Key Concept: Ratio Simplification, Proportional Reasoning

c) 3:2
[Solution Description]
To simplify the ratio 9:6, we divide both numbers by their greatest common divisor (GCD).
1. GCD of 9 and 6 is 3.
2. Divide both terms by 3: $\frac{9}{3} : \frac{6}{3} = 3:2$.
The simplified ratio is 3:2.

Key Concept: Proportional Reasoning

d) The image remains similar to the original.
[Solution Description]
When both the width and height of an image are scaled by the same multiplicative factor, the proportions remain unchanged, keeping the image visually similar to the original. This is known as proportional reasoning.
For example, if a rectangle has dimensions 60 mm (width) and 40 mm (height), scaling both by $\frac{1}{2}$ results in new dimensions of 30 mm and 20 mm respectively. The aspect ratio remains the same ($\frac{60}{40} = \frac{30}{20} = 1.5$).
Hence, the image appears similar when width and height change proportionally.

Key Concept: Proportional Scaling and Visual Similarity

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
Let us analyze the assertion and reason step-by-step:
- **Assertion**: If Image X is scaled down such that both width and height are multiplied by the same factor (2/3), the resulting image will maintain its visual similarity because the proportions (width:height ratio) remain unchanged. This matches the key idea from the syllabus that proportional change preserves similarity.
- **Reason**: The reasoning correctly explains that keeping the scaling factor identical for both dimensions maintains the aspect ratio. This aligns with the conclusion in the syllabus that proportional changes prevent distortion.
Thus, both Assertion and Reason are true, and Reason correctly justifies the Assertion.

Key Concept: Non-Proportional Change

a) The image becomes wider and shorter
[Solution Description]
Non-proportional changes distort the image.
Let the original width $W$ and height $H$.
New width = $W + 0.5W = 1.5W$.
New height = $H - 0.25H = 0.75H$.
Since scaling factors differ ($1.5 \neq 0.75$), the image appears stretched horizontally and compressed vertically.

Key Concept: Verification of Similarity

b) $2 : 1$
[Solution Description]
The ratio of width to height for Image B is $40 : 20$. Dividing both terms by their HCF (20), we get:
$\frac{40}{20} : \frac{20}{20} = 2 : 1$
The simplest form of the ratio for Image B is $2 : 1$.

Key Concept: Width-height ratio, Simplest form

a) 72:48
[Solution Description]
First, we simplify the given ratio 48:32 by dividing both terms by their HCF 16 to get 3:2. Now we check which option has the same simplest form.
Option a) 72:48 simplifies to 3:2 (dividing by 24)
Option b) 64:32 simplifies to 2:1 (dividing by 32)
Option c) 90:45 simplifies to 2:1 (dividing by 45)
Option d) 120:100 simplifies to 6:5 (dividing by 20)
Only option a) has the same simplest form as the given image.

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 scaling both width and height by the same multiplicative factor (0.8) preserves similarity. According to the syllabus, images remain similar only if their dimensions change proportionally (same multiplicative factor). Thus, the assertion is true. The reason correctly explains why multiplicative scaling preserves similarity, as it ensures proportional changes in dimensions. Therefore, both assertion and reason are true, and the reason correctly explains the assertion.

Key Concept: Proportional Scaling

b) The image looks smaller or larger but maintains its shape.
[Solution Description]
Proportional scaling means both the width and height of the image are multiplied by the same factor. This preserves the shape of the image, making it look similar but different in size.

Key Concept: Ratios, Proportionality

b) 6 : 9
[Solution Description]
First, simplify the given ratio $8 : 12$ by dividing both terms by their HCF, which is 4:
$8 : 12 = 2 : 3$
Now, check each option to see if it simplifies to $2 : 3$:
\begin{itemize}

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: 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: Ratio Application

b) 25
[Solution Description]
Given the ratio $3 : 5$ for apples to oranges, let the number of oranges be $x$. Using proportionality: $\frac{3}{5} = \frac{15}{x}$. Cross-multiply to get $3x = 75$. Solving, $x = 25$. Thus, there are 25 oranges.

