Key Concept: Ratio Comparison

c) 30
[Solution Description]
Given the ratio of teachers to students is 1:34.
Let the number of teachers be $t$.
Therefore, $\frac{t}{1020} = \frac{1}{34}$.
Solving for $t$ gives $t = \frac{1020}{34} = 30$ teachers.

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: Proportional Change

c) 30 mm
[Solution Description]
To maintain similarity, the scaling factor for width and height must be the same.
Scaling factor = $\frac{\text{New Width}}{\text{Original Width}} = \frac{45}{60} = \frac{3}{4}$.
New Height = Original Height × Scaling Factor = $40 \times \frac{3}{4} = 30$ mm.

Key Concept: Ratios

d) 60 mm
[Solution Description]
Given the simplified ratio $3 : 2$ for Image D, let the height be $h$. We can write the proportion as $\frac{3}{2} = \frac{90}{h}$. Solving for $h$:
$3h = 180$
$h = 60$
The height of Image D is 60 mm.

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: 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: Multiplicative vs. Additive Changes

a) Multiply both width and height by 1.5
[Solution Description]
For the image to remain similar, both width and height must be scaled by the same multiplicative factor. The original ratio of width to height is $50 : 30$, which simplifies to $5 : 3$.
Option a): Multiplies width by 1.5 to get 75 mm and height by 1.5 to get 45 mm. The new ratio is $75 : 45 = 5 : 3$ (similar).
Option b): Adds 10 mm to width (60 mm) and subtracts 5 mm from height (25 mm). The new ratio is $60 : 25 = 12 : 5$ (not similar).
Option c): Multiplies width by 2 (100 mm) but adds 10 mm to height (40 mm). The new ratio is $100 : 40 = 5 : 2$ (not similar).
Option d): Adds 20 mm to width (70 mm) and height (50 mm). The new ratio is $70 : 50 = 7 : 5$ (not similar).

Key Concept: Proportional Ratios

c) $12 : 18$
[Solution Description]
A ratio proportional to $4 : 6$ must simplify to the same form or satisfy $ad = bc$. The simplest form of $4 : 6$ is $2 : 3$. Check each option:
- $8 : 12$ simplifies to $2 : 3$ (proportional).
- $6 : 9$ simplifies to $2 : 3$ (proportional).
- $10 : 15$ simplifies to $2 : 3$ (proportional).
- $12 : 18$ simplifies to $2 : 3$ (proportional).
Among the given options, $12 : 18$ is correct.

Key Concept: Application of Proportional Reasoning

c) 20 spoons
[Solution Description]
Given the ratio $5 : 8$ (sugar to lemonade), let the required sugar be $x$ spoons for 32 glasses. The proportional relationship is:
$5 : 8 :: x : 32$
Cross-multiplying gives:
$8x = 5 \times 32$
Solving for $x$:
$x = \frac{160}{8} = 20$
She would need 20 spoons of sugar.

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: Proportional Ratios

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
First, simplify the ratio $8 : 12$ by dividing both terms by their HCF, which is 4. This gives $\frac{8}{4} : \frac{12}{4} = 2 : 3$. Next, simplify the ratio $10 : 15$ by dividing both terms by their HCF, which is 5. This gives $\frac{10}{5} : \frac{15}{5} = 2 : 3$. Since both simplified ratios are equal ($2 : 3$), the assertion is true. The reason correctly states that both ratios simplify to $2 : 3$, and this is why they are proportional. Thus, the reason is not only true but also the correct explanation for the assertion.

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: 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: Assertion-Reason

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To simplify the ratio $6:4$, we find the HCF of 6 and 4, which is 2. Dividing both terms by 2 gives $3:2$. Thus, the Assertion is true. The Reason correctly explains that simplifying a ratio involves dividing both terms by their HCF. Therefore, both Assertion and Reason are true, and the Reason is the correct explanation of the Assertion.

Key Concept: Interpretation of Ratios

c) 15 cups
[Solution Description]
The given ratio of sugar to flour is $3 : 5$. For every 3 cups of sugar, 5 cups of flour are needed. If 9 cups of sugar are used, we need to find the corresponding amount of flour.
First, determine the scaling factor:
$\text{Scaling factor} = \frac{9}{3} = 3$
Multiply the flour part of the ratio by this scaling factor:
$5 \times 3 = 15$
Thus, 15 cups of flour are needed.

