Decimals are a natural extension of the Indian place value system that allows us to represent decimal fractions (fractions with denominators 10, 100, 1000, and so on).
**Key Concept:** A decimal number like 27.53 represents:
So we can write: 27.53 = 20 + 7 + 0.5 + 0.03 = 2 × 10 + 7 × 1 + 5 × (1/10) + 3 × (1/100)
Any fraction with denominator 10, 100, 1000, etc., can be easily converted to a decimal by using the place value system.
**Method to Convert Fractions to Decimals:**
For a fraction like 254/1000:
**Rule for Dividing by Powers of 10:**
When dividing any number by 10, 100, 1000, etc., we can simply move the decimal point to the left.
**Steps:**
1. Write the dividend as it is and place a decimal point at the end
2. Count the number of zeroes in the divisor
3. Move the decimal point left by the same number of places as the count of zeroes. Add zeroes in front if needed.
**Examples:**
**Real-Life Example:** Jonali buys spices at the market:
---
Multiplying decimals is essentially the same as multiplying their equivalent fractions. We can convert decimals to fractions, perform the multiplication, and convert back to decimals.
**Example 1: Cost Calculation**
Arshad buys 5 pens at ₹9.5 per pen. What is the total cost?
Solution:
**Example 2: Distance Calculation**
A car travels 12.5 km per litre of petrol. What distance is covered with 7.5 litres?
Solution:
**Example 3: Calculating Total Distance Walked**
Ajay's school is 827 m (0.827 km) from his home. He walks to school and back, 6 days a week. How much does he walk in a week?
Solution:
**Example 4: Area of Rectangle**
Find the area of a rectangle with length 13.3 cm and width 5.7 cm.
Solution:
**Key Rule: The Decimal Point Rule for Multiplication**
When multiplying two decimals directly:
1. Ignore the decimal points and multiply the numbers as whole numbers
2. Count the total number of digits after the decimal point in both the multiplier and multiplicand
3. Place the decimal point in the product so that the number of digits after the decimal point equals the sum counted in step 2
**Important Pattern:**
| Multiplication | Decimal Places in First Number | Decimal Places in Second Number | Decimal Places in Product | Total Check |
|---|---|---|---|---|
| 9.5 × 5 | 1 | 0 | 1 | 1 + 0 = 1 ✓ |
| 12.5 × 7.5 | 1 | 1 | 2 | 1 + 1 = 2 ✓ |
| 1.64 × 6 | 2 | 0 | 2 | 2 + 0 = 2 ✓ |
| 5.7 × 13.35 | 1 | 2 | 3 | 1 + 2 = 3 ✓ |
**Example: Finding 5.96 × 24.8**
Solution:
**Example 5: Finding 5.8 × 1.24**
Solution:
**Verification using fractions:**
5.8 × 1.24 = (58/10) × (124/100) = 7192/1000 = 7.192 ✓
**Key Observation:** Unlike counting numbers, the product of decimals is NOT always greater than both numbers being multiplied.
**Three Situations:**
**Situation 1: Both numbers > 1**
Example: 3.4 × 6.5 = 22.1
**Situation 2: Both numbers between 0 and 1**
Example: 0.75 × 0.4 = 0.3
**Situation 3: One number > 1 and one between 0 and 1**
Example: 0.75 × 5 = 3.75
**Rule:** When you multiply a number by a decimal between 0 and 1, the product becomes smaller than the original number.
When multiplying a decimal by 10, 100, 1000, etc., the decimal point moves to the RIGHT.
**Examples:**
**Problem 1: Understanding Tenths, Hundredths**
(a) 6 × 4 tenths = 24 tenths = 2.4
(b) 7 × 0.3 = 2.1
(c) 9 × 5 hundredths = 45 hundredths = 0.45
**Problem 2: Finding Products**
(a) 27.34 × 6
(b) 4.23 × 3.7
(c) 0.432 × 0.23
**Problem 3: Cloth Needed for Shirts**
Thejus needs 1.65 m of cloth for one shirt. How much for 3 shirts?
Solution:
**Problem 4: Meenu's Shopping**
Meenu bought 4 notebooks at ₹15.50 each and 3 erasers at ₹2.75 each.
