Multiplication of Decimals
Decimals are everywhere in our daily lives. The price of a pencil might be Rs. 3.50, your height might be 1.52 metres, and the temperature today might be 32.5°C. When we need to find the total cost of 6 pencils at Rs. 3.50 each, or calculate the area of a room that is 4.5 m long and 3.2 m wide, we multiply decimals.
Multiplication of decimals is an important topic in Class 7 NCERT Maths. The good news is that multiplying decimals is very similar to multiplying whole numbers. The only extra step is keeping track of the decimal point in the answer. Once you learn where to place the decimal point, you will find decimal multiplication straightforward and easy.
In this chapter, we will learn how to multiply a decimal by a whole number, a decimal by another decimal, and a decimal by powers of 10 (like 10, 100, 1000). We will use many real-life examples involving money, measurements, and shopping to make the learning practical and fun. By the end, you will be able to handle any decimal multiplication problem with confidence.
Here is the key idea: when you multiply decimals, first ignore the decimal points and multiply the numbers as if they were whole numbers. Then count the total number of decimal places in both the original numbers, and place the decimal point in the answer so that it has the same total number of decimal places. Let us see this in action!
Why does counting decimal places work? Consider 0.3 x 0.2. In fraction form, this is 3/10 x 2/10 = 6/100 = 0.06. The first number has 1 decimal place (tenths) and the second has 1 decimal place (tenths). Tenths times tenths give hundredths, which have 2 decimal places. So the total decimal places (1 + 1 = 2) tells us the answer should have 2 decimal places: 0.06. This pattern always works, and it is the foundation of the decimal place counting method.
This topic builds on your knowledge of multiplying whole numbers and your understanding of decimals from Class 6. It will also prepare you for more advanced topics like percentage calculations, scientific notation, and algebraic expressions involving decimals in higher classes.
What is Multiplication of Decimals?
A decimal number is a number that has a whole number part and a fractional part separated by a decimal point. For example, in 3.75, the whole number part is 3 and the fractional part is 75 (meaning 75 hundredths).
Multiplication of decimals follows a simple rule:
Step 1: Ignore the decimal points and multiply the numbers as whole numbers.
Step 2: Count the total number of decimal places in both numbers being multiplied.
Step 3: Place the decimal point in the product so that it has the same total number of decimal places.
Example to understand: To find 2.3 x 1.5:
- Ignore decimals: 23 x 15 = 345
- Count decimal places: 2.3 has 1 decimal place, 1.5 has 1 decimal place. Total = 2 decimal places.
- Place the decimal: 345 becomes 3.45 (2 decimal places from the right).
So 2.3 x 1.5 = 3.45.
Multiplying by Powers of 10:
- Multiplying by 10: move the decimal point 1 place to the right. Example: 3.45 x 10 = 34.5
- Multiplying by 100: move the decimal point 2 places to the right. Example: 3.45 x 100 = 345
- Multiplying by 1000: move the decimal point 3 places to the right. Example: 3.45 x 1000 = 3450
Multiplication of Decimals Formula
Multiplying Decimals - Step-by-Step Method:
1. Multiply as whole numbers (ignore decimal points)
2. Count total decimal places in both factors
3. Put decimal point in product with that many decimal places
Multiplying a Decimal by a Whole Number:
Multiply as usual. The product has the same number of decimal places as the decimal factor.
Multiplying by Powers of 10:
| Multiply by | Move decimal point | Example |
|---|---|---|
| 10 | 1 place right | 4.56 x 10 = 45.6 |
| 100 | 2 places right | 4.56 x 100 = 456 |
| 1000 | 3 places right | 4.56 x 1000 = 4560 |
Multiplying by 0.1, 0.01, 0.001:
| Multiply by | Move decimal point | Example |
|---|---|---|
| 0.1 | 1 place left | 4.56 x 0.1 = 0.456 |
| 0.01 | 2 places left | 4.56 x 0.01 = 0.0456 |
| 0.001 | 3 places left | 4.56 x 0.001 = 0.00456 |
Types and Properties
Here are the different types of decimal multiplication problems:
Type 1: Decimal x Whole Number
Multiply normally, keeping the same number of decimal places as the decimal number. Example: 3.7 x 4 = 14.8. (37 x 4 = 148, then 1 decimal place gives 14.8). Think of it as: if one book costs Rs. 3.70, then 4 books cost Rs. 14.80.
