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: Indian Place Value System

b) 1 hundred
[Solution Description] In the Indian place value system, 10 tens equal 1 hundred as per the rule that each place value is 10 times larger than the previous one on the right.

Key Concept: Concept of Indian Place Value System

c) $\frac{1}{16}$
[Solution Description]
If a unit is divided into 16 equal parts, each part represents $\frac{1}{16}$ of the whole unit. This requires dividing the whole unit by the number of equal parts:
$\frac{1}{16}$
Thus, each part is $\frac{1}{16}$ of the unit.

Key Concept: Fractional Parts and Unit Division

a) $\frac{1}{500}$
[Solution Description]
One-hundredth of a unit is denoted as $\frac{1}{100}$. When this is divided into 5 equal parts, each part represents:
$\frac{1}{100} \div 5 = \frac{1}{100} \times \frac{1}{5} = \frac{1}{500}$
Therefore, each new part is $\frac{1}{500}$ of the original unit.

Key Concept: Understanding unit conversions

c) 63 mm
[Solution Description]
We need to convert $6 \frac{3}{10}$ cm to millimeters. Knowing that 1 cm equals 10 mm:
Convert the fractional part $\frac{3}{10}$ cm to millimeters:
$\frac{3}{10}$ cm = 0.3 cm
Add this to 6 cm: $6 + 0.3 = 6.3$ cm
Now, convert to millimeters:
$6.3 \text{ cm} = 6.3 \times 10 \text{ mm} = 63 \text{ mm}.$

Key Concept: Measuring small differences

c) 35 mm
[Solution Description]
To convert centimeters to millimeters, we use the fact that 1 cm = 10 mm. Therefore, for the given screw:
The fractional part $\frac{5}{10}$ cm is equivalent to 0.5 cm.
So, the total length in cm is $3 + 0.5 = 3.5$ cm.
Now converting to millimeters:
$3.5 \text{ cm} = 3.5 \times 10 \text{ mm} = 35 \text{ mm}.$

Key Concept: Measuring small differences

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.
[Solution Description]
Assertion is about the importance of using a scale with smaller divisions to measure slight differences accurately. For instance, in the case of toy screws, even a minor variation can affect their functionality. Reason provides a fact regarding unit conversion, which is crucial for measuring and comparing different lengths. While both statements are true and related to measurements, the reason does not directly explain why smaller scale divisions help differentiate screw sizes.

Key Concept: Understanding the concept of a tenth part

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.
[Solution Description]
To solve this assertion-reason type question:
- Firstly, analyze the Assertion: "A length of $4 \frac{3}{10}$ units can be expressed as 4 units plus three one-tenths." This statement is correct because $4 \frac{3}{10}$ can be split into 4 whole units and 3 tenths, i.e., $4 + \frac{3}{10}$.
- Next, examine the Reason: "The Indian place value system allows expressing decimals by separating the whole number and fractional parts using a decimal point." This statement is also true, as it accurately describes how decimals are represented in the Indian place value system.
- Now, determine if the Reason explains the Assertion. While the reason provides information on how decimals are expressed, it does not directly explain why $4 \frac{3}{10}$ can be broken down into 4 units plus three one-tenths. The division into whole numbers and fractions is more about understanding fractions than the specific mechanics of the decimal point used in the place value system.
Thus, both statements are true, but the Reason is NOT the correct explanation of the Assertion.

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

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.
[Solution Description]
The assertion states that the length of a pencil is $3 \frac{4}{10}$ units, equivalent to 34 one-tenths units. This conversion is correct because one whole unit is made up of 10 one-tenths, so 3 units and four-tenths would indeed make 34 one-tenths units:
$3 \times 10 + 4 = 30 + 4 = 34$.
The reason explains why dividing a unit into ten equal parts results in tenths, aligning with the Indian place value system, where each place is ten times the previous place. Given this alignment, both the Assertion and Reason are true, but the Reason is explaining the concept rather than directly justifying the specific expression in the Assertion.
Therefore, the correct option here is b).

Key Concept: Conversion of units, tenths and hundredths understanding

b) $7.5$ cm
[Solution Description]
We know that:
$1 \text{ cm} = 10 \text{ mm}$
Therefore:
$75 \text{ mm} = \frac{75}{10} \text{ cm} = 7.5 \text{ cm}$
In the Indian place value system, $7.5$ cm represents $7$ whole units and $5$ tenths.

Key Concept: Understanding tenths and comparing fractions using decimal conversion

b) $1 \frac{2}{10}$
[Solution Description]
Convert all fractions to decimals first:
\begin{align*}
\frac{5}{10} &= 0.5
1 \frac{2}{10} &= 1.2
\frac{15}{10} &= 1.5
0.8 &= 0.8
\end{align*}
Now order them: $0.5$, $0.8$, $1.2$, $1.5$. Thus, $1.2$ comes third in this sequence.

Key Concept: Representation Understanding

b) 98
[Solution Description]
We need to express $9 \frac{8}{10}$ meters solely as tenths. Start by representing the whole number 9 as tenths: $9 = 9 \times \frac{10}{10} = \frac{90}{10}$. Then add the fractional part:
$\frac{90}{10} + \frac{8}{10} = \frac{98}{10}$
So, $9 \frac{8}{10}$ meters is expressed as 98 tenths.

