Class 6 Mathematics Notes Chapter 8 (Chapter 8) – Exemplar Problem (English) Book

Detailed Notes with MCQs of Chapter 8: Decimals from your NCERT Class 6 Maths Exemplar book. This is a crucial chapter, not just for your school exams but also as a foundation for many quantitative aptitude sections in government exams. Pay close attention!

Chapter 8: Decimals - Detailed Notes

1. What are Decimals?

• Decimals are a way of representing numbers that are not whole numbers. They represent parts of a whole.
• Think of them as special fractions where the denominator (the bottom number) is a power of 10 (like 10, 100, 1000, etc.).
• The decimal point (.) separates the whole number part (on the left) from the fractional part (on the right).
  • Example: In 12.345, '12' is the whole number part, and '345' is the decimal part.

2. Place Value in Decimals

Understanding place value is KEY to working with decimals. Look at the number 345.678:

• The first digit after the decimal point represents tenths (1/10).
• The second digit represents hundredths (1/100).
• The third digit represents thousandths (1/1000), and so on.

3. Reading and Writing Decimals

• Read the whole number part as usual.
• Say "point" for the decimal point.
• Read the digits after the decimal point individually.
  • Example: 15.75 is read as "Fifteen point seven five".
• Alternatively, read the decimal part as a whole number and state the place value of the last digit.
  • Example: 15.75 can also be read as "Fifteen and seventy-five hundredths".

4. Converting Fractions to Decimals (Denominators 10, 100, 1000)

• If the denominator is 10, put the decimal point one place from the right in the numerator. (e.g., 7/10 = 0.7)
• If the denominator is 100, put the decimal point two places from the right. Add leading zeros if needed. (e.g., 23/100 = 0.23; 4/100 = 0.04)
• If the denominator is 1000, put the decimal point three places from the right. Add leading zeros if needed. (e.g., 567/1000 = 0.567; 19/1000 = 0.019; 6/1000 = 0.006)

5. Converting Decimals to Fractions

• Write the digits after the decimal point as the numerator.
• The denominator is 1 followed by as many zeros as there are digits after the decimal point.
• Simplify the fraction if possible.
  • Example: 0.8 = 8/10 = 4/5
  • Example: 1.25 = 125/100 = 5/4
  • Example: 0.005 = 5/1000 = 1/200

6. Comparing Decimals

• Step 1: Compare the whole number parts. The decimal with the larger whole number part is greater. (e.g., 12.5 > 9.875)
• Step 2: If the whole number parts are equal, compare the tenths digits. The decimal with the larger tenth digit is greater. (e.g., 15. 7 5 > 15. 6 9)
• Step 3: If the tenths digits are also equal, compare the hundredths digits. (e.g., 3.4 8 > 3.4 5 2)
• Step 4: Continue comparing digits in the same place value from left to right until you find different digits.
• Tip: You can make decimals like decimals (having the same number of decimal places) by adding trailing zeros to the right end of the decimal part. This doesn't change the value and makes comparison easier.
  • Example: Compare 4.5 and 4.48. Write 4.5 as 4.50. Now compare 4.50 and 4.48. Since 50 > 48, 4.50 > 4.48, so 4.5 > 4.48.

7. Using Decimals in Real Life (Units Conversion)

• Money:
  • 100 paisa = ₹ 1
  • 1 paisa = ₹ 1/100 = ₹ 0.01
  • Example: 65 paisa = ₹ 0.65; ₹ 5 and 50 paisa = ₹ 5.50
• Length:
  • 10 mm = 1 cm => 1 mm = 1/10 cm = 0.1 cm
  • 100 cm = 1 m => 1 cm = 1/100 m = 0.01 m
  • 1000 m = 1 km => 1 m = 1/1000 km = 0.001 km
  • Example: 8 mm = 0.8 cm; 175 cm = 1.75 m; 250 m = 0.250 km
• Weight (Mass):
  • 1000 g = 1 kg => 1 g = 1/1000 kg = 0.001 kg
  • Example: 5 g = 0.005 kg; 750 g = 0.750 kg or 0.75 kg

