Class 7 Notes

Class 7 Mathematics Notes Chapter 2 (Fractions and Decimals) – Mathematics Book

Philoid

Philoid

— 9 min read

Mathematics
Alright class, let's get down to business with Chapter 2: Fractions and Decimals. This is a fundamental chapter, and understanding these concepts thoroughly is crucial not just for your Class 7 exams but also forms the base for many quantitative aptitude questions in government exams. Pay close attention!

Chapter 2: Fractions and Decimals - Detailed Notes

1. Introduction to Fractions

  • Definition: A fraction represents a part of a whole or a collection. It is written in the form p/q, where 'p' is the Numerator (the number of parts considered) and 'q' is the Denominator (the total number of equal parts the whole is divided into). The denominator (q) cannot be zero.
  • Types of Fractions:
    • Proper Fraction: Numerator is less than the denominator (p < q). Represents a quantity less than one whole. Example: 2/3, 5/8, 1/10.
    • Improper Fraction: Numerator is greater than or equal to the denominator (p ≥ q). Represents a quantity equal to or greater than one whole. Example: 5/3, 7/7, 11/4.
    • Mixed Fraction (or Mixed Number): A combination of a whole number and a proper fraction. Represents a quantity greater than one whole. Example: 1 ¾ (which is 1 + 3/4), 2 ⅕ (which is 2 + 1/5).
  • Conversion:
    • Improper to Mixed: Divide the numerator by the denominator. The quotient is the whole number part, the remainder is the new numerator, and the denominator stays the same. Example: 11/4 -> 11 ÷ 4 = 2 with remainder 3 -> 2 ¾.
    • Mixed to Improper: Multiply the whole number by the denominator and add the numerator. This result becomes the new numerator, and the denominator stays the same. Example: 2 ¾ -> (2 * 4 + 3) / 4 = (8 + 3) / 4 = 11/4.

2. Equivalent Fractions

  • Fractions that represent the same value or proportion of the whole, even though they have different numerators and denominators.
  • How to find: Multiply or divide both the numerator and the denominator of a fraction by the same non-zero number.
    • Example: 1/2 = (1×2)/(2×2) = 2/4 = (1×5)/(2×5) = 5/10.
    • Example: 6/9 = (6÷3)/(9÷3) = 2/3.
  • Simplest Form (or Lowest Term): A fraction is in its simplest form when its numerator and denominator have no common factor other than 1 (their HCF is 1). To reduce a fraction to its simplest form, divide both the numerator and denominator by their Highest Common Factor (HCF). Example: 12/18 -> HCF(12, 18) = 6 -> (12÷6)/(18÷6) = 2/3.

3. Comparison of Fractions

  • Like Fractions (Same Denominator): The fraction with the larger numerator is greater. Example: 5/7 > 3/7.
  • Unlike Fractions (Different Denominators):
    • Method 1 (LCM): Convert the fractions into equivalent fractions with a common denominator (usually the LCM of the original denominators). Then compare the numerators. Example: Compare 2/3 and 3/4. LCM(3, 4) = 12. -> 2/3 = (2×4)/(3×4) = 8/12; 3/4 = (3×3)/(4×3) = 9/12. Since 9 > 8, 9/12 > 8/12, therefore 3/4 > 2/3.
    • Method 2 (Cross-Multiplication): For fractions a/b and c/d, compare a×d and b×c.
      • If a×d > b×c, then a/b > c/d.
      • If a×d < b×c, then a/b < c/d.
      • If a×d = b×c, then a/b = c/d.
        Example: Compare 2/3 and 3/4. -> 2×4 = 8; 3×3 = 9. Since 8 < 9, 2/3 < 3/4.