Key Concept: Mixed number decomposition, Understanding tenths and hundredths

c) 5 whole, 6 tenths, and 7 hundredths
[Solution Description]
The decimal number $5.67$ can be decomposed into its whole number part and fractional parts:
$5.67 = 5 + 0.67$
Breaking down $0.67$, we have:
$0.67 = \frac{67}{100} = \frac{60}{100} + \frac{7}{100} = \frac{6}{10} + \frac{7}{100}$
Thus, $5.67$ can be expressed as:
$5.67 = 5 + \frac{6}{10} + \frac{7}{100}$
Hence, $5.67$ is expressed as 5 whole numbers, 6 tenths, and 7 hundredths.

Key Concept: Fractional addition, Conversion to mixed number

b) $8 \frac{1}{5}$
[Solution Description]
We need to find the sum of $4 \frac{7}{10}$ and $3 \frac{5}{10}$.
Convert both mixed numbers to improper fractions:
$4 \frac{7}{10} = \frac{(4 \times 10) + 7}{10} = \frac{47}{10}$
$3 \frac{5}{10} = \frac{(3 \times 10) + 5}{10} = \frac{35}{10}$
Add these fractions:
$\frac{47}{10} + \frac{35}{10} = \frac{47 + 35}{10} = \frac{82}{10}$
Convert the result back to a mixed number:
$\frac{82}{10} = 8 \frac{2}{10}$
Simplify the fractional part:
$\frac{2}{10} = \frac{1}{5}$
So, the simplified form is:
$8 \frac{1}{5}$

Key Concept: Comparing decimal points, Conceptual complexity

a) $8.902$
[Solution Description] To find the greatest number, compare each by place value:
Compare units digit:
- All numbers have $8$ as their units digit.
Compare tenths digit:
- $8.902$: 9 tenths
- $8.092$: 0 tenths
- $8.29$: 2 tenths
- $8.209$: 2 tenths
The highest tenths value is $9$ for $8.902$,
making it the greatest number.
Hence, $8.902$ is greater than all other given numbers.

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: Comparing decimal numbers

b) 1.52
[Solution Description]
To find the greatest decimal number, compare each digit starting from the left:
1. The ones place for all numbers ($1.25$, $1.52$, $1.23$, $1.35$) is $1$. Therefore, move to the tenths place.
2. Compare tenths: For $1.25$, it's $2$; for $1.52$, it's $5$; for $1.23$, it's $2$; and for $1.35$, it's $3$.
3. The largest value in the tenths place is $5$ in $1.52$. So, $1.52$ is the greatest.

Key Concept: Understanding and Converting Fractions to Decimals

c) 0.35
[Solution Description] To convert the fraction $\frac{7}{20}$ into a decimal, we can express it as an equivalent fraction with a denominator that is a power of 10.
First, find a number that when multiplied by 20 gives a power of 10, such as 100. So, multiply both numerator and denominator of $\frac{7}{20}$ by 5:
$\frac{7}{20} \times \frac{5}{5} = \frac{35}{100}$
Now, expressing $\frac{35}{100}$ in decimal form:
$\frac{35}{100} = 0.35$
So, the decimal equivalent of $\frac{7}{20}$ is $0.35$.

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: Writing fractions in decimal form

c) 0.6
[Solution Description] To convert $\frac{3}{5}$ into a decimal, divide 3 by 5. We perform long division:
3 divided by 5 is 0 with a remainder of 3. Bringing down a zero, 30 divided by 5 gives 6. Therefore, $\frac{3}{5} = 0.6$.

Key Concept: Converting tenths to hundredths

c) $350$ hundredths
[Solution Description]
To convert $3.5$ to hundredths, recognize that the number $3.5$ can be expressed as $3 \frac{5}{10}$. Since $1$ tenth equals $10$ hundredths, multiply the $\frac{5}{10}$ by $\frac{10}{10}$ to get an equivalent fraction in terms of hundredths:
$3 \frac{5}{10} = 3 \frac{50}{100}$
Therefore, $3.5$ is equal to $350$ hundredths when converted from a decimal.
Thus, $3.5$ is equal to $350$ hundredths.

Key Concept: Dividing tenths into hundredths

Correct Answer option Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that 0.4 units can also be expressed in terms of one-hundredths.
To verify this, let's convert 0.4 to one-hundredths.
Since $1$ unit $= 100$ one-hundredths and $1$ tenth $= 10$ one-hundredths, the decimal $0.4$
implies $4$ tenths or $40$ one-hundredths because:
$0.4 = \frac{4}{10} = \frac{4 \times 10}{100} = \frac{40}{100} = 40 \text{ one-hundredths}$
Hence, the Assertion is true.
The Reason correctly explains that when a tenth is divided by 10, it results in one-hundredths.
Therefore, both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

Key Concept: Dividing tenths into hundredths

c) Assertion is true, but Reason is false.
[Solution Description]
The assertion states that a tenth is equivalent to ten hundredths. According to the syllabus, $1 \text{ tenth} = 10 \text{ one-hundredths}$, which makes the assertion true.
The reason states that each one-tenth can be divided into ten equal parts called thousandths. However, dividing a tenth into ten equal parts results in hundredths, not thousandths. Therefore, the reason is false.
Hence, Assertion is true, but Reason is false.

