\frac{a}{b}$ is $\frac{b}{a}$; zero has no reciprocal.
Takeaway 8
To divide by a fraction, multiply by its reciprocal: $\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}$.
Takeaway 9
Complex multi-tiered fractions are evaluated as $(a/b) / (c/d) = (ad)/(bc)$.
Takeaway 10
Follow the BODMAS rule strictly: evaluate "Of" before division and multiplication.

Check Your Understanding (Diagnostic Practice Questions)

Diagnostic questions testing core conceptual clarity. Answers are hidden initially — solve each problem first, then click to reveal the step-by-step verified solution.

1
Simplify: $3\frac{1}{4} - \left[ 1\frac{1}{4} + \left\{ 1\frac{1}{2} - \left( \frac{1}{2} - \frac{1}{4} \right) \right\} \right]$.
Reveal Answer & Explanation
Answer: Step 1: Convert all mixed fractions to improper fractions:
$$3\frac{1}{4} = \frac{13}{4}, \quad 1\frac{1}{4} = \frac{5}{4}, \quad 1\frac{1}{2} = \frac{3}{2}$$
Step 2: Evaluate round brackets $\left( \frac{1}{2} - \frac{1}{4} \right) = \frac{2 - 1}{4} = \frac{1}{4}$.
Step 3: Evaluate curly braces $\left\{ \frac{3}{2} - \frac{1}{4} \right\} = \frac{6 - 1}{4} = \frac{5}{4}$.
Step 4: Evaluate square brackets $\left[ \frac{5}{4} + \frac{5}{4} \right] = \frac{10}{4} = \frac{5}{2}$.
Step 5: Perform final subtraction:
$$\frac{13}{4} - \frac{5}{2} = \frac{13 - 10}{4} = \mathbf{\frac{3}{4}}$$.
Convert mixed fractions to improper fractions first, then resolve brackets from innermost to outermost.
2
Arrange the fractions in ascending order: $\frac{2}{3}, \frac{5}{6}, \frac{7}{9}, \frac{11}{18}$.
Reveal Answer & Explanation
Answer: Step 1: Find the LCM of denominators $3, 6, 9, 18$:
$$\text{LCM}(3, 6, 9, 18) = 18$$
Step 2: Convert each fraction to an equivalent fraction with denominator $18$:
$$\frac{2}{3} = \frac{2 \times 6}{3 \times 6} = \frac{12}{18}$$
$$\frac{5}{6} = \frac{5 \times 3}{6 \times 3} = \frac{15}{18}$$
$$\frac{7}{9} = \frac{7 \times 2}{9 \times 2} = \frac{14}{18}$$
$$\frac{11}{18} = \frac{11}{18}$$
Step 3: Compare numerators: $11 < 12 < 14 < 15$.
Ascending order: $\mathbf{\frac{11}{18} < \frac{2}{3} < \frac{7}{9} < \frac{5}{6}}$.
Find LCM of denominators (18), convert each fraction, and order by numerators: $11 < 12 < 14 < 15$.
3
Evaluate using BODMAS: $\frac{4}{5} \text{ of } \left( \frac{3}{8} + \frac{5}{12} \right) \div \frac{19}{24}$.
Reveal Answer & Explanation
Answer: Step 1: Simplify inside brackets (LCM of $8$ and $12$ is $24$):
$$\frac{3}{8} + \frac{5}{12} = \frac{3 \times 3 + 5 \times 2}{24} = \frac{9 + 10}{24} = \frac{19}{24}$$
Step 2: Evaluate "of" (multiplication):
$$\frac{4}{5} \text{ of } \frac{19}{24} = \frac{4}{5} \times \frac{19}{24} = \frac{1 \times 19}{5 \times 6} = \frac{19}{30}$$
Step 3: Perform division by multiplying by the reciprocal:
$$\frac{19}{30} \div \frac{19}{24} = \frac{19}{30} \times \frac{24}{19} = \frac{24}{30} = \mathbf{\frac{4}{5}}$$.
Resolve the bracket first ($19/24$), evaluate "of" ($19/30$), and then divide by multiplying by the reciprocal ($24/19$).
4
A cord of length $14\frac{2}{5}\text{ meters}$ is cut into equal pieces of length $1\frac{1}{5}\text{ meters}$ each. How many pieces are obtained?
