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Topics

3.1 The Need for Smaller Units

When standard units are not precise enough, they are divided into smaller parts.

• Measurement Accuracy: Measuring screws that look the same but have different lengths (e.g., $2\frac{7}{10}$ cm vs. $3\frac{2}{10}$ cm).
• Object Measurement: Using a scale to measure everyday items like a pen or sharpener to get exact lengths.

3.2 A Tenth Part

Dividing one unit into 10 equal parts ($1/10$).

• Reading Tenths: $4\frac{1}{10}$ is read as "four and one-tenth".
• Adding Tenths: Calculating total arm length by adding lower arm ($\frac{27}{10}$ units) and upper arm ($\frac{36}{10}$ units) to get $\frac{63}{10}$ or $6\frac{3}{10}$ units.
• Subtracting Tenths: Finding a finger's length by subtracting the palm (6.7 units) from the total hand length (12.4 units), requiring splitting a unit into 10 tenths ($11\frac{14}{10} - 6\frac{7}{10} = 5\frac{7}{10}$).

3.3 A Hundredth Part

Dividing one-tenth into 10 equal parts, resulting in 100 parts per unit ($1/100$).

• Measurement: A folded paper length of $4\frac{4}{10}\frac{5}{100}$, read as 4 units, 4 tenths, and 5 hundredths.
• Sum of Hundredths: $15\frac{3}{10}\frac{4}{100} + 2\frac{6}{10}\frac{8}{100} = 18\frac{2}{100}$.
• Difference of Hundredths: $25\frac{9}{10} - 6\frac{4}{10}\frac{7}{100} = 19\frac{4}{10}\frac{3}{100}$.

3.4 Decimal Place Value

Using a decimal point to separate whole numbers and fractional parts.

• Notation Comparison: 705 (7 hundreds, 5 ones), 70.5 (7 tens, 5 tenths), and 7.05 (7 units, 5 hundredths).
• Reading Multi-digit Decimals: 0.274 is read as "zero point two seven four".
• Large Quantity Tenths: 234 tenths is written as 23.4.

3.5 Units of Measurement

• Length: $5\text{ mm} = 0.5\text{ cm}$; $15\text{ cm} = 0.15\text{ m}$; $1\text{ mm} = 0.001\text{ m}$.
• Weight: $5\text{ g} = 0.005\text{ kg}$; $254\text{ g} = 0.254\text{ kg}$; $1\text{ mg} = 0.001\text{ g}$.
• Currency: $75\text{ paise} = ₹0.75$; $1\text{ paisa} = ₹0.01$.

3.6 Locating and Comparing Decimals

• Number Line: Locating 1.4 between the whole numbers 1 and 2 by taking the fourth division of 10.
• Trailing Zeros: 0.2, 0.20, and 0.200 all represent the same value (2 tenths).
• Comparison: Comparing 6.456 and 6.465; though both have 6 units and 4 tenths, 6.465 is larger because 6 hundredths is greater than 5 hundredths.
• Proximity: Determining that 1.01 is closer to 1 than 0.9 is.

3.7 Addition and Subtraction of Decimals

Standard procedures for whole numbers apply if place values are aligned.

• Addition: $2.7\text{ m} + 3.5\text{ m} = 6.2\text{ m}$.
• Subtraction: $3.5\text{ m} - 2.7\text{ m} = 0.8\text{ m}$.
• Detailed Place Value Sum: $75.345 + 86.691 = 162.036$.
• Sequences: Identifying a pattern in 4.4, 4.8, 5.2, 5.6 (adding 0.4).

3.8 More on the Decimal System

• Historical Notation: In 950 CE, 0.059375 was represented as $0\text{\textbrokenbar}059375$.
• Measurement Disasters: Air Canada flight miscalculating 22,300 pounds of fuel instead of kilograms.
• Non-Decimal Notation: Cricket overs "5.5" meaning 5 overs and 5 balls, rather than 5.5 in a true decimal system.

Answer key

3.1 - The Need for smaller units

Page 47: Measuring the Screws

Question 1

In the following figure, screws are placed above a scale. Measure them and write their length in the space provided.

