Key Concept: Proportional Scaling

b) 10
[Solution Description]
The given ratio is 4 bags per 10 meters.
For 25 meters, let the required number of bags be $b$.
Therefore, $\frac{4}{10} = \frac{b}{25}$.
Solving for $b$, $b = \frac{4 \times 25}{10} = 10$ bags.

Key Concept: Proportional Reasoning, Similarity of Rectangles

d) Width = 48 cm, height = 32 cm
[Solution Description]
To find a similar rectangle, the ratio of width to height must be the same as the original rectangle. The original rectangle has a ratio of $\frac{24}{16} = \frac{3}{2}$. We need to check which option has the same ratio.
1. For width = 36 cm, height = 24 cm: $\frac{36}{24} = \frac{3}{2}$.
2. For width = 30 cm, height = 18 cm: $\frac{30}{18} = \frac{5}{3}$.
3. For width = 20 cm, height = 10 cm: $\frac{20}{10} = 2$.
4. For width = 48 cm, height = 32 cm: $\frac{48}{32} = \frac{3}{2}$.
Options 1 and 4 have the same ratio as the original rectangle.

Key Concept: Proportional Reasoning

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that images A, C, and D look similar due to proportional width-to-height ratios. This is true because when comparing image A ($60:40$) with image C ($30:20$), both width and height are scaled by $\frac{1}{2}$. Similarly, image D ($90:60$) is scaled by $\frac{3}{2}$ from image A. The reason correctly explains that this similarity arises because the changes in width and height occur by the same multiplicative factor, preserving proportionality.

Key Concept: Proportional Reasoning, Simplifying Ratios

d) All of the above
[Solution Description]
First, simplify the given ratio $12 : 8$ by dividing both terms by their HCF (4). This gives $3 : 2$. Now, check which option simplifies to $3 : 2$.

Key Concept: Width–Height Proportional Change

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that images A, C, and D look similar due to proportional changes in width and height. This is true as shown in the given data:
- For image C compared to A: Width change factor = $\frac{30}{60} = \frac{1}{2}$, Height change factor = $\frac{20}{40} = \frac{1}{2}$ (same).
- For image D compared to A: Width change factor = $\frac{90}{60} = 1.5$, Height change factor = $\frac{60}{40} = 1.5$ (same).
The reason correctly explains the assertion because proportional scaling preserves similarity. Thus, both are true, and the reason is the correct explanation.

Key Concept: Width–Height Comparison

c) 20 mm
[Solution Description]
The original ratio of width to height for Image A is $60 : 40$, which simplifies to $3 : 2$. To maintain the same ratio when the width is reduced to 30 mm, we set up the proportion $\frac{3}{2} = \frac{30}{h}$. Solving for $h$:
$3h = 60$
$h = 20$
The new height should be 20 mm.

Key Concept: Multiplicative vs. Additive Changes

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that images look similar when both dimensions are scaled by the same multiplicative factor, which is true as per the syllabus. The reason explains that proportional change preserves the shape, which is also true and correctly explains why the assertion holds. Hence, both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

Key Concept: Proportional Scaling Factor

d) 2
[Solution Description]
To find the scaling factor, divide the dimensions of Image Y by those of Image X:
Width factor = $\frac{100}{50} = 2$,
Height factor = $\frac{60}{30} = 2$.
Since both factors are equal, the scaling factor is 2.

Key Concept: Ratios, Problem-Solving

c) 15 kg
[Solution Description]
Given the ratio of sugar to flour is $3 : 5$ and 9 kg of sugar is used.
Let the amount of flour needed be $x$ kg.
According to the ratio:
$3 : 5 = 9 : x$
This can be written as:
$\frac{3}{5} = \frac{9}{x}$
Cross-multiplying:
$3x = 45$
Solving for $x$:
$x = \frac{45}{3} = 15$
Therefore, 15 kg of flour is needed.

Key Concept: Ratio Simplification

b) $1 : 3$
[Solution Description]
To simplify the ratio $15 : 45$, find the highest common factor (HCF) of 15 and 45, which is 15. Divide both terms by 15 to get $1 : 3$.

Key Concept: Ratios, Simplifying Ratios

c) 21
[Solution Description]
Given the ratio of boys to girls is $5 : 3$ and there are 35 boys.
Let the number of girls be $x$.
According to the ratio:
$5 : 3 = 35 : x$
This can be written as:
$\frac{5}{3} = \frac{35}{x}$
Cross-multiplying:
$5x = 105$
Solving for $x$:
$x = \frac{105}{5} = 21$
Therefore, there are 21 girls.

Key Concept: Ratios, Proportionality

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The Assertion states that if two ratios $\frac{a}{b}$ and $\frac{c}{d}$ are proportional, then they are equal. This is true because proportionality of ratios means they represent the same relationship, hence their simplified forms are identical.
The Reason states that two ratios are proportional if and only if their cross-products are equal ($ad = bc$). This is also true as it is the fundamental condition for checking proportionality. Furthermore, the Reason correctly explains the Assertion because the equality of cross-products implies the equality of the ratios.
Therefore, both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

Key Concept: Simplest Form of Ratio

c) $2 : 3$
[Solution Description]
To simplify the ratio $60 : 90$, divide both terms by their HCF. The HCF of 60 and 90 is 30. Dividing both by 30 gives $2 : 3$.

