---
In our daily life, we often need to measure things with great accuracy. Sometimes, using whole units (like centimeters) is not enough to measure small differences precisely.
**Real-life Example:** Sonu's mother was fixing a toy with a screw. Two screws looked almost identical to Sonu, but they were actually of different lengths. When measured on a scale, one screw was 2 7/10 cm and the other was 3 2/10 cm. This small difference was crucial for the toy to be fixed properly. Without measuring to one-tenth of a centimeter, Sonu's mother could not have identified which screw was correct.
When we use a scale to measure, the space between two consecutive whole numbers (like between 2 cm and 3 cm) is divided into 10 equal parts. Each of these parts represents 1/10 (one-tenth) of a centimeter.
**How to read 2 7/10 cm:**
**How to read 3 2/10 cm:**
The unit length (1 cm) is divided into 10 equal smaller parts when exact measurements are required. This allows us to measure quantities between whole units with precision.
**Practice:** Measure a pen, sharpener, or any object using a centimeter scale and express your measurement as a mixed number with tenths (e.g., 7 4/10 cm).
---
A **tenth** is created when we divide 1 unit into 10 equal parts. Each part is called **one-tenth** and is written as 1/10.
**Important Relationship:**
Any length can be expressed in two different ways:
**Example:** A pencil's length is 3 4/10 units
**Way 1 - Mixed Number Form:**
3 4/10 means "3 units and 4 one-tenths"
= (3 × 1) + (4 × 1/10) units
**Way 2 - Pure Fractional Form:**
34 one-tenths = 34 × (1/10) = 34/10 units
Both representations show the **same length**.
| Notation | Reading | Meaning |
|----------|---------|---------|
| 4 1/10 | Four and one-tenth | 4 units + 1 tenth |
| 4/10 | Four-tenths | 4 tenths total |
| 41/10 | Forty-one tenths | 41 tenths total = 4 1/10 |
| 41 1/10 | Forty-one and one-tenth | 41 units + 1 tenth |
**Example - USB Cable:**
The length is shown as:
Both mean the exact same measurement.
To compare and arrange numbers with tenths, convert them to the same form.
**Example Problem:** Arrange these in increasing order:
**Solution Process:**
First, convert all to mixed numbers or pure fractions:
**Increasing Order:**
9/10, 4/10, 1 7/10, 6 7/10, 7 6/10, 10 5/10, 13, 13 1/10
Wait - let me correct: 9/10 > 4/10, so: 4/10, 9/10, 1 7/10, 6 7/10, 7 6/10, 10 5/10, 130/10, 13 1/10
**Problem:** Sonu's lower arm is 2 7/10 units, and his upper arm is 3 6/10 units. Find the total length.
**Method 1 - Adding Units and Tenths Separately:**
```
2 7/10
+ 3 6/10
________
```
Step 1: Add the whole units: 2 + 3 = 5 units
Step 2: Add the tenths: 7/10 + 6/10 = 13/10
Step 3: Convert excess tenths: 13/10 = 10/10 + 3/10 = 1 + 3/10
Step 4: Add the new unit: 5 + 1 = 6 units
Step 5: Result: 6 3/10 units
**Complete Working:**
(2 + 3) + (7/10 + 6/10)
= 5 + 13/10
= 5 + (10/10 + 3/10)
= 5 + 1 + 3/10
= 6 3/10
**Method 2 - Converting to Pure Fractions:**
Convert both numbers to one-tenths:
Add: 27/10 + 36/10 = 63/10
Convert back: 63/10 = 60/10 + 3/10 = 6 + 3/10 = 6 3/10
**Both methods give the same answer: 6 3/10 units**
**Problem:** Shylaja's hand length is 12 4/10 units. Her palm is 6 7/10 units. Find the length of her longest finger.
Calculate: (12 4/10) - (6 7/10)
**Method 1 - Regrouping/Borrowing:**
```
12 4/10
---------
```
We cannot subtract 7/10 from 4/10 directly.