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: Proportional Ratios

a) $8 : 12$
[Solution Description]
To determine which ratio is proportional to $4 : 6$, first simplify $4 : 6$ to its simplest form. The HCF of 4 and 6 is 2. Dividing both terms by 2 gives:
$\frac{4}{2} : \frac{6}{2} = 2 : 3$
Now, check each option by simplifying it to see if it matches $2 : 3$.
Option a) simplifies to $8 : 12$ → $\frac{8}{4} : \frac{12}{4} = 2 : 3$. This matches the simplified form of $4 : 6$.
The other options do not simplify to $2 : 3$.

Key Concept: Simplest Form of Ratios

b) $3 : 2$
[Solution Description]
To find the simplest form of the ratio $90 : 60$, divide both terms by their Highest Common Factor (HCF). The HCF of 90 and 60 is 30. Dividing both terms by 30 gives:
$\frac{90}{30} : \frac{60}{30} = 3 : 2$
Thus, the simplest form of the ratio $90 : 60$ is $3 : 2$.

Key Concept: Definition and Simplest Form of Ratios, Proportionality

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
First, simplify the ratio $12:18$ by dividing both terms by their HCF (6): $\frac{12}{6} : \frac{18}{6} = 2:3$. Similarly, simplify $20:30$ by dividing both terms by their HCF (10): $\frac{20}{10} : \frac{30}{10} = 2:3$. Since both simplified forms are identical, the ratios are proportional. The assertion is true. The reason states that two ratios are proportional if their simplest forms are identical, which is correct and directly explains the assertion.

Key Concept: Maintaining Proportions, Problem Solving

b) 25 cups
[Solution Description] The given ratio is $5:7$ (flour to water). To find the required flour for 35 cups of water, set up the proportion $5:7 :: x:35$. The factor of change is $\frac{35}{7} = 5$. Multiply the flour quantity by this factor: $5 \times 5 = 25$ cups of flour.

Key Concept: Proportional Ratios

d) $25 : 18$
[Solution Description]
To determine if ratios are proportional, we simplify them to their simplest form. The given ratio is $5:3$. For another ratio to be proportional, it must simplify to the same ratio ($5:3$) when reduced.
Let's check each option:
\begin{itemize}

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: Simplifying Ratios

b) $2 : 3$
[Solution Description]
To simplify the ratio $30 : 45$, find the Highest Common Factor (HCF) of 30 and 45, which is 15. Divide both terms by 15: $30 \div 15 = 2$ and $45 \div 15 = 3$. The simplified ratio is $2 : 3$.

Key Concept: Proportional Ratios

d) $5 : 7$
[Solution Description]
First, simplify $8 : 12$ by dividing both terms by their HCF (which is 4), giving $2 : 3$. Now check each option:
\begin{itemize}
- Option a) $4 : 6$ simplifies to $2 : 3$ (proportional).
- Option b) $10 : 15$ simplifies to $2 : 3$ (proportional).
- Option c) $6 : 9$ simplifies to $2 : 3$ (proportional).
- Option d) $5 : 7$ does not simplify to $2 : 3$ (not proportional).
\end{itemize}
All except option d) are proportional to $8 : 12$.

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: Dividing a quantity in a given ratio, Proportional reasoning

a) Rs.4,000 and Rs.5,600
[Solution Description]
Total parts = $5 + 7 = 12$. First part = $\frac{5}{12} \times 9600 = 4000$. Second part = $\frac{7}{12} \times 9600 = 5600$. Thus, the amounts are Rs.4,000 and Rs.5,600.

Key Concept: Dividing quantities in a ratio

c) Rs.1,600 and Rs.2,000
[Solution Description]
To divide Rs.3,600 in the ratio $4 : 5$, first add the parts of the ratio: $4 + 5 = 9$. Then calculate each part:
$\text{First part} = \frac{4}{9} \times 3600 = Rs.1,600 \\ \\ \text{Second part} = \frac{5}{9} \times 3600 = Rs.2,000$
So, the amounts are Rs.1,600 and Rs.2,000.

Key Concept: Simplifying Ratios

b) $3 : 5$
[Solution Description]
To simplify the ratio $45 : 75$, find the HCF of 45 and 75, which is 15. Divide both terms by 15 to get the simplest form:
$\frac{45}{15} : \frac{75}{15} = 3 : 5$
So, the simplest form of the ratio is $3 : 5$.

