Class 6 Mathematics: Master the World of Decimals!

50**, measured your height as 1.35 meters, or seen a running race won by 0.02 seconds?

Where do these numbers with little dots come from? They are called Decimals!

In your NCERT Class 6 Math journey, understanding decimals is like unlocking a superpower—it lets us talk about parts of a whole with ultimate precision. Let’s dive in and explore the magical world of decimals step-by-step!

1. What is a Decimal Number?

Imagine you have a full slab of delicious chocolate bar. You divide it into 10 equal parts.

• If you eat 1 part, you have eaten $\frac{1}{10}$ (one-tenth) of the chocolate.
• In decimal language, we write $\frac{1}{10}$ as $0.1$ (read as "zero point one").

If you eat 3 parts, you have eaten $\frac{3}{10}$ of the chocolate, which we write as $0.3$.

💡 Teacher's Definition: A decimal number consists of two main parts separated by a decimal point (.):

1. Whole Number Part (to the left of the decimal point)
2. Decimal Part / Fractional Part (to the right of the decimal point)

Example: $25.7$

• $25$ is the Whole Number Part.
• $.$ is the Decimal Point.
• $7$ is the Fractional Part (representing $\frac{7}{10}$).

2. Understanding Place Value in Decimals

You already know place values for whole numbers: Ones, Tens, Hundreds, Thousands, and so on. As you move from right to left, each place becomes $10$ times bigger.

When we move to the right of the decimal point, each place becomes $10$ times smaller!

Here is how the decimal place value chart looks:

Breaking Down $143.258$:

• $1$ Hundreds $= 100$
• $4$ Tens $= 40$
• $3$ Ones $= 3$
• $2$ Tenths $= \frac{2}{10} = 0.2$
• $5$ Hundredths $= \frac{5}{100} = 0.05$
• $8$ Thousandths $= \frac{8}{1000} = 0.008$

Expanded Form:
$143.258 = 100 + 40 + 3 + \frac{2}{10} + \frac{5}{100} + \frac{8}{1000}$

3. How to Compare Decimal Numbers

Suppose your friend says, "I have ₹5.8, and you have ₹5.75. So, you have more money!" Is your friend correct? Let’s find out!

To compare any two decimal numbers, follow these 3 simple steps:

Step-by-Step Rule for Comparison:

1. Compare the Whole Number Part first: The number with the larger whole number part is greater.
   • Example: $12.35 > 9.89$ because $12 > 9$.
2. If the Whole Number Parts are equal, compare the Tenths digit:
   • Example: $5.8$ vs $5.75$.
   • Both have $5$ in the ones place.
   • Look at the tenths place: $8$ tenths vs $7$ tenths.
   • Since $8 > 7$, $5.8 > 5.75$! (Your friend was wrong!)
3. If the Tenths digits are also equal, compare the Hundredths digit, and so on.

✏️ Pro-Tip (Like Decimals): Add extra zeros at the end of a decimal number to make comparison easy—it doesn't change the value!

• $5.8$ is the exact same as $5.80$.
• Now comparing $5.80$ and $5.75$ is super easy: $80$ hundredths is bigger than $75$ hundredths!

4. Basic Operations on Decimals

A. Addition of Decimals

Adding decimals is just like adding regular whole numbers. The Golden Rule is:
👉 Line up the decimal points vertically!

Example: Add $14.25$ and $7.8$

Step 1: Write the numbers one below the other so that the decimal points align vertically.
Step 2: Fill empty spaces with zero so both numbers have the same number of decimal digits (like decimals).
Step 3: Add normally from right to left and place the decimal point directly below in the answer.

   1 4 . 2 5
+  0 7 . 8 0   <-- (Added 0 to make 7.8 into 7.80)
-------------
   2 2 . 0 5
-------------

Answer: $22.05$

B. Subtraction of Decimals

Just like addition, subtraction requires lining up the decimal points!

Example: Subtract $12.35$ from $20.5$

Step 1: Write the bigger number on top and line up the decimal points.
Step 2: Add a trailing zero to $20.5$ to make it $20.50$.
Step 3: Subtract as usual with borrowing.

   1 10 . 4 10
   2  0 . 5  0
-  1  2 . 3  5
---------------
   0  8 . 1  5
---------------

Answer: $8.15$

5. Quick Recap Checklist ✔️

• A decimal point separates whole numbers from parts of a whole.
• Moving right from the decimal point gives: Tenths ($\frac{1}{10}$) $\rightarrow$ Hundredths ($\frac{1}{100}$) $\rightarrow$ Thousandths ($\frac{1}{1000}$).
• Trailing zeros at the very end of a decimal number do not change its value ($0.5 = 0.50 = 0.500$).
• Golden Rule for $+ / -$: ALWAYS line up the decimal points before adding or subtracting!