Looking closely at the image provided for the first set of screws:

• First Scale: It only shows integer markings ($0, 1, 2, 3$). The tip of the screw is past $2$, roughly between $2$ and $3$.

• Second Scale: It includes half-unit or fewer subdivisions, showing it lies past $2.5$.

• Third Scale: The unit length between two consecutive numbers is divided into $10$ equal parts (each division represents $\frac{1}{10}$ or $0.1$). The red dotted line aligns perfectly with the 7th mark after $2$.

• Answer for the First Screw: $2\frac{7}{10}\text{ cm}$ (or $2.7\text{ cm}$)

Looking closely at the image provided for the second set of screws:

• Third Scale: The red dotted line aligns perfectly with the 2nd mark after $3$.
• Answer for the Second Screw: $3\frac{2}{10}\text{ cm}$ (or $3.2\text{ cm}$)

Question 2

Which scale helped you measure the length of the screws accurately? Why?

• Answer: The third scale helped measure the length most accurately.

• Reason: Because it has the finest precision, where each unit length is divided into $10$ equal parts. This allows us to read the exact number of tenths ($\frac{1}{10}$) past the whole number, leaving no room for guessing.

Question 3

What is the meaning of $2\frac{7}{10}\text{ cm}$ (the length of the first screw)?

• Answer: It means a whole length of $2$ units plus $7$ one-tenths of a unit. On a standard ruler where the unit is in centimeters, this represents $2\text{ cm}$ and $7\text{ mm}$.

3.2 - A Tenth Part

Page 49: Comparing and Ordering Tenths

Question 1

Arrange these lengths in increasing order:

$\frac{9}{10},\; 1\frac{7}{10},\; \frac{130}{10},\; 13\frac{1}{10},\; 10\frac{5}{10},\; 7\frac{6}{10},\; 6\frac{7}{10},\; \frac{4}{10}$

To arrange them easily, let's look at all of them as improper fractions or improper values over $10$:

• $\frac{4}{10}$

• $\frac{9}{10}$

• $1\frac{7}{10} = \frac{17}{10}$

• $6\frac{7}{10} = \frac{67}{10}$

• $7\frac{6}{10} = \frac{76}{10}$

• $10\frac{5}{10} = \frac{105}{10}$

• $\frac{130}{10}$

• $13\frac{1}{10} = \frac{131}{10}$

Increasing Order (Smallest to Largest):

\frac{4}{10} < \frac{9}{10} < 1\frac{7}{10} < 6\frac{7}{10} < 7\frac{6}{10} < 10\frac{5}{10} < \frac{130}{10} < 13\frac{1}{10}

Page 50: Adding and Subtracting Tenths

Question 1: Addition Problems

Perform the following additions:

(a) $4\frac{3}{10} + 3\frac{5}{10}$

• Separate the whole numbers and fractional parts:

  $(4 + 3) + \left(\frac{3}{10} + \frac{5}{10}\right) = 7 + \frac{8}{10}$

• Answer: $7\frac{8}{10}$ (or $7.8$)

(b) $8\frac{7}{10} + 5\frac{6}{10}$

• Add the components:

  $(8 + 5) + \left(\frac{7}{10} + \frac{6}{10}\right) = 13 + \frac{13}{10}$

• Simplify the improper fraction $\frac{13}{10} = 1\frac{3}{10}$:

  $13 + 1 + \frac{3}{10} = 14\frac{3}{10}$

• Answer: $14\frac{3}{10}$ (or $14.3$)

(c) $12\frac{9}{10} + \frac{4}{10}$

• Add the components:

  $12 + \left(\frac{9}{10} + \frac{4}{10}\right) = 12 + \frac{13}{10}$

• Simplify $\frac{13}{10} = 1\frac{3}{10}$:

  $12 + 1 + \frac{3}{10} = 13\frac{3}{10}$

• Answer: $13\frac{3}{10}$ (or $13.3$)

Question 2: Subtraction Problems

Perform the following subtractions:

(a) $7\frac{8}{10} - 5\frac{5}{10}$

• Subtract the components directly:

  $(7 - 5) + \left(\frac{8}{10} - \frac{5}{10}\right) = 2 + \frac{3}{10}$

• Answer: $2\frac{3}{10}$ (or $2.3$)

(b) $8\frac{2}{10} - 3\frac{7}{10}$

• Since $\frac{2}{10}$ is smaller than $\frac{7}{10}$, borrow $1$ from the whole number $8$:

  $8\frac{2}{10} = 7 + 1 + \frac{2}{10} = 7\frac{12}{10}$

• Now subtract:

  $(7 - 3) + \left(\frac{12}{10} - \frac{7}{10}\right) = 4 + \frac{5}{10}$

• Answer: $4\frac{5}{10}$ (or $4.5$)

(c) $10 - 4\frac{3}{10}$

• Borrow $1$ from $10$ to make it a fraction:

  $10 = 9\frac{10}{10}$

• Now subtract:

  $(9 - 4) + \left(\frac{10}{10} - \frac{3}{10}\right) = 5 + \frac{7}{10}$

• Answer: $5\frac{7}{10}$ (or $5.7$)

Page 51: Honeybee Length, Shylaja's Hand, and Number Patterns

Question 4

The lengths of the body parts of a honeybee are given. Find its total length.

• Head: $2\frac{3}{10}\text{ units}$ * Thorax: $5\frac{4}{10}\text{ units}$ * Abdomen: $7\frac{5}{10}\text{ units}$

To find the total length, we add the whole numbers together and the fractional tenths together:

$\text{Total Length} = \left(2 + \frac{3}{10}\right) + \left(5 + \frac{4}{10}\right) + \left(7 + \frac{5}{10}\right)$

$\text{Total Length} = (2 + 5 + 7) + \left(\frac{3 + 4 + 5}{10}\right)$

$\text{Total Length} = 14 + \frac{12}{10}$

Since $\frac{12}{10} = 1\frac{2}{10}$:

$\text{Total Length} = 14 + 1 + \frac{2}{10} = 15\frac{2}{10}\text{ units} \text{ (or } 15.2\text{ units)}$

Question 5

The length of Shylaja’s hand is $12\frac{4}{10}\text{ units}$, and her palm is $6\frac{7}{10}\text{ units}$, as shown in the picture. What is the length of the longest (middle) finger?

To find the length of the middle finger, we subtract the palm length from the total hand length:

$\text{Finger Length} = 12\frac{4}{10} - 6\frac{7}{10}$

Convert $12\frac{4}{10}$ into a form that makes subtraction easier:

$12\frac{4}{10} = 11 + 1 + \frac{4}{10} = 11\frac{14}{10}$

Now, subtract:

$\text{Finger Length} = 11\frac{14}{10} - 6\frac{7}{10} = (11 - 6) + \left(\frac{14 - 7}{10}\right) = 5\frac{7}{10}\text{ units} \text{ (or } 5.7\text{ units)}$

Question 6: Extend the Patterns

Extend the pattern for the following rows:

(a) $4,\; 4\frac{3}{10},\; 4\frac{6}{10},\; \dots$

• Rule: Increasing by $\frac{3}{10}$ each time.
• Answer: $4\frac{9}{10},\; 5\frac{2}{10},\; 5\frac{5}{10},\; 5\frac{8}{10}$

(b) $8\frac{2}{10},\; 8\frac{7}{10},\; 9\frac{2}{10},\; \dots$

• Rule: Increasing by $\frac{5}{10}$ each time.
• Answer: $9\frac{7}{10},\dots 10\frac{2}{10},\dots 10\frac{7}{10},\dots 11\frac{2}{10}$

(c) $7\frac{6}{10},\; 8\frac{7}{10},\; \dots$

• Rule: Increasing by $1\frac{1}{10}$ (or $\frac{11}{10}$) each time.
• Answer: $9\frac{8}{10},\dots 10\frac{9}{10},\dots 12,\dots 13\frac{1}{10}$

(d) $5\frac{7}{10},\; 5\frac{3}{10},\; \dots$

• Rule: Decreasing by $\frac{4}{10}$ each time.
• Answer: $4\frac{9}{10},\dots 4\frac{5}{10},\dots 4\frac{1}{10},\dots 3\frac{7}{10}$