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: 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: A Hundredth Part

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that the length of the folded paper is 4 units and 45 one-hundredths, which is equivalent to $4 + \frac{45}{100}$. This means the paper is divided into smaller parts where every part represents a hundredth of a unit. Thus, each one-tenth of a unit is made up of 10 one-hundredths because $0.1 = \frac{10}{100}$. Therefore, both the assertion and reason are true, and since recognizing the division of a unit into tenths and then further into hundredths explains how measurements like 4 units and 45 one-hundredths are possible, the reason is the correct explanation for the assertion.

Key Concept: Decimal Place Value System, Decimal Equivalence

c) $2.7$
[Solution Description] To solve this problem, we need to perform the subtraction first.
The number given is $x = 2.705$. We need to subtract $0.005$ from $x$:
$2.705 - 0.005 = 2.700$
Now, we identify decimal equivalence. The number $2.700$ can be simplified further because trailing zeros do not affect the value, therefore:
$2.700 = 2.7$
Therefore, the equivalent number is $2.70$, but it is represented as $2.7$, which means our answer is an option that represents $2.7$.

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: Reading Decimal Numbers

b) Three point two zero five
[Solution Description]
To read 3.205, identify each part:
- 3 is the whole number part.
- 2 is in the tenths place,
- 0 is in the hundredths place,
- 5 is in the thousandths place.
The number is read as "three point two zero five."

Key Concept: Decimal System and Place Value

c) 23.45
[Solution Description] To find the decimal representation, we convert each place value into its equivalent decimal form:
- Two tens: $2 \times 10 = 20$
- Three units: $3 \times 1 = 3$
- Four one-tenths: $4 \times \frac{1}{10} = 0.4$
- Five one-hundredths: $5 \times \frac{1}{100} = 0.05$
Now add these values together:
$20 + 3 + 0.4 + 0.05 = 23.45$
Thus, the decimal representation is $23.45$.

Key Concept: Writing and Reading Decimal Numbers

c) 2.75
[Solution Description] To convert the fraction $\frac{11}{4}$ into a decimal, divide $11$ by $4$:
- Divide $11$ by $4$: $11 \div 4 = 2.75$
To understand this in terms of place values:
- $2$ is in the ones place
- $0.7$ (or $7 \times \frac{1}{10}$) represents the tenths place
- $0.05$ (or $5 \times \frac{1}{100}$) represents the hundredths place
Thus, $\frac{11}{4}$ equals $2.75$ in decimal form.

Key Concept: Reading Decimal Numbers, Comparing Decimal Numbers

c) 8.540
[Solution Description]
To find the greatest value among these decimals, we compare them digit by digit from left to right:
- All numbers have the same whole number part ($8$).
- The tenth's place for all numbers is compared: $5 > 0$, so $8.504$, $8.540$, and $8.450$ are greater than $8.045$.
- Next, compare the hundredth's place for $8.504$, $8.540$, and $8.450$: $5 = 5 = 5$.
- Finally, compare the thousandth's place where: $4 8.504$.
Therefore, the greatest number is $8.540$.

Key Concept: Subtraction of Decimals

b) 0.75
[Solution Description]
To find the difference between $1.00$ and $0.25$, align the decimals and subtract:
$1.00 - 0.25 = 0.75$
Thus, the result is $0.75$.

Key Concept: Decimal Place Value, Conversion of fractions to decimals, Multi-step Solutions

c) $1.45$
[Solution Description]
First, we convert $\frac{3}{4}$ into its decimal form. We know that:
$\frac{3}{4} = 0.75$
Next, we add $7$ tenths ($0.7$) to this decimal.
$0.75 + 0.7 = 1.45$
Thus, the resulting decimal after increasing by $7$ tenths is $1.45$.

Key Concept: Comparing decimal numbers, Non-Straightforward Path, Advanced Reasoning

c) $A$ is larger by $0.03$
[Solution Description]
First, convert both numbers to their decimal forms:
- For $A$:
$A = 5 + 0.56 = 5.56$
- For $B$:
$B = 6 - 0.47 = 5.53$
Compare the decimals:
Since $5.56 > 5.53$, $A$ is larger than $B$.
Now, calculate the difference between $A$ and $B$:
$5.56 - 5.53 = 0.03$
Thus, $A$ is larger than $B$ by $0.03$.

Key Concept: Converting Fractions to Decimals

b) 0.54
[Solution Description] To convert the fraction $\frac{54}{100}$ into a decimal, we divide 54 by 100. This gives us 0.54.

Key Concept: Comparison of Decimal Values, Conceptual Reasoning

c) 6.95
[Solution Description] To determine the smallest number, we must compare each number's decimal values:
- $7.15$: Consists of $7$ units and $15$ hundredths.
- $8.04$: Consists of $8$ units and $4$ hundredths.
- $6.95$: Consists of $6$ units and $95$ hundredths.
- $7.105$: Consists of $7$ units and $105$ thousandths.
Since $6.95$ has the smallest whole number part ($6$), it is the smallest number overall regardless of other digits in the decimal places.
Hence, the smallest number is $6.95$.