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: Comparing Ratios, Proportionality Check

b) Yes, because both simplify to $1:3$
[Solution Description] Simplify both ratios to their simplest form. For Project X, the ratio is $15:45$. The HCF of 15 and 45 is 15, so simplifying gives $\frac{15}{15} : \frac{45}{15} = 1:3$. For Project Y, the ratio is $10:30$. The HCF of 10 and 30 is 10, so simplifying gives $\frac{10}{10} : \frac{30}{10} = 1:3$. Since both simplified ratios are equal ($1:3$), the ratios are proportional.

Key Concept: Problem Solving with Ratios

b) 25
[Solution Description]
Given the ratio of boys to girls is $3 : 5$. Let the number of boys be $3x$ and girls be $5x$. Since there are 15 boys, $3x = 15 \Rightarrow x = 5$. Number of girls $= 5x = 5 \times 5 = 25$.

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: 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

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. The factors of 12 are 1, 2, 3, 4, 6, and 12. The factors of 18 are 1, 2, 3, 6, 9, and 18. The highest common factor (HCF) of 12 and 18 is 6. Dividing both terms of the ratio $12 : 18$ by 6, we get $\frac{12}{6} : \frac{18}{6} = 2 : 3$. Therefore, both Assertion and Reason are true, and Reason correctly explains Assertion.

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: Simplifying ratios

b) $3 : 5$
[Solution Description]
To simplify the ratio $45 : 75$, first find the HCF of 45 and 75. The HCF is 15. Now divide both terms by 15:
$45 \div 15 = 3 \\ \\ 75 \div 15 = 5$
So, the simplified ratio is $3 : 5$.

Key Concept: Proportional Ratios

a) Yes
[Solution Description]
Simplify both ratios to their simplest forms. For $16 : 24$, the HCF is 8:
$\frac{16}{8} : \frac{24}{8} = 2 : 3$
For $20 : 30$, the HCF is 10:
$\frac{20}{10} : \frac{30}{10} = 2 : 3$
Both ratios simplify to $2 : 3$, so they are proportional.

Key Concept: Age ratios, Changing ratios

c) 3 : 4
[Solution Description]
At Rahul's age 6, sister's age is $6 \times 2 = 12$ years.
Age difference is always $12 - 6 = 6$ years.
When Rahul is 18, sister will be $18 + 6 = 24$ years.
The ratio will be $18 : 24$.
Simplifying by dividing both terms by 6 gives $3 : 4$.

Key Concept: Proportional Ratios

c) 15
[Solution Description]
For ratios to be proportional, the cross-products must be equal. Thus, $5 \times 21 = 7 \times x$, which simplifies to $105 = 7x$. Solving for $x$ gives $x = 15$.

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: 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: 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: 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) Yes, both simplify to $3 : 2$
[Solution Description]
To check proportionality, simplify both ratios and compare their simplest forms.
For $54 : 36$:
Step 1: Find the HCF of 54 and 36. The HCF is 18.
Step 2: Divide both terms by 18.
$54 \div 18 = 3$
$36 \div 18 = 2$
Simplified ratio: $3 : 2$.
For $90 : 60$:
Step 1: Find the HCF of 90 and 60. The HCF is 30.
Step 2: Divide both terms by 30.
$90 \div 30 = 3$
$60 \div 30 = 2$
Simplified ratio: $3 : 2$.
Since both ratios simplify to $3 : 2$, they are proportional.

Key Concept: Equivalence of ratios in simplest form

d) 40 : 56
[Solution Description]
Simplify $25 : 35$ to its simplest form: divide both terms by 5 to get $5 : 7$. Now check each option:
- $10 : 14$ simplifies to $5 : 7$ (proportional).
- $30 : 42$ simplifies to $5 : 7$ (proportional).
- $50 : 70$ simplifies to $5 : 7$ (proportional).
- $40 : 56$ simplifies to $5 : 7$ (proportional).
However, $40 : 56$ is proportional as well, but since all options seem proportional, rechecking: The correct non-proportional ratio would be $15 : 21$ which simplifies to $5 : 7$ (all given options are proportional). Thus, the question may need adjustment, but assuming the intent was to test recognition of simplified forms.

Key Concept: HCF Calculation

c) 16
[Solution Description]
The factors of 48 are: 1, 2, 3, 4, 6, 8, 12, 16, 24, 48. The factors of 64 are: 1, 2, 4, 8, 16, 32, 64. The highest common factor (HCF) of 48 and 64 is 16.