(e) $13\frac{5}{10},\; 13,\; 12\frac{5}{10},\; \dots$

• Rule: Decreasing by $\frac{5}{10}$ (half a unit) each time.
• Answer: $12,\dots 11\frac{5}{10},\dots 11,\dots 10\frac{5}{10}$

(f) $11\frac{5}{10},\; 10\frac{4}{10},\; 9\frac{3}{10},\; \dots$

• Rule: Decreasing by $1\frac{1}{10}$ (or $\frac{11}{10}$) each time.
• Answer: $8\frac{2}{10},\dots 7\frac{1}{10},\dots 6,\dots 4\frac{9}{10}$

3.3 A Hundredth Part

Page 53: A Hundredth Part

Question 7

The length of a sheet of paper was $8\frac{9}{10}\text{ units}$. It is folded in half along its length. What is its length now?

Folding the paper in half along its length divides the length measurement by $2$:

$\text{New Length} = \frac{8\frac{9}{10}}{2} = \frac{8}{2} + \frac{\frac{9}{10}}{2}$

$\text{New Length} = 4 + \frac{9}{20}$

To change this to hundredths to keep with the chapter's focus:

$\frac{9}{20} = \frac{9 \times 5}{20 \times 5} = \frac{45}{100}$

• Answer: $4\frac{45}{100}\text{ units}$ (or $4.45\text{ units}$)

Page 55: Identifying Longest & Shortest / Fraction Sums

Question 8

In each group, identify the longest and the shortest lengths.

To easily compare, let's write all fractions with a common denominator of $100$:

Question 9

What will be the sum of $15 + \frac{3}{10} + \frac{4}{100}$ and $2 + \frac{6}{10} + \frac{8}{100}$?

Group the whole numbers, tenths, and hundredths together:

$\text{Sum} = (15 + 2) + \left(\frac{3}{10} + \frac{6}{100}\right) + \left(\frac{4}{100} + \frac{8}{100}\right)$

Convert everything nicely into shared fractional values:

$\text{Whole numbers} = 17$

$\text{Tenths} = \frac{3 + 6}{10} = \frac{9}{10}$

$\text{Hundredths} = \frac{4 + 8}{100} = \frac{12}{100}$

Since $\frac{12}{100} = \frac{10}{100} + \frac{2}{100} = \frac{1}{10} + \frac{2}{100}$, carry the $\frac{1}{10}$ over to the tenths place:

$\text{Total Tenths} = \frac{9}{10} + \frac{1}{10} = \frac{10}{10} = 1 \text{ whole}$

$\text{Remaining Hundredths} = \frac{2}{100}$

Add the carried whole number to $17$:

$\text{Total Sum} = 17 + 1 + \frac{2}{100} = 18\frac{2}{100} \text{ (or } 18.02\text{)}$

Page 58: Figure it Out — Fractions Sums and Differences

Question 1: Find the sums and differences

To make adding and subtracting these mixed fractions easy, we can convert all fractional components to have a common denominator of $100$ (hundredths).

(a) $\frac{3}{10} + 3\frac{4}{100}$**

• Convert to hundredths: $\frac{3}{10} = \frac{3 \times 10}{10 \times 10} = \frac{30}{100}$

• Combine with the mixed number:

  $3 + \frac{30}{100} + \frac{4}{100} = 3 + \frac{34}{100}$

• Answer: $3\frac{34}{100}$ (or $3.34$)

(b) $9\frac{5}{10}\frac{7}{100} + 2\frac{1}{10}\frac{3}{100}$

Let's group the components into whole units, tenths, and hundredths:

• Whole numbers: $9 + 2 = 11$

• Tenths: $\frac{5}{10} + \frac{1}{10}= \frac{6}{10}$

• Hundredths: $\frac{7}{100} + \frac{3}{100} = \frac{10}{100}$

• Combine:

  $11 + \frac{6}{10} + \frac{10}{100} = 11 + \frac{60}{100} + \frac{10}{100}$

• Answer: **$11\frac{70}{100} = 11\frac{7}{10}$

(c) $15\frac{6}{10} \frac{4}{100} + 14\frac{3}{10} \frac{6}{100}$

• Group the whole numbers: $15 + 14 = 29$

• Group the tenths: $\frac{6}{10} + \frac{3}{10} = \frac{9}{10} = \frac{90}{100}$

• Group the hundredths: $\frac{4}{100} + \frac{6}{100} = \frac{10}{100}$

• Add fractional components together: $\frac{90}{100} + \frac{10}{100} = \frac{100}{100} = 1\text{ whole unit}$

• Add the carried whole unit to your whole number total:

  $29 + 1 = 30$

• Answer: $30$

(d) $7\frac{7}{100} - \frac{4}{100}$

• Subtract the hundredths directly:

  $7 + \left(\frac{7}{100} - \frac{4}{100}\right) = 7 + \frac{3}{100}$

• Answer: $7\frac{3}{100}$ (or $7.03$)

(e) $8\frac{6}{100} - 5\frac{3}{100}$

• Subtract the components:

  $(8 - 5) + \left(\frac{6}{100} - \frac{3}{100}\right) = 3 + \frac{3}{100}$

• Answer: $3\frac{3}{100}$ (or $3.03$)

(f) $12\frac{6}{10} \frac{2}{100} - \frac{9}{10}\frac{9}{100}$

Given Expression:

$\left(12\frac{6}{10} + \frac{2}{100}\right) - \left(\frac{9}{10} + \frac{9}{100}\right)$

Step 1: Simplify the first bracket

Convert everything into hundredths to find a common denominator:

• $12\frac{6}{10} = 12 + \frac{6 \times 10}{10 \times 10} = 12 + \frac{60}{100} = 12\frac{60}{100}$

• Add the parts together:

  $12\frac{60}{100} + \frac{2}{100} = 12\frac{62}{100}$

Step 2: Simplify the second bracket

Convert the tenths to hundredths:

• $\frac{9}{10} = \frac{9 \times 10}{10 \times 10} = \frac{90}{100}$

• Add the parts together:

  $\frac{90}{100} + \frac{9}{100} = \frac{99}{100}$

Step 3: Subtract the simplified brackets

Now substitute back into the original problem:

$12\frac{62}{100} - \frac{99}{100}$

Since $\frac{62}{100}$ is smaller than $\frac{99}{100}$, borrow $1$ from the whole number $12$:

• $12\frac{62}{100} = 11 + 1 + \frac{62}{100} = 11 + \frac{100}{100} + \frac{62}{100} = 11\frac{162}{100}$

Now, perform the final subtraction:

$11\frac{162}{100} - \frac{99}{100} = 11 + \left(\frac{162 - 99}{100}\right) = 11\frac{63}{100}$

Final Answers:

• Fraction form: $11\frac{63}{100}$

• Decimal form: $11.63$

3.4 Decimal Place Value

Page 61 — "How Big?" In-Text Exercises

(a) How many thousandths make one unit?

• Understanding: "One unit" is written as $1$, and "one thousandth" is written as $\frac{1}{1000}$. We need to see how many times $\frac{1}{1000}$ goes into $1$.

• Calculation:

  $1 \div \frac{1}{1000} = 1 \times 1000 = 1000$

• Answer: $1000$ thousandths make one unit.

(b) How many thousandths make one tenth?

• Understanding: "One tenth" is $\frac{1}{10}$ and "one thousandth" is $\frac{1}{1000}$.

• Calculation:

  $\frac{1}{10} \div \frac{1}{1000} = \frac{1}{10} \times 1000 = 100$

• Answer: $100$ thousandths make one tenth.

(c) How many thousandths make one hundredth?

• Understanding: "One hundredth" is $\frac{1}{100}$ and "one thousandth" is $\frac{1}{1000}$.

• Calculation:

  $\frac{1}{100} \div \frac{1}{1000} = \frac{1}{100} \times 1000 = 10$

• Answer: $10$ thousandths make one hundredth.

(d) How many tenths make one ten?