Step 1: Borrow 1 unit from 12, converting it to 10/10
Step 2: Now we have 11 units and (10/10 + 4/10) = 11 units and 14/10
```
11 14/10
---------
```
Step 3: Subtract tenths: 14/10 - 7/10 = 7/10
Step 4: Subtract units: 11 - 6 = 5
Step 5: Result: 5 7/10 units
**Method 2 - Converting to Pure Fractions:**
Subtract: 124/10 - 67/10 = 57/10 = 5 7/10
**Answer: 5 7/10 units**
Identify the pattern in each sequence and extend it:
**Example:** 4, 4 3/10, 4 6/10, _____, _____, _____, _____
**Pattern:** Each term increases by 3/10
**Sequence:** 4, 4 3/10, 4 6/10, 4 9/10, 5 2/10, 5 5/10, 5 8/10
**Example:** A Celestial Pearl Danio fish is 2 4/10 cm long, and a Philippine Goby is 9/10 cm long. Find the difference.
2 4/10 - 9/10
Convert 2 4/10 to tenths: 2 4/10 = 24/10
Subtract: 24/10 - 9/10 = 15/10 = 1 5/10 cm
**The Celestial Pearl Danio is 1 5/10 cm longer than the Philippine Goby.**
---
When we need even more precise measurements than tenths allow, we divide each tenth into 10 equal parts. This gives us **hundredths**.
**Relationship:**
If a unit is divided into 10 equal parts (tenths), and each tenth is further divided into 10 equal parts (hundredths), then:
**Example:** When a sheet of paper that is 8 9/10 units long is folded in half, its new length falls between 4 4/10 and 4 5/10 units. To measure this precisely, we need hundredths.
The length becomes: **4 4/10 5/100** (read as "4 units and 4 one-tenths and 5 one-hundredths")
The same length can be written in three different ways:
**Example:** Wire length = 1 1/10 4/100
**Way 1 - Mixed Form with Tenths and Hundredths:**
1 1/10 4/100 means 1 unit + 1 tenth + 4 hundredths
**Way 2 - Mixed Form with Only Hundredths:**
1 14/100 means 1 unit + 14 hundredths
(Because 1/10 = 10/100, so 1/10 + 4/100 = 10/100 + 4/100 = 14/100)
**Way 3 - Pure Fraction Form:**
114/100 means 114 hundredths
(All parts expressed in hundredths)
**Key Conversions:**
| Notation | Reading | Meaning |
|----------|---------|---------|
| 0.34 | Zero point three four | 3 tenths + 4 hundredths |
| 3.07 | Three point zero seven | 3 units + 0 tenths + 7 hundredths |
| 2.5 | Two point five | 2 units + 5 tenths (50 hundredths) |
| 0.99 | Zero point nine nine | 9 tenths + 9 hundredths |
**Problem:** In the group {3/10, 3/100, 33/100}, identify longest and shortest.
**Solution:**
Convert all to hundredths:
Arrange: 3/100 < 30/100 < 33/100
**Shortest:** 3/100
**Longest:** 33/100
**Problem:** Find the sum: 15 3/10 4/100 + 2 6/10 8/100
**Method 1 - Adding Parts Separately:**
```
15 3/10 4/100
+ 2 6/10 8/100
_________________
```
Step 1: Add units: 15 + 2 = 17
Step 2: Add tenths: 3/10 + 6/10 = 9/10
Step 3: Add hundredths: 4/100 + 8/100 = 12/100
Step 4: Convert 12/100 to tenths and hundredths: 12/100 = 10/100 + 2/100 = 1/10 + 2/100
Step 5: Add to tenths: 9/10 + 1/10 = 10/10 = 1 unit
Step 6: Final answer: 17 + 1 unit + 2/100 = **18 2/100**
**Method 2 - Converting All to Hundredths:**
Convert everything to hundredths first:
Add: 1534/100 + 268/100 = 1802/100
Convert back: 1802/100 = 1800/100 + 2/100 = 18 + 2/100 = **18 2/100**
**Both methods give: 18 2/100**
**Problem:** Find: 25 9/10 - 6 4/10 7/100
**Step-by-Step Solution:**
```
25 9/10 0/100
__________________
```
Step 1: We cannot subtract 7/100 from 0/100, so borrow from tenths
Step 2: 9/10 = 8/10 + 10/100, so now we have 25 8/10 10/100
```
25 8/10 10/100
__________________
```
Step 3: Subtract hundredths: 10/100 - 7/100 = 3/100
Step 4: Subtract tenths: 8/10 - 4/10 = 4/10
Step 5: Subtract units: 25 - 6 = 19
Step 6: Answer: **19 4/10 3/100**
To compare numbers like 3 6/10, 3 6/100, and 3 6/10 6/100:
**Convert to a common denominator (hundredths):**
**Increasing order:** 3 6/100 < 3 6/10 < 3 6/10 6/100
---
The **decimal system** is based on the number 10. The word "decimal" comes from the Latin word "decem" (meaning ten), which is related to the Sanskrit word "daśha" (meaning ten). Similar words for 10 exist across Indian languages like Hindi, Marathi, Gujarati, Odia, and others.