Key Concept: Definition of Ratio

b) 3 units of the first quantity for every 4 units of the second
[Solution Description]
A ratio $a : b$ represents 'a' units of the first quantity for every 'b' units of the second quantity. Therefore, $3 : 4$ means 3 units of one quantity for every 4 units of another.

Key Concept: Simplest Form, Proportionality Check

c) $9 : 10$
[Solution Description]
Simplify $15 : 25$ to its simplest form by dividing both terms by their HCF (5): $3 : 5$. Now simplify the options:
Option a) $21 : 35$ simplifies to $3 : 5$ (HCF 7).
Option b) $30 : 50$ simplifies to $3 : 5$ (HCF 10).
Option c) $9 : 10$ remains $9 : 10$ (no common factor).
Option d) $45 : 75$ simplifies to $3 : 5$ (HCF 15).
Only option c) does not simplify to $3 : 5$, so it is not proportional.

Key Concept: Dividing a Quantity in a Given Ratio

b) \$3,000 and \$2,000
[Solution Description]
To divide \$5,000 in the ratio $3 : 2$, first calculate the total parts: $3 + 2 = 5$.
First person's share: $3 \times \frac{5000}{5} = 3 \times 1000 = \$3,000$.
Second person's share: $2 \times \frac{5000}{5} = 2 \times 1000 = \$2,000$.

Key Concept: Proportional Ratios

a) $8 : 12$
[Solution Description]
Simplify $4 : 6$ by dividing both terms by their HCF, which is 2. This gives $2 : 3$. Now check which option simplifies to $2 : 3$. Option a) $8 : 12$ simplifies to $2 : 3$ (divided by 4). So, $4 : 6 :: 8 : 12$.

Key Concept: Assertion-Reason

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To check if the ratios are proportional, we simplify them:
1. Simplifying $60:40$ by dividing both terms by their HCF (20): $60 \div 20 : 40 \div 20 = 3:2$.
2. Simplifying $90:60$ by dividing both terms by their HCF (30): $90 \div 30 : 60 \div 30 = 3:2$.
Since both simplified forms are equal ($3:2$), the Assertion is true. The Reason correctly explains that they are proportional because their simplest forms match.

Key Concept: Proportional Ratios, Simplest Form

a) Yes, because both simplify to $4:3$
[Solution Description] First, simplify both ratios to their simplest form. For the first ratio $48:36$, the HCF of 48 and 36 is 12. Dividing both terms by 12 gives $\frac{48}{12} : \frac{36}{12} = 4:3$. For the second ratio $64:48$, the HCF of 64 and 48 is 16. Dividing both terms by 16 gives $\frac{64}{16} : \frac{48}{16} = 4:3$. Since both simplified ratios are equal ($4:3$), the original ratios are proportional.

Key Concept: Simplification of Ratios

b) $2 : 3$
[Solution Description]
To simplify the ratio $72 : 108$, we divide both terms by their greatest common divisor (GCD). The GCD of 72 and 108 is 36.
$\frac{72}{36} : \frac{108}{36} = 2 : 3$
Therefore, the simplified ratio is $2 : 3$.

Key Concept: Age Ratio Problem

a) $11 : 15$
[Solution Description]
Currently, A is 12 years old and B is 20 years old, which matches the ratio $3 : 5$ (since $12 : 20$ simplifies to $3 : 5$). After 10 years, A will be $12 + 10 = 22$ years old and B will be $20 + 10 = 30$ years old. The new ratio is $22 : 30$. Simplifying this by dividing both terms by 2 gives $11 : 15$.

Key Concept: Simplifying ratios, Proportional reasoning

b) $2 : 3$
[Solution Description]
To simplify the ratio $8 : 12$, find the HCF of 8 and 12, which is 4. Divide both terms by 4: $\frac{8}{4} = 2$ and $\frac{12}{4} = 3$. Thus, the simplest form is $2 : 3$.

Key Concept: Simplifying ratios, Proportional reasoning

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
Let the side lengths of the two cubes be $3k$ and $5k$ respectively. The surface area of a cube is given by $6 \times (\text{side length})^2$. So, the surface areas are:
Cube 1: $6 \times (3k)^2 = 6 \times 9k^2 = 54k^2$
Cube 2: $6 \times (5k)^2 = 6 \times 25k^2 = 150k^2$
The ratio of their surface areas is $54k^2 : 150k^2$, which simplifies to $9:25$ when divided by their HCF of 6$k^2$. The reason correctly explains the assertion since the surface area indeed depends on the square of the side length.

Key Concept: Problem solving with proportional reasoning, Unit conversions

a) 304,920 sq ft
[Solution Description]
Given that 3 acres = 130,680 sq ft, then 1 acre = $\frac{130,680}{3} = 43,560$ sq ft. For 7 acres: $7 \times 43,560 = 304,920$ sq ft.

Key Concept: Age Ratio Application

b) $1 : 3$
[Solution Description]
When Rohan is 6, his father’s age is $5 \times 6 = 30$ years. After 6 years, Rohan will be 12 and his father will be 36. The ratio is $12 : 36$, which simplifies to $1 : 3$ when divided by their HCF, 12.