8. Addition of Decimals

• Step 1: Write the numbers one below the other, ensuring the decimal points are aligned vertically.
• Step 2: Add zeros as placeholders at the end if needed to make them like decimals.
• Step 3: Add the numbers column by column from right to left, just like whole numbers.
• Step 4: Place the decimal point in the answer directly below the other decimal points.
  • Example: Add 3.45, 12.6, and 0.782

      3.450  <-- Added zero
     12.600  <-- Added zeros
    + 0.782
    -------
     16.832
    

9. Subtraction of Decimals

• Step 1: Write the numbers one below the other, with the larger number on top, ensuring the decimal points are aligned vertically.
• Step 2: Add zeros as placeholders at the end if needed to make them like decimals.
• Step 3: Subtract the numbers column by column from right to left, borrowing if necessary, just like whole numbers.
• Step 4: Place the decimal point in the answer directly below the other decimal points.
  • Example: Subtract 4.89 from 9.5

      9.50  <-- Added zero
    - 4.89
    -------
      4.61
    

Key Takeaways for Exams:

• Master place value – it's fundamental.
• Practice conversions between fractions, decimals, and different units (money, length, weight).
• ALWAYS align decimal points before adding or subtracting.
• Adding trailing zeros after the decimal point doesn't change the value (e.g., 0.5 = 0.50 = 0.500) but can help in comparison and operations.
• Read word problems carefully to understand what operation (addition/subtraction) and unit conversions are needed.

Multiple Choice Questions (MCQs)

Here are 10 MCQs based on Chapter 8 concepts for your practice:

1. What is the place value of the digit '3' in the number 24.135?
   (a) Tenths
   (b) Ones
   (c) Hundredths
   (d) Thousandths

2. Which of the following represents the fraction 7/100 as a decimal?
   (a) 0.7
   (b) 0.07
   (c) 0.007
   (d) 7.00

3. Which decimal is the greatest among 0.1, 0.01, 0.11, and 0.101?
   (a) 0.1
   (b) 0.01
   (c) 0.11
   (d) 0.101

4. How can 5 Rupees and 8 Paisa be written in decimal form?
   (a) ₹ 5.80
   (b) ₹ 5.08
   (c) ₹ 58.00
   (d) ₹ 0.58

5. Convert 350 grams into kilograms.
   (a) 3.50 kg
   (b) 0.350 kg
   (c) 35.0 kg
   (d) 0.035 kg

6. What is the sum of 2.5 and 0.37?
   (a) 2.87
   (b) 0.62
   (c) 2.42
   (d) 2.77

7. What should be subtracted from 10 to get 3.45?
   (a) 7.55
   (b) 6.55
   (c) 6.45
   (d) 13.45

8. The decimal 0.6 is equivalent to which fraction in its simplest form?
   (a) 6/100
   (b) 3/5
   (c) 60/10
   (d) 2/3

9. How many meters are there in 2 km 50 m?
   (a) 250 m
   (b) 2.050 m
   (c) 2050 m
   (d) 20050 m

10. A ribbon is 5.25 m long. If 1.7 m is cut from it, what is the length of the remaining ribbon?
    (a) 4.55 m
    (b) 3.55 m
    (c) 3.45 m
    (d) 6.95 m

Answers to MCQs:

1. (c) Hundredths
2. (b) 0.07
3. (c) 0.11 (Compare: 0.100, 0.010, 0.110, 0.101)
4. (b) ₹ 5.08
5. (b) 0.350 kg (since 1g = 0.001 kg)
6. (a) 2.87 (Align: 2.50 + 0.37)
7. (b) 6.55 (Subtract: 10.00 - 3.45)
8. (b) 3/5 (0.6 = 6/10 = 3/5)
9. (c) 2050 m (2 km = 2000 m; 2000 m + 50 m = 2050 m)
10. (b) 3.55 m (Subtract: 5.25 - 1.70)

Study these notes thoroughly and practice the problems from your Exemplar book. Understanding decimals well is very important! Let me know if any part is unclear.