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, simplify both ratios to their simplest form:
\begin{itemize}

Key Concept: Ratios in Simplest Form

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 need to simplify both ratios:
1. For $18:12$, divide both terms by HCF of 18 and 12 which is 6. This gives $3:2$.
2. For $27:18$, divide both terms by HCF of 27 and 18 which is 9. This gives $3:2$.
Since both simplified ratios are equal ($3:2$), they are proportional. Also, the Reason correctly explains the condition for proportionality of ratios as it states that two ratios are proportional if their simplest forms are equal.

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: Proportionality using simplest forms

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
First, simplify the ratio $4:6$ by dividing both terms by their HCF 2, giving $2:3$. Next, simplify $10:15$ by dividing both terms by their HCF 5, which also gives $2:3$. Since both simplified ratios are equal ($2:3$), they are proportional. Therefore, both Assertion and Reason are true, and Reason correctly explains why the Assertion is true.

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: 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: 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: Proportional Reasoning, Unit Conversions

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The Assertion states that the distance covered is proportional to the time taken when speed is constant. This is true because speed ($v$) is given by $v = \frac{d}{t}$. If speed is constant, then $d$ must be directly proportional to $t$.
The Reason explains this relationship correctly by defining speed as $\frac{d}{t}$ and stating that for constant speed, distance is directly proportional to time.
Thus, both Assertion and Reason are true, and Reason correctly explains the Assertion.

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 Comparison in Real-life Application

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that the price per mL decreases with increasing volume, which is true because larger sizes generally have lower per-unit costs due to reduced packaging and handling expenses. The reason correctly explains this by mentioning economies of scale, which refers to cost advantages gained through increased production efficiency. Thus, both are true, and the reason correctly explains the assertion.

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 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: Ratio comparison, Unit conversion

b) 6
[Solution Description]
First, convert 7000 square yards to acres: $7000 \div 4840 \approx 1.446$ acres.
The ratio of bags to acres is $12 : 3$ or $4 : 1$.
For 1.446 acres, the number of bags needed is $4 \times 1.446 \approx 5.785$.
Since bags must be whole numbers, round up to 6 bags.

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: 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: 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: Proportional Ratios

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 to their simplest forms. For $12 : 18$, the HCF of 12 and 18 is 6. Dividing both terms by 6 gives $2 : 3$. For $20 : 30$, the HCF of 20 and 30 is 10. Dividing both terms by 10 also gives $2 : 3$. Since both simplified ratios are equal, they are proportional. The reason correctly explains the assertion.

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: 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: Assertion-Reason

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. For $16 : 20$, dividing both terms by their HCF (which is 4) gives $4 : 5$. Since both ratios simplify to $4 : 5$, they are proportional. The Reason correctly explains why the Assertion is true because proportionality is determined by comparing the simplest forms of the ratios.

Key Concept: Trairasika — The Rule of Three

c) \$160
[Solution Description]
Using the Rule of Three, we can set up the proportion as $5 : 100 :: 8 : x$. Cross-multiplying gives $5x = 800$, so $x = \frac{800}{5} = 160$. Thus, 8 books cost \$160.

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: Comparative proportional analysis

b) Factory A is more efficient
[Solution Description]
Factory A good units = $500 \times 0.75 = 375$ in 3 hours → 125/hour.
Factory B good units = $800 \times 0.70 = 560$ in 5 hours → 112/hour.
Factory A has higher efficiency (125 vs 112 good units/hour).

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 $5 : 250 :: 8 : x$, cross multiplication gives $5x = 250 \times 8$. Solving for $x$, we get $x = \frac{2000}{5} = 400$. Here, Assertion correctly applies the Rule of Three and derives the cost of 8 kg rice as \$400. The Reason explains why the method works mathematically, stating that proportional ratios must satisfy $ad = bc$. Both statements are true, and the Reason justifies 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: 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: Cross Multiplication Method

b) 15
[Solution Description]
Given the proportion $3:4 :: x:20$, cross multiplying yields:
$3 \times 20 = 4 \times x$
Solving for $x$:
$x = \frac{3 \times 20}{4} = \frac{60}{4} = 15$

Key Concept: Compound Proportion

d) 11 pumps
[Solution Description]
This is a compound proportion problem with three variables. Let $x$ be the required pumps. The relationship is:
$\frac{4 \text{ pumps} \times 8 \text{ hrs} \times 2 \text{ days}}{1 \text{ tank}} = \frac{x \text{ pumps} \times 6 \text{ hrs} \times 1 \text{ day}}{1 \text{ tank}}$
Simplifying:
$4 \times 8 \times 2 = x \times 6 \times 1 \Rightarrow 64 = 6x \Rightarrow x = \frac{64}{6} \approx 10.\overline{6}$
Since we can't have a fraction of a pump, we round up to 11 pumps.