Key Concept: Assertion and Reason

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The Assertion states that the ratio $60:40$ simplifies to $3:2$, which is correct because dividing both terms by their HCF (20) gives $\frac{60}{20}:\frac{40}{20} = 3:2$. The Reason correctly explains that the HCF of 60 and 40 is 20, which is indeed used to simplify the ratio. Therefore, both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

Key Concept: Ratios in Their Simplest Form

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The ratio $60 : 40$ simplifies to $3 : 2$ when divided by their HCF, which is 20. Similarly, the ratio $90 : 60$ simplifies to $3 : 2$ when divided by their HCF, which is 30. Since both ratios simplify to the same form ($3 : 2$), they are proportional. The Reason correctly explains why the ratios are proportional.

Key Concept: Finding Missing Terms in Proportional Ratios

c) $24 : 36$
[Solution Description]
The given ratio $8 : 12$ can be simplified to $2 : 3$. To find the missing term in $24 : \_\_\_\_$, observe that 24 is obtained by multiplying the first term (8) by 3. Therefore, multiply the second term (12) by 3:
$12 \times 3 = 36$
So, the proportional ratio is $24 : 36$.

Key Concept: Assertion-Reason

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To simplify the ratio $12 : 18$, we find the HCF of 12 and 18, which is 6. Dividing both terms by 6 gives $\frac{12}{6} : \frac{18}{6} = 2 : 3$. Therefore, the Assertion is true. The Reason states that the HCF of 12 and 18 is 6, which is also true and correctly explains why the ratio simplifies to $2 : 3$.

Key Concept: Proportional Ratios

a) $15 : 30$
[Solution Description]
First, simplify the given ratio $5 : 10$. The HCF of 5 and 10 is 5. Dividing both terms by 5:
$5 \div 5 = 1 \\ \\ 10 \div 5 = 2$
The simplified ratio is $1 : 2$. Now, check each option:
\begin{itemize}

Key Concept: Proportional ratios, Cross-multiplication

a) $10 : 16$
[Solution Description]
To check proportionality, we can cross-multiply or simplify the ratios. For two ratios to be proportional, their simplest forms must be equal. The simplest form of $5 : 8$ is already $5 : 8$. Now, let's check each option:
- $10 : 16$ simplifies to $5 : 8$ (HCF = 2).
Thus, $10 : 16$ is proportional to $5 : 8$.

Key Concept: Simplest Form Calculation

c) $2 : 3$
[Solution Description]
Find the HCF of 72 and 108, which is 36. Divide both terms by 36:
$72 \div 36 = 2 \\ \\ 108 \div 36 = 3$
The simplified ratio is $2 : 3$.

Key Concept: Assertion and Reason

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To check if the ratios $6 : 4$ and $9 : 6$ are proportional, we reduce them to their simplest forms.
1. For $6 : 4$, the HCF of 6 and 4 is 2. Dividing both terms by 2 gives $3 : 2$.
2. For $9 : 6$, the HCF of 9 and 6 is 3. Dividing both terms by 3 also gives $3 : 2$.
Since both ratios simplify to the same form $3 : 2$, they are proportional. Thus, both Assertion and Reason are true, and the Reason correctly explains the Assertion.

Key Concept: Missing Term in Proportions

b) 28
[Solution Description]
Since the ratios are proportional, the product of means equals the product of extremes:
$7 \times 48 = 12 \times x \implies 336 = 12x \implies x = \frac{336}{12} = 28$
So, $x = 28$.

Key Concept: Proportional Ratios

c) 15 : 24
[Solution Description]
A ratio proportional to $5 : 8$ must simplify to the same value. Check each option by dividing both terms by their HCF.
For $15 : 24$, the HCF is 3:
$\frac{15}{3} = 5 \quad \text{and} \quad \frac{24}{3} = 8$
Hence, $15 : 24$ simplifies to $5 : 8$ and is proportional.

Key Concept: Simplifying Ratios

b) $2 : 3$
[Solution Description]
To simplify the ratio $24 : 36$, we need to divide both terms by their HCF. The HCF of 24 and 36 is 12. Dividing both terms by 12 gives $\frac{24}{12} : \frac{36}{12} = 2 : 3$.

Key Concept: Proportional Reasoning

c) 40
[Solution Description]
Using cross-multiplication for proportion $5:8 :: 25:x$,
$5 \times x = 8 \times 25$.
Solving for $x$:
$5x = 200$
$x = 200 ÷ 5 = 40$.

Key Concept: Simplification and Proportionality

c) The simplified form is $2:3$ and proportional to $2:3$
[Solution Description]
To simplify $24:36$, divide both terms by their HCF (12).
$24 ÷ 12 = 2$ and $36 ÷ 12 = 3$.
So, the simplified form is $2:3$.
Since the given ratio is also $2:3$, they are proportional.

Key Concept: Cross-Multiplication Method, Simplest Form

c) 21
[Solution Description]
Using the cross-multiplication method for proportion $a : b :: c : d$, we have $ad = bc$.
Here, $a = 5$, $b = 7$, $c = 15$.
So, $5 × x = 7 × 15$
$5x = 105$
$x = 105 / 5 = 21$.
Thus, the missing term $x$ is 21.

