Key Concept: Concept of Precision with Smaller Units

c) 6730
[Solution Description]
The expression $6 \frac{7}{10} \frac{3}{100}$ represents 6 full units, 7 tenths, and 3 hundredths. Converting these to one-thousandths:
- 6 units becomes $6 \times 1000 = 6000$ thousandths
- 7 tenths becomes $7 \times 100 = 700$ thousandths
- 3 hundredths becomes $3 \times 10 = 30$ thousandths
Totaling them gives:
$6000 + 700 + 30 = 6730$
Therefore, the result is 6730 one-thousandths.

Key Concept: The Need for Smaller Units

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.
[Solution Description]
The assertion states that dividing a unit into smaller parts increases measurement precision. This is true because by having smaller increments, we can measure lengths more accurately.
The reason states that when a unit is divided into ten equal parts, it results in one-tenth. This is also true and aligns with how the Indian place value system extends below one, where division by 10 provides one-tenth.
Both statements are true; however, the reason does not directly explain why precision is enhanced merely by knowing one-tenth as a division result. Therefore, while both are true, the reason is not the correct explanation of the assertion.

Key Concept: Dividing Units and Place Value System

b) $0.03$ units
[Solution Description]
To solve this, we first note that dividing a length into 10 equal parts gives one-tenth ($\frac{1}{10}$) of the whole unit. Dividing each one-tenth into 10 equal further parts results in parts of size $\frac{1}{100}$ of the original unit. Therefore, if we combine three of these smaller portions:
$\text{Value of 3 parts} = 3 \times \frac{1}{100} = \frac{3}{100}$
Thus, three smaller parts make up $\frac{3}{100}$ of the original unit.

Key Concept: Extended Place Value and Fraction Conversion

b) 5670
[Solution Description]
The given measurement can be expressed in decimal form as:
$5\frac{6}{10}\frac{7}{100} = 5 + \frac{6}{10} + \frac{7}{100} = 5.67$
To convert it into one-thousandths:
$5.67 = \frac{567}{100}$
Converting to one-thousandths involves multiplying by 10 to get:
$\frac{567}{100} \times \frac{10}{10} = \frac{5670}{1000}$
So, there are 5670 one-thousandths in this measurement.

Key Concept: Understanding Metric Conversions

b) 0.56 m
[Solution Description] To convert centimeters to meters, use the conversion: $1$ m = $100$ cm. Divide the given length in cm by 100 to find the equivalent length in meters.
$56 \text{ cm} = \frac{56}{100} \text{ m} = 0.56 \text{ m}$
Thus, the length of the string is $0.56$ m.

Key Concept: Practical Application, Unit Conversion

c) 32 mm
[Solution Description] The screw length is given as $3 \frac{2}{10}$ cm, which translates to centimeters as:
$3 + \frac{2}{10} = 3.2 \text{ cm}$
To convert centimeters to millimeters, we use the conversion factor $1 \text{ cm} = 10 \text{ mm}$:
$3.2 \times 10 = 32 \text{ mm}$
The length of the screw in millimeters is 32 mm.

Key Concept: Measuring small differences

c) Assertion is true, but Reason is false.
[Solution Description]
Measuring small differences requires precision, which is achieved by using scales with smaller divisions. If we have a screw length of $2 \frac{7}{10}$ cm, this implies moving from 0 to 2 and then taking seven parts of $\frac{1}{10}$, which means the scale is divided into 10 equal parts per whole number. Hence, a scale with 10 subdivisions is sufficient for measuring $2 \frac{7}{10}$ cm, not 100 as stated in the reason. Therefore, the assertion is true as it correctly describes the need for detailed measurement tools, but the reason is false because it inaccurately describes the required level of subdivision for the given measurement.

Key Concept: Converting Fractions to Tenths

c) $\frac{28}{10}$
[Solution Description]
First, convert both mixed numbers to improper fractions with a denominator of 10.
For $4 \frac{5}{10}$:
$4 \frac{5}{10} = \frac{(4 \times 10 + 5)}{10} = \frac{45}{10}$
For $1 \frac{7}{10}$:
$1 \frac{7}{10} = \frac{(1 \times 10 + 7)}{10} = \frac{17}{10}$
Subtract the two fractions:
$\frac{45}{10} - \frac{17}{10} = \frac{45 - 17}{10} = \frac{28}{10}$
So, the result is $\frac{28}{10}$.

Key Concept: Division Into Smaller Parts, Place Values

c) 43
[Solution Description] The given rope measures $4 \frac{3}{10}$ meters. We can express this as:
$4 + \frac{3}{10}$
This means we have $4$ whole units and an additional $3$ one-tenths. In terms of converting everything into one-tenths:
$4$ whole units are equal to
$4 \times 10 = 40 \text{ one-tenths}$
Adding the existing three one-tenths:
$40 + 3 = 43 \text{ one-tenths}$
Therefore, the rope has $\boxed{43}$ one-tenths.

Key Concept: Place Values, Extending System Beyond Unit

b) 2500
[Solution Description] To determine how many times $1000$ one-thousandths must be combined to reach $2.5$ kilograms, consider:
Each kilogram consists of $1000$ thousandths, so:
$1 \, \text{kg} = 1000 \times \frac{1}{1000} \, \text{kg}$
Therefore, to represent $2.5$ kg in terms of thousandths, convert it entirely to thousandths:
$2.5 \, \text{kg} = 2.5 \times 1000 = 2500 \, \text{one-thousandths}$
Thus, you will need a total of $\boxed{2500}$ such one-thousandths to create $2.5$ kilograms.

Key Concept: Understanding fractions of units

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
In this assertion, we are given a length expressed in fractional form: $5 \frac{1}{10}$, which can also be written as $5 + \frac{1}{10}$ or $\frac{51}{10}$. This format is correct and follows the concept of expressing numbers in terms of whole units and fractions of a unit.
The reason provided states that the unit system divides into tenths due to its alignment with the Indian place value system. The Indian place value system indeed emphasizes divisions by ten, such as tens, hundreds, etc., thus extending naturally to tenths for smaller units. While both statements are true, the Reason specifically explains why tenths are used, directly linking to the structure of the Indian place value system. Therefore, the Reason correctly explains the Assertion.
Hence, the correct option is: Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

Key Concept: The role of 10 in the Indian place value system and its extension to fractions

d) 10
[Solution Description] According to the syllabus, 1 unit is equal to 10 one-tenths. This is derived from the fact that multiplying 10 by $\frac{1}{10}$ results in 1 whole unit:
$10 \times \frac{1}{10} = 1$

Key Concept: Comparing decimal numbers

c) $3.04$, $3.14$, $3.4$, $3.41$
[Solution Description]
To arrange these numbers in increasing order, we need to compare them from smallest to largest:
Comparing decimals involves looking at each digit starting from the left:
- Compare the integer parts: All numbers have "3" as an integer part.
- Compare the tenths place:
- $3.04$: Tenths = 0
- $3.14$: Tenths = 1
- $3.4$: Tenths = 4
- $3.41$: Tenths = 4
- $3.04$ has the smallest tenths digit followed by $3.14$.
- Both $3.4$ and $3.41$ have tenths digit as 4, so we then compare hundredths:
- $3.4$ = $3.40$
- $3.41$
Thus, the correct order is: $3.04$, $3.14$, $3.4$, $3.41$.