Solution:
**Problem 5: Stack of Rupee Coins**
Thickness of one rupee coin = 1.45 mm. Total height of 36 coins stacked = ?
Solution:
**Problem 6: Price of Oranges**
Price of 1 kg oranges = ₹56.50. Price of 2.250 kg = ?
Solution:
**Problem 7: Notebook Business**
Dwarakanath buys notebooks at ₹23.6 wholesale and sells at ₹30. Profit on 50 books?
Solution:
**Problem 8: Using Known Products**
Given: 18 × 12 = 216. Find:
(a) 18 × 1.2 = 21.6
(b) 18 × 0.12 = 2.16
(c) 1.8 × 1.2 = 2.16
(d) 0.18 × 0.12 = 0.0216
(e) 0.018 × 0.012 = 0.000216
(f) 1.8 × 12 = 21.6
**Products Less Than 1:** Parts (b), (d), and (e) have products less than 1
**Problem 9: Predicting When Product < 1**
Without calculating, identify which products are less than 1:
(a) 7 × 0.6 = 4.2 (NOT less than 1, because 7 > 1)
(b) 0.7 × 0.6 (LESS than 1, because both numbers are between 0 and 1)
(c) 0.7 × 6 (NOT less than 1, because 6 > 1)
(d) 0.07 × 0.06 (LESS than 1, because both numbers are between 0 and 1)
**Rule:** Product is less than 1 when BOTH factors are between 0 and 1.
**Problem 10: Multiplying by Powers of 10**
Complete the table by multiplying by 10, 100, and 1000 (move decimal right):
| Number | × 10 | × 100 | × 1000 |
|---|---|---|---|
| 5.7 | 57 | 570 | 5700 |
| 23.02 | 230.2 | 2302 | 23020 |
| 0.92 | 9.2 | 92 | 920 |
| 0.306 | 3.06 | 30.6 | 306 |
| 24.67 | 246.7 | 2467 | 24670 |
---
**Rule: Moving the Decimal Point Left**
When dividing a decimal by 10, 100, 1000, etc., move the decimal point to the LEFT by as many places as there are zeroes in the divisor.
**Examples:**
| Decimal | ÷ 10 | ÷ 100 | ÷ 1000 | ÷ 10000 |
|---|---|---|---|---|
| 18.7 | 1.87 | 0.187 | 0.0187 | 0.00187 |
| 21.1 | 2.11 | 0.211 | 0.0211 | 0.00211 |
| 0.13 | 0.013 | 0.0013 | 0.00013 | 0.000013 |
| 2.146 | 0.2146 | 0.02146 | 0.002146 | 0.0002146 |
| 0.0058 | 0.00058 | 0.000058 | 0.0000058 | 0.00000058 |
**Example 6: Cutting a Ribbon**
Anuja has a 3.9 m ribbon to cut into 10 equal pieces. Length of each piece?
Solution - Method 1 (Converting to Fractions):
Solution - Method 2 (Moving Decimal):
If cut into 100 equal pieces:
Converting to other units:
**When the denominator is NOT 10, 100, 1000, etc.:**
We need to find an equivalent fraction with a denominator of the form 10, 100, 1000, etc.
**Example 7: Sharing Ribbon Between Two People**
Neenu and Anu share 29 m of ribbon equally. How much does each get?
Solution:
**Example: Sharing Among 4 Friends**
Each of 4 friends gets 29 ÷ 4 = 29/4 m
Solution:
**When we cannot express the denominator in the form 10, 100, 1000, etc., we use long division.**
**Key Concept:** In long division, we can:
This allows us to find decimal quotients for any division.
**Example 8: Whole Number Division (Recap)**
Find 1324 ÷ 4
Solution using long division:
```
331
------
4 | 1324
12↓
--
12
12
--
04
4
--
0
```
Process breakdown:
**Example 9: Division with Decimal Quotient**
Find 1325 ÷ 4
Solution using long division:
```
331.25
---------
4 | 1325.00
12
--
12
12
--
05
4
--
10
8
--
20
20
--
0
```
Process breakdown (using place value):
**Verification:** Multiply back: 331.25 × 4 = 1325 ✓
**Alternative verification using fractions:**
**Example 10: Another Example - Finding 237 ÷ 8**
```
29.625
---------
8 | 237.000
16
--
77
72
--
50
48
--
20
16
--
40
40
--
0
```
Process breakdown:
**Key Principle:** When the dividend (number being divided) is already a decimal, we follow the same long division process. The decimal point in the quotient is placed when we move from ones to tenths (if there are ones), or directly when dividing the decimal part.