Reveal Answer & Explanation
Answer: Total length $= 14\frac{2}{5} = \frac{14 \times 5 + 2}{5} = \frac{72}{5}\text{ m}$.
Length of each piece $= 1\frac{1}{5} = \frac{1 \times 5 + 1}{5} = \frac{6}{5}\text{ m}$.
$$\text{Number of pieces} = \frac{\text{Total Length}}{\text{Length of each piece}} = \frac{72}{5} \div \frac{6}{5}$$
$$= \frac{72}{5} \times \frac{5}{6} = \frac{72}{6} = \mathbf{12 \text{ pieces}}$$.
Convert mixed fractions to improper ($72/5$ and $6/5$), and divide: $(72/5) \times (5/6) = 12$.
5
Simplify the complex fraction: $\frac{2 + \frac{1}{3}}{3 - \frac{1}{2}}$.
Reveal Answer & Explanation
Answer: Step 1: Simplify the numerator:
$$\text{Numerator} = 2 + \frac{1}{3} = \frac{6 + 1}{3} = \frac{7}{3}$$
Step 2: Simplify the denominator:
$$\text{Denominator} = 3 - \frac{1}{2} = \frac{6 - 1}{2} = \frac{5}{2}$$
Step 3: Divide numerator by denominator:
$$\frac{\frac{7}{3}}{\frac{5}{2}} = \frac{7}{3} \div \frac{5}{2} = \frac{7}{3} \times \frac{2}{5} = \mathbf{\frac{14}{15}}$$.
Simplify the top to $7/3$, simplify the bottom to $5/2$, and divide: $(7/3) \times (2/5) = 14/15$.
6
Out of an income of $\text{Rs. } 45,000$, a person spends $\frac{1}{3}$ on food, $\frac{1}{5}$ on rent, and $\frac{2}{15}$ on education. How much money is left as savings?
Reveal Answer & Explanation
Answer: Step 1: Total fraction of income spent:
$$\text{Fraction Spent} = \frac{1}{3} + \frac{1}{5} + \frac{2}{15}$$
LCM of $3, 5, 15$ is $15$:
$$= \frac{5 + 3 + 2}{15} = \frac{10}{15} = \frac{2}{3}$$
Step 2: Fraction of income saved:
$$\text{Fraction Saved} = 1 - \frac{2}{3} = \frac{1}{3}$$
Step 3: Total savings amount:
$$\text{Savings} = \frac{1}{3} \times 45,000 = \mathbf{\text{Rs. } 15,000}$$.
Sum the fractions spent: $1/3 + 1/5 + 2/15 = 2/3$. The remaining fraction is $1/3$. Calculate $1/3$ of 45,000.
7
Explain why the reciprocal of a proper fraction is always an improper fraction, while the reciprocal of an improper fraction is always a proper fraction.
Reveal Answer & Explanation
Answer:

• A proper fraction has $N < D$. When we take its reciprocal, it flips to $\frac{D}{N}$. Since the new numerator ($D$) is strictly greater than the new denominator ($N$), it satisfies the definition of an improper fraction ($> 1$). (e.g., reciprocal of $\frac{3}{7}$ is $\frac{7}{3}$).
• An improper fraction has $N > D$. Its reciprocal flips to $\frac{D}{N}$, where the new numerator ($D$) is smaller than the new denominator ($N$), which is a proper fraction ($< 1$). (e.g., reciprocal of $\frac{5}{2}$ is $\frac{2}{5}$).


Reciprocal flips numerator and denominator: if $N < D$, flipping gives $D/N$ which is greater than 1.
8
Find the reciprocal of $4\frac{3}{7}$.
Reveal Answer & Explanation
Answer: Step 1: Convert the mixed fraction into an improper fraction:
$$4\frac{3}{7} = \frac{(4 \times 7) + 3}{7} = \frac{28 + 3}{7} = \frac{31}{7}$$
Step 2: Invert the improper fraction to find its reciprocal:
$$\text{Reciprocal} = \mathbf{\frac{7}{31}}$$.
Convert to improper fraction ($31/7$) and flip it ($7/31$).
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