Key Concept: Converting decimals to tenths

a) $\frac{47}{10}$
[Solution Description] The decimal number 4.7 can be broken down into a whole number and a decimal fraction:
- Whole number: $4$
- Decimal fraction: $0.7$
To convert $0.7$ to a fraction of tenths:
$0.7 = \frac{7}{10}$.
Therefore, $4.7$ as a fraction is:
$4 + \frac{7}{10} = \frac{40}{10} + \frac{7}{10} = \frac{47}{10}$.

Key Concept: Representing measurements as fractions of tenths

c) Assertion is true, but Reason is false.
[Solution Description]
Assertion is true because $4 \frac{1}{10}$ is equivalent to $4 + \frac{1}{10} = \frac{40}{10} + \frac{1}{10} = \frac{41}{10}$. This process converts the mixed number into an improper fraction.
Reason is false because in the decimal system, a unit divided into 100 equal parts represents hundredths, but this has no relation to converting a number from a mixed number to an improper fraction using tenths. Hence, the reason does not justify the assertion.

Key Concept: Understanding the concept of a tenth part in measurements

c) Assertion is true, but Reason is false.
[Solution Description]
Assertion: The statement is about expressing the length in terms of whole units and tenths. For $4 \frac{3}{10}$ meters, it is indeed correct to say it is equivalent to 4 meters plus three-tenths of a meter because $\frac{3}{10}$ represents three one-tenth units.
Reason: This reason incorrectly describes the conversion process from fractions to decimals. To convert $\frac{3}{10}$ to a decimal, you do not multiply by 10 but simply divide 3 by 10, resulting in 0.3.
Therefore, while the Assertion is true, the Reason does not correctly describe the conversion from fractions to decimals.

Key Concept: Comparing and Ordering Numbers with Tenths

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To compare the decimal numbers $6.465$ and $6.456$, we need to assess each digit starting from the left.
- The units digit in both numbers is 6.
- The tenths digit in both numbers is 4.
Next, compare the hundredths digit:
- In $6.465$, the hundredths digit is 6.
- In $6.456$, the hundredths digit is 5.
Since 6 is greater than 5, $6.465$ is greater than $6.456$.
The Reason provided states that "In the decimal system, digits to the right of the decimal point have decreasing place value from tenths to hundredths to thousandths." This statement is correct and explains why comparing digits from left to right helps determine which decimal is larger.
Therefore, both Assertion and Reason are true, and the Reason correctly explains the Assertion.

Key Concept: Identifying the closest decimal numbers to a given value

b) $3.56$
[Solution Description] We will calculate the absolute distance of each option from $3.57$:
- Distance from $3.56$: $|3.57 - 3.56| = 0.01$
- Distance from $3.50$: $|3.57 - 3.50| = 0.07$
- Distance from $3.67$: $|3.57 - 3.67| = 0.10$
- Distance from $3.65$: $|3.57 - 3.65| = 0.08$
The smallest distance is $0.01$ which corresponds to $3.56$.

Key Concept: Comparing and ordering numbers with tenths, Identifying closest decimal number

d) Assertion is false, but Reason is true.
[Solution Description]
The assertion states that when comparing decimals, only the smallest place value's digit decides which number is larger. This is incorrect because to compare decimal numbers, one starts with the most significant digits (leftmost) and moves right; the decision of which number is larger comes as soon as there's a difference in corresponding places, starting from leftmost. Meanwhile, the reason correctly describes the process: once a differing digit is found while moving from left to the right, these subsequent digits have lesser significance. Thus, it is true that further digits do not impact the result of the comparison after a difference is identified. Therefore, the Assertion is false, but the Reason is true.

Key Concept: Understanding Place Values

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
In the Indian place value system, each place value is indeed 10 times larger than the one immediately to its right, which is a fundamental aspect of the base-10 numeral system. This means that as you move from right to left, the place values increase by a factor of 10.
The reason correctly describes this sequence, starting from units and increasing through tens, hundreds, thousands, etc., following the powers of 10. Therefore, both the assertion and the reason are true statements, and the reason provides the correct explanation for the assertion.

Key Concept: Performing operations with decimals - Addition

b) 6.01
[Solution Description] To add $2.45$ and $3.56$, align the numbers by their decimal points and proceed with the addition column-wise:
Starting from the rightmost digit:
$5 + 6 = 11 \quad (\text{write 1, carry over 1})$
$4 + 5 + 1 = 10 \quad (\text{write 0, carry over 1})$
$2 + 3 + 1 = 6$
Hence, the sum is $6.01$.

Key Concept: Understanding the concept of dividing a unit into smaller parts to achieve more precise measurements

b) 1000
[Solution Description]
To find out how many one-thousandths make up a single unit, we start by understanding the hierarchy:
- $1$ unit = $10$ tenths
- $1$ tenth = $10$ hundredths
- $1$ hundredth = $10$ thousandths
Therefore, $1$ unit = $10 \times 10 \times 10 = 1000$ one-thousandths.
Hence, there are $1000$ one-thousandths in $1$ unit.

Key Concept: Conversion of tenths to hundredths

b) $7.80$
[Solution Description] Given the number $7.8$, which means there are 7 units and 8 tenths. To convert $8$ tenths into hundredths, we multiply by $10$ since $1$ tenth is equal to $10$ hundredths. So, $8$ tenths equals $80$ hundredths. Therefore, $7.8$ can be expressed as $7 \frac{80}{100}$ or equivalently $7.80$ in decimal form.

Key Concept: Comparing decimals

c) $7.5$
[Solution Description]
To compare the numbers, express them both in terms of hundredths. The number $7.48$ is already in hundredths. For $7.5$, rewrite it using hundredths:
$7.5 = 7 \frac{50}{100}$
Now we compare $7.48$ and $7.50$. Since $48$ is less than $50$, $7.48$ is smaller than $7.50$.
Hence, $7.5$ is greater than $7.48$.

Key Concept: Addition and Subtraction of Decimals, Dividing tenths into hundredths

b) 10.3 hundredths
[Solution Description]
First, determine the total amount of water used from the tank:
$$
25.6 - 15.3 = 10.3 \text{ liters}
$$
This 10.3 liters is distributed evenly among 100 smaller tanks. We calculate the reduction in each small tank:
$$
\frac{10.3}{100} = 0.103 \text{ liters}
$$
Converting 0.103 liters to hundredths gives:
$$
0.103 \text{ liters} = 10.3 \text{ hundredths of a liter}
$$
Thus, the reduction per smaller tank is 10.3 hundredths of a liter.