Key Concept: Decimal Notation and Place Value

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.
[Solution Description] The Assertion states that the number 7.05 is read as "seven point zero five." This reading is correct because it reflects the expression of the number in terms of units and hundredths: $7 \times 1 + 5 \times \frac{1}{100}$. The Reason, however, claims that the digit to the right of the decimal point represents tenths. While this is true when there is only one digit after the decimal point, in 7.05, the first digit after the decimal represents tenths and the second digit represents hundredths. Therefore, both statements are true, but the reason does not correctly explain the assertion since it omits the complete explanation for two digits after the decimal.

Key Concept: A Peek Beyond the Point

c) $\frac{1}{8}$
[Solution Description]
When a unit is divided into 8 equal parts, each part represents one-eighth of the whole unit:
$\frac{1}{8} \text{ of a unit}$
This means each part is $\frac{1}{8}$th of the whole unit.

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

b) 1.5 cm
[Solution Description] We know that $1 \text{ mm} = 0.1 \text{ cm}$. Hence, for 15 mm:
$15 \times 0.1 = 1.5 \text{ cm}.$
Thus, 15 millimeters is equal to 1.5 centimeters.

Key Concept: Length Conversion

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.
[Solution Description]
To convert millimeters to centimeters, we use the fact that each centimeter equals $10 \text{ mm}$. Therefore, for $5 \text{ mm}$:
$5 \text{ mm} = \frac{5}{10} \text{ cm} = 0.5 \text{ cm}$
The assertion that $5 \text{ mm} = 0.5 \text{ cm}$ is correct because when you divide $5 \text{ mm}$ by $10$, you get $0.5 \text{ cm}$.
The reason given is also true: $1 \text{ cm}$ equals $10 \text{ mm}$. This forms the basis for converting millimeters to centimeters. However, the reason is not the explanation of why $5 \text{ mm}$ equals $0.5 \text{ cm}$; it's just a known conversion factor.
Hence, both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.

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: Conversion of Units from mm to cm

b) 2.5 cm
[Solution Description]
To convert millimeters to centimeters, we use the conversion factor $1 \text{ cm} = 10 \text{ mm}$.
Therefore, for 25 mm, we calculate:
$25 \text{ mm} = \frac{25}{10} \text{ cm} = 2.5 \text{ cm}$
The answer is 2.5 cm.

Key Concept: Units of Measurement: Centimeters ↔ Meters

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that 250 cm equals 2.5 m. To verify this, we use the conversion factor that 1 cm = 0.01 m, or equivalently, you divide the number of cm by 100 to get meters:
$250\, \text{cm} \times \frac{1\, \text{m}}{100\, \text{cm}} = 2.5\, \text{m}$
Therefore, the Assertion is true.
The Reason provided is the method for converting centimeters to meters, which involves dividing the centimeter value by 100:
$x\, \text{cm} \rightarrow \frac{x}{100}\, \text{m}$
This is a correct and relevant explanation for the conversion process used in the Assertion.
Thus, both the Assertion and the Reason are true, and the Reason is the correct explanation of the Assertion.

Key Concept: Decimal Representation and Conversion

a) 0.186
[Solution Description] Use the conversion factor $1\, \text{m} = 100\, \text{cm}$ to convert centimeters to meters. Divide the length in centimeters by 100:
$18.6\, \text{cm} = \frac{18.6}{100}\, \text{m}$
$18.6\, \text{cm} = 0.186\, \text{m}$
Thus, the pencil is 0.186 meters long.

Key Concept: Conversion from Meters to Centimeters

c) 720
[Solution Description] To convert meters to centimeters, multiply the length in meters by 100:
$7.2\, \text{m} = 7.2 \times 100\, \text{cm}$
$7.2\, \text{m} = 720\, \text{cm}$
Therefore, the distance run by the athlete is 720 cm.

Key Concept: Weight Conversion

b) 0.25 kg
[Solution Description] To convert grams to kilograms, we divide the number of grams by 1000 because $1 \text{ kg} = 1000 \text{ g}$. So, for 250 grams:
$250 \text{ g} = \frac{250}{1000} \text{ kg} = 0.25 \text{ kg}$
Thus, 250 grams is equal to 0.25 kilograms.

Key Concept: Multiplication and Conversion

c) 0.5 kg
[Solution Description]
Determine the total weight in grams first:
$4 \times 125 \text{ g} = 500 \text{ g}$
Convert this total weight to kilograms:
$500 \text{ g} = \frac{500}{1000} \text{ kg} = 0.5 \text{ kg}$
Therefore, the total required weight is 0.5 kg.

Key Concept: Weight Conversion

b) 0.75 kg
[Solution Description] To convert grams to kilograms, we divide the number of grams by 1000. For 750 grams:
$750 \text{ g} = \frac{750}{1000} \text{ kg} = 0.75 \text{ kg}$
Hence, a packet weighing 750 grams is equivalent to 0.75 kilograms.