Key Concept: Ratios in Their Simplest Form, Proportional Ratios

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To determine whether the assertion and reason are correct:
1. Simplify the ratio $14 : 21$ by dividing both terms by their HCF, which is 7. This gives the simplified form as $2 : 3$.
2. Similarly, simplify the ratio $6 : 9$ by dividing both terms by their HCF, which is 3. This also gives the simplified form as $2 : 3$.
3. Since both simplified forms are equal ($2 : 3$), the ratios $14 : 21$ and $6 : 9$ are indeed proportional.
4. The Reason states that two ratios are proportional if their terms change by the same multiplicative factor. This is true, as seen here where both $14 : 21$ and $6 : 9$ reduce to the same simplest form. Hence, the Reason correctly explains the Assertion.
Therefore, both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

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: 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: 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: Simplifying Ratios Using HCF

a) $2 : 3$
[Solution Description]
To simplify the ratio $84 : 126$, first find the HCF of 84 and 126.
Step 1: Find the factors of 84. The factors are 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84.
Step 2: Find the factors of 126. The factors are 1, 2, 3, 6, 7, 9, 14, 18, 21, 42, 63, 126.
Step 3: Identify the highest common factor (HCF). The common factors are 1, 2, 3, 6, 7, 14, 21, 42. The HCF is 42.
Step 4: Divide both terms by the HCF.
$84 \div 42 = 2$
$126 \div 42 = 3$
So, the simplified ratio is $2 : 3$.

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: Using HCF for Simplification

a) $2 : 3$
[Solution Description]
To simplify $120 : 180$, follow these steps:
Step 1: Find the HCF of 120 and 180.
The prime factorization of 120 is $2^3 \times 3 \times 5$.
The prime factorization of 180 is $2^2 \times 3^2 \times 5$.
The HCF is the product of the lowest powers of common primes: $2^2 \times 3 \times 5 = 60$.
Step 2: Divide both terms by the HCF.
$120 \div 60 = 2$
$180 \div 60 = 3$
The simplified ratio is $2 : 3$.

Key Concept: Simplifying Ratios

c) 2 : 3
[Solution Description]
To find the simplest form, divide both terms by their HCF. The HCF of 84 and 126 is 42. Dividing both terms gives:
$\frac{84}{42} = 2 \quad \text{and} \quad \frac{126}{42} = 3$
Thus, the simplest form is $2 : 3$.

Key Concept: Proportional Ratios

a) Yes, because both simplify to $1 : 2$
[Solution Description]
First, simplify both ratios. For $5 : 10$, the HCF is 5, so the simplified form is $1 : 2$. For $15 : 30$, the HCF is 15, so the simplified form is also $1 : 2$. Since both simplified ratios are equal, they are 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 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: Simplest Form of a Ratio

c) $2 : 3$
[Solution Description]
To find the simplest form, divide both terms by their HCF. The HCF of 60 and 90 is 30. Dividing both terms by 30: $\frac{60}{30} = 2$, $\frac{90}{30} = 3$. The simplest form is $2 : 3$.

Key Concept: Problem-Solving with Ratios

c) 6
[Solution Description]
The ratio of flour to eggs is $4:6$, which simplifies to $2:3$.
For 9 eggs, we set up the proportion $2:3 :: x:9$.
Cross-multiplying gives:
$3x = 18$
$x = 6$.
Hence, 6 cups of flour are needed.

Key Concept: Problem Solving with Proportionality

c) 30 teachers
[Solution Description]
Given the ratio $1 : 34$, for every 1 teacher there are 34 students. To find the number of teachers for 1020 students, solve $1/34 = x/1020$. Cross-multiplying: $34x = 1020$. Dividing both sides by 34: $x = 30$. So, there are 30 teachers.

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: Proportional Division

c) \$450 and \$750
[Solution Description]
To divide \$1,200 in the ratio 3:5:
$\text{Total parts} = 3 + 5 = 8 \\ \\ \text{First person's share} = \frac{3}{8} \times 1200 = 450 \\ \\ \text{Second person's share} = \frac{5}{8} \times 1200 = 750$
The shares are \$450 and \$750.

Key Concept: Proportional reasoning, unit conversion

c) 3 days
[Solution Description]
First find production rate: 1200 units/8 hours = 150 units/hour.
Daily production with 9 hours: 150 × 9 = 1350 units/day.
Days needed for 3150 units: 3150 ÷ 1350 = 2.33 days → 3 full days required (can't have partial day).

