In our daily life, we often need to measure things more precisely than just using whole numbers. When a carpenter needs to cut wood exactly or when we buy 1.5 kg of vegetables, we use parts of units. This chapter will help you understand how to work with these fractional parts using decimals, which makes calculations much easier and more accurate than working with fractions.
3.1 The Need for Smaller Units
Why Smaller Units Matter
When we measure things in our daily life, sometimes we need to be very precise. Think about a carpenter making furniture or a doctor giving medicine – small differences can matter a lot! This is why we need smaller units of measurement.
Here are some imp points about why smaller units are needed:
- Small differences in measurements can have big impact on how things work
- To get exact measurements, we need smaller units
- We can divide standard units into smaller parts to be more precise
Measuring with Precision
When we look at a ruler or measuring tape, we see small lines between the numbers. These help us measure things more accurately.
How to read precise measurements:
- The unit length between two consecutive numbers is divided into 10 equal parts
- If something measures 2 7/10 cm, we say “two and seven-tenth centimeters”
- For even more precise measurements, we need even smaller divisions
Questions and Answers
Q: Why was the unit divided into smaller parts to measure the screws? A: The screws looked almost the same but had slightly different lengths. To measure and show this small difference accurately, the unit needed to be divided into smaller parts (tenths).
Q: What is the meaning of 2 7/10 cm? A: It means 2 cm and 7 parts of 1/10 cm. We go from 0 to 2 and then take seven parts of 1/10.
3.2 A Tenth Part
Understanding Tenths
A tenth is a very imp concept in decimals. Let me explain it simply:
- One-tenth = 1/10 of a unit
- 10 one-tenths = 1 unit
- 10 × 1/10 = 1 unit
Reading Decimal Numbers
Different ways to read decimal numbers:
| Number | How to Read |
|---|---|
| 4 1/10 | ‘four and one-tenth’ |
| 4/10 | ‘four one-tenths’ or ‘four-tenths’ |
| 41/10 | ‘forty-one one-tenths’ or ‘forty-one tenths’ |
| 41 1/10 | ‘forty-one and one-tenth’ |
Converting Between Forms
We can change between different forms of the same number:
- 3 4/10 units = (3 × 1) + (4 × 1/10) units
- 34 × 1/10 = 34/10 = 10/10 + 10/10 + 10/10 + 4/10
- 34 one-tenths = 3 units and 4 one-tenths
Addition with Tenths
When adding numbers with tenths:
- Add whole numbers and tenths separately
- Convert excess tenths to units when needed
Example: 2 7/10 + 3 6/10 = (2 + 3) + (7/10 + 6/10) = 5 + 13/10 = 6 3/10
Since 13/10 = 1 unit + 3/10, we get 5 + 1 + 3/10 = 6 3/10
Subtraction with Tenths
For subtraction with tenths:
- Subtract tenths from tenths, units from units
- When you don’t have enough tenths, borrow 1 unit = 10 tenths
Example: 12 4/10 – 6 7/10 = 11 14/10 – 6 7/10 = 5 7/10
Questions and Answers
Q: Arrange these lengths in increasing order: 9/10, 1 7/10, 130/10, 13 1/10, 10 5/10, 7 6/10, 6 7/10, 4/10 A: 4/10, 9/10, 6 7/10, 7 6/10, 10 5/10, 1 7/10, 130/10, 13 1/10
Q: Arrange the following lengths in increasing order: 4 1/10, 4/10, 41/10, 41 1/10 A: 4/10, 4 1/10, 41/10, 41 1/10
Q: What is the total length of Sonu’s arm if lower arm is 2 7/10 units and upper arm is 3 6/10 units? A: 2 7/10 + 3 6/10 = (2 + 3) + (7/10 + 6/10) = 5 + 13/10 = 5 + 1 3/10 = 6 3/10 units
Q: Find the total length of a honeybee: Head: 2 3/10 units, Thorax: 5 4/10 units, Abdomen: 7 5/10 units A: 2 3/10 + 5 4/10 + 7 5/10 = (2 + 5 + 7) + (3/10 + 4/10 + 5/10) = 14 + 12/10 = 14 + 1 2/10 = 15 2/10 units
Q: If Shylaja’s hand is 12 4/10 units and her palm is 6 7/10 units, what is the length of the longest finger? A: 12 4/10 – 6 7/10 = 11 14/10 – 6 7/10 = 5 7/10 units
Q: What is the difference between Celestial Pearl Danio (2 4/10 cm) and Philippine Goby (9/10 cm)? A: 2 4/10 – 9/10 = 1 14/10 – 9/10 = 1 5/10 = 1.5 cm
3.3 A Hundredth Part
Understanding Hundredths
Sometimes we need even smaller parts than tenths. That’s when hundredths come in:
- Each one-tenth divided into 10 parts = one-hundredth
- 1 hundredth = 1/100 of a unit
- 100 one-hundredths = 1 unit
- 10 one-hundredths = 1 one-tenth
Reading Hundredths
Different ways to read numbers with hundredths:
| Number | How to Read |
|---|---|
| 1 1/10 4/100 | ‘one and one-tenth and four-hundredths’ |
| 1 14/100 | ‘one and fourteen-hundredths’ |
| 114/100 | ‘one hundred and fourteen-hundredths’ |
Comparing and Ordering
To compare decimal numbers:
- Compare digit by digit from left to right
- Remember place values: units > tenths > hundredths
- You can mark positions on number line for visual comparison
Addition with Hundredths
When adding with hundredths:
- Add units, tenths, and hundredths separately
- Convert excess hundredths to tenths: 10 hundredths = 1 tenth
Example: 15 3/10 4/100 + 2 6/10 8/100 = 18 2/100
Subtraction with Hundredths
For subtraction with hundredths:
- Borrow from higher place values when needed
- Remember: 1 unit = 10 tenths, 1 tenth = 10 hundredths
- Work systematically from right to left
Questions and Answers
Q: What is the length of a folded paper if original length was 8 9/10 units? A: Half of 8 9/10 = 4 4/10 5/100 units (4 units and 4 tenths and 5 hundredths)
Q: How many one-hundredths make one-tenth? A: 10 one-hundredths = 1 one-tenth
Q: Can we say that 4 4/10 5/100 is the same as 4 units and 45 one-hundredths? A: Yes, because 4/10 = 40/100, so 4 4/10 5/100 = 4 40/100 5/100 = 4 45/100
Q: What is the sum of 15 3/10 4/100 and 2 6/10 8/100? A: Method 1: (15 + 2) + (3/10 + 6/10) + (4/100 + 8/100) = 17 + 9/10 + 12/100 = 17 + 10/10 + 2/100 = 18 2/100 Method 2: Converting to hundredths: 1534/100 + 268/100 = 1802/100 = 18 2/100
Q: What is the difference: 25 9/10 – 6 4/10 7/100? A: 25 9/10 – 6 4/10 7/100 = 25 8/10 10/100 – 6 4/10 7/100 = 19 4/10 3/100
3.4 Decimal Place Value
Why Base 10 System
The Indian place value system uses base 10, which means:
- Each place value is 10 times the next smaller place
- This is a natural extension to fractional parts
- It makes calculations easier
Place Value Pattern
The pattern continues both ways from the units place:
10,000 ← 1000 ← 100 ← 10 ← 1 → 1/10 → 1/100 → 1/1000
Each position is multiplied or divided by 10 as we move left or right.
Extended Place Values
Some imp relationships:
- 1/1000 = one-thousandth
- 1000 one-thousandths = 1 unit
- 100 one-thousandths = 1 one-tenth
- 10 one-thousandths = 1 one-hundredth
Historical Context
The decimal system has interesting history:
- Based on number 10
- ‘Decem’ means ten in Latin
- Related to Sanskrit ‘daśha’ meaning 10
- Ancient Indian mathematicians developed these concepts
Questions and Answers
Q: Can we not split a unit into 4 equal parts, 5 equal parts, 8 equal parts, or any other number of equal parts instead? A: Yes, we can. But we split into 10 parts because of the special role that 10 plays in the Indian place value system.
Q: What will the fraction be when 1/100 is split into 10 equal parts? A: It will be 1/1000, i.e., a thousand such parts make up a unit.
Q: How many thousandths make one unit? A: 1000 thousandths = 1 unit
Q: How many thousandths make one tenth? A: 100 thousandths = 1 tenth
Q: How many thousandths make one hundredth? A: 10 thousandths = 1 hundredth
Q: How many tenths make one ten? A: 100 tenths = 1 ten
Q: How many hundredths make one ten? A: 1000 hundredths = 1 ten
3.5 Decimal Notation
Decimal Point Usage
The decimal point (.) is very imp – it separates whole numbers from fractional parts:
| Number | Meaning |
|---|---|
| 705 | 7 hundreds + 0 tens + 5 ones |
| 70.5 | 7 tens + 0 ones + 5 tenths |
| 7.05 | 7 ones + 0 tenths + 5 hundredths |
Reading Decimal Numbers
How to read decimal numbers properly:
- 70.5 = ‘seventy point five’
- 7.05 = ‘seven point zero five’
- 0.274 = ‘zero point two seven four’
Place Value Table
| Hundreds | Tens | Units | . | Tenths | Hundredths |
|---|---|---|---|---|---|
| 100 | 10 | 1 | . | 1/10 | 1/100 |
Each position represents a specific value. We write quantities in decimal form using these place values.