Type 2: Decimal x Decimal
Multiply as whole numbers, then count the total decimal places. Example: 0.6 x 0.3 = 0.18. (6 x 3 = 18, total 2 decimal places gives 0.18). This type requires careful counting of decimal places.
Type 3: Decimal x 10, 100, or 1000
Simply shift the decimal point to the right by 1, 2, or 3 places respectively. Example: 5.678 x 100 = 567.8. This is the easiest type and is very useful for converting between units.
Type 4: Decimal x 0.1, 0.01, or 0.001
Shift the decimal point to the left by 1, 2, or 3 places respectively. Example: 45.6 x 0.01 = 0.456. This is like dividing by 10, 100, or 1000.
Type 5: Problems with Trailing Zeros
Sometimes the product of the whole number multiplication ends with a zero, and you need to be careful with decimal placement. Example: 0.5 x 0.4 = 0.20 = 0.2. (5 x 4 = 20, 2 decimal places gives 0.20, which simplifies to 0.2).
Type 6: Problems Needing Extra Zeros
Sometimes you need to add zeros to the left to have enough decimal places. Example: 0.02 x 0.3 = 0.006. (2 x 3 = 6, but you need 3 decimal places, so add zeros: 0.006). Another example: 0.01 x 0.01 = 0.0001 (1 x 1 = 1, but 2 + 2 = 4 decimal places needed, so the answer is 0.0001).
Estimation Tip:
Before calculating, estimate whether your answer should be larger or smaller than the factors. If both factors are greater than 1, the product is larger than either. If one factor is less than 1, the product is less than the other factor. If both are less than 1, the product is less than both. Use this to check whether your answer makes sense. For example, 3.5 x 0.2 should be less than 3.5 (since 0.2 is less than 1). The answer 0.7 confirms this.
Solved Examples
Example 1: Multiplying a Decimal by a Whole Number
Problem: Find: 4.5 x 6
Solution:
Step 1: Multiply as whole numbers: 45 x 6 = 270
Step 2: Count decimal places: 4.5 has 1 decimal place. 6 has 0 decimal places. Total = 1.
Step 3: Place the decimal: 270 becomes 27.0 = 27.0
Answer: 4.5 x 6 = 27.0 (or simply 27)
Example 2: Multiplying Two Decimals
Problem: Find: 3.2 x 1.5
Solution:
Step 1: Multiply as whole numbers: 32 x 15 = 480
Step 2: Count decimal places: 3.2 has 1, and 1.5 has 1. Total = 2.
Step 3: Place the decimal: 480 becomes 4.80 = 4.8
Answer: 3.2 x 1.5 = 4.8
Example 3: Multiplying Decimals with More Decimal Places
Problem: Find: 2.54 x 0.3
Solution:
Step 1: Multiply as whole numbers: 254 x 3 = 762
Step 2: Count decimal places: 2.54 has 2, and 0.3 has 1. Total = 3.
Step 3: Place the decimal: 762 becomes 0.762
Answer: 2.54 x 0.3 = 0.762
Example 4: Multiplying Small Decimals (Adding Leading Zeros)
Problem: Find: 0.04 x 0.2
Solution:
Step 1: Multiply as whole numbers: 4 x 2 = 8
Step 2: Count decimal places: 0.04 has 2, and 0.2 has 1. Total = 3.
Step 3: We need 3 decimal places in the answer. 8 is a single digit, so we write it as 0.008.