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: 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: Ratio Comparison

c) 20 spoons
[Solution Description]
The given ratio is $8 : 12$ for sugar to glasses. To find the sugar needed for 30 glasses, set up the proportion:
$\frac{8}{12} = \frac{x}{30}$. Cross-multiplying gives $12x = 240$, so $x = 20$. Thus, 20 spoons of sugar are needed.

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: Proportional Reasoning

c) 20 kg
[Solution Description]
The ratio of fertilizer to land is $8 : 2$, which simplifies to $4 : 1$. For 5 acres, the required fertilizer is $4 \times 5 = 20$ kg.

Key Concept: Ratio Comparison

b) 100 seconds
[Solution Description]
First, convert the bucket's capacity to mL for consistent units: $4 \text{ liters} = 4000 \text{ mL}$. The ratio of volumes is $800 : 4000$, which simplifies to $1 : 5$. Since the flow rate is constant, the time taken will also be in the ratio $1 : 5$. Given that 800 mL takes 20 seconds, 4000 mL will take $20 \times 5 = 100$ seconds.

Key Concept: Identifying proportional relationships

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To check if the ratios $4 : 5$ and $16 : 20$ are proportional, we simplify both ratios to their simplest form.
The ratio $4 : 5$ is already in its simplest form as the HCF of 4 and 5 is 1.
For the ratio $16 : 20$, we find the HCF of 16 and 20, which is 4. Dividing both terms by 4, we get $4 : 5$.
Since both ratios simplify to $4 : 5$, they are proportional.
The assertion is true, and the reason correctly explains why the assertion is true because it states the condition for proportionality (simplest forms being equal).

Key Concept: Identifying proportional relationships, Applying factor of change

c) 25 liters
[Solution Description]
First, find the petrol consumption per kilometer: $\frac{15 \text{ liters}}{180 \text{ km}} = 0.0833$ liters/km. Then, multiply this rate by the new distance: $0.0833 \text{ liters/km} \times 300 \text{ km} = 25$ liters.

Key Concept: Problem Solving with Proportional Reasoning, Identifying proportional relationships

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
First, check if Assertion is true: Simplify $4 : 5$ which is already in simplest form. Now simplify $36 : 45$ by dividing both terms by their HCF (9): $\frac{36}{9} : \frac{45}{9} = 4 : 5$. Since both ratios simplify to $4 : 5$, the Assertion is true. Next, check Reason: The condition for proportionality is $a \times d = b \times c$ for ratios $a : b$ and $c : d$. For $4 : 5$ and $36 : 45$, $4 \times 45 = 180$ and $5 \times 36 = 180$. Thus, the Reason correctly explains why the ratios are proportional. Hence, both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

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: Simplifying Ratios

b) $2 : 3$
[Solution Description]
To simplify the ratio $24 : 36$, we find the HCF of 24 and 36, which is 12. Divide both terms by 12:
$24 ÷ 12 = 2$
$36 ÷ 12 = 3$
The simplified ratio is $2 : 3$.

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 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: Profit Sharing

b) \$8,000
[Solution Description]
The investment ratio is $4 : 5 : 6$.
Total parts = $4 + 5 + 6 = 15$.
Share of the friend who invested the least (4 parts):
$\text{Share} = \frac{4}{15} \times 30000 = 8000$.

Key Concept: Comparative Proportion

b) No, the second shop is cheaper
[Solution Description]
To check proportionality, compare the cost per notebook for both shops.
First shop: Cost per notebook = $\frac{\$120}{3} = \$40$ per notebook.
Second shop: Cost per notebook = $\frac{\$190}{5} = \$38$ per notebook.
Since $\$40 \neq \$38$, the two ratios are not proportional. The second shop offers notebooks at a cheaper rate (\$38 vs. \$40), so it is the better deal.

Key Concept: Trairasika — The Rule of Three

c) 56
[Solution Description]
The proportion is $3 : 24 :: 7 : x$. Cross-multiplying gives $3x = 168$, so $x = \frac{168}{3} = 56$. Thus, the machine will produce 56 toys in 7 hours.