Key Concept: Understanding One-Hundredths

c) $\frac{1}{200}$
[Solution Description] Since the line segment is divided into 200 equal parts, each part is $\frac{1}{200}$ of the whole. Therefore, one part represents $\frac{1}{200}$.

Key Concept: Converting Tenths to Hundredths

b) $60$ hundredths
[Solution Description] Each tenth is equivalent to $10$ hundredths. Hence, $6$ tenths equals $6 \times 10 = 60$ hundredths.

Key Concept: Expressing Measurements

b) $3$ units and $79$ hundredths
[Solution Description] The ribbon is $3$ units, $7$ tenths, and $9$ hundredths long. We convert tenths to hundredths: $7 \times 10 = 70$. Thus, the total length is $3$ units and $79$ hundredths.

Key Concept: Converting Fractions to Decimals

c) 0.47
[Solution Description] The fraction $\frac{47}{100}$ means 47 one-hundredths.
To convert this fraction into a decimal form, write the numerator as is and place it two places after the decimal point since the denominator has two zeros:
Therefore, $\frac{47}{100} = 0.47$.

Key Concept: Decimal Place Value

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.
[Solution Description] Assertion is true as the number $7.205$ is accurately expressed using its place values: $7$ units, $2$ tenths, $0$ hundredths, and $5$ thousandths, which translates to $7 \times 1 + 2 \times \frac{1}{10} + 0 \times \frac{1}{100} + 5 \times \frac{1}{1000}$. Reason is also true because in the decimal system, the place immediately to the right of the decimal point is the tenths place, followed by the hundredths place, the thousandths place, etc., each representing a decreasing power of one-tenth. However, while the reason explains how place values work, it does not specifically justify the assertion about expressing $7.205$, hence it is not the correct explanation for the assertion.

Key Concept: Decimal Place Value, Converting fractions to decimal form, Understanding the relationship between units, tenths, hundredths

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To determine if the Assertion and Reason are true, let's convert $\frac{3}{4}$ into its decimal equivalent.
1. Divide the numerator by the denominator:
- $\frac{3}{4} = 3 \div 4$
2. Perform the division:
- $3 \div 4 = 0.75$
Thus, $\frac{3}{4}$ is indeed equivalent to $0.75$, making the assertion true.
Now, let's analyze the Reason statement:
Every fraction can theoretically be converted to a decimal by performing the division of the numerator by the denominator. This means that for any fraction $\frac{a}{b}$, where $b \neq 0$, we can find a decimal representation by executing this division.
Therefore, both the Assertion and Reason are true and the Reason correctly explains the Assertion.

Key Concept: Decimal Notation

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description] The assertion states that in the decimal number 3.47, the digit 4 represents four-tenths. This is correct because the first digit after the decimal point represents tenths.
The reason given is that each position in a decimal number has a place value that is ten times the place value of the position to its right. This is also true as it explains how the place values decrease by powers of ten moving from left to right after the decimal point. For example, if we consider the digits in the number 3.47:
- The '3' is in the units place.
- The '4' is in the tenths place and represents $4 \times \frac{1}{10}$.
- The '7' is in the hundredths place and represents $7 \times \frac{1}{100}$.
Therefore, both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

Key Concept: Decimal System and Place Value

b) 100
[Solution Description]
To find out how many thousandths make up one tenth, we need to understand the relationship between tenths and thousandths in the decimal system.
One tenth is represented as $0.1$ and one thousandth is represented as $0.001$.
To find how many thousandths fit into one tenth:
$0.1 \div 0.001 = 100$
Hence, 100 thousandths make one tenth.

Key Concept: Decimal System and Place Value, Extending the Place Value System

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.
[Solution Description]
The assertion states that each digit's place value in the decimal system is based on its position relative to others and aligns with the principle of base 10, meaning every digit to the left represents a place value ten times greater than the one to its right. This concept is correct within the structure of both whole numbers and fractions.
The reason accurately describes an aspect of extending the Indian place value system into decimal fractions: dividing units further into smaller parts like breaking $1$ hundredth into $10$ thousandths. However, while this statement is true, it does not directly explain the positional determination of place values for digits in the assertion.
Therefore, both statements are true, but the reason does not correctly explain the assertion.

Key Concept: Decimal Place Value, Conversion of fractions to decimal form

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description] To convert the fraction $\frac{1}{4}$ into a decimal, you divide the numerator by the denominator:
$\frac{1}{4} = 1 \div 4$
Performing this division, we get:
\begin{align*}
1 \div 4 &= 0.25
\end{align*}
Therefore, $\frac{1}{4}$ is equal to 0.25 in decimal form.
Both the assertion and reason are true statements. The assertion states a fact about converting the specific fraction $\frac{1}{4}$ into its decimal representation of 0.25. The reason explains the general process for converting any fraction into a decimal, which is through division of the numerator by the denominator. Hence, the reason correctly explains the assertion.

Key Concept: Understanding Decimal Place Value

c) 0.45
[Solution Description]
To convert $45$ hundredths to a decimal, we place the digits $45$ in the tenths and hundredths places respectively, after the decimal point. This means $45$ hundredths is represented as $0.45$.

Key Concept: Comparison of Decimal Numbers

b) 3.287
[Solution Description]
To compare decimal numbers, start from the leftmost digit (the highest place value) and compare each digit sequentially until one decimal number has a larger digit than the other at some place value.
Consider the decimals: 3.274, 3.287, 3.265, and 3.286.
- Compare the tenths: All have 3 in the units place, move to tenths.
- Compare lesser places: All have 2 in the tenths place, move to hundredths.
- Compare the hundredths:
- 3.274 has 7 in the hundredths
- 3.287 has 8 in the hundredths
- 3.265 has 6 in the hundredths
- 3.286 has 8 in the hundredths
Since 3.287 and 3.286 both have 8 in the hundredths, we check the next place (thousandths).
- 3.287 has 7 in thousandths
- 3.286 has 6 in thousandths
Thus, 3.287 is the largest number.

Key Concept: Decimal Place Value

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.
[Solution Description]
The assertion correctly states that 7.05 means $7 \times 1 + 5 \times \frac{1}{100}$, as per our understanding of decimal notation: 7 units and 5 hundredths. The reason given is a fundamental property of the decimal system, where each place represents a power of ten relative to its neighbors. However, while both statements are true, the reason does not directly explain why the assertion is correct; rather, it describes how the whole decimal system works without specific reference to the numbers given in the assertion.