Key Concept: Rupee ─ Paise Conversion, Conceptual Complexity

c) 17
[Solution Description]
Let the number of 50 paise coins be $x$. Then, the number of 25 paise coins is $2x$.
The value of x 50 paise coins is:
$50x \, \text{paise} = 0.50x \, \text{rupees}$
The value of 2x 25 paise coins is:
$25(2x) = 50x \, \text{paise} = 0.25(2x) \, \text{rupees}$
According to the problem, these together make up Rs.8.50:
$0.50x + 0.25(2x) = 8.50$
Simplify and solve:
$0.50x + 0.50x = 8.50$
$x = 8.50$
Thus, the number of 25 paise coins is:
$2x = 2 \times 8.50 = 17$
So, there are 17 (as 2x) 25 paise coins.

Key Concept: Direct Application of Formula

b) 75
[Solution Description] Since $1 \text{ rupee} = 100 \text{ paise}$, to find out how many paise Rs.0.75 corresponds to, we multiply by 100:
$0.75 \times 100 = 75 \text{ paise}$
Hence, Rs.0.75 is equal to 75 paise.

Key Concept: Rupee ─ Paise Conversion, Multi-step Solutions

c) 275
[Solution Description]
To find the change in paise that a person receives after purchasing items, we start by calculating the total cost of the items in rupees. Each item costs Rs.0.75, so for 3 items:
$\text{Total Cost} = 3 \times 0.75 = 2.25 \, \text{rupees}$
The person pays with a Rs.5 note, so the change they should receive is:
$\text{Change in Rupees} = 5 - 2.25 = 2.75 \, \text{rupees}$
Since $1 \, \text{rupee} = 100 \, \text{paise}$, we convert the change to paise:
$\text{Change in Paise} = 2.75 \times 100 = 275 \, \text{paise}$
Therefore, the person receives 275 paise as change.

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: The Need for Smaller Units, Real-World Applications

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.
[Solution Description]
The assertion states that precise measurements are essential for mechanical objects like toys and electronics to function correctly. This is true because any mismatch in dimensions can lead to improper fitting or malfunction.
The reason explains that using smaller units offers more precise length representations, which helps in accurately comparing and assembling parts. This statement is also true.
However, while both statements are true, the reason does not directly explain why precision is necessary for the functionality of mechanical objects. It only provides a way to achieve precision.
Therefore, both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.

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]
The assertion states that precise measurements are critical for tasks such as assembling toys due to minor size variations impacting the outcome. This is correct since even small discrepancies in component dimensions can cause issues during assembly.
The reason given supports this by explaining why Sonu's mother chose different screws, highlighting that their unique lengths played a vital role in the repair process.
Both statements are true and related; however, the reason elaborates on an example rather than directly explaining the necessity of measurement precision outlined in the assertion. Therefore, both the assertion and the reason are true, but the reason does not serve as the direct explanation for the assertion.

Key Concept: Locating decimal numbers on a number line

d) At the seventh mark after 3
[Solution Description]
To locate 3.7 on a number line between 3 and 4, divide the segment between these two whole numbers into ten equal parts. Each division represents one-tenth of the unit. The decimal 3.7 is seven-tenths from 3, so it would be located at the seventh mark after 3.

Key Concept: Identifying decimal numbers on a number line

c) 6.5
[Solution Description] On this given number line:
- The label ‘A’ corresponds to 6.2,
- The label ‘B’ to 6.5,
- The label ‘C’ to 6.8.
Based on the sequence provided, the letter ‘B’ falls exactly at the midpoint between 6.2 and 6.8, indicating it is representing 6.5.

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: Locating and Comparing Decimals

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To solve this problem, we need to understand how decimals are compared:
1. **Compare Place Values**: Start from the leftmost digit (the highest place value). Compare each digit moving towards the right until a difference is found.
2. **Identify the Greater Digit**: The number with the larger digit at the first position where they differ is the greater one.
3. **Irrelevance of Subsequent Digits**: Once you find the first difference, subsequent digits will not change which number is larger since the decision point has already been reached.
Now let's apply this understanding:
- Consider two decimal numbers, say $A = 1.234$ and $B = 1.245$.
- Starting from the leftmost side, both share '1' in the units place.
- Moving to the tenths place, both have '2'.
- At the hundredths place, $4 > 3$, so $1.245$ is greater than $1.234$.
- Here, further decimals beyond this point don't alter the result since we've determined the larger number by the first differing place value.
Since the reason correctly explains that any digits after the differing point do not impact the determination of which decimal is greater, both assertion and reason are true, and the reason justifies the assertion.

Key Concept: Comparing Decimals

b) $4.7$
[Solution Description] To determine which decimal number is greater, we compare the digits starting from the highest place value. All three numbers have 4 in the units place, so we move to the tenths place:
- $4.007$: Tenths digit is 0.
- $4.7$: Tenths digit is 7.
- $4.07$: Tenths digit is 0.
The number $4.7$ has the largest tenths digit. Therefore, $4.7$ is the greatest among them.