Key Concept: Trairasika — The Rule of Three

c) 18
[Solution Description]
The problem involves inverse proportion, so we set it as $6 \times 12 = x \times 4$. Solving for $x$ gives $x = \frac{72}{4} = 18$. Thus, 18 workers are needed to complete the task in 4 days.

Key Concept: Proportion Application

b) 7.5 days
[Solution Description]
First, let’s represent the given data proportionally. Here, the number of workers and the number of days are inversely proportional because more workers would take fewer days for the same amount of work. So, we model it as $5 : 12 :: 8 : x$ where $x$ is the unknown number of days.
Now, using cross multiplication:
$5 \times 12 = 8 \times x$
$60 = 8x$
$x = \frac{60}{8} = 7.5$
Thus, 8 workers will take 7.5 days to build the wall.

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: Trairasika — The Rule of Three

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
Given the proportion $a : b :: c : d$. According to the Rule of Three, if two ratios are proportional, then their cross products are equal: $ad = bc$. Here, $a = 5$, $b = 10$, and $c = 15$. Substituting these values into the equation:
$5 \times d = 10 \times 15$
$5d = 150$
$d = \frac{150}{5} = 30$.
Therefore, the Assertion is true. The Reason states that for proportional ratios, the product of the means ($b$ and $c$) equals the product of the extremes ($a$ and $d$), which is also true and correctly explains why the value of $d$ is 30.

Key Concept: Mixing ratios, Proportionality

b) 5 liters
[Solution Description]
Current mixture has $(5/8)\times40 = 25$ liters alcohol and $(3/8)\times40 = 15$ liters water. Let $x$ liters of water be added. New ratio is:
$\frac{25}{15 + x} = \frac{5}{4}$
Cross multiplying:
$25 \times 4 = 5 \times (15 + x)$
$100 = 75 + 5x \Rightarrow 5x = 25 \Rightarrow x = 5 \text{ liters}$

Key Concept: Rule of Three Application

b) 300 km
[Solution Description]
The problem can be represented as $2 : 120 :: 5 : x$. Using the Rule of Three:
$2 \times x = 120 \times 5$
Solving for $x$:
$2x = 600 \implies x = \frac{600}{2} = 300$
Thus, the car will travel 300 km in 5 hours.

Key Concept: Rule of Three Proportion

b) 10 workers
[Solution Description]
We use the rule of three to solve this problem. Let the number of workers be $x$. The work remains constant, so we have an inverse proportion between workers and days.
Given:
5 workers $\times$ 20 days = $x$ workers $\times$ 10 days.
To find $x$, divide both sides by 10 days:
$(5 \times 20) / 10 = x$.
Simplifying:
$100 / 10 = 10$.
Therefore, 10 workers are needed.

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: Cross Multiplication

c) 8
[Solution Description]
Using the cross-multiplication method for proportions:
Given proportion is $4 : 9 :: x : 18$.
This can be written as $\frac{4}{9} = \frac{x}{18}$.
Cross-multiplying gives $4 \times 18 = 9 \times x$.
Calculating:
$72 = 9x$.
Solving for $x$:
$x = 72 / 9 = 8$.
Thus, the value of $x$ is 8.

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: Proportionality in Time and Work

c) 6 days
[Solution Description]
Here, more workers imply less time (inverse proportion). Let $d$ be the required days. By inverse proportion rule:
$3 \times 10 = 5 \times d \implies d = \frac{3 \times 10}{5} = 6$
5 workers will take 6 days.

Key Concept: Rule of Three and Proportionality

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The Assertion claims that the Rule of Three can solve direct proportionality problems, which is true as it is designed for such cases. The Reason correctly describes the mathematical relationship in the Rule of Three, where the product of the first (pramāṇa) and fourth (ichchhāphala) terms equals the product of the second (phala) and third (ichchhā) terms. Since the Reason accurately explains why the Rule of Three works for proportional relationships, it validates the Assertion.

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: Sharing in given ratio

b) 12 and 6
[Solution Description]
Total chocolates = 18
Ratio = 2:1
Total parts = 2 + 1 = 3
Size of each part = $18 \div 3 = 6$
First friend's share = $2 \times 6 = 12$ chocolates
Second friend's share = $1 \times 6 = 6$ chocolates.