Key Concept: Rule of Three, Multi-step Application

c) 103:249
[Solution Description]
Take 1 liter from each solution to make 2 liters total.
From Solution A (3:8 ratio):
Acid = $(3/(3+8)) \times 1 = 3/11$ L
Water = $(8/11) \times 1 = 8/11$ L
From Solution B (5:11 ratio):
Acid = $(5/(5+11)) \times 1 = 5/16$ L
Water = $(11/16) \times 1 = 11/16$ L
Total acid in Solution C: $3/11 + 5/16 = (48 + 55)/176 = 103/176$ L
Total water in Solution C: $8/11 + 11/16 = (128 + 121)/176 = 249/176$ L
Simplifying the ratio (divide both by 103/176):
Acid : Water = 1 : (249/103) ≈ 103:249

Key Concept: Proportional Ratios

b) 40
[Solution Description]
Model the problem as the proportion $5 : 25 :: 8 : x$.
Step 1: Write the proportion and use the Rule of Three:
$5 \times x = 25 \times 8$
Step 2: Multiply the known terms:
$5x = 200$
Step 3: Solve for $x$ by dividing both sides by 5:
$x = \frac{200}{5} = 40$
The machine can produce 40 toys in 8 hours.

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: 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: Unit Conversion in Rule of Three

c) 90 m$^2$
[Solution Description]
Given: Pramāṇa = 2.5 L (a), Phala = 30 m$^2$ (b), Ichchhā = 7.5 L (c). Using the Rule of Three:
$d = \frac{b \times c}{a} = \frac{30 \times 7.5}{2.5} = 90$
7.5 liters will cover 90 m$^2$.

Key Concept: Rule of Three and Proportionality

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
Here, the assertion calculates $ichchhāphala$ using Āryabhaṭa's formula: $ichchhāphala = \frac{phala \times ichchhā}{pramāṇa} = \frac{12 \times 8}{4} = 24$. This matches the rule mentioned in the reason, which describes the proportionality relationship $pramāṇa : phala :: ichchhā : ichchhāphala$. Therefore, both are true, and the reason correctly explains the assertion.

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, Mixed Numbers

c) 30 liters
[Solution Description]
Convert mixed numbers to improper fractions: Pramāṇa = $\frac{13}{4}$ kg, Phala = $\frac{15}{2}$ liters, Ichchhā = 13 kg. Using the Rule of Three:
$ichchhāphala = \frac{phala \times ichchhā}{pramāṇa} = \frac{\frac{15}{2} \times 13}{\frac{13}{4}} = \frac{195/2}{13/4} = \frac{195 \times 4}{2 \times 13} = \frac{780}{26} = 30 \text{ liters}$

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: Ratio Division, Investment Sharing

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To verify Assertion (A), we use the formula for division in a given ratio $m:n$. Here, $x = Rs.12,000$, $m = 5$, and $n = 3$.
First, calculate the total parts: $5 + 3 = 8$.
Now, find the size of each part: $\frac{12,000}{8} = 1,500$.
Partner A's share: $5 \times 1,500 = 7,500$.
Thus, Assertion (A) is true.
Reason (R) states the correct formula for dividing a quantity into a given ratio, which was used to verify Assertion (A). Hence, both Assertion and Reason are true, and Reason correctly explains Assertion.

Key Concept: Ratio division, profit sharing

c) Rs.10,000
[Solution Description]
First, find the ratio of their investments:
Prashanti : Bhuvan = 1,20,000 : 80,000 = 3 : 2 (simplified).
Total parts = 3 + 2 = 5.
Each part's value = Rs.25,000 / 5 = Rs.5,000.
Bhuvan's share = 2 × Rs.5,000 = Rs.10,000.

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: Dividing quantities in a given ratio, adjusting ratios

b) 5 liters
[Solution Description]
First, determine the current amounts of milk and water in the mixture.
Total parts in the original ratio 7:5 = 7 + 5 = 12 parts.
Amount of milk = $\frac{7}{12} \times 60 = 35$ liters.
Amount of water = $\frac{5}{12} \times 60 = 25$ liters.
Let $x$ be the liters of water to be added to change the ratio to 7:6.
The new amount of water will be $25 + x$, while milk remains 35 liters.
According to the new ratio:
$\frac{35}{25 + x} = \frac{7}{6}$.
Cross-multiplying:
$35 \times 6 = 7 \times (25 + x)$.
$210 = 175 + 7x$.
$35 = 7x$.
$x = 5$.
Therefore, 5 liters of water must be added.

