Chapter 4: Operations on Fractions

1. 🔑 Important Keywords and Definitions

• Fraction: A number that represents a part of a whole.
  Example:

  $\frac{2}{3}, \frac{5}{8}$

  32,85

• Proper Fraction: Numerator < Denominator.
  Example:

  $\frac{3}{5}$

  53

• Improper Fraction: Numerator ≥ Denominator.
  Example:

  $\frac{7}{4}$

  47

• Mixed Fraction: A whole number with a proper fraction.
  Example:

  $2\frac{1}{3}$

  231

• Equivalent Fractions: Fractions that represent the same value.
  Example:

  $\frac{2}{4} = \frac{1}{2}$

  42=21

• Like Fractions: Fractions with the same denominator.

• Unlike Fractions: Fractions with different denominators.

• Reciprocal: Inverse of a fraction.
  Example: Reciprocal of

  $\frac{3}{4}$

  43 is

  $\frac{4}{3}$

  34

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2. 🧠 Key Concepts and Explanations

• Fractions can be added, subtracted, multiplied, and divided.

• For addition/subtraction:

  • Convert to like fractions (same denominator).

• For multiplication:

  • Multiply numerators and denominators directly.

• For division:

  • Multiply by the reciprocal of the second fraction.

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3. 📏 Formulas and Rules

To Add/Subtract Fractions:

1. Find LCM of denominators.

2. Convert to like fractions.

3. Add or subtract numerators.

4. Simplify if needed.

To Multiply Fractions:

1. Multiply numerators and denominators.

2. Simplify the result.

To Divide Fractions:

1. Take reciprocal of the divisor.

2. Multiply with the dividend.

3. Simplify the answer.

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5. ✅ Examples with Full Solutions

Example 1:

$\frac{1}{3} + \frac{2}{3} = \frac{1+2}{3} = \frac{3}{3} = 1$

31+32=31+2=33=1

Example 2:

$\frac{2}{5} + \frac{3}{10}$

52+103
LCM of 5 and 10 = 10
Convert:

$\frac{2}{5} = \frac{4}{10}$

52=104

$\frac{4}{10} + \frac{3}{10} = \frac{7}{10}$

104+103=107

Example 3:

$\frac{4}{7} \times \frac{2}{5} = \frac{8}{35}$

74×52=358

Example 4:

$\frac{3}{4} \div \frac{2}{5} = \frac{3}{4} \times \frac{5}{2} = \frac{15}{8} = 1\frac{7}{8}$

43÷52=43×25=815=187

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6. ⚠️ Common Mistakes to Avoid

• Adding unlike fractions without converting to same denominator.

• Forgetting to take the reciprocal in division.

• Not simplifying the final answer.

• Mixing up rules of multiplication and addition.

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7. ✍️ Practice Questions

1. Add:

   $\frac{3}{5} + \frac{2}{10}$

   53+102

2. Subtract:

   $\frac{7}{8} – \frac{3}{8}$

   87−83

3. Multiply:

   $\frac{5}{6} \times \frac{2}{3}$

   65×32

4. Divide:

   $\frac{4}{9} \div \frac{2}{3}$

   94÷32

5. Convert to like fractions:

   $\frac{2}{3}, \frac{5}{6}$

   32,65

6. Write the reciprocal of:

   $\frac{7}{9}$

   97

7. Express

   $1\frac{3}{4}$

   143 as an improper fraction.

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8. 📊 Conceptual Diagrams

• Fraction strips or pie diagrams to show parts of a whole.

• Number line showing comparison of fractions.

• Flowchart for selecting the correct operation.

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9. 💡 Word Problems Section

Q: A recipe uses

$\frac{3}{4}$

43 cup of sugar. How much sugar is needed for 2 such recipes?
A:

$\frac{3}{4} \times 2 = \frac{6}{4} = 1\frac{1}{2}$

43×2=46=121 cups

Q: Raju walked

$\frac{2}{3}$

32 km in the morning and

$\frac{1}{6}$

61 km in the evening. How far did he walk in total?
A: LCM = 6 →

$\frac{2}{3} = \frac{4}{6}$

32=64

$\frac{4}{6} + \frac{1}{6} = \frac{5}{6}$

64+61=65 km

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10. 📝 Important Points / Quick Revision

• Always simplify your final answer.

• Convert mixed fractions to improper fractions before solving.

• Use LCM to make denominators equal.

• Multiplication and division are easier than addition and subtraction.

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11. 🔗 Connections to Other Chapters

• Helps in Decimals and Ratios.

• Required in Algebra and Mensuration.

• Applied in real life: recipes, construction, time management.

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12. 🎯 Extra Tips or Tricks

• Always cross-check numerators and denominators.

• For quick LCM, use factor method.

• Practice with real-life situations to build fluency.