Key Concept: Dividing a mixture in ratio

c) 28 kg
[Solution Description]
Total weight = 40 kg
Ratio = 7:3
Total parts = 7 + 3 = 10
Size of each part = $40 \div 10 = 4$ kg
Flour quantity = $7 \times 4 = 28$ kg.

Key Concept: Sharing in a given ratio

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To divide 60 objects in the ratio of 5:1:
1. Total parts = $5 + 1 = 6$.
2. Value of each part = $\frac{60}{6} = 10$.
3. First person's share = $5 \times 10 = 50$ objects.
4. Second person's share = $1 \times 10 = 10$ objects.
Thus, Assertion is true because the shares are correctly calculated using the given ratio. Reason is also true and correctly explains how the shares are determined.

Key Concept: Sharing in Given Ratio

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that when 20 sweets are shared in the ratio 3:2, one gets 12 and another gets 8. This is correct because:
1. Total parts = 3 + 2 = 5 parts
2. Value of each part = $20 \div 5 = 4$ sweets
3. First share = $3 \times 4 = 12$ sweets
4. Second share = $2 \times 4 = 8$ sweets
The reason correctly explains the method to find total parts (m+n) which is used to divide any quantity in given ratio. Therefore, both assertion and reason are true and reason correctly explains the assertion.

Key Concept: Assertion-Reason

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.
[Solution Description]
The Assertion states that to divide \Rs.5,000 in the ratio $3 : 2$, we use the formula $\frac{x}{m + n}$.
The Reason explains that the formula $\frac{x}{m + n}$ helps find the size of each part when dividing a quantity in a given ratio.
Here, both Assertion and Reason are true. However, while the formula $\frac{x}{m + n}$ is indeed used to find the size of one part, it alone does not fully explain how the division is done in the ratio $3 : 2$. The complete process involves multiplying this value by the respective ratio terms ($3$ and $2$). Hence, the Reason does not correctly explain the Assertion.

Key Concept: Dividing quantities in a given ratio

b) 100 mL
[Solution Description]
Given ratio $1 : 4$, total parts = $1 + 4 = 5$.
$\text{Quantity of salt} = \frac{1}{5} \times 500 = 100 \text{ mL}$
The quantity of salt is 100 mL.

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

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.
[Solution Description]
To solve this, let's analyze the assertion and reason step by step.
1. The assertion states that for a ratio $3 : 5$, the larger part is $\frac{5}{8}$ of the total quantity. This is correct because:
Total parts = $3 + 5 = 8$.
Larger part corresponds to $5$ parts, so its fraction is $\frac{5}{8}$.
2. The reason claims that for any ratio $m : n$ where $n > m$, the larger part is $\frac{n}{m + n}$. This is also correct mathematically. However, the reason does not correctly explain the assertion because it generalizes the concept but doesn't directly justify why the specific case ($3 : 5$) holds true.
Therefore, both assertion and reason are true, but the reason is not the correct explanation of the assertion.

Key Concept: Dividing a Quantity in Given Ratio

b) 24 and 36
[Solution Description]
To divide 60 marbles in the ratio 2:3:
Total parts = $2 + 3 = 5$.
Size of each part = $\frac{60}{5} = 12$ marbles.
First friend's share = $2 \times 12 = 24$ marbles.
Second friend's share = $3 \times 12 = 36$ marbles.

Key Concept: Ratio division method

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To verify the assertion:
1. Given quantity = 60 kg, ratio = 4:1.
2. Total parts = 4 + 1 = 5.
3. Size of each part = $\frac{60}{5} = 12$ kg.
4. Larger part = $4 \times 12 = 48$ kg (matches assertion).
The reason provides the correct formula for calculating the larger part, which explains how 48 kg was obtained. Therefore, both are true, and the reason correctly explains the assertion.

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: Sharing in a given ratio

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.
[Solution Description]
To divide Rs.500 in the ratio 2:3:
1. Total parts = 2 + 3 = 5
2. Value of each part = Rs.500 / 5 = Rs.100
3. Larger share = 3 × Rs.100 = Rs.300
The Assertion is true as the larger share is indeed Rs.300. The Reason is also true because the total parts are 5. However, the Reason does not explain why the larger share is Rs.300; it only states the total parts. Hence, Reason is NOT the correct explanation of Assertion.