Converting to Decimal Form
Examples of conversion:
- 234 tenths = 234/10 = 23.4
- 234 hundredths = 234/100 = 2.34
- Distribute digits according to their place values
Questions and Answers
Q: Can the quantity 4 2/10 be written as 42 (skipping the 1/10 in 2 × 1/10)? A: No, because we wouldn’t know if 42 means 4 tens and 2 units or 4 units and 2 tenths.
Q: Write 2 ones, 3 tenths and 5 hundredths in decimal form A: 2.35
Q: Write 1 ten and 5 tenths in decimal form A: 10.5
Q: Write 4 ones and 6 hundredths in decimal form A: 4.06
Q: Write 1 hundred, 1 one and 1 hundredth in decimal form A: 101.01
Q: Write 8/100 and 9/10 in decimal form A: 0.08 and 0.9 = 0.98
Q: Write 234 hundredths in decimal form A: 234/100 = 2.34
Q: Write 105 tenths in decimal form A: 105/10 = 10.5
3.6 Units of Measurement
Length Conversion
Imp conversions for length:
| From | To | Conversion |
|---|---|---|
| 1 cm | mm | 10 mm |
| 1 mm | cm | 0.1 cm |
| 1 m | cm | 100 cm |
| 1 cm | m | 0.01 m |
| 1 mm | m | 0.001 m |
Weight Conversion
Imp conversions for weight:
| From | To | Conversion |
|---|---|---|
| 1 kg | g | 1000 g |
| 1 g | kg | 0.001 kg |
| 1 g | mg | 1000 mg |
| 1 mg | g | 0.001 g |
Currency Conversion
For Indian currency:
- 1 rupee = 100 paise
- 1 paisa = 1/100 rupee = 0.01 rupee
- 75 paise = 75/100 rupee = 0.75 rupee
Questions and Answers
Q: How many cm is 1 mm? A: 1 mm = 1/10 cm = 0.1 cm
Q: How many cm is 5 mm? A: 5 mm = 5/10 cm = 0.5 cm
Q: How many cm is 12 mm? A: 12 mm = 10 mm + 2 mm = 1 cm + 2/10 cm = 1.2 cm
Q: How many mm is 5.6 cm? A: 5.6 cm = 56 mm (since each cm has 10 mm)
Q: How many m is 10 cm? A: 10 cm = 1/10 m = 0.1 m
Q: How many m is 15 cm? A: 15 cm = 15/100 m = 0.15 m
Q: How many mm does 1 meter have? A: 1 m = 1000 mm
Q: Can we write 1 mm = 1/1000 m? A: Yes, 1 mm = 1/1000 m = 0.001 m
Q: How many kilograms is 5 g? A: 5 g = 5/1000 kg = 0.005 kg
Q: How many kilograms is 254 g? A: 254 g = 254/1000 kg = 0.254 kg
3.7 Locating and Comparing Decimals
Number Line Representation
To show decimals on a number line:
- Divide unit segments into 10 equal parts
- Each division represents one-tenth
- Further divide for hundredths and thousandths
Comparing Decimals
Steps to compare decimal numbers:
- Compare digits from left to right
- Start with highest place value
- Stop when digits differ – larger digit means larger number
Equivalent Decimals
Some decimals have same value:
- 0.2 = 0.20 = 0.200 (same value)
- Adding zeros to right doesn’t change value
- But 0.2 ≠ 0.02 ≠ 0.002 (different values)
Finding Closest Decimals
To find which decimal is closest:
- Calculate distance from target number
- Smaller distance means closer value
- Use number line or subtraction to find out
Questions and Answers
Q: Does 0.2 = 0.20 = 0.200? A: Yes, they are all equal as they represent the same quantity (2 tenths). Adding zeros to the right doesn’t change the value.