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: 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: Decimal ordering based on different place value positions

a) $12.3564, 12.3546, 12.3465, 12.3456$
[Solution Description]
To sort these numbers in descending order, commence with examination of each integer's place values sequentially from left to right:
- Observe the whole number '12' common across all.
- Analyze tenths position:
- All have '3'.
- At hundredths:
- $12.3456$ has '4'
- $12.3546$ has '5'
- $12.3465$ has '4'
- $12.3564$ has '5'
Ascertain thousandths for ties involving '5':
- $12.3546$ has '4'
- $12.3564$ has '6'
Therefore, the descending sequence is:
1. $12.3564$
2. $12.3546$
3. $12.3465$
4. $12.3456$

Key Concept: Comparing decimal numbers using place value and number line visualization

b) $3.465$
[Solution Description]
To determine which decimal is the largest, we need to compare them digit by digit, starting from the leftmost digit:
- All numbers have the same whole number part, '3'.
- Moving to the tenths place:
- All numbers have '4' in the tenths place.
- Next, examine the hundredths place:
- $3.456$ has '5'
- $3.465$ has '6'
- $3.546$ has '4'
- $3.564$ has '6'
As two numbers, $3.465$ and $3.564$, share the highest value '6', we move on to the thousandths place for distinction:
- $3.465$ has '5'
- $3.564$ has '4'
Since $5 > 4$, $3.465$ is larger than $3.564$. Hence, $3.465$ is the largest number among all given options.

Key Concept: Digit-by-digit place value comparison, locating and comparing decimals

d) Assertion is false, but Reason is true.
[Solution Description]
Let's compare the two decimal numbers: $3.456$ and $3.465$. We follow the process of digit-by-digit comparison from the most significant to the least significant.
1. **Units Place**: Both numbers have '3' in the units place, so they are equal here.
2. **Tenths Place**: Both numbers also have '4' in the tenths place, remaining equal at this point.
3. **Hundredths Place**: Here, $3.456$ has a '5', and $3.465$ has a '6'. Since '6' is greater than '5', $3.465$ is actually greater than $3.456$.
Therefore, the assertion that $3.456$ is greater than $3.465$ is false.
As for the reason, it correctly states that we start by comparing the leftmost non-zero digit when comparing decimal numbers. However, the given logic behind the assertion was incorrect since it didn't consider all the digits accurately.
Hence, the assertion is false, but the reason is true.

Key Concept: Place Value Comparison of Decimals

c) $4.908$
[Solution Description]
Compare the digits of each number:
- Both numbers have $4$ units.
- Both have $9$ tenths.
- The first number has $0$ hundredths, while the second has $8$ hundredths.
Because $8 > 0$, $4.908$ is greater than $4.809$.

Key Concept: Decimal Place Value, Closest Decimals, Digit-by-digit Place Value Comparison

c) $3.645$
[Solution Description] Calculate the absolute difference between each option and $3.456$:
- Difference with $3.465$ is $\mid 3.456 - 3.465 \mid = 0.009$
- Difference with $3.546$ is $\mid 3.456 - 3.546 \mid = 0.09$
- Difference with $3.645$ is $\mid 3.456 - 3.645 \mid = 0.189$
- Difference with $3.354$ is $\mid 3.456 - 3.354 \mid = 0.102$
The largest difference is with $3.645$, hence it is the farthest from $3.456$.

Key Concept: Estimating Sums of Decimals

b) Between 21 and 23
[Solution Description] According to Sonu's observation, the sum of two decimal numbers will be greater than the sum of their whole number parts and less than 2 more than the sum of their whole number parts.
Find the sum of the whole number parts:
\begin{align*}
12 + 9 = 21
\end{align*}
Add 2 to this sum for the upper estimate:
\begin{align*}
21 + 2 = 23
\end{align*}
Therefore, the estimated range for the sum is between $21$ and $23$.

Key Concept: Advanced Reasoning, Non-Straightforward Path

b) \$7.78
[Solution Description] The initial amount spent on groceries is \$13.47.
Since David received a \$1.25 discount, the effective cost becomes:
$13.47 - 1.25 = 12.22$
David pays with a \$20 bill, so the change he should receive is:
$20.00 - 12.22 = 7.78$

Key Concept: Addition and Subtraction of Decimals

c) Assertion is true, but Reason is false.
[Solution Description]
Let's analyze both statements:
Assertion (A) states that adding a decimal number to the whole number part always increases the total value. This is generally true because any positive addition, whether it's a whole number or a fraction (decimal), will increase the total value.
Reason (R) says that the fractional part of a decimal is irrelevant in determining the change in value when added to a whole number. This is false because the fractional part has its own contribution to the sum. For example, if we add $2$ (whole number) to $0.5$, the result is $2.5$. The fractional part ($0.5$) directly affects the sum, contributing half a unit more than the whole number itself.
Hence, while Assertion (A) is true, Reason (R) is false as it incorrectly claims that the fractional part doesn't matter.
Therefore, the correct option is "Assertion is true, but Reason is false."

Key Concept: Subtracting Decimal Numbers

c) $5.9$
[Solution Description]
To subtract $4.5$ from $10.4$, we align the decimal points and perform the subtraction:
$\begin{array}{c@{}c@{}c@{}c@{}c} \\ & 10.4 \\ \\ - & 4.5 \\ \hline \\ & 5.9 \\ \\ \end{array}$
Therefore, the result of the subtraction is $5.9$.