Key Concept: Profit sharing and ratio application

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.
[Solution Description]
To verify the assertion, we calculate the shares using the given ratio 3:2. Total parts = 3 + 2 = 5. Partner A's share = $\frac{3}{5} \times 10000 = 6000$. Partner B's share = $\frac{2}{5} \times 10000 = 4000$. Thus, the assertion is true. The reason correctly states the methodology for dividing the profit based on the ratio, but it does not specifically explain why Partner A gets \Rs.6,000; it states the general method.

Key Concept: Business Investment

b) \Rs.6,000
[Solution Description]
Investment ratio = 50000:30000 = 5:3.
Total parts = 5 + 3 = 8.
Smaller investor's share = $\frac{3}{8} \times 16000 = \Rs.6,000$.

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: Profit Sharing

c) \Rs.7,200
[Solution Description]
To divide \Rs.12,000 in the ratio 2:3:
Total parts = 2 + 3 = 5.
Each part = $\frac{12000}{5} = 2400$.
Second partner's share = $3 \times 2400 = \Rs.7,200$.

Key Concept: Unit Conversions - Area

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.
[Solution Description]
1 hectare = 10,000 square metres (given). From the syllabus, we know that 1 acre = 43,560 square feet. However, the Assertion states a direct conversion between hectares and acres (1 hectare = 2.471 acres), which is also correct. The Reason provides definitions of hectare and acre in terms of square metres and square feet but does not directly explain why 1 hectare equals 2.471 acres. Thus, both Assertion and Reason are true, but the Reason does not correctly explain the Assertion.

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 - Length

c) 32.81 feet
[Solution Description]
Using the conversion factor: $1 \text{ metre} = 3.281 \text{ feet}$. To convert 10 metres to feet:
$10 \text{ metres} \times 3.281 \text{ feet/metre} = 32.81 \text{ feet}$
Hence, 10 metres is equal to 32.81 feet.

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

b) 36 degrees Fahrenheit
[Solution Description]
The relationship between Celsius and Fahrenheit is given by:
$\Delta F = \frac{9}{5} \times \Delta C$.
Here, $\Delta C = 20^{\circ}C$.
So, $\Delta F = \frac{9}{5} \times 20 = 36^{\circ}F$.

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: 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: Area Conversion

c) 217,800 square feet
[Solution Description]
To convert acres to square feet, multiply the number of acres by the conversion factor. Here, 1 acre = 43,560 square feet. Therefore:
$5 \text{ acres} \times 43,560 \text{ square feet/acre} = 217,800 \text{ square feet}$

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.

Key Concept: Unit Conversions, Proportional Reasoning

c) 27.55 litres
[Solution Description]
First, convert the area of the field from square feet to acres. The area is $800 \times 600 = 480,000$ square feet. Since $1$ acre = $43,560$ square feet, the area in acres is $\frac{480,000}{43,560} \approx 11.02$ acres.
Next, use the given ratio: $5$ litres for $2$ acres. The amount of diesel needed for $11.02$ acres can be calculated using proportionality:
$\text{Diesel} = \left( \frac{5 \text{ litres}}{2 \text{ acres}} \right) \times 11.02 \text{ acres} \approx 27.55 \text{ litres}$
Therefore, approximately $27.55$ litres of diesel are needed.

Key Concept: Unit Conversion Consistency

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The Assertion states that converting area measurements to acres is necessary for accurate cost calculation when fertilizing a field. This is true because using consistent units (e.g., acres) allows proper ratio application. The Reason correctly explains that proportional reasoning demands uniform units to prevent errors, which directly supports the Assertion. Thus, both are true, and the Reason justifies the Assertion.

Key Concept: Age Ratio, Proportion Check

c) 8 years
[Solution Description]
Let the number of years after which the ratio becomes $3 : 5$ be $x$. At that time:
- Harmain's age will be $4 + x$.
- Brother's age will be $12 + x$.
According to the problem:
$\frac{4 + x}{12 + x} = \frac{3}{5}$
Cross-multiplying:
$5(4 + x) = 3(12 + x) \Rightarrow 20 + 5x = 36 + 3x \Rightarrow 2x = 16 \Rightarrow x = 8$
Therefore, after $8$ years, the ratio of their ages will be $3 : 5$.