Key Concept: Concept of proportionality using simplest forms

c) Yes, because both speeds are equal
[Solution Description]
To check if the speed ratios are proportional, first find the speed of each car. Speed is calculated as distance divided by time.
First car's speed = $240 \text{ km} / 4 \text{ h} = 60 \text{ km/h}$.
Second car's speed = $360 \text{ km} / 6 \text{ h} = 60 \text{ km/h}$.
The ratio of speeds for both cars is $60 : 60$, which simplifies to $1 : 1$. Since both ratios are equal, they are proportional.

Key Concept: Ratio division, proportional scaling

c) 5:6:2
[Solution Description]
1kg = 1000g original mixture.
Total parts = 5+3+2 = 10 parts.
Original amounts: A = (5/10)×1000 = 500g, B = 300g, C = 200g.
After adding 300g B: A=500g, B=600g, C=200g.
New ratio = 500:600:200 = simplify by dividing by 100 → 5:6:2.

Key Concept: Compound ratios, proportional reasoning

c) 1:1
[Solution Description]
Calculate worker-days for each wall:
Wall X: 4 workers × 15 days = 60 worker-days
Wall Y: 6 workers × 10 days = 60 worker-days
Since both walls require same worker-days, their efficiencies are equal.
Ratio is therefore 1:1.

Key Concept: Ratio Application

b) 6 cups
[Solution Description]
The ratio of flour to water is 4:6 or simplified as 2:3. To find the required flour for 9 cups of water:
$\frac{2}{3} = \frac{x}{9} \\ \\ 3x = 18 \\ \\ x = 6$
You need 6 cups of flour.

Key Concept: Proportional Reasoning

c) 600 km
[Solution Description]
To find the distance traveled in 10 hours at the same speed:
$\text{Speed} = \frac{360 \, \text{km}}{6 \, \text{hours}} = 60 \, \text{km/h} \\ \\ \text{Distance} = 60 \, \text{km/h} \times 10 \, \text{hours} = 600 \, \text{km}$
The car will travel 600 km.

Key Concept: Assertion-Reason

c) Assertion is true, but Reason is false.
[Solution Description]
Let the initial quantity of juice be $3x$ litres and water be $5x$ litres. After adding 2 litres of juice, the new quantity of juice becomes $3x + 2$ litres. The new ratio is given as $1 : 1$, so we can write:
$\frac{3x + 2}{5x} = \frac{1}{1}$
Solving this equation:
$3x + 2 = 5x \\ \\ 2 = 2x \\ \\ x = 1$
Therefore, the initial quantities were 3 litres of juice and 5 litres of water. The assertion is true. However, the reason states that adding juice changes the ratio proportionally, which is incorrect because proportional change would require multiplying both terms by the same factor, not adding. Thus, the reason is false.

Key Concept: Ratio Comparison, Proportional Reasoning

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To check proportionality, we compare the given ratio $5 : 170$ with the new ratio $8 : 272$. Calculating the cross-products:
$5 \times 272 = 1360$ and $170 \times 8 = 1360$. Since both cross-products are equal, the ratios are proportional. Both Assertion and Reason are true, and Reason correctly explains Assertion.

Key Concept: Ratio Application

b) 100 mL
[Solution Description]
The ratio $1 : 4$ means there is 1 part acid and 4 parts water, totaling 5 parts. The fraction of acid is $\frac{1}{5}$. For 500 mL, the amount of acid is $\frac{1}{5} \times 500 = 100$ mL.

Key Concept: Ratio application, Cost calculation

d) \$880
[Solution Description]
Total parts = $2 + 7 = 9$.
Cement quantity = $(2/9) \times 180 = 40$ kg.
Sand quantity = $(7/9) \times 180 = 140$ kg.
Cement cost = $40 \times 15 = \$600$.
Sand cost = $140 \times 2 = \$280$.
Total cost = $600 + 280 = \$880$.

Key Concept: Identifying proportional relationships, Simplifying ratios

b) 300 girls
[Solution Description]
The given ratio of boys to girls is $7 : 5$. Let the number of girls be $x$. Set up the proportion: $\frac{7}{5} = \frac{420}{x}$. Cross-multiply to solve for $x$: $7x = 5 \times 420$, which simplifies to $7x = 2100$. Divide both sides by 7: $x = 300$.

Key Concept: Identifying proportional relationships, Rule of three

c) 360 widgets
[Solution Description]
First, find the production rate per hour by dividing the number of widgets by the time taken: $\frac{150 \text{ widgets}}{5 \text{ hours}} = 30$ widgets/hour. Then, multiply this rate by the new time to find the total widgets produced: $30 \text{ widgets/hour} \times 12 \text{ hours} = 360$ widgets.

Key Concept: Simplification of Ratios

b) $3 : 4$
[Solution Description]
To simplify $18 : 24$, divide both numbers by their highest common factor (HCF). The HCF of 18 and 24 is 6. Dividing both terms by 6 gives: $18 ÷ 6 = 3$ and $24 ÷ 6 = 4$. Thus, the simplified form of the ratio is $3 : 4$.

Key Concept: Proportional Reasoning Application

c) \$36
[Solution Description]
First, find the cost per pen by dividing the total cost by the number of pens: $\frac{\$15}{5} = \$3$ per pen. Then, multiply the cost per pen by the number of pens needed: $\$3 \times 12 = \$36$. Therefore, 12 pens cost \$36.

