Division of Decimals
After learning how to multiply decimals, the natural next step is division of decimals. Division of decimals answers questions like: "If 6.5 metres of cloth costs Rs. 325, what is the cost per metre?" or "If you walk 3.75 km in 1.5 hours, what is your speed?" These calculations come up all the time in real life.
Division of decimals might seem difficult at first, but there are simple methods that make it easy. The key idea is to convert the division problem into one that involves whole numbers. You can do this by moving the decimal point in both the divisor and the dividend by the same number of places. Once you have whole numbers, you divide normally. Another approach is to convert decimals to fractions, divide, and convert back.
In Class 7 NCERT Maths, you will learn three main types: dividing a decimal by a whole number, dividing a decimal by another decimal, and dividing a whole number by a decimal. We will also revisit how dividing by 10, 100, and 1000 simply shifts the decimal point to the left. With clear steps, plenty of examples, and real-world problems, you will master decimal division in this chapter.
Think of decimal division as the opposite of decimal multiplication. If 4 pencils cost Rs. 9.60, the price per pencil is 9.60 / 4 = Rs. 2.40. If a jug holds 1.5 litres and each glass holds 0.3 litres, the number of glasses you can fill is 1.5 / 0.3 = 5. These are questions you encounter in daily life, in shops, in science experiments, and in your maths textbook.
One interesting thing about dividing by decimals less than 1: the answer is always bigger than the number you started with. For example, 6 / 0.5 = 12. This seems surprising, but think of it this way: how many half-litre bottles can you fill from 6 litres? The answer is 12, which is larger than 6. Keeping this idea in mind will help you avoid mistakes and develop a strong number sense.
What is Division of Decimals?
Division of decimals means splitting a decimal number into equal parts or finding how many times one decimal fits into another. It is the inverse (opposite) of multiplication of decimals.
The main strategy for dividing decimals is to make the divisor a whole number by moving the decimal point to the right. You then move the decimal point in the dividend by the same number of places. After that, you divide as you would with whole numbers.
Method: Making the Divisor a Whole Number
Step 1: If the divisor is a decimal, count how many decimal places it has.
Step 2: Move the decimal point in BOTH the divisor and the dividend to the right by that many places.
Step 3: Now the divisor is a whole number. Divide normally.
Step 4: Place the decimal point in the quotient directly above where it appears in the dividend.
Example: 4.56 / 0.3
- The divisor 0.3 has 1 decimal place.
- Move decimal in both: 4.56 becomes 45.6, and 0.3 becomes 3.
- Now divide: 45.6 / 3 = 15.2
So 4.56 / 0.3 = 15.2.
Dividing by 10, 100, 1000:
- Dividing by 10: move decimal 1 place to the left. Example: 45.6 / 10 = 4.56
- Dividing by 100: move decimal 2 places left. Example: 45.6 / 100 = 0.456
- Dividing by 1000: move decimal 3 places left. Example: 45.6 / 1000 = 0.0456
Division of Decimals Formula
Dividing a Decimal by a Whole Number:
Divide normally, placing the decimal point in the quotient directly above the decimal point in the dividend.
Dividing a Decimal by a Decimal:
Move the decimal point in both divisor and dividend to the right until the divisor becomes a whole number. Then divide normally.
Dividing by Powers of 10:
| Divide by | Move decimal point | Example |
|---|---|---|
| 10 | 1 place left | 56.7 / 10 = 5.67 |
| 100 | 2 places left | 56.7 / 100 = 0.567 |
| 1000 | 3 places left | 56.7 / 1000 = 0.0567 |
Alternative Method (Using Fractions):
Convert the decimal to a fraction, perform the division, and convert back to a decimal.
Example: 0.6 / 0.2 = (6/10) / (2/10) = 6/2 = 3
Types and Properties
There are several types of decimal division problems:
Type 1: Decimal / Whole Number
This is the simplest type. Divide normally and place the decimal point in the quotient above its position in the dividend. Example: 15.6 / 4. Divide 156 by 4 = 39. Since the decimal was after the first digit (15.6), the answer is 3.9.