**Key Feature:** Each place value is **10 times** the place value to its right, and **1/10** of the place value to its left.
In the whole number system:
```
10,000 → 1,000 → 100 → 10 → 1
× 10 × 10 × 10 × 10
÷ 10 ÷ 10 ÷ 10 ÷ 10
```
**Example:** In the number 281:
To represent numbers smaller than 1, we divide 1 unit into 10 equal parts, getting **1/10 (tenths)**.
Further dividing 1/10 into 10 parts gives **1/100 (hundredths)**.
This pattern continues:
```
... → 100 → 10 → 1 → 1/10 → 1/100 → 1/1000 → ...
× 10 × 10 × 10 × 10 × 10
÷ 10 ÷ 10 ÷ 10 ÷ 10 ÷ 10
```
**Complete Place Value Chart:**
| Place Name | Value |
|-----------|-------|
| Hundreds | 100 |
| Tens | 10 |
| Units/Ones | 1 |
| Tenths | 1/10 |
| Hundredths | 1/100 |
| Thousandths | 1/1000 |
| Ten-thousandths | 1/10,000 |
To distinguish where the whole number ends and the fractional part begins, we use a **decimal point (.)** as a separator.
**Examples:**
| Number | Meaning | Reading |
|--------|---------|---------|
| 705 | 7×100 + 0×10 + 5×1 | Seven hundred and five |
| 70.5 | 7×10 + 0×1 + 5×(1/10) | Seventy point five |
| 7.05 | 7×1 + 0×(1/10) + 5×(1/100) | Seven point zero five |
| Decimal | Hundreds | Tens | Units | Tenths | Hundredths |
|---------|----------|------|-------|--------|------------|
| 705 | 7×100 | 0×10 | 5×1 | — | — |
| 70.5 | — | 7×10 | 0×1 | 5×(1/10) | — |
| 7.05 | — | — | 7×1 | 0×(1/10) | 5×(1/100) |
**Example 1:** Number 3.45
This means:
**Total value:** 3 + 4/10 + 5/100 = 3 + 0.4 + 0.05
**Rule:** Read the digits after the decimal point individually, not as a complete number.
| Decimal | Reading | Why? |
|---------|---------|------|
| 0.274 | Zero point two seven four | NOT "zero point two hundred seventy-four" |
| 70.5 | Seventy point five | Short form of "seventy and five-tenths" |
| 7.05 | Seven point zero five | We must say "zero" for the missing tens place after decimal |
**Important:** 0.274 means:
NOT 274 hundredths (which would be written as 2.74).
**Example 1:** How many tenths in 2 3/10?
**Example 2:** How to write 234 tenths in decimal form?
**Using a place value table:**
| Hundreds | Tens | Units | Tenths |
|----------|------|-------|--------|
| 2 | 3 | 4 | |
Since 234 tenths has 23 complete units (230/10 = 23) and 4 tenths remaining:
| Tens | Units | Tenths |
|------|-------|--------|
| 2 | 3 | 4 |
**Result: 23.4**
**Example 3:** Write 234 hundredths in decimal form.
**Example 4:** Write 105 tenths in decimal form.
Important questions about fractional parts:
1. **How many thousandths make one unit?**
Answer: 1000 (because 1000 × 1/1000 = 1)
2. **How many thousandths make one tenth?**
Answer: 100 (because 100 × 1/1000 = 100/1000 = 1/10)
3. **How many thousandths make one hundredth?**
Answer: 10 (because 10 × 1/1000 = 10/1000 = 1/100)
4. **How many tenths make one ten?**
Answer: 100 (because 10 × 10 = 100 tenths)
5. **How many hundredths make one ten?**
Answer: 1000 (because 10 × 100 = 1000 hundredths)
---
**Relationship:** 1 cm = 10 mm
Therefore: **1 mm = 1/10 cm = 0.1 cm**
#### Converting Millimeters to Centimeters
**Example 1:** Convert 5 mm to cm
**Example 2:** Convert 12 mm to cm
**Example 3:** Convert 56 mm to cm
#### Converting Centimeters to Millimeters
**Example:** Convert 5.6 cm to mm
#### Practice Conversions (mm ↔ cm)
| Millimeters | Centimeters |
|-------------|-------------|
| 12 mm | 1.2 cm |
| 56 mm | 5.6 cm |
| 70 mm | 7.0 cm |
| 9 mm | 0.9 cm |
| 134 mm | 13.4 cm |
| 2036 mm | 203.6 cm |
**Relationship:** 1 m = 100 cm
Therefore: **1 cm = 1/100 m = 0.01 m**
#### Converting Centimeters to Meters
**Example 1:** Convert 10 cm to m
**Example 2:** Convert 15 cm to m
**Example 3:** Convert 36 cm to m
**Example 4:** Convert 4 cm to m
**Example 5:** Convert 325 cm to m
#### Converting Meters to Centimeters
**Example:** Convert 0.89 m to cm
**Practice Conversions (cm ↔ m)**
| Centimeters | Meters |
|-------------|--------|
| 36 cm | 0.36 m |
| 50 cm | 0.50 m |
| 89 cm | 0.89 m |
| 4 cm | 0.04 m |
| 325 cm | 3.25 m |
| 207 cm | 2.07 m |
**Relationship Chain:**
**Conversion:** **1 mm = 1/1000 m = 0.001 m**
**Relationship:** 1 kg = 1000 g
Therefore: **1 g = 1/1000 kg = 0.001 kg**
#### Converting Grams to Kilograms
**Example 1:** Convert 5 g to kg
**Example 2:** Convert 10 g to kg
**Example 3:** Convert 254 g to kg
**Breaking it down:**
**Market/Cooking Contexts:**
| Conversion | Relationship |
|-----------|--------------|
| Millimeters to Centimeters | 1 mm = 0.1 cm |
| Centimeters to Meters | 1 cm = 0.01 m |
| Millimeters to Meters | 1 mm = 0.001 m |
| Grams to Kilograms | 1 g = 0.001 kg |
**Key Understanding:** Each conversion uses powers of 10, which is why the decimal system is so useful for measurements.
---
**Steps:**
1. Add units separately
2. Add tenths separately
3. Add hundredths separately
4. If tenths exceed 9, convert 10 tenths to 1 unit
5. If hundredths exceed 9, convert 10 hundredths to 1 tenth
**Formula:**
(A B C) + (D E F) = (A+D) (B+E) (C+F) with conversions as needed
where A and D are units, B and E are tenths, C and F are hundredths.
**Steps:**
1. If we cannot subtract from tenths, borrow from units (1 unit = 10 tenths)
2. If we cannot subtract from hundredths, borrow from tenths (1 tenth = 10 hundredths)
3. Subtract units, then tenths, then hundredths
**Property:** Each place value = 10 × (place value to its right) = (1/10) × (place value to its left)
---
1. **Confusing 0.5 with 0.05**
2. **Reading decimal numbers incorrectly**
3. **Forgetting to write zero in the tenths place**
Q1. Sonu's mother needed screws of different sizes. This teaches us that measuring small differences requires:
Answer: A — The story shows that standard units are too large for precise measurements, so we divide 1 cm into 10 tenths to measure accurately.
Q2. How many one-tenths are there in 27/10?
Answer: C — 27/10 means 27 one-tenths, since each 1/10 is one one-tenth.
Q3. Read the measurement 3 5/10 cm aloud:
Answer: B — A mixed number like 3 5/10 is read as 'three and five one-tenths' or 'three and five-tenths.'
Q4. Which represents the smallest length?
Answer: B — 3/100 is much smaller than 3/10 because hundredths are 10 times smaller than tenths.