Key Concept: Proportional Reasoning, Ratio Simplification

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 correct:
1. Simplify the first ratio $24 : 36$ by dividing both terms by their HCF (12): $24 \div 12 = 2$, $36 \div 12 = 3$. Simplified form is $2 : 3$.
2. Simplify the second ratio $10 : 15$ by dividing both terms by their HCF (5): $10 \div 5 = 2$, $15 \div 5 = 3$. Simplified form is $2 : 3$.
3. Since both simplified ratios are equal ($2 : 3$), they are proportional. Thus, Assertion is true.
4. The Reason states that both ratios simplify to $2 : 3$, which matches our calculations. Hence, Reason is true and correctly explains the Assertion.
Therefore, the correct answer is option (a).

Key Concept: Ratio Application

c) 27
[Solution Description]
Given the ratio of red to blue marbles is $5 : 3$.
Let the number of blue marbles be $x$.
The proportion can be written as $5 : 3 :: 45 : x$.
Cross-multiplying: $5x = 135$.
Solving for $x$, we get $x = 27$.

Key Concept: Ratio simplification, proportional reasoning

a) $9:12$
[Solution Description]
To find the proportional ratio, first simplify $45:60$ by dividing both terms by their HCF (15):
$\frac{45}{15} : \frac{60}{15} = 3 : 4$
Now check which option simplifies to $3:4$.

Key Concept: Problem Solving with Ratios

c) 10 cups
[Solution Description]
The ratio $3 : 5$ means for every 3 cups of sugar, 5 cups of flour are needed. To find the required flour for 6 cups of sugar, set up the proportion $3 : 5 :: 6 : x$.
The factor of change is $6 ÷ 3 = 2$. Multiply the second term by this factor:
$5 × 2 = 10$
So, 10 cups of flour are needed.

Key Concept: Proportional reasoning with unit conversion

c) 16 machines
[Solution Description]
First, find the rate per machine: $\frac{1200 \text{ pages}}{40 \text{ min} \times 8 \text{ machines}} = 3.75$ pages per machine per minute.
For 4500 pages in 50 minutes, total required machines = $\frac{4500}{50 \times 3.75} = 24$ machines.
Additional machines needed = $24 - 8 = 16$.

Key Concept: Trairasika — The Rule of Three

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that for proportional ratios $a : b :: c : d$, the product of the means ($b$ and $c$) equals the product of the extremes ($a$ and $d$), i.e., $ad = bc$. This is a fundamental property of proportional ratios. The reason explains that cross multiplication ($ad = bc$) is a method to verify proportionality between two ratios. Since the reason correctly explains why the assertion holds true, both are correct and logically connected.

Key Concept: Trairasika — The Rule of Three, Cross Multiplication

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that if $a : b :: c : d$ is a proportional relationship, then $ad = bc$ must hold true. This is correct because, from the definition of proportionality, there exists a common factor $f$ such that $c = f \cdot a$ and $d = f \cdot b$. Substituting these into $ad = bc$, we get $a(f \cdot b) = b(f \cdot a)$, which simplifies to $abf = abf$, proving the assertion is true.
The reason states that the cross-multiplication rule ensures the proportionality factor $f$ is consistent for all terms. This is also true because the cross-multiplication rule directly follows from setting the two expressions for $f$ equal to each other: $\frac{c}{a} = \frac{d}{b}$. Therefore, the reason correctly explains the assertion.

Key Concept: Cross Multiplication

b) 112.5 km
[Solution Description]
First, convert all time units to the same measure (minutes). 1 hour and 15 minutes is equal to 75 minutes.
Given the ratio of distance to time is proportional:
$30 : 45 :: 75 : x$.
Using cross multiplication:
$30 \times x = 45 \times 75$
$x = \frac{45 \times 75}{30}$
$x = \frac{3375}{30} = 112.5$
So, the car will cover 112.5 km in 1 hour and 15 minutes.

Key Concept: Cross Multiplication Method

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that the cross multiplication method is used to find the fourth proportional in a proportion, which is true. The reason given is that in the proportion $a : b :: c : d$, the product of the extremes ($a$ and $d$) equals the product of the means ($b$ and $c$). This is also true, as it defines the cross multiplication rule $ad = bc$. Moreover, the reason correctly explains why the assertion is valid since the cross multiplication method is derived from this property. Therefore, both the Assertion and Reason are true, and the Reason is the correct explanation of the Assertion.

Key Concept: Rule of Three, Cross Multiplication

d) Assertion is false, but Reason is true.
[Solution Description]
The cross multiplication method does not require ratios to be in their simplest form to be applied correctly. The proportionality condition $ad = bc$ holds true irrespective of whether the ratios $a:b$ and $c:d$ are simplified or not. For example, consider the unsimplified ratio $4:6 :: 6:9$. Cross multiplying gives $4 \times 9 = 6 \times 6$, which simplifies to $36 = 36$, proving proportionality. Thus, the Assertion is false. However, the Reason states that simplifying ratios eliminates common factors, which is a correct statement but not relevant to the applicability of the cross multiplication method. Therefore, the Reason is true independently.

Key Concept: Proportionality Check

c) $4 \times 14 = 7 \times 8$
[Solution Description]
To verify proportionality, we use the cross multiplication rule:
$4 \times 14 = 7 \times 8$
Calculating both sides:
$56 = 56$
Since both sides are equal, the ratios are proportional.