Key Concept: Conversion and Examples

b) 0.45
[Solution Description] To convert the fraction $\frac{45}{100}$ into a decimal, divide the numerator by the denominator: $45 \div 100 = 0.45$. Here, '4' is in the tenths place and '5' is in the hundredths place after the decimal point.

Key Concept: Decimal Place Value, Indian Place Value System

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.
[Solution Description]
The assertion claims that the decimal number 70.5 can be broken down into the form $7 \times 10 + 5 \times \frac{1}{10}$. This is true because:
- The digit before the decimal point represents tens: $7 \times 10$.
- The digit after the decimal point represents tenths: $5 \times \frac{1}{10}$.
Therefore, the assertion correctly describes the representation of the number 70.5 using decimal place value.
The reason states that the decimal point is used to separate the whole number part from the fractional part of a number. This statement is also true. In decimal notation, the point/period serves to distinguish between integer values and fractions.
While both statements are true, the reason given does not directly explain how the expression $7 \times 10 + 5 \times \frac{1}{10}$ represents 70.5. Therefore, although each statement is accurate individually, they do not have a direct explanatory relationship.

Key Concept: Units of Measurement - Length Conversion

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.
[Solution Description]
To determine the correctness of each statement, let's first evaluate them:
Assertion: "1 cm is equal to 10 mm." This statement is true as per the basic conversion factor between centimeters and millimeters.
Reason: "The unit of length, centimeter (cm), is larger than the millimeter (mm)." This statement is also true because a centimeter is indeed larger than a millimeter since $1$ cm contains $10$ mm.
Now, let's assess whether the reason correctly explains the assertion:
While both statements are true, the reason does not directly explain why 1 cm equals 10 mm. Instead, the reason provides an additional piece of information about the relative sizes of the units but does not justify the numerical relationship given in the assertion.
Therefore, both the assertion and the reason are true, but the reason is NOT the correct explanation of the assertion.

Key Concept: Length Conversion, Real-World Application

b) 7
[Solution Description]
To solve this problem, we first convert the total length of the string from meters to centimeters because it's easier to work with whole numbers when dealing with division.
The given length of the string in meters is $3 \frac{3}{10}$ meters.
This can be converted to meters as follows:
$3 + \frac{3}{10} = 3.3 \text{ meters}$
Since $1 \text{ meter} = 100 \text{ cm}$, we multiply by 100 to convert to centimeters:
$3.3 \times 100 = 330 \text{ cm}$
Now, divide the total length in centimeters by the length of one piece to find how many full pieces can be obtained:
$\frac{330}{45} = 7.333$
Since only full pieces are possible, we round down to the nearest whole number, which gives us 7 full pieces.

Key Concept: Length Conversion

a) $2500 \text{ mm}$
[Solution Description]
To convert centimeters to millimeters, we use the conversion factor: $1 \text{ cm} = 10 \text{ mm}$. Therefore, for $250 \text{ cm}$:
$250 \text{ cm} \times 10 \text{ mm/cm} = 2500 \text{ mm}$
Thus, the length of the rope in millimeters is $2500 \text{ mm}$.

Key Concept: Conversion and Dimensional Analysis

b) 129
[Solution Description]
First, convert the diameter of the rod from centimeters to millimeters using the conversion factor $1 \text{ cm} = 10 \text{ mm}$. Thus,
$0.9 \text{ cm} = 0.9 \times 10 = 9 \text{ mm}$
Next, determine how many hairs are needed by dividing the diameter of the rod by the thickness of one hair:
$\frac{9 \text{ mm}}{0.07 \text{ mm}} \approx 128.57$
Since we need whole hairs, round up to get 129 hairs. Therefore, it requires approximately 129 hairs.

Key Concept: Units of Measurement: Millimeters ↔ Centimeters

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To verify the assertion, we need to convert 25 mm to centimeters. We know that $1 \text{ cm} = 10 \text{ mm}$.
Therefore, to convert millimeters to centimeters:
$25 \text{ mm} = \frac{25}{10} \text{ cm} = 2.5 \text{ cm}$
Thus, the Assertion is true as 25 mm indeed equals 2.5 cm.
Now, examining the Reason, it states that each centimeter consists of 10 millimeters. This statement is correct and provides the basis for converting millimeters to centimeters by dividing by 10.
Hence, both the Assertion and Reason are true, and the Reason correctly explains why the Assertion is true.
Therefore, the correct answer is option a).

Key Concept: Composite Conversion

b) 9 cm 2 mm
[Solution Description] First, we find the number of complete centimeters, which is obtained by dividing by 10:
$\text{Complete cm} = \left\lfloor \frac{92}{10} \right\rfloor = 9 \text{ cm}.$
The remaining millimeters are found by computing $92 - (9 \times 10)$:
$\text{Remaining mm} = 92 - 90 = 2 \text{ mm}.$
So, the rod's length is 9 cm 2 mm.

Key Concept: Conversion between centimeters and meters

b) 2.5 meters
[Solution Description]
To convert centimeters to meters, we use the conversion factor that 1 meter is equivalent to 100 centimeters. Thus, we divide the given centimeters by 100:
$250\, \text{cm} = \frac{250}{100}\, \text{m} = 2.5\, \text{m}$
Therefore, 250 cm is equal to 2.5 meters.

Key Concept: Conversion Concepts and Units of Measurement

d) Assertion is false, but Reason is true.
[Solution Description]
The Assertion states that 250 centimeters is equivalent to 25 meters. To verify this, we use the conversion factor from the Reason:
$1 \text{ meter} = 100 \text{ centimeters}$
Therefore, to convert centimeters to meters:
$250 \text{ cm} = \frac{250}{100} = 2.5 \text{ m}$
Since $2.5$ meters is not equal to $25$ meters, the Assertion is false.
The Reason is true because it correctly describes how to convert centimeters into meters using division by $100$.
Hence, the correct answer is that the Assertion is false, but the Reason is true.

Key Concept: Decimal Representation, Multi-step Solutions

c) 12.56 $\text{m}$
[Solution Description]
First, convert the diameter from centimeters to meters. The conversion factor is $1 \text{ m} = 100 \text{ cm}$.
Convert the diameter:
$400 \text{ cm} = \frac{400}{100} \text{ m} = 4.0 \text{ m}$
Next, calculate the circumference of the circle using the formula $C = \pi \times d$, where $d$ is the diameter.
$C = 3.14 \times 4.0 \text{ m} = 12.56 \text{ m}$
Therefore, the circumference of the track is $12.56 \text{ m}$.

Key Concept: Weight Conversion: Grams to Kilograms

b) 0.75
[Solution Description]
To convert grams to kilograms, we use the conversion factor where $1 \text{ kg} = 1000 \text{ g}$. Therefore, to convert 750 grams into kilograms:
$750 \text{ g} = \frac{750}{1000} \text{ kg} = 0.75 \text{ kg}$
The weight of the packet in kilograms is 0.75 kg.