Think of it as: if 4 friends share Rs. 15.60 equally, each gets Rs. 3.90.
Type 2: Decimal / Decimal
Make the divisor a whole number by moving both decimal points. Example: 7.2 / 0.9. Move both by 1 place: 72 / 9 = 8. So 7.2 / 0.9 = 8.
Another example: 3.45 / 0.15. Move both by 2 places: 345 / 15 = 23. So 3.45 / 0.15 = 23.
Type 3: Whole Number / Decimal
Make the divisor a whole number by moving decimal points. Example: 6 / 0.3. Move both by 1 place: 60 / 3 = 20. So 6 / 0.3 = 20. This tells us that twenty 0.3-sized pieces make up 6.
Type 4: Dividing by 10, 100, 1000
Simply move the decimal point to the left. Example: 250 / 100 = 2.50 = 2.5. This is useful for converting units (like paise to rupees, or cm to metres).
Type 5: Division Giving a Non-Terminating Decimal
Sometimes the division does not end neatly. Example: 10 / 3 = 3.333... In such cases, we either round the answer to a specified number of decimal places or express it as a fraction (10/3 = 3 1/3). Another example: 1 / 3 = 0.333..., and 2 / 7 = 0.285714285714... (a repeating decimal).
Type 6: Division Requiring Extra Decimal Places in the Dividend
Sometimes you need to add trailing zeros to the dividend to continue dividing. Example: 5 / 8 = 0.625. You write 5 as 5.000 and divide 50 by 8 = 6 remainder 2, then 20 by 8 = 2 remainder 4, then 40 by 8 = 5. Result: 0.625.
Important Observation: When you divide a number by a decimal less than 1, the quotient is larger than the dividend. For example, 5 / 0.5 = 10. This makes sense: you are asking how many half-portions fit into 5, and the answer is 10. Conversely, dividing by a number greater than 1 always gives a result smaller than the dividend (e.g., 10 / 2.5 = 4, which is less than 10).
Solved Examples
Example 1: Dividing a Decimal by a Whole Number
Problem: Find: 18.6 / 3
Solution:
Step 1: The divisor (3) is already a whole number.
Step 2: Divide as usual: 186 / 3 = 62
Step 3: Place the decimal: 18.6 has 1 decimal place, so 62 becomes 6.2.
Answer: 18.6 / 3 = 6.2
Example 2: Dividing a Decimal by a Decimal (1 Decimal Place)
Problem: Find: 8.4 / 0.7
Solution:
Step 1: The divisor 0.7 has 1 decimal place.
Step 2: Move decimal in both by 1 place: 8.4 becomes 84, and 0.7 becomes 7.
Step 3: Divide: 84 / 7 = 12
Answer: 8.4 / 0.7 = 12
Example 3: Dividing a Decimal by a Decimal (2 Decimal Places)
Problem: Find: 6.25 / 0.25
Solution:
Step 1: The divisor 0.25 has 2 decimal places.
Step 2: Move decimal in both by 2 places: 6.25 becomes 625, and 0.25 becomes 25.
Step 3: Divide: 625 / 25 = 25
Answer: 6.25 / 0.25 = 25
Example 4: Dividing a Whole Number by a Decimal
Problem: Find: 9 / 0.3
Solution:
Step 1: The divisor 0.3 has 1 decimal place.
Step 2: Move decimal in both by 1 place: 9 becomes 90, and 0.3 becomes 3.
Step 3: Divide: 90 / 3 = 30
Answer: 9 / 0.3 = 30
Example 5: Dividing by 10, 100, and 1000
Problem: Find: (a) 85.4 / 10, (b) 85.4 / 100, (c) 85.4 / 1000
Solution:
(a) Move decimal 1 place left: 85.4 / 10 = 8.54
(b) Move decimal 2 places left: 85.4 / 100 = 0.854
(c) Move decimal 3 places left: 85.4 / 1000 = 0.0854
Answer: (a) 8.54, (b) 0.854, (c) 0.0854
Example 6: Using the Fraction Method
Problem: Find: 0.8 / 0.04 using fractions.