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 quantities in a given ratio

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To divide Rs.6,000 in the ratio 3:2, we calculate the total number of parts as $3 + 2 = 5$. Each part is worth $\frac{6000}{5} = Rs.1,200$. A's share is $3 \times 1200 = Rs.3,600$, which matches the Assertion. The Reason correctly states the formula for finding the first part of a division in a given ratio $m:n$, which is $m \times \frac{x}{m + n}$. Therefore, both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

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: 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: 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: 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: Ratio division, Profit sharing

b) \$9,000
[Solution Description]
Total ratio parts = $5 + 3 = 8$. Each part corresponds to $\frac{24000}{8} = 3000$. Bina's share is $3 \times 3000 = 9000$.

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: Mixture Problem

b) 20 kg
[Solution Description]
The ratio of sugar to flour is $2 : 5$. Total parts = $2 + 5 = 7$. Each part = $\frac{70}{7} = 10$ kg. Sugar quantity = $2 \times 10 = 20$ 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: Ratio Adjustment

b) 10 liters
[Solution Description]
Let the initial quantities be $4x$ (red) and $3x$ (blue). After adding 5 liters of blue, the new ratio is $4x : (3x + 5) = 4 : 5$. Solving: $\frac{4x}{3x + 5} = \frac{4}{5}$. Cross-multiplying gives $20x = 12x + 20$, so $8x = 20$ and $x = 2.5$. Initial red paint = $4 \times 2.5 = 10$ liters.

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: Area Conversion

b) 1.67 hectares
[Solution Description]
First, calculate the area in square feet:
$300 \, \text{ft} \times 600 \, \text{ft} = 180{,}000 \, \text{sq ft}$
Convert square feet to square metres using $1 \, \text{sq m} = 10.764 \, \text{sq ft}$:
$\frac{180{,}000}{10.764} = 16{,}722.5 \, \text{sq m}$
Now, convert square metres to hectares using $1 \, \text{hectare} = 10{,}000 \, \text{sq m}$:
$\frac{16{,}722.5}{10{,}000} = 1.67225 \, \text{hectares}$
The area is approximately 1.67 hectares.

Key Concept: Temperature conversion, Multi-step reasoning

c) $19^\circ \text{F}$
[Solution Description]
First, convert $-10^\circ \text{C}$ to Fahrenheit using the formula: $\frac{9}{5} \times (-10) + 32 = -18 + 32 = 14^\circ \text{F}$.
Then add the instrument error of $+5^\circ \text{F}$: $14 + 5 = 19^\circ \text{F}$.
The final temperature reading will be $19^\circ \text{F}$.

Key Concept: Area conversion, Proportional reasoning

c) 20.66 tonnes
[Solution Description]
First, calculate the area of the plot in square feet: $300 \text{ ft} \times 600 \text{ ft} = 180,000 \text{ sq ft}$.
Convert this to acres using the given conversion factor ($1 \text{ acre} = 43,560 \text{ sq ft}$).
Divide the total area by the number of square feet in an acre: $\frac{180,000}{43,560} \approx 4.132 \text{ acres}$.
Multiply this by the manure rate per acre: $4.132 \times 5 = 20.66 \text{ tonnes}$.
The farmer needs approximately 20.66 tonnes of manure.

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: 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:

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

b) 5,000,000 cubic centimeters
[Solution Description]
Convert liters to milliliters first:
$1 \text{ L} = 1000 \text{ mL}$.
So, $5000 \text{ L} = 5000 \times 1000 \text{ mL} = 5,000,000 \text{ mL}$.
Since $1 \text{ mL} = 1 \text{ cc}$, the volume in cubic centimeters is also $5,000,000 \text{ cc}$.

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: Work Rate Conversion

b) 30 hours
[Solution Description]
First, calculate the time taken by the oxen for 20 acres: $20 \times 6 = 120\,\text{hours}$. Since the tractor is 4 times faster, its time is:
$\text{Tractor's time} = \frac{120}{4} = 30\,\text{hours}$

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: 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:
$\text{Time taken} = \frac{15 \times 10,000}{500} = 300\,\text{seconds}$