4. Operations on Fractions

  • Addition and Subtraction:
    • Like Fractions: Add or subtract the numerators and keep the common denominator. Example: 2/5 + 1/5 = (2+1)/5 = 3/5. Example: 4/7 - 1/7 = (4-1)/7 = 3/7.
    • Unlike Fractions: First, find the LCM of the denominators. Convert each fraction to an equivalent fraction with the LCM as the denominator. Then, add or subtract the numerators and keep the common denominator.
      Example: 1/2 + 1/3 -> LCM(2, 3) = 6. -> (1×3)/(2×3) + (1×2)/(3×2) = 3/6 + 2/6 = (3+2)/6 = 5/6.
      Example: 3/4 - 1/6 -> LCM(4, 6) = 12. -> (3×3)/(4×3) - (1×2)/(6×2) = 9/12 - 2/12 = (9-2)/12 = 7/12.
    • Mixed Fractions: Convert to improper fractions first, then perform addition/subtraction. Or, add/subtract the whole parts and fractional parts separately (be careful with borrowing in subtraction).
  • Multiplication:
    • Fraction by a Whole Number: Multiply the whole number by the numerator, keeping the denominator the same. Example: 3 × (2/5) = (3×2)/5 = 6/5.
    • Fraction by a Fraction: Multiply the numerators together and the denominators together. (Numerator × Numerator) / (Denominator × Denominator). Simplify the result if possible. Example: (2/3) × (4/5) = (2×4)/(3×5) = 8/15.
    • The word 'of' often means multiplication: Example: 1/2 of 10 means (1/2) × 10 = 10/2 = 5.
  • Division:
    • Reciprocal of a Fraction: To find the reciprocal, interchange the numerator and the denominator. The reciprocal of a/b is b/a (where a, b ≠ 0). The product of a fraction and its reciprocal is 1. Example: Reciprocal of 3/4 is 4/3.
    • Dividing Fractions: 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).
      Example: (2/3) ÷ (4/5) = (2/3) × (5/4) = (2×5)/(3×4) = 10/12 = 5/6 (simplified).
    • Dividing a Whole Number by a Fraction: Multiply the whole number by the reciprocal of the fraction. Example: 5 ÷ (2/3) = 5 × (3/2) = 15/2.
    • Dividing a Fraction by a Whole Number: Multiply the fraction by the reciprocal of the whole number (reciprocal of 'n' is 1/n). Example: (3/4) ÷ 2 = (3/4) ÷ (2/1) = (3/4) × (1/2) = 3/8.

5. Introduction to Decimals

  • Definition: Decimals are a way of writing fractions with denominators that are powers of 10 (like 10, 100, 1000, etc.). The decimal point separates the whole number part from the fractional part.
  • Place Value: The place value of digits to the right of the decimal point are tenths (1/10), hundredths (1/100), thousandths (1/1000), and so on.
    Example: In 23.456
    • 2 is in tens place (20)
    • 3 is in ones place (3)
    • 4 is in tenths place (4/10 or 0.4)
    • 5 is in hundredths place (5/100 or 0.05)
    • 6 is in thousandths place (6/1000 or 0.006)
  • Conversion:
    • Decimal to Fraction: Write the decimal number without the point as the numerator. The denominator is 1 followed by as many zeros as there are digits after the decimal point. Simplify the fraction. Example: 0.75 = 75/100 = 3/4. Example: 2.5 = 25/10 = 5/2.
    • Fraction to Decimal: Divide the numerator by the denominator. Example: 3/4 = 3 ÷ 4 = 0.75. Example: 1/8 = 1 ÷ 8 = 0.125.

6. Comparison of Decimals

  • Compare the whole number parts first. The decimal with the larger whole number part is greater.
  • If the whole number parts are equal, compare the digits starting from the tenths place, moving rightwards. The first place where the digits differ determines which decimal is greater.
  • You can add trailing zeros after the last digit to the right of the decimal point to make the number of decimal places equal for easier comparison. Example: Compare 2.5 and 2.45. -> Compare 2.50 and 2.45. Since 5 > 4 in the tenths place, 2.5 > 2.45. Example: Compare 0.07 and 0.1. -> Compare 0.07 and 0.10. Since 0 < 1 in the tenths place, 0.07 < 0.1.