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 of Decimal Numbers

c) 4.00 kg
[Solution Description] To find the total weight of fruits Ravi bought, we need to add the weights of apples and bananas. The weights are given as decimals:
First, line up the decimal numbers:
$2.35 + 1.65$
Start adding from the rightmost digit (hundredths place):
- Hundredths: $5 + 5 = 10$; write 0 and carry over 1.
- Tenths: $3 + 6 + 1 = 10$; write 0 and carry over 1.
- Units: $2 + 1 + 1 = 4$.
Therefore, the total weight is $4.00 \, \text{kg}$.

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 subtract decimals accurately, aligning their decimal points ensures that each digit is correctly placed under its corresponding place value. For instance, when subtracting $3.45$ from $5.6$, write:
$\begin{array}{c@{}c@{}c@{}c} \\ & 5.60 \\ \\ - & 3.45 \\ \\ \hline \\ & 2.15 \\ \\ \end{array}$
Here, adding an extra zero to make $5.6$ into $5.60$ helps align the numbers properly, ensuring correct subtraction of hundredths, tenths, and units separately. Hence both the assertion and reason statements are true, and the reason perfectly explains why the alignment is needed.

Key Concept: Aligning decimal points

c) 20.24
[Solution Description]
Align the decimals for 12.34 and 7.9 before adding:
$\begin{array}{c@{}c@{}c@{}c@{}c} \\ & 12.34 \\ \\ + & 7.90 \\ \hline \\ & 20.24 \\ \\ \end{array}$
The result after addition is 20.24.

Key Concept: Aligning decimal points, Real-World Application

b) \$10.44
[Solution Description]
First, add up the cost of the items before considering the discount:
Book's price rounded: \$4.57 (rounded from \$4.568)
Pen's price: \$1.56
Total cost of items:
$4.57 + 1.56 = 6.13$
Subtract the discount voucher rounded to the nearest hundredth:
$0.70$
Final cost after applying discount:
$6.13 - 0.70 = 5.43$
Subtract this cost from Sarah's initial amount:
$15.87 - 5.43 = 10.44$
Thus, Sarah’s remaining balance is \$10.44.

Key Concept: Addition and Subtraction of Decimals, Place value method of addition/subtraction

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.
[Solution Description]
The assertion states that in the place value method for adding decimals, we ensure alignment based on decimal places to maintain accuracy. This is a true statement as it's fundamental for correct addition results.
The reason provides insight into another method involved: converting decimals to fractions (with base ten denominators) to facilitate operations like addition. For instance, $3.45$ can be expressed as $\frac{345}{100}$, aiding in manual calculations without errors from misalignment.
Both statements are true; however, the reason does not directly explain the process described in the assertion about aligning by decimal places. Instead, it refers to an alternative approach involving fractions.

Key Concept: Place Value Method of Addition/Subtraction

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that when adding two decimal numbers, aligning the decimal points ensures correct addition based on place value. This is valid because aligning the decimals allows for the effective summation of each respective unit, tenth, hundredth, etc., as they are directly above or below their corresponding counterparts.
The reason explains why aligning decimal points is crucial: it allows all digits in a column to represent consistent positional values (ones with ones, tenths with tenths, hundredths with hundredths, and so forth). This explanation is true, as alignment is necessary for correctly interpreting each digit's contribution to the total sum based on its position relative to the decimal point.
Both the Assertion and Reason are true statements. Furthermore, the Reason provides a direct explanation for why the Assertion is accurate, making the alignment of decimal points critical for maintaining proper positional values during addition.

Key Concept: Place value method of addition, Decimal Sequences

c) $12.2$
[Solution Description] The sequence progresses by adding $0.7$ each time:
Term 1: $0.7$
Term 2: $1.4$
Term 3: $2.1$
Term 4: $2.8$
Let's calculate:
Term 5: $2.8 + x = 15$
Solving for $x$:
$x = 15 - 2.8 = 12.2$
Therefore, the additional number required is $12.2$.

Key Concept: Decimal Notation Over Time, A Peek Beyond the Point

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.
[Solution Description]
The assertion states that the decimal point was standardized in the 16th century by John Napier, which is true. In fact, John Napier did contribute significantly to this notation becoming popular along with others like Christopher Clavius.
As for the reason, it explains why the decimal notation could naturally extend existing systems. The historical development and widespread acceptance of the decimal system are grounded in its ability to seamlessly integrate fractional components into the established integer-based system.
While both statements are true, the reason given doesn't directly explain why the decimal point itself was adopted as standard during the 16th century; rather, it provides background on the foundational compatibility of the decimal system with previous numerical understanding.
Thus, while both the assertion and reason are true, the reason does not directly explain the assertion's context about the adoption of the decimal point.

Key Concept: Decimal System, Unit Conversion, A Peek Beyond the Point

b) $\frac{3}{8}$ and 375 one-thousandths
[Solution Description]
Convert 0.375 to a fraction:
Start by recognizing that 0.375 has three decimal places, so it is out of 1000:
$0.375 = \frac{375}{1000}$
Simplifying the fraction by dividing both numerator and denominator by their greatest common divisor, which is 125:
$\frac{375 \div 125}{1000 \div 125} = \frac{3}{8}$
Hence, 0.375 is equivalent to $\frac{375}{1000}$ or simplified to $\frac{3}{8}$, representing 375 one-thousandths.

