📝 Chapter Notes & Revision

Fractions and Decimals

🏫 MP BoardClass 7Mathematics

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Class 7 Mathematics

Chapter: Fractions and Decimals

1. Introduction to Fractions (भिन्न)

A fraction is a number representing a part of a whole. It is written in the form a/b, where a is the Numerator (अंश) and b is the Denominator (हर, b ≠ 0).

Types of Fractions:

  • Proper Fraction (उचित भिन्न): Numerator is less than the denominator (Numerator < Denominator). Example: 3/5.
  • Improper Fraction (विषम भिन्न): Numerator is greater than or equal to the denominator (Numerator ≥ Denominator). Example: 7/4.
  • Mixed Fraction (मिश्रित भिन्न): Combination of a whole number and a proper fraction. Example: 2 ¾.
    • Conversion: Mixed Fraction = Whole + (Numerator / Denominator) = (Whole × Denominator + Numerator) / Denominator.
  • Equivalent Fractions (तुल्य भिन्न): Fractions that represent the same value.
    • Rule: Multiply or divide both numerator and denominator by the same non-zero number. Example: 1/2 = 2/4 = 3/6.
  • Like Fractions (समान भिन्न): Fractions with the same denominators. Example: 2/7, 5/7.
  • Unlike Fractions (असमास भिन्न / असमान भिन्न): Fractions with different denominators. Example: 1/3, 2/5.

2. Operations on Fractions (भिन्न पर संक्रियाएँ)

A. Addition and Subtraction (जोड़ और घटाना)

  • Like Fractions: Add or subtract the numerators while keeping the denominator the same.
    • a/c + b/c = (a + b) / c
    • a/c - b/c = (a - b) / c
  • Unlike Fractions: First, find the LCM of the denominators to convert them into like fractions, then add or subtract.

B. Multiplication of Fractions (भिन्न का गुणा)

To multiply two fractions:

  1. Multiply their numerators together.
  2. Multiply their denominators together.
  • a/b × c/d = (a × c) / (b × d)
  • Fraction as an operator "of": "Of" means multiplication (×). Example: 1/2 of 4 = 1/2 × 4 = 2.

C. Division of Fractions (भिन्न का भाग)

  • Reciprocal (व्युत्क्रम): The reciprocal of a non-zero fraction a/b is b/a. (Product of a fraction and its reciprocal is always 1).
  • Rule for Division: To divide one fraction by another, multiply the first fraction by the reciprocal of the second fraction.
    • (a/b) ÷ (c/d) = a/b × d/c = (a × d) / (b × c)

3. Decimals (दशमलव)

A decimal number is another way of writing a fraction whose denominator is a power of 10 (like 10, 100, 1000, etc.). It consists of a Whole Number Part and a Decimal Part, separated by a Decimal Point (दशमलव बिंदु).

  • Example: 34.56
    • 3 is in the Tens place (3 × 10)
    • 4 is in the Ones place (4 × 1)
    • 5 is in the Tenths place (5 / 10 or 0.5)
    • 6 is in the Hundredths place (6 / 100 or 0.06)

Conversion:

  • Fraction to Decimal: Convert the denominator to 10, 100, etc., or use long division. (Example: 3/4 = 75/100 = 0.75).
  • Decimal to Fraction: Remove the decimal point and place 1 followed by as many zeros as there are digits after the decimal point in the denominator, then simplify. (Example: 0.6 = 6/10 = 3/5).

4. Operations on Decimals (दशमलव पर संक्रियाएँ)

A. Addition and Subtraction

  1. Write the numbers vertically lining up the decimal points.
  2. Add or subtract as whole numbers, keeping the decimal point in the same column.
  3. Fill empty decimal places with zeros if necessary.

B. Multiplication of Decimals

  1. Multiply the numbers ignoring the decimal point (as whole numbers).
  2. Count the total number of decimal places in both numbers combined.
  3. Place the decimal point in the product to give it that same number of decimal places, counting from right to left.
  • Multiplication by 10, 100, 1000: Move the decimal point to the right by as many places as there are zeros.
    • 2.34 × 10 = 23.4
    • 2.34 × 100 = 234

C. Division of Decimals

  1. Division by 10, 100, 1000: Move the decimal point to the left by as many places as there are zeros.
    • 45.2 ÷ 10 = 4.52
    • 45.2 ÷ 100 = 0.452
  2. Division of a Decimal by a Whole Number: Divide normally, placing the decimal point in the quotient directly above the decimal point in the dividend.
  3. Division of a Decimal by a Decimal: Shift the decimal point of both the divisor and the dividend to the right by the same number of places until the divisor becomes a whole number, then divide.

5. Units of Measurement (माप की इकाइयाँ)

Useful for decimal conversions in word problems:

  • 1 rupee = 100 paise $\rightarrow$ 1 paisa = ₹ 1/100 = ₹ 0.01
  • 1 cm = 10 mm $\rightarrow$ 1 mm = 1/10 cm = 0.1 cm
  • 1 m = 100 cm $\rightarrow$ 1 cm = 1/100 m = 0.01 m
  • 1 km = 1000 m $\rightarrow$ 1 m = 1/1000 km = 0.001 km
  • 1 kg = 1000 g $\rightarrow$ 1 g = 1/1000 kg = 0.001 kg