Answer: 0.04 x 0.2 = 0.008
Example 5: Multiplying by 10, 100, and 1000
Problem: Find: (a) 6.35 x 10, (b) 6.35 x 100, (c) 6.35 x 1000
Solution:
(a) Move decimal 1 place right: 6.35 x 10 = 63.5
(b) Move decimal 2 places right: 6.35 x 100 = 635
(c) Move decimal 3 places right: 6.35 x 1000 = 6350
Answer: (a) 63.5, (b) 635, (c) 6350
Example 6: Multiplying by 0.1 and 0.01
Problem: Find: (a) 72.5 x 0.1, (b) 72.5 x 0.01
Solution:
(a) Move decimal 1 place left: 72.5 x 0.1 = 7.25
(b) Move decimal 2 places left: 72.5 x 0.01 = 0.725
Answer: (a) 7.25, (b) 0.725
Example 7: Word Problem: Cost of Fruits
Problem: Apples cost Rs. 85.50 per kg. Reema buys 2.5 kg. How much does she pay?
Solution:
Step 1: Total cost = 85.50 x 2.5
Step 2: Multiply as whole numbers: 8550 x 25 = 213750
Step 3: Total decimal places: 85.50 has 2, and 2.5 has 1. Total = 3.
Step 4: Place decimal: 213750 becomes 213.750 = 213.75
Answer: Reema pays Rs. 213.75.
Example 8: Word Problem: Area of a Rectangle
Problem: A rectangular garden is 12.5 m long and 8.4 m wide. Find its area.
Solution:
Step 1: Area = length x width = 12.5 x 8.4
Step 2: Multiply as whole numbers: 125 x 84 = 10500
Step 3: Total decimal places: 12.5 has 1, and 8.4 has 1. Total = 2.
Step 4: Place decimal: 10500 becomes 105.00 = 105
Answer: The area of the garden is 105 m2.
Example 9: Word Problem: Distance Covered
Problem: A cyclist rides at 15.5 km per hour. How far does the cyclist ride in 2.5 hours?
Solution:
Step 1: Distance = Speed x Time = 15.5 x 2.5
Step 2: Multiply as whole numbers: 155 x 25 = 3875
Step 3: Total decimal places: 1 + 1 = 2
Step 4: Place decimal: 3875 becomes 38.75
Answer: The cyclist rides 38.75 km.
Example 10: Word Problem: Price Per Unit
Problem: A cloth costs Rs. 125.75 per metre. What is the cost of 0.8 m of cloth?
Solution:
Step 1: Cost = 125.75 x 0.8
Step 2: Multiply as whole numbers: 12575 x 8 = 100600
Step 3: Total decimal places: 2 + 1 = 3
Step 4: Place decimal: 100600 becomes 100.600 = 100.60
Answer: The cost of 0.8 m of cloth is Rs. 100.60.
Real-World Applications
Multiplication of decimals is used in countless everyday situations:
Shopping: When items are priced with decimals (Rs. 49.99) and you buy multiple quantities, you multiply decimals. If petrol costs Rs. 105.50 per litre and you fill 15.5 litres, the total cost is 105.50 x 15.5 = Rs. 1,635.25.
Measurements: In construction and tailoring, measurements often involve decimals. The area of a room that is 4.5 m by 3.8 m is 4.5 x 3.8 = 17.1 m2. This is needed to buy the right amount of tiles or carpet.
Science: Scientists constantly multiply decimals. The mass of a molecule, the speed of light, chemical concentrations, all use decimal multiplication. For example, if an experiment uses 0.25 litres of a solution and you need to repeat it 4.5 times, you need 0.25 x 4.5 = 1.125 litres.
Money and Finance: Calculating interest, taxes, and discounts involves decimal multiplication. A 7.5% discount on an item worth Rs. 800 is 800 x 0.075 = Rs. 60.
Unit Conversions: Converting between metres and kilometres (multiply by 0.001) or between grams and kilograms (multiply by 0.001) uses decimal multiplication. For example, 250 m = 250 x 0.001 = 0.25 km.
Cooking: Recipes with decimal measurements need to be scaled up or down. If a recipe uses 1.5 cups of flour for 4 servings and you want 6 servings (1.5 times), you need 1.5 x 1.5 = 2.25 cups.
Key Points to Remember
- To multiply decimals, first multiply as whole numbers (ignoring the decimal points).