Q5. A paper sheet measures 8 9/10 units. When folded in half, its length falls between which two tenths?
Answer: B — Half of 8 9/10 is 4 45/100, which lies between 4 4/10 (=4 40/100) and 4 5/10 (=4 50/100).
Q6. Shylaja's hand length is 12 4/10 units and palm length is 6 7/10 units. What is her middle finger length? (Hint: Subtract palm from hand length.)
Answer: A — 12 4/10 − 6 7/10: Cannot subtract 7/10 from 4/10, so borrow 1 unit to get 11 14/10 − 6 7/10 = 5 7/10.
Q7. In the place value system, if ones place has value 1, tenths place has value:
Answer: A — Each place to the right is 1/10 of the previous place, so tenths = 1/10 of ones place.
Q8. A Celestial Pearl Danio is 2 4/10 cm long and a Philippine Goby is 9/10 cm long. How much longer is the Danio?
Answer: A — 2 4/10 − 9/10 = 1 14/10 − 9/10 = 1 5/10 cm. (Converting 2 4/10 to 1 14/10 by borrowing 1 unit.)
Q9. When adding 15 3/10 4/100 and 2 6/10 8/100, why do we group units, tenths, and hundredths separately?
Answer: B — Just as we add ones with ones and tens with tens in whole numbers, we add hundredths with hundredths, tenths with tenths, and units with units in decimals.
Q10. If we split a one-tenth into 10 equal parts, what fraction of the unit is each part?
Answer: B — One-tenth divided into 10 parts means 1/10 ÷ 10 = 1/100 of the unit.
How many one-tenths make 1 unit?
10 one-tenths equal 1 unit.
What does 2 7/10 cm mean?
2 cm and 7 one-tenths of a cm, or 27/10 cm total.
How many one-hundredths are in 1 one-tenth?
10 one-hundredths make 1 one-tenth.
Write 45 one-hundredths in two other ways.
45/100 can be written as 4 5/10 (converting 40/100 to tenths) or 0 45/100.
Why do we split units into 10 parts instead of other numbers?
Because 10 is the base of our Indian place value system, making calculations consistent and easy.
If Sonu's lower arm is 2 7/10 units and upper arm is 3 6/10 units, what is the total?
Total is 6 3/10 units: (2+3) units and (7+6) tenths = 5 13/10 = 6 3/10.
Read 15 3/10 4/100 aloud.
Fifteen and three-tenths and four-hundredths, or fifteen and thirty-four-hundredths.
When subtracting 6 7/10 from 12 4/10, why do we need to borrow?
Because we cannot subtract 7 tenths from 4 tenths, so we borrow 1 unit (= 10 tenths) first.
How many one-hundredths equal 1 unit?
100 one-hundredths equal 1 unit.
Convert 34/10 into mixed number form.
34/10 = 3 4/10 (three and four-tenths).
What is the length of the Sonu's arm if his lower arm is 2 7/10 units and upper arm is 3 6/10 units? Write your answer as a mixed number. [1 mark]
Add units separately from tenths. If tenths exceed 10, convert 10 tenths to 1 unit.
A pencil is 7 8/10 units long. A sharpener is 2 3/10 units long. Find the total length of both objects. Show your working. [2 marks]
Add the units together and add the tenths together separately. Remember, 10 tenths = 1 unit.
Shylaja's total hand length is 12 4/10 units. Her palm measures 6 7/10 units. Calculate the length of her middle finger by subtracting palm from hand length. Show all steps, including borrowing if needed. [3 marks]
Cannot subtract 7/10 from 4/10 directly. Borrow 1 unit (convert to 10/10) from the whole number part first. Then subtract tenths from tenths and units from units.
A honeybee has three body parts: Head = 2 3/10 units, Thorax = 5 4/10 units, and Abdomen = 7 5/10 units. (a) Find the total length of the honeybee. (b) If a fly is 15 8/10 units long, how much longer is the fly than the honeybee? Show all working and explain each step. [5 marks]
(a) Add all three parts by grouping units and tenths separately. Watch for tenths that sum to 10 or more and convert to units. (b) Subtract the honeybee's total from the fly's length; borrow if the tenths part of honeybee is larger than fly's tenths.
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