Solution:
Step 1: Convert to fractions: 0.8 = 8/10 and 0.04 = 4/100
Step 2: Divide: (8/10) / (4/100) = 8/10 x 100/4
Step 3: Cross-cancel: 8 and 4 (cancel to 2 and 1), 100 and 10 (cancel to 10 and 1)
Step 4: Multiply: 2/1 x 10/1 = 20
Answer: 0.8 / 0.04 = 20
Example 7: Word Problem: Cost Per Metre
Problem: 3.5 metres of ribbon costs Rs. 52.50. What is the cost per metre?
Solution:
Step 1: Cost per metre = Total cost / Length = 52.50 / 3.5
Step 2: Make divisor whole: move both by 1 place. 525.0 / 35
Step 3: Divide: 525 / 35 = 15
Answer: The cost per metre is Rs. 15.
Example 8: Word Problem: Speed Calculation
Problem: A car travels 156.8 km in 3.2 hours. What is the speed of the car?
Solution:
Step 1: Speed = Distance / Time = 156.8 / 3.2
Step 2: Make divisor whole: move both by 1 place. 1568 / 32
Step 3: Divide: 1568 / 32 = 49
Answer: The speed of the car is 49 km/hr.
Example 9: Word Problem: Sharing Money
Problem: Rs. 87.50 is shared equally among 5 children. How much does each child get?
Solution:
Step 1: Each child's share = 87.50 / 5
Step 2: Divide: 8750 / 5 = 1750
Step 3: Place decimal: 2 decimal places, so 17.50
Answer: Each child gets Rs. 17.50.
Example 10: Word Problem: Unit Conversion
Problem: Convert 450 cm to metres.
Solution:
Step 1: 1 m = 100 cm, so divide by 100.
Step 2: 450 / 100 = 4.50 = 4.5 m
Answer: 450 cm = 4.5 m
Real-World Applications
Division of decimals is essential in many real-life situations:
Shopping and Unit Price: When you know the total cost and the quantity, you divide to find the price per unit. If 2.5 kg of mangoes costs Rs. 375, the price per kg is 375 / 2.5 = Rs. 150. Comparing prices per unit is how smart shoppers find the best deals.
Speed, Distance, and Time: To find speed, divide distance by time. If you cycle 12.6 km in 0.9 hours, your speed is 12.6 / 0.9 = 14 km/hr. To find time, divide distance by speed. To find distance, multiply speed by time. All these formulae frequently involve decimals.
Sharing and Distribution: When money or items need to be divided equally, you use decimal division. Rs. 255.60 shared among 4 people gives 255.60 / 4 = Rs. 63.90 each. If a group of friends splits a restaurant bill of Rs. 1,347.50 among 5 people, each pays 1347.50 / 5 = Rs. 269.50.
Unit Conversions: Converting centimetres to metres (divide by 100), grams to kilograms (divide by 1000), and paise to rupees (divide by 100) all involve decimal division. For example, 575 paise = 575 / 100 = Rs. 5.75. Similarly, 2350 grams = 2350 / 1000 = 2.35 kg.
Average Calculations: Finding the average of decimal numbers requires division. If a student scores 85.5, 92.0, and 78.5 in three tests, the average is (85.5 + 92.0 + 78.5) / 3 = 256 / 3 = 85.33 (approximately). Averages of heights, weights, and temperatures are commonly calculated using decimal division.
Science and Measurement: Scientists divide measurements to find rates, densities, and averages. If a plant grows 4.5 cm in 1.5 weeks, its growth rate is 4.5 / 1.5 = 3 cm per week. Density is calculated as mass / volume. If a metal block has mass 15.6 g and volume 2.4 cm3, its density is 15.6 / 2.4 = 6.5 g/cm3.