Key Concept: Combined Weight Conversion

c) 1
[Solution Description]
First, find the total weight in grams:
$300 \text{ g} + 700 \text{ g} = 1000 \text{ g}$
Now, convert the total grams to kilograms using $1 \text{ kg} = 1000 \text{ g}$:
$1000 \text{ g} = \frac{1000}{1000} \text{ kg} = 1 \text{ kg}$
The combined weight of both objects is 1 kg.

Key Concept: Conversion and Addition

c) 1.70 kg
[Solution Description]
To find the total weight in kilograms, we must first convert all weights to kilograms. We know:
$250 \text{ g} = \frac{250}{1000} \text{ kg} = 0.25 \text{ kg}$
$450 \text{ g} = \frac{450}{1000} \text{ kg} = 0.45 \text{ kg}$
The item already weighs 1 kg.
Now we sum them up:
$0.25 \text{ kg} + 0.45 \text{ kg} + 1 \text{ kg} = 1.70 \text{ kg}$
Therefore, the total weight of the package is 1.70 kg.

Key Concept: Paise to Rupee Conversion

c) Rs.2.50
[Solution Description] To convert paise to rupees, we divide the number of paise by 100 since $1 \text{ paisa} = \frac{1}{100} \text{ rupee}$. Thus, for 250 paise:
$\frac{250}{100} = 2.50 \text{ rupees}$
So, 250 paise is equivalent to Rs.2.50.

Key Concept: Rupee to Paise Conversion

c) 125
[Solution Description] To convert rupees to paise, we multiply the amount in rupees by 100 because $1 \text{ rupee} = 100 \text{ paise}$. Therefore, for Rs.1.25:
$1.25 \times 100 = 125 \text{ paise}$
So, there are 125 paise in Rs.1.25.

Key Concept: Paise to Rupee Conversion

c) Rs.2.25
[Solution Description] To convert paise into rupees, divide the paise by 100 since $1 \text{ rupee} = 100 \text{ paise}$.
Calculation:
$\frac{225}{100} = 2.25 \text{ rupees}$
Therefore, 225 paise is equal to Rs.2.25.

Key Concept: Estimation of Sums with Decimals

b) Between $22$ and $24$
[Solution Description]
Firstly, calculate the sums of whole number parts:
- $17 + 5 = 22$
Then add an estimation factor as described:
- The sum should be greater than $22$ and less than $(17 + 1) + (5 + 1) = 24$.
Thus, the sum is estimated to lie between $22$ and $24$.
So, the sum would be within the range $22$ to $24$.

Key Concept: Estimating Sums and Differences

c) 20.00
[Solution Description] First add the whole number parts:
$12 + 7 = 19$
Next, consider adding one to each whole number part:
$12 + 1 + 7 + 1 = 21$
Thus, the estimated sum is between 19 and 21. Performing the actual addition:
$12.45 + 7.55 = 20.00$
The estimation confirms the calculated sum since it lies between 19 and 21.

Key Concept: Units of Measurement: Examples with Real-World Objects

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.
[Solution Description]
In the assertion, it is mentioned that precise measurement is important for practical tasks, which is true because even small differences in measurements can affect the quality and functionality of objects being assembled or fixed.
In the reason, expressing measurements using fractional units indeed allows for accuracy and precision in measurements. Both statements are true; however, while the reason supports the importance of accurate measurement, it does not directly explain why precise measurement is specifically crucial in the context of fixing a toy with screws. Therefore, the reason supports but does not fully explain the need identified in the assertion.

Key Concept: Locating specific decimal numbers, Creating number lines for decimal numbers

a) 0.69
[Solution Description]
To find the total distance covered from Q to R, R to S, and S to T, sum the individual distances between consecutive points.
Distance from Q to R:
\begin{align*}
D_{QR} &= 4.49 - 4.07,
&= 0.42.
\end{align*}
Distance from R to S:
\begin{align*}
D_{RS} &= 4.56 - 4.49,
&= 0.07.
\end{align*}
Distance from S to T:
\begin{align*}
D_{ST} &= 4.76 - 4.56,
&= 0.20.
\end{align*}
Total distance covered:
\begin{align*}
\text{Total Distance} &= D_{QR} + D_{RS} + D_{ST},
&= 0.42 + 0.07 + 0.20,
&= 0.69.
\end{align*}
Hence, the total distance covered is 0.69 units.

Key Concept: Placing decimal numbers on a number line, Use place value to compare decimals

d) 2.625
[Solution Description]
To find the point 'P' equidistant from 2.35 and 2.90, we need to determine the midpoint between these two numbers.
The formula for calculating the midpoint between two numbers $a$ and $b$ is:
$\text{Midpoint} = \frac{a + b}{2}$
Substituting the values of 2.35 and 2.90 into the formula:
\begin{align*}
\text{Midpoint} &= \frac{2.35 + 2.90}{2},
&= \frac{5.25}{2},
&= 2.625.
\end{align*}
Thus, the decimal representation of point 'P' is 2.625.

Key Concept: Understanding the placement of decimal numbers on a number line

c) 3.7
[Solution Description] The number line is divided into 10 equal parts between the integers 3 and 4. Each part represents 0.1 units. To locate 3.7, we start from 3 and count seven parts forward:
3 + 0.1 $\times$ 7 = 3.7.
Therefore, the seventh division starting from 3 is at 3.7.

Key Concept: Comparing Decimals Using Place Value

b) 0.365
[Solution Description]
To determine which decimal is the largest, we compare each digit from left to right starting with the highest place value:
- Compare: $0.356$, $0.365$, $0.354$, and $0.364$.
- All values have '0' in the units place.
- In the tenths place:
- $0.3 0.36$, so compare further.
- In the thousandths place:
- $\text{For } 0.365 \text{ and } 0.364$: $0.005 > 0.004$. Therefore, $0.365$ is the largest.
Hence, the correct answer is $0.365$.

Key Concept: Decimal Place Value System

b) 0.7
[Solution Description]
Converting a fraction to a decimal involves dividing the numerator by the denominator:
- $\frac{7}{10} = 7 \div 10 = 0.7$
Thus, the decimal form of $\frac{7}{10}$ is $0.7$.

Key Concept: Locating Closest Decimal

d) $1.999$
[Solution Description] To find the decimal closest to $2$, calculate the absolute difference from $2$ for each option:
- Difference of $1.98$ from $2$ is: $|1.98 - 2| = 0.02$
- Difference of $2.05$ from $2$ is: $|2.05 - 2| = 0.05$
- Difference of $2.01$ from $2$ is: $|2.01 - 2| = 0.01$
- Difference of $1.999$ from $2$ is: $|1.999 - 2| = 0.001$
The smallest difference is $0.001$ for $1.999$. Hence, $1.999$ is closest to $2$.