Q: Which of these are the same: 4.5, 4.05, 0.405, 4.050, 4.50, 4.005, 04.50? A: 4.5 = 4.50 = 04.50 and 4.05 = 4.050
Q: Which is larger: 6.456 or 6.465? A: 6.465 is larger because at the hundredths place, 6 > 5
Q: Which decimal number is greater: 1.23 or 1.32? A: 1.32 is greater because at the tenths place, 3 > 2
Q: Which decimal number is greater: 3.81 or 13.800? A: 13.800 is greater because at the tens place, 1 > 0
Q: Which decimal number is greater: 1.009 or 1.090? A: 1.090 is greater because at the tenths place, 0 < 9
Q: Among 0.9, 1.1, 1.01, and 1.11, which is closest to 1? A: 1.01 is closest to 1 (distance = 0.01)
Q: Which among 3.56, 3.65, 3.099 is closest to 4? A: 3.65 is closest to 4 (distance = 0.35)
Q: Which among 0.8, 0.69, 1.08 is closest to 1? A: 1.08 is closest to 1 (distance = 0.08)
3.8 Addition and Subtraction of Decimals
Standard Procedure
The standard way to add or subtract decimals:
- Align decimal points vertically
- Add/subtract as with whole numbers
- Carry/borrow between place values
- Place decimal point in same position in answer
Addition Examples
Simple addition examples:
- 2.7 + 3.5 = 6.2
- Add tenths to tenths, units to units
- Convert excess tenths to units when needed
Subtraction Examples
Simple subtraction examples:
- 3.5 – 2.7 = 0.8
- Borrow from higher place values when needed
- Keep place value alignment proper
Decimal Sequences
In decimal sequences:
- Identify the pattern in sequence
- Add/subtract same amount each term
- Example: 4.4, 4.8, 5.2, 5.6 (adding 0.4 each time)
Estimating Results
To check if your answer is reasonable:
- Sum of decimals lies between sum of whole parts and sum of whole parts + 2
- This helps verify calculations
- Useful for quick approximations
Questions and Answers
Q: Find the sums:
- (a) 5.3 + 2.6 = 7.9
- (b) 18 + 8.8 = 26.8
- (c) 2.15 + 5.26 = 7.41
- (d) 9.01 + 9.10 = 18.11
- (e) 29.19 + 9.91 = 39.10
- (f) 0.934 + 0.6 = 1.534
- (g) 0.75 + 0.03 = 0.78
- (h) 6.236 + 0.487 = 6.723
Q: Find the differences:
- (a) 5.6 – 2.3 = 3.3
- (b) 18 – 8.8 = 9.2
- (c) 10.4 – 4.5 = 5.9
- (d) 17 – 16.198 = 0.802
- (e) 17 – 0.05 = 16.95
- (f) 34.505 – 18.1 = 16.405
- (g) 9.9 – 9.09 = 0.81
- (h) 6.236 – 0.487 = 5.749
Q: Continue the sequence 4.4, 4.8, 5.2, 5.6, 6.0, … A: 6.4, 6.8, 7.2 (adding 0.4 each time)
Q: Identify the change and write next 3 terms for 4.4, 4.45, 4.5, … A: Adding 0.05 each time: 4.55, 4.6, 4.65
Q: Continue 25.75, 26.25, 26.75, … A: Adding 0.5 each time: 27.25, 27.75, 28.25
Q: Continue 10.56, 10.67, 10.78, … A: Adding 0.11 each time: 10.89, 11.00, 11.11
Q: Continue 13.5, 16, 18.5, … A: Adding 2.5 each time: 21, 23.5, 26
Q: Continue 8.5, 9.4, 10.3, … A: Adding 0.9 each time: 11.2, 12.1, 13.0
Q: Continue 5, 4.95, 4.90, … A: Subtracting 0.05 each time: 4.85, 4.80, 4.75
Q: Continue 12.45, 11.95, 11.45, … A: Subtracting 0.5 each time: 10.95, 10.45, 9.95
Q: Continue 36.5, 33, 29.5, … A: Subtracting 3.5 each time: 26, 22.5, 19
3.9 More on the Decimal System
Real-World Applications
Decimals are used everywhere in real life:
- Measurement precision in engineering
- Financial calculations
- Scientific measurements
- Medical dosages
Common Mistakes
Some common errors people make:
- Decimal point placement errors
- Unit conversion mistakes
- Misreading decimal values
- These can lead to serious problems
Deceptive Decimal Notation
Be careful with decimal-like notations:
- Time: 4.5 hours = 4 hours 30 minutes (not 4 hours 5 minutes)
- Sports: 5.5 overs = 5 overs 5 balls (not 5 overs 3 balls)
- Different systems use decimal-like notation differently
Historical Development
The decimal system has rich history:
- Ancient Indian mathematicians used decimal fractions
- Al-Uqlīdisī described decimal notation around 950 CE
- Various notations were used before standardization
- Modern decimal point notation is most popular now
Questions and Answers
Q: When will the bus reach if it arrives 4.5 hours post noon? A: 4:30 p.m. (4.5 hours = 4 hours 30 minutes, not 4 hours 5 minutes)
Q: What does 5.5 overs mean in cricket? A: 5 overs and 5 balls (since 1 over = 6 balls, 0.5 over = 3 balls, but this notation means 5/6 over)
Chapter-End Questions and Answers
Figure it Out Questions
Q1: Convert the following fractions into decimals:
- (a) 5/100 = 0.05
- (b) 16/1000 = 0.016
- (c) 12/10 = 1.2
- (d) 254/1000 = 0.254
Q2: Convert the following decimals into a sum of tenths, hundredths and thousandths:
- (a) 0.34 = 3/10 + 4/100
- (b) 1.02 = 1 + 0/10 + 2/100
- (c) 0.8 = 8/10
- (d) 0.362 = 3/10 + 6/100 + 2/1000
Q4: Arrange in descending order:
- (a) 11.10, 11.01, 1.101, 1.011, 1.01
- (b) 2.768, 2.698, 2.675, 2.567, 2.499
- (c) 4.678 g, 4.666 g, 4.656 g, 4.600 g, 4.595 g
- (d) 33.331 m, 33.313 m, 33.133 m, 33.31 m, 33.13 m
Q5: Using digits 1, 4, 0, 8, and 6:
- (a) Decimal closest to 30: 30.468 or 30.486
- (b) Smallest decimal between 100 and 1000: 104.68
Q6: Will a decimal number with more digits be greater than a decimal number with fewer digits? A: Not necessarily. For example, 0.9 (1 decimal digit) > 0.8999 (4 decimal digits)
Q7: Mahi purchases 0.25 kg beans, 0.3 kg carrots, 0.5 kg potatoes, 0.2 kg capsicums, and 0.05 kg ginger. Total weight? A: 0.25 + 0.3 + 0.5 + 0.2 + 0.05 = 1.3 kg
Q8: Pinto supplies 3.79 L, 4.2 L, and 4.25 L milk in first three days. Total in 6 days is 25 L. Find last three days’ supply. A: First three days: 3.79 + 4.2 + 4.25 = 12.24 L Last three days: 25 – 12.24 = 12.76 L
Q9: Tinku weighed 35.75 kg in January and 34.50 kg in February. Gain or loss? A: Loss of 35.75 – 34.50 = 1.25 kg
Q10: Extend pattern: 5.5, 6.4, 6.39, 7.29, 7.28, 6.18, 6.17, ___, ___ A: Pattern alternates between +0.9/-0.01 and +0.9/-1.1 Next terms: 5.07, 5.06
Q11: How many millimeters make 1 kilometer? A: 1 km = 1000 m = 1000 × 1000 mm = 1,000,000 mm
Q12: Insurance costs 45 paise per passenger. If 1 lakh people opt for insurance, total fee? A: 1,00,000 × 0.45 = ₹45,000
Q13: Which is greater?
- (a) 10/1000 = 0.01 or 1/10 = 0.1? Answer: 1/10
- (b) One-hundredth (0.01) or 90 thousandths (0.09)? Answer: 90 thousandths
- (c) One-thousandth (0.001) or 90 hundredths (0.9)? Answer: 90 hundredths
Q14: Write in decimal form:
- (a) 87 ones, 5 tenths, 60 hundredths = 87 + 0.5 + 0.6 = 88.1
- (b) 12 tens and 12 tenths = 120 + 1.2 = 121.2
- (c) 10 tens, 10 ones, 10 tenths, 10 hundredths = 100 + 10 + 1 + 0.1 = 111.1
- (d) 25 tens, 25 ones, 25 tenths, 25 hundredths = 250 + 25 + 2.5 + 0.25 = 277.75
Q16: Write fractions in decimal form:
- (a) 1/2 = 0.5
- (b) 3/2 = 1.5
- (c) 1/4 = 0.25
- (d) 3/4 = 0.75
- (e) 1/5 = 0.2
- (f) 4/5 = 0.8
Imp Concepts from This Chapter
Fundamental Principles
- Units are divided into 10 equal parts for precision
- Place value system extends to fractional parts
- Decimal point separates whole from fractional parts
Imp Relationships
- 1 unit = 10 tenths = 100 hundredths = 1000 thousandths
- 10 hundredths = 1 tenth
- 100 hundredths = 1 unit
Operations
- Addition and subtraction follow standard algorithms
- Always align decimal points for calculations
- Convert between different decimal forms as needed
Practical Applications
- Measurement conversions in daily life
- Money calculations
- Scientific notation
- Precision required in various fields
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