Fuel Efficiency: Cars measure fuel efficiency in km per litre. If a car travels 245.7 km on 15.5 litres of petrol, its fuel efficiency is 245.7 / 15.5 = 15.85 km/litre (approximately). This helps in planning fuel budgets for trips.
Tailoring and Construction: A tailor with 12.5 metres of fabric makes curtains that each require 1.25 metres. The number of curtains is 12.5 / 1.25 = 10. In construction, dividing lengths of pipes, beams, and tiles is a daily calculation.
Key Points to Remember
- To divide a decimal by a whole number, divide normally and place the decimal point in the quotient directly above its position in the dividend.
- To divide by a decimal, make the divisor a whole number by moving the decimal point in both the divisor and dividend to the right by the same number of places.
- Dividing by 10 moves the decimal 1 place left. By 100, 2 places left. By 1000, 3 places left.
- Dividing by a number less than 1 gives a quotient larger than the dividend. (Example: 5 / 0.5 = 10)
- An alternative method is to convert decimals to fractions, divide, and convert back.
- When the decimal point needs to be moved and there are not enough digits, add trailing zeros to the dividend.
- Always check your answer by multiplying: quotient x divisor should equal the dividend.
- Some decimal divisions produce non-terminating decimals; round as needed.
Practice Problems
- Find: 24.6 / 6
- Find: 9.45 / 0.5
- Find: 12 / 0.4
- Find: 7.56 / 0.12
- Find: 345.6 / 100
- If 4.8 litres of milk costs Rs. 264, what is the cost per litre?
- A runner covers 10.5 km in 1.5 hours. What is the runner's speed?
- Convert 2750 mm to metres by dividing by 1000.
Frequently Asked Questions
Q1. How do you divide a decimal by another decimal?
Move the decimal point in both the divisor and the dividend to the right until the divisor becomes a whole number. Then divide normally. For example, 4.5 / 0.9: move both 1 place right to get 45 / 9 = 5. So 4.5 / 0.9 = 5.
Q2. How do you divide a decimal by a whole number?
Divide as you normally would with whole numbers, but place the decimal point in the quotient directly above where it appears in the dividend. For example, 12.6 / 3: divide 126 by 3 to get 42, then place the decimal to get 4.2. So 12.6 / 3 = 4.2.
Q3. What happens when you divide by 10, 100, or 1000?
Dividing by 10 moves the decimal point 1 place to the left. Dividing by 100 moves it 2 places left. Dividing by 1000 moves it 3 places left. For example, 56.7 / 10 = 5.67, 56.7 / 100 = 0.567, and 56.7 / 1000 = 0.0567. This is the reverse of multiplying by powers of 10.
Q4. Why is the answer bigger when you divide by a decimal less than 1?
When you divide by a number less than 1, you are asking how many small portions fit into the number, and the answer is always more than the original number. For example, 6 / 0.5 = 12, because twelve half-portions make 6 wholes. The smaller the divisor, the larger the quotient.
Q5. Can you divide a whole number by a decimal?
Yes! Make the divisor a whole number by moving both decimal points. For example, 8 / 0.4: move both 1 place right to get 80 / 4 = 20. So 8 / 0.4 = 20. You can also think of it as: 8 / 0.4 = 8 x (10/4) = 80/4 = 20.
Q6. How do you check your answer in decimal division?
Multiply the quotient by the divisor. If you get the dividend, your answer is correct. For example, if 7.2 / 0.9 = 8, check: 8 x 0.9 = 7.2. Correct!
Q7. What if the decimal division does not end evenly?
Sometimes the division produces a non-terminating (repeating) decimal. For example, 10 / 3 = 3.333... In such cases, you can round to the required number of decimal places (e.g., 3.33 rounded to 2 decimal places) or express the answer as a fraction (10/3 or 3 1/3).
Q8. Can you convert decimal division to fraction division?
Yes! Convert both decimals to fractions, divide the fractions, and simplify. For example, 0.6 / 0.3 = (6/10) / (3/10) = (6/10) x (10/3) = 6/3 = 2. This gives the same answer as the decimal method.
