Decimals - Place value, comparison, and basic operations on decimal numbers
Class 6 Mathematics: Master the World of Decimals!
Welcome to another exciting lesson, my young mathematicians! Have you ever noticed price tags like ₹45.50, 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!
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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 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: $25.7$
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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:
| Hundreds ($100$) | Tens ($10$) | Ones ($1$) | Decimal Point (.) | Tenths ($\frac{1}{10}$) | Hundredths ($\frac{1}{100}$) | Thousandths ($\frac{1}{1000}$) |
| :---: | :---: | :---: | :---: | :---: | :---: | :---: |
| 1 | 4 | 3 | . | 2 | 5 | 8 |
Breaking Down $143.258$:
Expanded Form:
$$143.258 = 100 + 40 + 3 + \frac{2}{10} + \frac{5}{100} + \frac{8}{1000}$$
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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:
* *Example:* $12.35 > 9.89$ because $12 > 9$.
* *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!)
> ✏️ 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!
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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.
```text
1 4 . 2 5
+ 0 7 . 8 0 <-- (Added 0 to make 7.8 into 7.80)
-------------
2 2 . 0 5
-------------
```
Answer: $22.05$
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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.
```text
1 10 . 4 10
2 0 . 5 0
---------------
0 8 . 1 5
---------------
```
Answer: $8.15$
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5. Quick Recap Checklist ✔️
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📝 Practice Questions with Step-by-Step Solutions
Try solving these questions yourself first, then check the detailed solutions below!
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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:
$$200 + 40 + 5 = 245$$
* $\frac{3}{10} = 3\text{ tenths} = 0.3$
* $\frac{7}{100} = 7\text{ hundredths} = 0.07$
* Combined decimal part $= 0.37$
$$245 + 0.37 = \mathbf{245.37}$$
* The digit $7$ is two places to the right of the decimal point.
* Therefore, the place value of $7$ is $7\text{ Hundredths}$ or $\frac{7}{100}$ ($0.07$).
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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:
* In a race, the person who takes less time is the faster runner!
* 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, $12.680 > 12.609$.
* Since $12.609 < 12.680$, Sohan took less time and is the faster runner.
```text
1 2 . 6 8 0
---------------
0 0 . 0 7 1
---------------
```
Answer: Sohan ran faster by $0.071\text{ seconds}$.
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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)
```text
3 . 2 5
+ 2 . 5 0
-----------
5 . 7 5
-----------
```
Step 2: Find the remaining fruit after giving away $1.75\text{ kg}$ (Subtraction)
```text
5 . 7 5
-----------
4 . 0 0
-----------
```
Answer: Sunita is left with $4\text{ kg}$ (or $4.00\text{ kg}$) of fruit.
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*Great job completing this tutorial! Keep practicing, and remember: math is all around us, especially in the little details like decimals!*