- Then count the total number of decimal places in both numbers being multiplied.
- Place the decimal point in the product so it has that many decimal places from the right.
- Multiplying by 10 moves the decimal 1 place right; by 100, 2 places right; by 1000, 3 places right.
- Multiplying by 0.1 moves the decimal 1 place left; by 0.01, 2 places left; by 0.001, 3 places left.
- If the product does not have enough digits for the required decimal places, add leading zeros. (Example: 0.02 x 0.3 = 0.006)
- Multiplying a decimal by a whole number: the product has the same number of decimal places as the decimal factor.
- The product of two decimals less than 1 is smaller than either decimal. (Example: 0.5 x 0.3 = 0.15)
- Always simplify trailing zeros in decimal answers when appropriate (27.0 = 27).
Practice Problems
- Find: 7.8 x 9
- Find: 2.6 x 3.4
- Find: 0.05 x 0.4
- Find: 45.67 x 100
- Find: 8.25 x 0.01
- A book costs Rs. 235.50. Find the cost of 3.5 such books.
- A car travels at 65.5 km/hr. How far will it go in 3.2 hours?
- The length of a rectangular swimming pool is 25.5 m and its width is 10.4 m. Find the area.
Frequently Asked Questions
Q1. How do you multiply two decimal numbers?
First, ignore the decimal points and multiply the numbers as if they were whole numbers. Then, count the total number of decimal places in both numbers. Finally, place the decimal point in the product so that it has the same total number of decimal places. For example, 1.2 x 0.5: multiply 12 x 5 = 60, total decimal places = 1 + 1 = 2, so the answer is 0.60 = 0.6.
Q2. What happens when you multiply a decimal by 10, 100, or 1000?
Multiplying by 10 moves the decimal point 1 place to the right. Multiplying by 100 moves it 2 places right. Multiplying by 1000 moves it 3 places right. For example, 3.456 x 100 = 345.6. This is one of the easiest and most useful rules for decimals.
Q3. What happens when you multiply a decimal by 0.1 or 0.01?
Multiplying by 0.1 moves the decimal point 1 place to the left (same as dividing by 10). Multiplying by 0.01 moves it 2 places left (same as dividing by 100). For example, 45.6 x 0.1 = 4.56 and 45.6 x 0.01 = 0.456.
Q4. Why do we count decimal places when multiplying decimals?
When you multiply decimals, you are essentially multiplying fractions (tenths, hundredths, etc.). For example, 0.3 x 0.2 = 3/10 x 2/10 = 6/100 = 0.06. The 1 decimal place from 0.3 plus the 1 decimal place from 0.2 gives 2 decimal places in the answer. Counting decimal places is a shortcut for this fraction multiplication.
Q5. What if the product does not have enough digits for the decimal places?
Add leading zeros before the product digits. For example, 0.02 x 0.3: multiply 2 x 3 = 6, total decimal places = 3. The digit 6 alone gives only 1 digit, so add zeros: 0.006. Similarly, 0.01 x 0.01 = 0.0001 (1 with 4 decimal places needed, so add three leading zeros).
Q6. Is the product of two decimals less than 1 always smaller than either decimal?
Yes! When you multiply two positive numbers that are both less than 1, the result is always smaller than either number. For example, 0.5 x 0.4 = 0.2, which is less than both 0.5 and 0.4. This is similar to how the product of two proper fractions is less than either fraction.
Q7. How is multiplying decimals used in real life?
Multiplying decimals is used in shopping (calculating the total price of items), measurements (finding the area of a room), science (calculating quantities), money calculations (finding tax or interest amounts), and unit conversions (converting meters to kilometers). Any time you work with numbers that have decimal parts, multiplication of decimals comes into play.
Q8. Can you multiply a decimal by a fraction?
Yes! Convert the decimal to a fraction or the fraction to a decimal, then multiply. For example, 0.5 x 3/4: either convert 0.5 to 1/2 and multiply 1/2 x 3/4 = 3/8 = 0.375, or convert 3/4 to 0.75 and multiply 0.5 x 0.75 = 0.375. Both give the same answer.