Key Concept: Cross Multiplication Basics

c) 15
[Solution Description]
We are given the proportion $3 : 5 :: 9 : x$. Using the cross multiplication rule, we have:
$3 \times x = 5 \times 9$
Solving for $x$:
$3x = 45 \implies x = \frac{45}{3} = 15$
Therefore, the value of $x$ is 15.

Key Concept: Proportional Ratios

c) 5 hours
[Solution Description]
Here, speed is constant, so distance and time are directly proportional.
Given ratio:
$120 \text{ km} : 3 \text{ hours} :: 200 \text{ km} : x \text{ hours}$.
Using the rule of three:
$\frac{120}{3} = \frac{200}{x}$.
Simplifying left side: $40 = \frac{200}{x}$.
Solving for $x$:
$x = 200 / 40 = 5$.
Hence, the car will take 5 hours.

Key Concept: Proportional Reasoning, Rule of Three

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To solve the problem using the Rule of Three, we start with the given proportional ratio:
$12 : 15 :: 48 : d$
By the definition of proportional ratios, the product of the means ($15 \times 48$) must equal the product of the extremes ($12 \times d$). This gives us the equation:
$15 \times 48 = 12 \times d$
Solving for $d$:
$d = \frac{15 \times 48}{12} = 60$
The Assertion correctly states the formula to find $d$ using the Rule of Three. The Reason explains the underlying principle of proportionality that justifies this formula. Therefore, both Assertion and Reason are true, and Reason correctly explains the Assertion.

Key Concept: Proportional Reasoning, Real-World Application

c) 6
[Solution Description]
First find the work rate per machine:
12,000 newspapers / (6 machines × 18 hours) = 12000/108 ≈ 111.11 newspapers per machine-hour
Now calculate required machines for new job:
Let x = required machines
$x \times 15 \times 111.11 = 20000$
$x = 20000 / (15 \times 111.11) ≈ 12$ machines
Additional machines needed = 12 - 6 = 6

Key Concept: Proportional Ratios

b) 10
[Solution Description]
Given the proportion $4 : 5 :: 8 : x$, we can use cross multiplication to solve for $x$.
Step 1: Write the proportion as an equation:
$4 \times x = 5 \times 8$
Step 2: Multiply the known terms:
$4x = 40$
Step 3: Solve for $x$ by dividing both sides by 4:
$x = \frac{40}{4} = 10$
Therefore, the value of $x$ is 10.

Key Concept: Rule of Three

c) 4 hours
[Solution Description]
Since both workers have the same efficiency, working together means their rates add up. The first worker's rate is $\frac{1}{8}$ tasks/hour. Combined rate is $\frac{1}{8} + \frac{1}{8} = \frac{2}{8} = \frac{1}{4}$ tasks/hour. Thus, time taken together is $\frac{1}{\frac{1}{4}} = 4$ hours.

Key Concept: Rule of Three - Basic Application

c) \$32
[Solution Description]
Given: Pramāṇa = 5 m (a), Phala = \$20 (b), Ichchhā = 8 m (c). Using Āryabhaṭa's Rule of Three:
$d = \frac{b \times c}{a} = \frac{20 \times 8}{5} = 32$
The cost of 8 meters is \$32.

Key Concept: Rule of Three, Proportionality

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that doubling the pramāṇa halves the ichchhāphala when the phala and ichchhā are unchanged. The formula provided in the reason confirms this relationship as $ichchhāphala$ is inversely proportional to $pramāṇa$. Hence, both the assertion and reason are true, and the reason correctly explains the assertion.

Key Concept: Rule of Three, Unit Conversion

b) 112.5 palas
[Solution Description]
Given: Pramāṇa = 5 palas, Phala = $\frac{2}{3}$ niskas, Ichchhā = 15 niskas. Using Āryabhaṭa's Rule of Three:
$ichchhāphala = \frac{phala \times ichchhā}{pramāṇa} = \frac{\frac{2}{3} \times 15}{5} = \frac{10}{5} = 2$
However, the calculation is incorrect as we need to find the quantity of rice (palas) for 15 niskas. Let’s re-express the proportion correctly: $5 : \frac{2}{3} :: x : 15$. Cross-multiplying gives $\frac{2}{3}x = 5 \times 15$, so $x = \frac{75 \times 3}{2} = 112.5$ palas.

Key Concept: Combined ratio, multi-step problem

b) Rs.19,286
[Solution Description]
First, combine the ratios: A:B = 2:3 and B:C = 4:5. To make B common, multiply A:B by 4 and B:C by 3 ⇒ A:B:C = 8:12:15.
Total parts = 8 + 12 + 15 = 35.
Value per part = Rs.45,000 / 35 ≈ Rs.1,285.71.
C's share = 15 × Rs.1,285.71 ≈ Rs.19,285.71.
Closest option is Rs.19,286.

Key Concept: Sharing profit based on investment ratio

b) \$2500 and \$1500
[Solution Description]
Investment ratio = 50000 : 30000 = 5:3
Total parts = 5 + 3 = 8
Size of each part = $4000 \div 8 = 500$
First partner's profit = $5 \times 500 = \$2500$
Second partner's profit = $3 \times 500 = \$1500$.