7. Operations on Decimals

  • Addition and Subtraction:
    • Align the numbers vertically according to their decimal points.
    • Add or subtract as you would with whole numbers.
    • Place the decimal point in the answer directly below the decimal points in the numbers being added or subtracted. Add trailing zeros if needed to align.
      Example: 12.5 + 3.45 + 0.07 ->
      12.50
      3.45
    • 0.07
    16.02
    Example: 15.8 - 6.73 ->
    15.80
    • 6.73
    9.07
  • Multiplication:
    • By 10, 100, 1000, etc.: Shift the decimal point to the right by the number of zeros in the multiplier. Example: 2.345 × 10 = 23.45; 2.345 × 100 = 234.5.
    • By a Whole Number or Another Decimal: Multiply the numbers as if they were whole numbers (ignore the decimal points initially). Count the total number of digits after the decimal point in both factors. Place the decimal point in the product so that it has this total number of decimal places (counting from the right).
      Example: 2.5 × 0.3 -> Multiply 25 × 3 = 75. Total decimal places = 1 (in 2.5) + 1 (in 0.3) = 2. So, the answer is 0.75.
      Example: 1.2 × 0.15 -> Multiply 12 × 15 = 180. Total decimal places = 1 (in 1.2) + 2 (in 0.15) = 3. So, the answer is 0.180 or 0.18.
  • Division:
    • By 10, 100, 1000, etc.: Shift the decimal point to the left by the number of zeros in the divisor. Example: 234.5 ÷ 10 = 23.45; 234.5 ÷ 100 = 2.345.
    • By a Whole Number: Divide as with whole numbers. Place the decimal point in the quotient directly above the decimal point in the dividend.
      Example: 7.5 ÷ 3 -> 7.5 / 3 = 2.5.
    • By a Decimal: Make the divisor a whole number by shifting its decimal point to the right. Shift the decimal point in the dividend by the same number of places to the right (add zeros if necessary). Then divide as in division by a whole number.
      Example: 7.75 ÷ 0.25 -> Shift decimal 2 places right in both: 775 ÷ 25. -> 775 / 25 = 31.
      Example: 1.2 ÷ 0.03 -> Shift decimal 2 places right in both: 120 ÷ 3. -> 120 / 3 = 40.

Key Takeaways for Exams:

  • Master conversions: Mixed <-> Improper, Fraction <-> Decimal.
  • LCM is essential for adding/subtracting unlike fractions.
  • 'Of' means multiplication.
  • Division by a fraction means multiplication by its reciprocal.
  • Align decimal points carefully for addition/subtraction.
  • Count total decimal places for multiplication.
  • Shift decimal points correctly for division by decimals and powers of 10.
  • Practice word problems involving money, length, weight, etc., using fractions and decimals.

Multiple Choice Questions (MCQs)

Here are 10 MCQs to test your understanding. Choose the correct option.

  1. Which of the following is an improper fraction?
    (a) 7/8
    (b) 1 3/4
    (c) 8/7
    (d) 0.5

  2. The equivalent fraction of 3/5 with denominator 20 is:
    (a) 12/20
    (b) 15/20
    (c) 9/20
    (d) 6/20

  3. What is the sum of 2/3 and 1/4?
    (a) 3/7
    (b) 3/12
    (c) 11/12
    (d) 8/12

  4. Calculate: 5 ÷ (1/2)
    (a) 5/2
    (b) 10
    (c) 2/5
    (d) 5.5

  5. The value of 0.5 × 0.03 is:
    (a) 1.5
    (b) 0.15
    (c) 0.015
    (d) 0.0015

  6. Convert the decimal 2.35 into a fraction in its simplest form.
    (a) 235/100
    (b) 47/20
    (c) 235/10
    (d) 47/50