**Example 11: Dividing Decimal by Whole Number**
Problem: A shopkeeper has 9.5 kg of sugar to pack equally in 4 bags. Weight of each bag?
Solution: 9.5 ÷ 4
```
2.375
---------
4 | 9.500
8
--
15
12
--
30
28
--
20
20
--
0
```
Process:
**Example 12: Dividing Small Decimal by Whole Number**
Problem: What is 0.06 ÷ 5?
Solution:
```
0.012
---------
5 | 0.060
0
--
00
00
--
06
5
--
10
10
--
0
```
Process:
**Important Rule:** When we have zero ones and zero tenths but non-zero hundredths, we must write 0.0 before the result. This shows the decimal place is at the correct position.
**Problem 1: Converting Denominator Method**
(a) 18/5
(b) 415/4
(c) 1217/2
(d) 4827/8
**Problem 2: Multiple Choice Division**
(a) 1526 ÷ 4 = ?
(b) 3567 ÷ 8 = ?
**Problem 3: Seeing the Pattern**
(a) 132 ÷ 4 = 33
(b) 13.2 ÷ 4 = 3.3
(c) 1.32 ÷ 4 = 0.33
(d) 0.132 ÷ 4 = 0.033
**Rule:** When the dividend becomes 1/10 of its previous value, the quotient also becomes 1/10 of its previous value. The decimal point moves left in both cases by the same amount.
**Problem 4: Another Pattern**
(a) 126 ÷ 8 = 15.75
(b) 12.6 ÷ 8 = 1.575
(c) 1.26 ÷ 8 = 0.1575
(d) 0.126 ÷ 8 = 0.01575
(e) 0.0126 ÷ 8 = 0.001575
**Observation:** As the dividend decreases by a factor of 10 each time, the quotient also decreases by a factor of 10. The decimal moves left in the quotient accordingly.
---
**Key Principle:** When the divisor is a decimal, we must convert it to a whole number before performing long division.
**Method:**
1. Count the number of decimal places in the decimal divisor
2. Multiply BOTH the dividend and divisor by 10, 100, 1000, etc. (use the same power of 10) to make the divisor a whole number
3. Perform long division as usual
**Why this works:** Multiplying both dividend and divisor by the same number doesn't change the quotient.
Mathematical basis: a ÷ b = (a × k) ÷ (b × k) where k is any non-zero number
**Example 13: Average Speed Calculation**
Problem: Ravi traveled 126 km in 2.5 hours. What was his average speed?
Solution: 126 ÷ 2.5
Step 1: Identify decimal places in divisor
Step 2: Multiply both numbers by 10 to eliminate the decimal
Step 3: Perform long division
```
50.4
-------
25 | 1260.0
125
---
100
100
---
0
```
Process:
Step 4: Place decimal point
**When divisor has 2 or more decimal places:**
Multiply both dividend and divisor by 100, 1000, etc., as needed.
**Example:** If divisor is 2.45 (2 decimal places)
**Example:** If divisor
Q1. What does the decimal 0.347 represent as a fraction?
Answer: A — 0.347 has 3 decimal places, so it represents 347 parts out of 1000.
Q2. When you divide 45.6 by 100, what is the result?
Answer: C — Dividing by 100 means moving the decimal point 2 places to the left: 45.6 → 0.456.
Q3. What is 7.3 × 2 in decimal form?
Answer: B — 7.3 × 2 = 14.6; the product has 1 decimal place because 7.3 has 1 and 2 has 0.
Q4. If 15 × 12 = 180, what is 1.5 × 1.2?
Answer: B — 1.5 has 1 decimal place and 1.2 has 1 decimal place, so 1 + 1 = 2 places; 180 becomes 1.80 = 1.8.
Q5. Arjun bought 4 notebooks at ₹25.50 each. How much did he spend?