Key Concept: Visualizing decimal numbers on a number line

c) Seven marks past 1
[Solution Description] A number line divided into ten equal parts between 1 and 2 represents increments of $0.1$. Thus, positions are:
- $1.1$, $1.2$, ..., up to $2.0$
For $1.7$, it is seven-tenths above 1, or seventh marker past 1 ($1.0 + 0.7 = 1.7$).

Key Concept: Differentiating between decimal numbers with different place values

c) $0.025$
[Solution Description] To determine the smallest decimal number, compare the digits at each place value:
- For $0.205$: hundreds = 2
- For $0.250$: hundreds = 2
- For $0.025$: tens = 0, so smallest so far
- For $0.520$: fifties = 5
Therefore, $0.025$ is the smallest because it has 0 in the tenths place while others have larger digits there.

Key Concept: Identifying equivalent decimal numbers

d) 5.300 and 5.30
[Solution Description] An equivalent decimal has the same value even if it appears different because of trailing zeros. Let's analyze each:
- 5.300 and 5.30: Trailing zero doesn't affect value, so both are equal.
- 5.03 is not equal to the others as its hundredths digit differs.
- 5.003 has a thousandths digit making it unequal to others.
Thus, $5.300$ and $5.30$ are equivalent.

Key Concept: Locating and Comparing Decimals: Digit-by-digit place value comparison

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To compare two decimal numbers, we start by examining the most significant digits, which have the highest place value. If they are the same, we move to the next smaller place value until we find a difference.
Given the decimal numbers $1.23$ and $1.32$:
- First compare the units place: Both have $1$.
- Next, compare the tenths place: $1.23$ has $2$ and $1.32$ has $3$.
- Since $3 > 2$, $1.32$ is greater than $1.23$.
This confirms that the assertion is true: When comparing decimals, having a larger digit in a higher place value indicates a larger number. The reason provided is also correct because it exemplifies this rule using the given numbers. Hence, the reason explains why the assertion is valid.
Therefore, both the Assertion and Reason are true, and the Reason correctly explains the Assertion.

Key Concept: Digit-by-digit place value comparison

c) $5.743$
[Solution Description]
To determine which number is greater, we compare each digit from left to right.
- Both numbers have $5$ units.
- Both have $7$ tenths.
- For hundredths, both have $3$.
- For thousandths, the first number has $4$, and the second number has $3$.
Since $4 > 3$, $5.743$ is greater than $5.734$.

Key Concept: Locating and Comparing Decimals: Digit-by-digit (place value comparison

c) Assertion is true, but Reason is false.
[Solution Description]
To compare two decimal numbers like $6.465$ and $6.456$, we use digit-by-digit comparison starting from the leftmost digit:
1. Compare the whole number part: Both have $6$, so they are equal in this aspect.
2. Compare the tenths position: Both have $4$ in the tenths place, so they are still equal here.
3. Compare the hundredths position: Both have $5$ in the hundredths place, maintaining equality.
4. Compare the thousandths position: $6.465$ has $6$ while $6.456$ has $5$ at this place. Since $6 > 5$, $6.465$ is greater than $6.456$.
Therefore, the assertion that $6.465$ is greater than $6.456$ is true.
However, the reason states that smaller place values determine which number is larger, which is incorrect because it's actually the largest differing place value that determines the size of the numbers when comparing decimals.

Key Concept: Decimal Sequences

c) 4.5
[Solution Description] We can calculate the fifth term in the sequence by adding the constant increase to the initial term multiple times.
Start with the initial term, which is $3.5$, and add $0.25$ four times since we are looking for the fifth term:
\begin{align*}
\text{Fifth term} &= 3.5 + 4 \times 0.25
&= 3.5 + 1.0
&= 4.5
\end{align*}
Thus, the fifth term is $4.5$ meters.

Key Concept: Addition of Decimals

b) $20.00$
[Solution Description] To calculate the total of $12.56$ and $7.44$, align the numbers by their decimal points and add:
$\begin{array}{r} \\ 12.56 \\ \\ + 7.44 \\ \\ \hline \\ 20.00 \\ \\ \end{array}$
Hence, the total of $12.56$ and $7.44$ is $20.00$.

Key Concept: Subtraction of Decimals

b) $6.5$
[Solution Description] To find the difference between $9.8$ and $3.3$, subtract $3.3$ from $9.8$:
$\begin{array}{r} \\ 9.8 \\ \\ - 3.3 \\ \\ \hline \\ 6.5 \\ \\ \end{array}$
The difference between $9.8$ and $3.3$ is $6.5$.

Key Concept: Estimating Sums of Decimals

c) $34$
[Solution Description]
First, round each number to the nearest whole number:
- $25.936$ rounds to $26$.
- $8.202$ rounds to $8$.
Now, add the rounded numbers:
$26 + 8 = 34$
So, the estimated sum is $34$.

Key Concept: Addition and Subtraction of Decimals, Place Value Calculation

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.
[Solution Description]
Let's first understand the assertion. The addition of $45.763$ and $54.237$ can be broken down into the sum of their whole number parts and decimal parts.
- Whole number part: $45 + 54 = 99$
- Decimal part: $0.763 + 0.237 = 1.000$
Thus, the total sum becomes:
$45.763 + 54.237 = 99 + 1.000 = 100.000$
Therefore, the assertion is true since the result $100.000$ is indeed greater than $99$, which is the sum of the whole numbers.
Now let's examine the reason.
- Adding the decimal parts gives us $1.000$, which means it added exactly one unit over the sum of the whole numbers.
Hence, both the assertion and the reason are correct, but the reason is NOT explaining the assertion correctly as it states "less than two units" rather than precisely indicating the additional amount.

Key Concept: Adding Decimal Numbers

b) $7.9$
[Solution Description]
To find the sum of $5.3$ and $2.6$, we align the decimal points and add each digit starting from the rightmost digit:
$\begin{array}{c@{}c@{}c@{}c@{}c} \\ & 5.3 \\ \\ + & 2.6 \\ \hline \\ & 7.9 \\ \\ \end{array}$
Thus, the sum is $7.9$.

Key Concept: Aligning decimal points, Numerical and Analytical Difficulty

c) \$174.33
[Solution Description]
Calculate the cost of Fabric A:
$3.25 \times 24.675 = 80.19375$
Rounded to the nearest cent: \$80.19
Calculate the cost of Fabric B:
$4.75 \times 19.85 = 94.1375$
Rounded to the nearest cent: \$94.14
Add these values to get the total cost:
$80.19 + 94.14 = 174.33$
Hence, the customer spends \$174.33 in total.