📝 Practice Questions with Step-by-Step Solutions

Try solving these questions yourself first, then check the detailed solutions below!

Question 1 (Place Value & Expansion)

Write the following expanded form as a single decimal number and state the place value of the digit $7$:

$200 + 40 + 5 + \frac{3}{10} + \frac{7}{100}$

Solution:

1. Combine the Whole Number Part: $200 + 40 + 5 = 245$

2. Combine the Fractional Part:

   • $\frac{3}{10} = 3\text{ tenths} = 0.3$
   • $\frac{7}{100} = 7\text{ hundredths} = 0.07$
   • Combined decimal part $= 0.37$

3. Put Whole Number and Decimal Parts Together: $245 + 0.37 = \mathbf{245.37}$

4. Identify the Place Value of $7$:

   • The digit $7$ is two places to the right of the decimal point.
   • Therefore, the place value of $7$ is $7\text{ Hundredths}$ or $\frac{7}{100}$ ($0.07$).

Question 2 (Comparing Decimals)

Rohan ran a race in $12.68$ seconds, while Sohan ran the same race in $12.609$ seconds. Who ran faster and by how much time?

Solution:

1. Understand the problem:

   • In a race, the person who takes less time is the faster runner!

2. Compare $12.68$ and $12.609$:

   • Convert both numbers to like decimals by adding a zero:
     • Rohan's time $= 12.680\text{ seconds}$
     • Sohan's time $= 12.609\text{ seconds}$
   • Compare whole numbers: $12 = 12$
   • Compare tenths: $6 = 6$
   • Compare hundredths: $8 > 0$
   • Therefore, $12.680 > 12.609$.

3. Determine who is faster:

   • Since $12.609 < 12.680$, Sohan took less time and is the faster runner.

4. Find the time difference (Subtraction):

   1 2 . 6 8 0
-  1 2 . 6 0 9
---------------
   0 0 . 0 7 1
---------------

Answer: Sohan ran faster by $0.071\text{ seconds}$.

Question 3 (Real-World Word Problem)

Sunita bought $3.25\text{ kg}$ of apples and $2.5\text{ kg}$ of mangoes. She gave $1.75\text{ kg}$ of fruit to her friend. How many kilograms of fruit are left with Sunita?

Solution:

Step 1: Find the total weight of fruits bought (Addition)

• Weight of apples $= 3.25\text{ kg}$
• Weight of mangoes $= 2.50\text{ kg}$ (made into like decimal)

   3 . 2 5
+  2 . 5 0
-----------
   5 . 7 5
-----------

• Total fruit bought $= \mathbf{5.75\text{ kg}}$

Step 2: Find the remaining fruit after giving away $1.75\text{ kg}$ (Subtraction)

• Total weight $= 5.75\text{ kg}$
• Weight given away $= 1.75\text{ kg}$

   5 . 7 5
-  1 . 7 5
-----------
   4 . 0 0
-----------

Answer: Sunita is left with $4\text{ kg}$ (or $4.00\text{ kg}$) of fruit.

Great job completing this tutorial! Keep practicing, and remember: math is all around us, especially in the little details like decimals!

Common Student Mistakes to Avoid

1. Sign Errors in Algebraic Calculations: Mistakes in distributing negative signs across brackets or when transferring terms across the equals sign.
2. Formula Misapplication: Memorizing formulas without checking required units or conditions (e.g. using diameter instead of radius).
3. Skipping Intermediate Steps: Jumping directly to final numerical answers without showing step-by-step mathematical working, leading to partial credit loss in board exams.
4. Incorrect Unit Conversions: Forgetting to convert parameters into uniform SI units (e.g., cm to meters or minutes to seconds) before computing.

Exam Preparation & Frequently Asked Questions (FAQ)

Q1. How should I revise Decimals for the Class 6 Mathematics examination?

Focus on mastering core textbook definitions, practicing 3-4 numerical problems daily with pen and paper, and reviewing previous year CBSE/NCERT board exam questions.

Q2. What are the key concepts that carry maximum marks in this chapter?

Pay special attention to core definitions, step-by-step derivations, solved textbook examples, and practical real-world applications outlined in your NCERT curriculum.

Q3. How can I avoid losing marks in long answer questions?

Always structure your answers with clear subheadings, write step-by-step working for numerical problems, state given values clearly, and highlight your final answers with correct SI units.