Answer: A — 25.50 × 4 = 102.00 = ₹102 (2 decimal places in 25.50, 0 in 4, so 2 total; 2550 ÷ 100 = 25.5, and 25.5 × 4 = 102).
Q6. A ribbon is 12.8 m long and is cut into 8 equal pieces. What is the length of each piece?
Answer: A — 12.8 ÷ 8 = 1.6 m because 128 ÷ 80 = 1.6 (or use 12.8 = 128/10, then 128/10 ÷ 8 = 128/80 = 1.6).
Q7. When is the product of two decimals less than both factors?
Answer: B — Example: 0.5 × 0.4 = 0.2, which is smaller than both 0.5 and 0.4.
Q8. Priya walks 1.25 km daily. How far does she walk in 6 days?
Answer: C — 1.25 × 6: multiply 125 × 6 = 750, then place decimal 2 places from right = 7.50 = 7.5 km.
Q9. The thickness of a coin is 2.05 mm. What is the height of 10 coins stacked?
Answer: A — 2.05 × 10 = 20.5 mm (move decimal 1 place right when multiplying by 10); this equals 2.05 cm but the best answer in same units is 20.5 mm.
Q10. Which multiplication will give a product less than 1?
Answer: B — 0.4 × 0.8 = 0.32, which is less than 1 because both factors are between 0 and 1; in option A, 2.5 × 0.6 = 1.5 > 1.
What does 27.53 mean in expanded form?
2 tens + 7 ones + 5 tenths + 3 hundredths.
How do you multiply 5.96 × 24.8?
Multiply 596 × 248 = 147,808, then place decimal after 3 places (2+1) to get 147.808.
What is the rule for dividing by 10, 100, or 1000?
Move the decimal point left by the number of zeros in the divisor.
Is the product 0.25 × 0.8 greater than both numbers?
No, the product 0.2 is less than both 0.25 and 0.8 because both factors are less than 1.
Convert the fraction 1/2 to a decimal.
Multiply numerator and denominator by 5 to get 5/10 = 0.5.
How many decimal places are in the product 1.64 × 6?
Two decimal places because 1.64 has 2 decimal places and 6 has 0, so 2 + 0 = 2.
What is 3.9 ÷ 10 in decimal form?
0.39 (move decimal point 1 place to the left).
If a pen costs ₹9.5 and you buy 5 pens, what is the total cost?
₹47.5 because 9.5 × 5 = 47.5.
When multiplying two decimals, where do you place the decimal point in the product?
Count the total number of decimal places in both factors and place the decimal that many places from the right in the product.
Is 2.25 × 8 = 18 greater than 2.25?
Yes, because 8 is greater than 1, so the product is larger than both factors.
Convert the decimal 0.8 into a fraction in simplest form. [1 mark]
0.8 = 8/10; then reduce by dividing both by their GCD. GCD of 8 and 10 is 2.
A car uses 5.5 litres of petrol to travel 100 km. How much petrol is needed to travel 250 km? Show your working. [2 marks]
Set up proportion: 5.5 litres → 100 km, so x litres → 250 km. Multiply: (5.5 × 250) ÷ 100 = 1375 ÷ 100 = 13.75 litres.
A bag of rice costs ₹245.50 and weighs 15.5 kg. If Meena wants to buy 3 bags, how much will she spend? Write your steps clearly. [3 marks]
Step 1: Multiply 245.50 × 3 by removing decimals first (24550 × 3 ÷ 100). Step 2: 24550 × 3 = 73650. Step 3: Place decimal 2 places from right = 736.50 = ₹736.50.
Ajay has 8.4 metres of cloth. He cuts it into 6 equal pieces for a craft project. (a) What is the length of each piece in metres? (b) If he needs 4 such pieces for one project, how much cloth does he need? (c) Express your answer in both decimals and fractions. Show all your work. [5 marks]
(a) Divide: 8.4 ÷ 6 = 84/10 ÷ 6 = 84/60 = 1.4 m. (b) Multiply: 1.4 × 4 = 5.6 m. (c) 5.6 m = 56/10 = 28/5 metres (in fraction, simplified form). Show fraction to decimal conversion using place value: 8.4 = 8 units + 4 tenths.
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