Key Concept: Decimal System and Notation

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
Assertion: "The number 70.5 is read as seventy point five, meaning seventy and five-tenths." This statement is true because in decimal notation, 70.5 means $7 \times 10 + 5 \times \frac{1}{10}$.
Reason: "Decimal notation uses a point to separate the whole number from its fractional part." This statement is also true. The point (or decimal separator) is used to distinguish between the integer part and the fractional part in a decimal number.
Both the Assertion and Reason are true, and the Reason correctly explains why the Assertion is true because it provides the mechanism (decimal point) by which the reading of numbers like 70.5 occurs.
Therefore, both the Assertion and Reason are true, and Reason is the correct explanation of Assertion.

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: Understanding deceptive decimal notation

d) 4:30 PM
[Solution Description] The message says 4.5 hours past noon. Decomposing 4.5, it means 4 full hours plus 0.5 hours. Convert 0.5 hours to minutes:
$0.5 \times 60 = 30 \text{ minutes}$
Adding these to the base time (noon):
$12:00 + 4 \, \text{hours} + 30 \, \text{minutes} = 4:30 \text{ PM}$
Hence, the bus arrives at 4:30 PM.

Key Concept: Decimal mishap, unit error

Correct Answer option c) 10.050 m
[Solution Description]
In historical notation, 10ˈ050 means the whole number part is 10 and the fractional part has three decimals, so it should be interpreted as 10.050.
Thus, the correct interpretation of the length is 10.050 m.

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: Understanding Euro Cents Conversion

b) 25000 euro cents
[Solution Description]
To determine the incorrect amount processed in euro cents, we convert euros to euro cents. Since 1 euro equals 100 euro cents, for €250 the conversion is:
$250 \text{ euros} \times 100 = 25000 \text{ euro cents}$
Therefore, the incorrect processed amount is 25000 euro cents.

Key Concept: Currency Conversion Error

a) 4500 euro cents
[Solution Description]
The mistake occurs due to using euro cents instead of euros. Since 1 euro = 100 euro cents, if the intended amount is €45, the incorrect amount sent out would be:
$45 \text{ euros} \times 100 = 4500 \text{ euro cents}$
Therefore, the incorrect amount sent out is 4500 euro cents.

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 Differences

b) Between 6 and 7
[Solution Description] The whole number parts of 15.75 and 8.26 are 15 and 8 respectively. The difference between these whole numbers is:
$15 - 8 = 7$
Since the decimals are subtracted, the result will always be less than 7 but more than one unit less than 7 (i.e., 6).
Therefore, the difference of the decimal numbers falls in the range from 6 to 7.

Key Concept: Decimal and Measurement Disasters

c) 99,338
[Solution Description] To convert kilograms to pounds, we use the formula:
$\text{Weight in pounds} = \frac{\text{Weight in kg}}{0.453}$
Substituting the given weight:
$\text{Weight in pounds} = \frac{45000}{0.453} \approx 99338$
Thus, approximately $99,338$ pounds should be loaded.

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: Decimal System

c) Assertion is true, but Reason is false.
[Solution Description]
The assertion correctly states that a proper understanding of decimal places is vital in medical dosage calculations. This understanding can prevent potentially dangerous dosing errors, such as giving a patient too much or too little medication. However, the reason provided is false as it wrongly equates $12.5$ mg with $125$ mg. Indeed, the placement of the decimal drastically changes the amount, making these two values very different: $12.5$ mg is actually ten times less than $125$ mg.

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: Deceptive Decimal Notation

a) 2
[Solution Description] In cricket, an over consists of 6 balls. The notation 8.4 means 8 complete overs and 4 balls. Since each over has 6 balls, 8.4 denotes 8 full overs plus 4 additional deliveries. Therefore, we calculate the number of balls left as follows:
Total balls in an over = 6
Balls remaining in this over = 6 - 4 = 2.
Hence, there are 2 balls left in this over.

Key Concept: Deceptive Decimal Notation

b) 12:48 p.m.
[Solution Description]
The starting time is 10:00 a.m. We need to add 2.8 hours to this time.
Let's separate 2.8 hours:
- 2 hours mean adding 2 to 10:00 a.m., giving us 12:00 p.m.
- 0.8 hours translates into dividing an hour into 10 equal parts, with 8 parts chosen. Each part corresponds to 6 minutes ($60 \div 10 = 6$).
- Therefore, 0.8 hours equals $8 \times 6 = 48$ minutes.
By adding these 48 minutes to 12:00 p.m., the workshop concludes at 12:48 p.m.

Key Concept: Deceptive Decimal Notations, Place Value Interpretation

a) $0$ cm
[Solution Description]
To solve this problem, we need to convert both lengths into the same unit and then find the difference.
1. Convert $3.4$ meters to centimeters:
Since $1$ meter = $100$ centimeters,
$3.4$ meters = $3.4 \times 100 = 340$ centimeters.
2. Convert $34$ decimeters to centimeters:
Since $1$ decimeter = $10$ centimeters,
$34$ decimeters = $34 \times 10 = 340$ centimeters.
Now compare the two quantities:
$340$ cm (interpreted length) - $340$ cm (actual required length) = $0$ cm.
Since the interpreted length equals the actual required length, there is no difference in the amount provided compared to what was needed.