• Understanding: "One ten" is $10$ and "one tenth" is $\frac{1}{10}$.

• Calculation:

  $10 \div \frac{1}{10} = 10 \times 10 = 100$

• Answer: $100$ tenths make one ten.

(e) How many hundredths make one ten?

• Understanding: "One ten" is $10$ and "one hundredth" is $\frac{1}{100}$.

• Calculation:

  $10 \div \frac{1}{100} = 10 \times 100 = 1000$

• Answer: $1000$ hundredths make one ten.

"Make a few more questions of this kind and answer them"

Here are sample self-generated questions to satisfy this textbook prompt:

Question 1: How many hundredths make one unit?

• Calculation: $1 \div \frac{1}{100} = 1 \times 100 = 100$

• Answer: $100$ hundredths make one unit.

Question 2: How many tenths make one unit?

• Calculation: $1 \div \frac{1}{10} = 1 \times 10 = 10$

• Answer: $10$ tenths make one unit.

Question 3: How many hundredths make one tenth?

• Calculation: $\frac{1}{10} \div \frac{1}{100} = \frac{1}{10} \times 100 = 10$

• Answer: $10$ hundredths make one tenth.

Question 4: How many thousandths make one ten?

• Calculation: $10 \div \frac{1}{1000} = 10 \times 1000 = 10,000$

• Answer: $10,000$ thousandths make one ten.

Page 63 –

Page 79: Decimals Comparisons & Applications

Question 10

Extend the pattern: $5.5,\; 6.4,\; 6.39,\; 7.29,\; 7.28,\; 6.18,\; 6.17,\; \dots,\; [cite_start]\dots$

Let’s track the differences alternating step-by-step:

1. $5.5 \to 6.4$ (Adds $0.9$)
2. $6.4 \to 6.39$ (Subtracts $0.01$)
3. $6.39 \to 7.29$ (Adds $0.9$)
4. $7.29 \to 7.28$ (Subtracts $0.01$)
5. $7.28 \to 6.18$ (Subtracts $1.1$ — Note: there is a structural pattern break or step down here)
6. $6.18 \to 6.17$ (Subtracts $0.01$)

Following the localized core alternating rule (Add $0.9$, then subtract $0.01$):

• Next step: $6.17 + 0.9 =$ $7.07$
• Following step: $7.07 - 0.01 =$ $7.06$

Answer: $7.07,\; 7.06$

Question 11

How many millimeters make $1\text{ kilometer}$?

• $1\text{ km} = 1,000\text{ meters}$
• $1\text{ meter} = 100\text{ centimeters}$
• $1\text{ cm} = 10\text{ millimeters}$

$1\text{ km} = 1,000 \times 100 \times 10\text{ mm} = 1,000,000\text{ mm}$

• Answer: $1,000,000\text{ mm}$ (One million millimeters)

Question 12

Indian Railways offers optional travel insurance for passengers who book e-tickets. It costs $45\text{ paise}$ per passenger. If $1\text{ lakh}$ ($100,000$) people opt for insurance in a day, what is the total insurance fee paid?

$\text{Total Fee in Paise} = 100,000 \times 45\text{ paise} = 4,500,000\text{ paise}$

Since 100paise = 1₹

$\text{Total Fee in Rupees}=\frac{4,500,000}{100}=₹45,000$

• Answer: $₹45,000$

Question 13

Which is greater?

(a) $\frac{10}{1000}$ or $\frac{1}{10}$

• Simplify $\frac{10}{1000} = \frac{1}{100}$. Since $\frac{1}{10} > \frac{1}{100}$:
• Answer: $\frac{1}{10}$ is greater.

(b) One-hundredth or $90$ thousandths?

• One-hundredth $= \frac{1}{100} = \frac{10}{1000}$.
• Compared with $\frac{90}{1000}$:
• Answer: $90$ thousandths is greater.

(c) One-thousandth or $90$ hundredths?

• One-thousandth $= \frac{1}{1000}$.
• $90$ hundredths $= \frac{90}{100} = \frac{900}{1000}$.
• Answer: $90$ hundredths is greater.