  7. Which is greater: 2/5 or 0.3?
    (a) 2/5
    (b) 0.3
    (c) They are equal
    (d) Cannot be compared

  8. A ribbon of length 5 ¼ m is cut into small pieces each of length ¾ m. How many pieces can be cut?
    (a) 5
    (b) 6
    (c) 7
    (d) 8

  9. What is the value of 43.07 ÷ 100?
    (a) 4307
    (b) 4.307
    (c) 0.4307
    (d) 430.7

  10. Find the value of: 3 ½ + 1 ¾ - 2 ¼
    (a) 3
    (b) 3 ½
    (c) 2 ¾
    (d) 3 ¼

Answer Key for MCQs:

  1. (c) 8/7 (Numerator > Denominator)
  2. (a) 12/20 (Multiply numerator and denominator by 4)
  3. (c) 11/12 (LCM is 12. 8/12 + 3/12 = 11/12)
  4. (b) 10 (5 × 2/1 = 10)
  5. (c) 0.015 (5 × 3 = 15. Total 3 decimal places)
  6. (b) 47/20 (235/100 = 47/20 after dividing by 5)
  7. (a) 2/5 (2/5 = 0.4, and 0.4 > 0.3)
  8. (c) 7 (Convert to improper: 21/4 ÷ 3/4 = 21/4 × 4/3 = 21/3 = 7)
  9. (c) 0.4307 (Shift decimal 2 places left)
  10. (d) 3 ¼ (Convert to improper: 7/2 + 7/4 - 9/4 = 14/4 + 7/4 - 9/4 = (14+7-9)/4 = 12/4 = 3. Wait, let's recheck calculation: 7/2 + 7/4 - 9/4 = 14/4 + (7-9)/4 = 14/4 - 2/4 = 12/4 = 3. Let's try whole/fraction parts: (3+1-2) + (1/2 + 3/4 - 1/4) = 2 + (2/4 + 3/4 - 1/4) = 2 + (2+3-1)/4 = 2 + 4/4 = 2 + 1 = 3. Hmm, let me re-read the question and options. 3 ½ + 1 ¾ - 2 ¼. 7/2 + 7/4 - 9/4 = 14/4 + 7/4 - 9/4 = (14+7-9)/4 = 12/4 = 3. Option (a) is 3. Let me re-evaluate the options. Option (d) is 3 ¼. Let me check the arithmetic again. 3 ½ = 3.5, 1 ¾ = 1.75, 2 ¼ = 2.25. So, 3.5 + 1.75 - 2.25 = 5.25 - 2.25 = 3.00. The answer is exactly 3. Option (a) is 3. Option (d) is 3 ¼ which is 13/4. Let me check the options again. Maybe I made a mistake in the question or options. Let's assume the question is correct. 12/4 = 3. The answer is 3. Option (a) is 3. Option (d) is 3 ¼. Let me stick with 3. Ah, wait, let me check the calculation one last time. (3 + 1 - 2) + (1/2 + 3/4 - 1/4) = 2 + (2/4 + 3/4 - 1/4) = 2 + (5/4 - 1/4) = 2 + 4/4 = 2 + 1 = 3. Okay, the answer is definitely 3. Option (a) seems correct. Let me assume option (d) was a typo in my generation and should have been 3. But sticking to the generated options, if the calculation is correct, (a) is the answer. Let me select (a). Correction: Re-reading the question and options. It's possible I intended a different calculation. Let's assume the answer key I might have had in mind was (d). Could 3 ½ + 1 ¾ - 2 ¼ lead to 3 ¼? (14+7-9)/4 = 12/4 = 3. No. Let's trust the math. The answer is 3. So (a) is correct. I will mark (a) as the correct answer.)

Corrected Answer Key:

  1. (c)
  2. (a)
  3. (c)
  4. (b)
  5. (c)
  6. (b)
  7. (a)
  8. (c)
  9. (c)
  10. (a)

Make sure you practice plenty of problems based on these concepts. Good luck with your preparation!

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