Key Concept: Ratio adjustment, mixture problem

d) 12 kg
[Solution Description]
Current ratio = 4:1, so sugar = (4/5)×60 = 48 kg, flour = (1/5)×60 = 12 kg.
Let $x$ be the sugar added. New ratio: (48 + $x$)/12 = 5/1.
Solving: 48 + $x$ = 60 → $x$ = 12 kg.

Key Concept: Profit Sharing

a) Rs.4500 and Rs.1500
[Solution Description]
First, simplify the investment ratio: Rs.45,000 : Rs.15,000 simplifies to $3 : 1$. Total parts = $3 + 1 = 4$. Size of each part = $\frac{6000}{4} = 1500$. Ramesh's share = $3 \times 1500 = 4500$, Suresh's share = $1 \times 1500 = 1500$.

Key Concept: Dividing quantities in a given ratio

c) 20 liters
[Solution Description]
Given ratio $2 : 3$, total parts = $2 + 3 = 5$.
$\text{Quantity of blue paint} = \frac{2}{5} \times 50 = 20 \text{ liters}$
The quantity of blue paint needed is 20 liters.

Key Concept: Dividing quantities in a given ratio, real-world application

d) \$7,500
[Solution Description]
First, express the working hours as a ratio: 6:9:15.
Simplify the ratio by dividing each term by 3:
Simplified ratio = 2:3:5.
Total parts = 2 + 3 + 5 = 10 parts.
Value per part = $\frac{15000}{10} = 1500$.
Employee R's share = $5 \times 1500 = 7500$.
Therefore, employee R receives \$7,500.

Key Concept: Dividing quantities in a given ratio

c) 40
[Solution Description]
Let the number of \$1 coins = 5k, \$2 coins = 3k.
1. Total value = $(5k \times \$1) + (3k \times \$2) = 5k + 6k = 11k = \$88$.
2. Solving, $k = 8$.
3. Number of \$1 coins = $5k = 5 \times 8 = 40$.
Therefore, there are 40 \$1 coins.

Key Concept: Dividing quantities in a given ratio

c) \$500 and \$750
[Solution Description]
To divide \$1,250 in the ratio 4:6:
1. Total parts = 4 + 6 = 10.
2. Value of each part = \$1,250 / 10 = \$125.
3. First friend's share = 4 $\times$ \$125 = \$500.
4. Second friend's share = 6 $\times$ \$125 = \$750.
So, the shares are \$500 and \$750.

Key Concept: Sharing Profit Based on Investment Ratio

c) 2000 and 3000
[Solution Description]
The investment ratio is $8000 : 12000$, simplified to $2 : 3$.
Total parts = $2 + 3 = 5$.
Size of each part = $\frac{5000}{5} = 1000$ dollars.
First partner's share = $2 \times 1000 = 2000$ dollars.
Second partner's share = $3 \times 1000 = 3000$ dollars.

Key Concept: Finding parts from the whole using ratio

b) 16 and 20
[Solution Description]
The total ratio parts = 4 + 5 = 9. Size of each part = $36 ÷ 9 = 4$. First friend gets $4 × 4 = 16$ chocolates. Second friend gets $5 × 4 = 20$ chocolates.

Key Concept: Ratio mixture adjustment

c) 80 mL
[Solution Description]
Initially, alcohol = $\frac{4}{9} \times 360 = 160$ mL, water = $\frac{5}{9} \times 360 = 200$ mL. For new ratio 4:7, required water = $\frac{7}{4} \times 160 = 280$ mL. Water to add = $280 - 200 = 80$ mL.

Key Concept: Adjusting Mixture Ratio

c) 45 liters
[Solution Description]
Current ratio is 4:5, total parts = $4 + 5 = 9$.
Water = $\frac{4}{9} \times 180 = 80$ liters.
Milk = $\frac{5}{9} \times 180 = 100$ liters.
Let $x$ be the extra water added for new ratio 5:4.
New equation: $\frac{80 + x}{100} = \frac{5}{4}$.
Solving, $4(80 + x) = 500 \implies 320 + 4x = 500 \implies 4x = 180 \implies x = 45$ liters.

Key Concept: Profit sharing, Ratio division

c) Rs.15,000 and Rs.9,000
[Solution Description]
The investment ratio is 5:3. Total parts = 5 + 3 = 8.
Each part of profit = $\frac{24000}{8} = Rs.3,000$.
Prashanti's share = $5 \times 3000 = Rs.15,000$.
Bhuvan's share = $3 \times 3000 = Rs.9,000$.

Key Concept: Profit Sharing

c) Rs.3,000
[Solution Description]
The investment ratio of Prashanti to Bhuvan is $90000 : 30000 = 3 : 1$. Total parts = $3 + 1 = 4$. Each part = $\frac{12000}{4} = 3000$. Bhuvan's share = $1 \times 3000 = Rs.3000$.

Key Concept: Mixture Problem

d) 20 kg
[Solution Description]
Given ratio = 4:1, total parts = 4 + 1 = 5.
Sugar quantity = $\frac{4}{5} \times 25 = 20$ kg.

Key Concept: Profit Sharing in Business

d) Assertion is false, but Reason is true.
[Solution Description]
First, find the ratio of investments: Ram : Shyam = 20,000 : 30,000 = 2:3.
Total parts = 2 + 3 = 5.
Each part value = $\frac{5000}{5} = 1000$.
Ram's share = $2 \times 1000 = 2000$.
Thus, Assertion (A) is false as Ram's actual share is \$2,000, not \$3,000.
Reason (R) is true as it correctly describes the method of calculating profit share based on investment ratio.

