What this chapter is about
This chapter takes you deeper into the world of decimal numbers. You already know decimals from earlier classes. Now you will learn to work with them in more detail — comparing decimals, placing them on a number line, and doing all four operations (addition, subtraction, multiplication and division) with them.
Decimals appear everywhere in daily life. When you buy vegetables for ₹23.50, check your height as 1.45 metres, or read the temperature as 32.5°C, you are using decimals. This chapter helps you handle these numbers with confidence.
After studying this chapter, you should be able to compare any two decimals, add or subtract decimals of different lengths, multiply decimals by whole numbers and by other decimals, and divide decimals. These skills will help you in measurement, money calculations and science.
Key ideas
- A decimal number has two parts separated by a decimal point: the whole number part (left of the point) and the decimal part (right of the point).
- Each place after the decimal point has a value: the first place is tenths (1/10), the second is hundredths (1/100), the third is thousandths (1/1000), and so on.
- Adding zeros at the end of the decimal part does not change the value: 3.5 = 3.50 = 3.500.
- To compare decimals, first compare the whole number parts. If they are equal, compare digit by digit after the point, moving left to right.
- When adding or subtracting decimals, line up the decimal points and fill empty places with zeros.
- To multiply a decimal by 10, 100 or 1000, move the decimal point to the right by 1, 2 or 3 places.
- To divide a decimal by 10, 100 or 1000, move the decimal point to the left by 1, 2 or 3 places.
- When multiplying two decimals, multiply as if they are whole numbers, then count the total decimal places in both numbers and put that many places in the answer.
Formulas and facts to remember
- Place values after the point: First place = tenths, second place = hundredths, third place = thousandths.
- Comparing decimals: Compare whole parts first; if equal, compare tenths, then hundredths, and so on.
- Adding/subtracting decimals: Align decimal points, add zeros if needed, then compute.
- Multiplying by 10, 100, 1000: Move the point right by 1, 2 or 3 places.
- Dividing by 10, 100, 1000: Move the point left by 1, 2 or 3 places.
- Multiplying two decimals: Total decimal places in the answer = sum of decimal places in the two numbers.
- Dividing decimals: Make the divisor a whole number by shifting points in both divisor and dividend equally.
Worked examples
Example 1: Comparing decimals
Which is greater: 4.08 or 4.8?
Step 1: Compare whole parts. Both have 4. They are equal.
Step 2: Compare tenths. In 4.08, the tenths digit is 0. In 4.8, the tenths digit is 8.
Step 3: Since 8 > 0, we have 4.8 > 4.08.
Answer: 4.8 is greater.
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Example 2: Subtracting decimals
Ravi had ₹50. He bought a notebook for ₹32.75. How much money is left?
Step 1: Write 50 as 50.00 so both numbers have the same decimal places.
Step 2: Subtract: ``` 50.00
- 32.75
------- 17.25 ```
Answer: ₹17.25 is left.
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Example 3: Multiplying decimals
Find 2.5 × 1.4.
Step 1: Ignore the decimal points and multiply 25 × 14 = 350.
Step 2: Count decimal places: 2.5 has 1 place, 1.4 has 1 place. Total = 2 places.
Step 3: Put 2 decimal places in 350. We get 3.50 or simply 3.5.
Answer: 2.5 × 1.4 = 3.5
Common mistakes
- Thinking 4.12 is greater than 4.9 because 12 > 9 → Compare place by place; 1 tenth is less than 9 tenths.
- Forgetting to align decimal points when adding, getting a wrong answer → Always line up the points first.
- Moving the decimal point the wrong way when multiplying or dividing by 10 → Multiplying makes the number bigger, so the point goes right; dividing makes it smaller, so the point goes left.
- Dropping the decimal point entirely after multiplication → Count total decimal places and place the point correctly.
- Writing 0.30 as smaller than 0.3 → They are equal; the trailing zero does not add value.
Quick revision
- Tenths, hundredths, thousandths are the first three places after the decimal point.
- To compare, go digit by digit from left to right after checking whole parts.
- Line up the decimal points before adding or subtracting.
- Multiplying by 10 shifts the point one place to the right.
- In decimal multiplication, add the decimal places of both numbers for the answer.
- Trailing zeros after the decimal point do not change a number's value.