Key Concept: Addition and Subtraction of Decimals, Aligning decimal points

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To understand why aligning decimal points is crucial, let's consider a decimal subtraction example: $5.34 - 2.67$.
1. **Align Decimal Points:**
$\begin{array}{c@{}c@{}c@{}c@{}c@{}c} \\ & 5 & . & 3 & 4 \\ \\ - & 2 & . & 6 & 7 \\ \\ \hline \\ \end{array}$
2. **Perform Subtraction Digit by Digit from Right to Left:**
- Start with the far right digit: $4 - 7$ requires borrowing since $4 < 7$.
- Borrow 1 from the tenths column:
$\begin{array}{c@{}c@{}c@{}c@{}c@{}c} \\ 4+10 = 14 & . & 3-1 = 2 & 5 & \\ \\ - & 7 & . & 6 & 2 \\ \\ \hline \\ \end{array}$
- Subtract: $14 - 7 = 7$, $2 - 6 = 6$ after another borrow,
- $5 - 2 = 3$.
3. **Write Down the Result**:
$5.34 - 2.67 = 2.67$
This shows when decimal points are aligned, each digit is correctly subtracted with the corresponding place value, ensuring accuracy. The reason complements the assertion as misaligned decimals would lead to errors.

Key Concept: Aligning decimal points for complex addition

b) $10.70$
[Solution Description] Align the decimal points and add $7.890$, $2.456$, and $0.354$:
$\begin{array}{c@{}c@{\;}c@{\;}c@{\;}c@{\;}c} \\ & 7 & . & 8 & 9 & 0 \\ \\ + & 2 & . & 4 & 5 & 6 \\ \\ + & 0 & . & 3 & 5 & 4 \\ \\ \hline \\ & 1 & 0 & . & 7 & 0 \\ \\ \end{array}$
Calculate each column starting from the right:
- Thousandths: $0 + 6 + 4 = 10$, write $0$, carry $1$,
- Hundredths: $9 + 5 + 5 = 19$, plus $1$ carried, equals $20$, write $0$, carry $2$,
- Tenths: $8 + 4 + 3 = 15$, plus $2$ carried, equals $17$, write $7$, carry $1$,
- Units: $7 + 2 + 0 = 9$, plus $1$ carried, equals $10$, write $0$, carry $1$,
- Tens: Carry the $1$ gives us a total of $10$.
The final sum is $10.70$.

Key Concept: Complex Subtraction, Decimal Notation

c) $25.10$ kg
[Solution Description] Calculate the total amount of fertilizer used:
$13.45 + 9.89 + 5.26 = 28.60 \text{ kg}$
Now subtract this total from the initial amount:
$53.70 - 28.60 = 25.10 \text{ kg}$
Thus, $25.10$ kg of fertilizer remains.

Key Concept: Place Value Method of Subtraction

d) $12.75$
[Solution Description]
To find the difference between $27.53$ and $14.78$ using the place value method, we subtract each column starting from the right.
\begin{align*}
&\,\,27.53
-&\,\,14.78
\hline
\end{align*}
Starting with the hundredths: $3 - 8$ requires borrowing. Borrowing $1$ from $5$ in the tenths gives us $13 - 8 = 5$.
For the tenths: After borrowing, it becomes $4 - 7 = -3$, so we borrow $1$ from the units making it $14 - 7 = 7$.
In the units column: $6 - 4 = 2$.
Finally, for the tens: $2 - 1 = 1$.
Thus, the result is $12.75$.

Key Concept: Place value method of addition

c) Assertion is true, but Reason is false.
[Solution Description]
The assertion states that aligning the decimal points while adding decimals ensures proper place value alignment. This is true because aligning the decimal points helps in arranging the digits according to their respective place values (units, tenths, hundredths, etc.).
The reason claims that decimal numbers are added by treating them as whole numbers without considering their fractional parts. This statement is false because when adding decimals, it is essential to consider both the whole number and the fractional part for accurate results.
Therefore, the correct response is: Assertion is true, but Reason is false.

Key Concept: Notation, Writing and Reading of Numbers

c) 3 ones, 5 tenths, and 6 hundredths
[Solution Description] The decimal number 3.56 can be expressed in terms of its place value as follows:
- The digit 3 represents 3 units or ones.
- The digit 5 is in the tenths place, so it represents $5 \times \frac{1}{10}$.
- The digit 6 is in the hundredths place, so it represents $6 \times \frac{1}{100}$.
Hence, the expression for the number would be:
$3 + \frac{5}{10} + \frac{6}{100} = 3 + 0.5 + 0.06 = 3.56$
This shows that 3.56 is equivalent to 3 ones, 5 tenths, and 6 hundredths.

Key Concept: More on the Decimal System

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description] Let's verify both statements:
Assertion: The statement claims that 1 unit consists of 100 hundredths. This is true because each whole number or "unit" can be divided into 10 tenths, each tenth into 10 hundredths, making $10 \times 10 = 100$ hundredths in a single unit.
Reason: It states that a hundredth is represented as $\frac{1}{100}$. This is correct because, by definition, a hundredth means one part out of a hundred, hence, it is written in fractional form as $\frac{1}{100}$.
Thus, both Assertion and Reason are true, and the Reason correctly explains why there are 100 hundredths in 1 unit, based on how we represent a hundredth fractionally.

Key Concept: Notation, Writing and Reading of Numbers

c) 3.46
[Solution Description]
Given are 3 ones, 4 tenths, and 6 hundredths. In decimal form, this translates to:
Ones: 3
Tenths: $0.4$
Hundredths: $0.06$
Adding these together gives:
$3 + 0.4 + 0.06 = 3.46$
Therefore, the decimal form is $3.46$.

Key Concept: Decimal and Measurement Disasters

c) 10
[Solution Description] The correct dose is 0.05 mg, but it was read as 0.5 mg. To find the factor by which the dose was increased:
$\text{Factor} = \frac{0.5}{0.05} = 10$
So, the dose was increased by a factor of 10.

Key Concept: Real-life application, estimation of sums

Correct Answer option c) Greater than 19 and less than 21
[Solution Description]
According to Sonu's claim, the sum should be greater than the sum of the whole number parts and less than two more than their sum.
Whole number parts:
$15 + 4 = 19$
Adding 1 to each:
$15 + 1 + 4 + 1 = 21$
So, according to Sonu’s claim, the sum of 15.372 and 4.689 should be greater than 19 and less than 21.
Calculating the exact sum:
$15.372 + 4.689 = 20.061$
The calculated sum indeed falls between 19 and 21, verifying Sonu's claim.

Key Concept: Decimal unit conversion, historical notation

Correct Answer option a) 25 mg
[Solution Description]
The notation $25^3$ implies that there are three decimal places, hence it represents 0.025 g.
To convert grams to milligrams, we use the conversion factor:
$1 \text{ g} = 1000 \text{ mg}$
Therefore, the weight in milligrams is:
$0.025 \text{ g} \times 1000 = 25 \text{ mg}$
Hence, the technician incorrectly assumed that 25 mg were required instead of 250 mg (since 0.25 g is actually 250 mg).