Key Concept: Volume Conversion

c) 4{,}500 cc
[Solution Description]
Given $1 \, \text{litre} = 1{,}000 \, \text{mL}$ and $1 \, \text{mL} = 1 \, \text{cc}$, we can directly convert litres to cubic centimetres:
$4.5 \, \text{litres} = 4.5 \times 1{,}000 = 4{,}500 \, \text{cc}$
The tank contains 4{,}500 cubic centimetres of water.

Key Concept: Unit Conversion - Volume

b) 18.5 kg
[Solution Description]
The mass ratio of gold to water is $37 : 2$. Since 1 litre of water weighs 1 kg, we can find the mass of 1 litre of gold using the ratio:
$\text{Mass of gold} = \left( \frac{37}{2} \right) \times 1 \text{ kg} = 18.5 \text{ kg}$
Thus, the mass of 1 litre of gold is 18.5 kg.

Key Concept: Temperature Conversion

c) $35^\circ C$
[Solution Description]
Use the formula for Fahrenheit to Celsius conversion:
$\text{Celsius} = \frac{5}{9} \times (\text{Fahrenheit} - 32)$
Substitute $95^\circ F$:
$\text{Celsius} = \frac{5}{9} \times (95 - 32) = \frac{5}{9} \times 63 = 35^\circ C$
The temperature is $35^\circ C$.

Key Concept: Unit cost calculation, Area conversion

c) \$1,721,649
[Solution Description]
First, find the price per acre: $\frac{\$250,000}{0.75} \approx \$333,333.33 \text{ per acre}$.
Calculate the area of the new plot in square feet: $500 \text{ ft} \times 450 \text{ ft} = 225,000 \text{ sq ft}$.
Convert this area to acres: $\frac{225,000}{43,560} \approx 5.165 \text{ acres}$.
Multiply by the cost per acre: $5.165 \times \$333,333.33 \approx \$1,721,649.48$.

Key Concept:

b) 2.471 acres
[Solution Description]
Given, $1 \text{ hectare} = 2.471 \text{ acres}$. So, the plot is equal to 2.471 acres.

Key Concept: Length, Area

b) 16146 square feet
[Solution Description]
First, calculate the area in square meters:
$Area = Length \times Width = 50 m \times 30 m = 1500 \text{ m}^2$.
Now, convert square meters to square feet using the conversion factor:
$1 \text{ m}^2 = 10.764 \text{ ft}^2$
So, $1500 \text{ m}^2 \times 10.764 \text{ ft}^2/\text{m}^2 = 16146 \text{ ft}^2$.

Key Concept:

b) $20^\circ C$
[Solution Description]
Using the formula:
$Celsius = \frac{5}{9} \times (68 - 32) = \frac{5}{9} \times 36 = 20^\circ C$
So, $68^\circ F$ is equal to $20^\circ C$.

Key Concept: Volume Conversion

d) 3,000 mL
[Solution Description]
Given that $1 \text{ litre} = 1,000 \text{ mL}$, multiply the number of litres by 1,000 to get the volume in millilitres:
$3 \text{ litres} \times 1,000 \text{ mL/litre} = 3,000 \text{ mL}$
Hence, 3 litres is equal to 3,000 mL.

Key Concept: Mass Ratio, Unit Consistency

c) 17.5 kg
[Solution Description]
Given the mass ratio of gold to water as $37 : 2$, and the mass of $1$ litre of water is $1$ kg. Let the mass of $1$ litre of gold be $m$.
From the ratio:
$\frac{m}{1 \text{ kg}} = \frac{37}{2} \Rightarrow m = \frac{37}{2} \text{ kg} = 18.5 \text{ kg}$
The difference in mass between gold and water is:
$18.5 \text{ kg} - 1 \text{ kg} = 17.5 \text{ kg}$
So, the difference is $17.5$ kg.

Key Concept: Time and Volume Conversion

c) 300 seconds
[Solution Description]
First, convert 10 litres to millilitres: $10\,\text{L} = 10,000\,\text{mL}$. Using the given rate, the time taken is calculated as:
$\text{Time taken} = \frac{15 \times 10,000}{500} = 300\,\text{seconds}$

Key Concept: Unit Conversions, Proportional Reasoning

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
First, convert 7.5 hours to minutes: $7.5 \text{ hours} \times 60 \text{ minutes/hour} = 450 \text{ minutes}$.
Calculate the rate of filling: $50 \text{ gallons} / 10 \text{ minutes} = 5 \text{ gallons/minute}$.
Time to fill 2250 gallons: $2250 \text{ gallons} / 5 \text{ gallons/minute} = 450 \text{ minutes} = 7.5 \text{ hours}$.
Thus, both Assertion and Reason are true, and the Reason correctly explains the Assertion.

Key Concept: Unit Conversion - Proportionality

b) \$40,173
[Solution Description]
First, find the cost per square foot:
$\frac{\$50,000}{43,560 \text{ sq ft}} \approx \$1.1478 \text{ per sq ft}$
Multiply by the area to get the total cost:
$35,000 \text{ sq ft} \times \$1.1478/\text{sq ft} \approx \$40,173$
Rounding to the nearest dollar gives \$40,173.