Key Concept: Correcting Decimal Misplacement in Payment Processing

d) €1,080,000
[Solution Description] The intended payment is €1.2 million, which equals €1,200,000. The actual payment was €120,000, leading to the following calculation:
$\text{Difference} = 1,200,000 - 120,000 = 1,080,000$
Therefore, the council sent €1,080,000 less than intended.

Key Concept: Decimal Notation in Financial Transactions

b) €3,105,000
[Solution Description] The intended payment was €3.45 million, which can be expressed as €3,450,000. Due to an error, it was processed as €345,000. To find the underpayment:
$\text{Underpayment} = 3,450,000 - 345,000 = 3,105,000$
Therefore, the company underpaid by €3,105,000.

Key Concept: Currency errors

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.
[Solution Description]
The assertion states that errors in decimal point placement can lead to significant financial discrepancies, which is true as highlighted by the example of Amsterdam City Council sending out incorrect housing benefits due to a programming error involving euro cents instead of euros.
The reason provided explains how the number 7.05 is broken down in terms of its place value, which is also true. It accurately describes the conversion of the decimal number into an expression based on Indian place value system:
$$7.05 = 7 \times 1 + 0 \times \frac{1}{10} + 5 \times \frac{1}{100}.$$
Although both statements are true, the explanation about the breakdown of the number 7.05 doesn't directly explain why errors in decimal point placement can cause financial discrepancies. Thus, the reason is not the correct explanation for the assertion.

Key Concept: Decimal and Measurement Disasters

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
In the assertion, it is stated that decimal errors in aviation fuel calculations can lead to dangerous situations like an aircraft running out of fuel mid-air. This is indeed true because incorrect calculations might result in insufficient fuel being loaded onto the aircraft, thus potentially leading to emergencies.
The reason discusses a specific incident with the Air Canada Boeing 767 where a miscalculation involving converting the necessary fuel from pounds to kilograms resulted in the wrong amount of fuel being loaded. This particular incident serves as evidence of how such decimal point errors can have serious consequences, supporting that these could be dangerous if not handled correctly.
Hence, both the Assertion and Reason are true, and the Reason provides a correct explanation of the Assertion. These errors highlight why precision in unit conversion is crucial.

Key Concept: Estimating Sums and Differences

c) The sum is between $43$ and $45$
[Solution Description] Let's first determine the whole number parts: $29$ and $14$.
The sum of these whole numbers is:
$29 + 14 = 43$
According to Sonu's observation, the sum should fall between $43$ and $45$.
Calculate the actual sum:
$29.784 + 14.215 = 43.999$
The calculated sum $43.999$ falls exactly within the estimated range $43 < 43.999 < 45$, validating Sonu’s observation.

Key Concept: Decimal and Measurement Disasters

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that decimal errors in unit conversion can cause significant issues, such as incorrect fuel loads for aircraft. This is true, as improper conversions can result in major discrepancies between intended and actual amounts.
The reason further explains why this occurs by noting a specific conversion error: $1 \text{ pound} \approx 0.453 \text{ kg}$. When using pounds incorrectly instead of kilograms, only about half the necessary fuel would be loaded because:
$\text{Fuel Loaded in Kilograms} = \text{Fuel Loaded in Pounds} \times 0.453$
For example, if 22,300 pounds are used due to an error rather than kilograms, it leads to
$22,300 \times 0.453 \approx 10,101.9 \text{ kg }$
This is significantly less than the intended amount since the correct calculation should have been 22,300 kg. Both statements are true, and Reason correctly explains Assertion.

Key Concept: Decimal Point Mistake

c) 10
[Solution Description]
The prescribed dose is 0.08 mg. The administered dose is 0.8 mg. To find how many times more the administered dose is compared to the prescribed dose, we divide the administered dose by the prescribed dose: $\frac{0.8}{0.08} = 10$. Therefore, the dose given was 10 times more than the prescribed amount.

Key Concept: Unit Conversion Error

c) 1000
[Solution Description]
First convert the prescribed dose from micrograms to milligrams: 25 mcg = $0.025$ mg. The received dose is 25 mg. To find how many times higher the given dose is compared to the prescribed dose, divide the received dose by the prescribed dose: $\frac{25}{0.025} = 1000$. Thus, the patient received a dose 1000 times higher than prescribed.

Key Concept: Rounding Error in Prescription

b) 96\%
[Solution Description]
To find the percentage of the prescribed dose received, use the formula:
Percentage received = $\left(\frac{\text{Actual dose}}{\text{Prescribed dose}}\right) \times 100$
Actual dose = 0.12 mg
Prescribed dose = 0.125 mg
Percentage received = $\left(\frac{0.12}{0.125}\right) \times 100 = 96\%$.
Hence, the infant received 96\% of the prescribed dose.

Key Concept: Decimal System, Deceptive Decimal Notations

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.
[Solution Description]
To determine the truthfulness of the assertion and reason:
- Assertion (A): "The decimal notation 4.5 signifies four and five-tenths, which equates to four hours and thirty minutes when applied to time." This is true. In the context of time, half an hour corresponds to thirty minutes, hence "five-tenths" of an hour represents 30 minutes.
- Reason (R): "In the decimal system, each digit represents a fraction of ten, meaning that 0.5 equals five-tenths or one-half of a whole." This statement is also true. The decimal system divides values by powers of ten, so 0.5 indeed means five-tenths.
Evaluation:
- Both statements are correct independently.
- However, the reason does not directly explain why 4.5 specifically results in 4 hours and 30 minutes; it only provides a general explanation about decimal fractions.
Therefore, both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.

Key Concept: Place Value Understanding, Advanced Reasoning

b) $122.5$ gallons
[Solution Description]
To find out how many gallons are actually in the tank, we must convert pounds back to gallons using the given approximation:
1. Given that $1$ gallon ≈ $6$ pounds, we calculate:
Number of gallons = $\frac{735}{6}$.
2. Perform the division:
$735 \div 6 = 122.5$
Therefore, the tank actually contains approximately $122.5$ gallons of gasoline.

Key Concept: Deceptive Decimal Notations

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
Assertion states that "5.5 overs" in cricket means 5 overs and 5 balls. To understand why this is correct, we need to understand the structure of an over in cricket. In cricket, each over consists of 6 deliveries or balls.
Therefore, when it says "5.5 overs", it implies:
- There are 5 complete overs, which equals $5 \times 6 = 30$ balls.
- The ".5" means 5 more balls since the decimal here relates to a fraction of an over where 1 over = 6 balls.
So, "5.5 overs" translates to 5 overs + 5 balls.
Reason states that each over consists of 6 deliveries, which is indeed true and explains why "5.5 overs" corresponds to 5 overs and 5 balls.
Therefore, both the assertion and reason are true, and the reason is the correct explanation for the assertion.