Key Concept: Proportional Change

c) 60 mm
[Solution Description]
To find the height of the similar rectangle when the width changes proportionally:
First, calculate the ratio of original width to height: $\frac{50}{30} = \frac{5}{3}$.
For the new rectangle, the width-to-height ratio must remain the same for similarity.
Let the new height be $h$. Then, $\frac{100}{h} = \frac{5}{3}$.
Solving for $h$, we get $h = \frac{100 \times 3}{5} = 60$ mm.

Key Concept: Proportional Change in Images

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
From the given measurements, Image A has width 60 mm and height 40 mm, while Image D has width 90 mm and height 60 mm. The scaling factor for width is $\frac{90}{60} = \frac{3}{2}$. The scaling factor for height is $\frac{60}{40} = \frac{3}{2}$. Since both dimensions change by the same factor ($\frac{3}{2}$), the images look similar. This matches the conclusion that proportional scaling preserves similarity. Thus, both Assertion and Reason are true, and the Reason correctly explains the Assertion.

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, Real-World Application

b) 25 cups
[Solution Description]
The given ratio is $5 : 3$ (flour : sugar). For 15 cups of sugar, multiply both terms of the ratio by 5 (since $3 \times 5 = 15$). Thus, flour needed = $5 \times 5 = 25$ cups.

Key Concept: Width-Height Ratio

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To determine similarity between images, we compare their width-to-height ratios in simplest form. For image A, the ratio is $60:40$, which simplifies to $3:2$. Similarly, for image C, the ratio is $30:20$, simplifying to $3:2$, and for image D, the ratio is $90:60$, also simplifying to $3:2$. Since all three ratios reduce to the same simplest form, the images are proportionally similar. Therefore, both the Assertion and Reason are true, and the Reason correctly explains the Assertion.

Key Concept: Width-Height Ratio Comparison

c) Width: 75 mm, Height: 45 mm
[Solution Description]
To determine similarity, the ratios of width-to-height must be equal. The original ratio is $\frac{50}{30} = \frac{5}{3}$. We check each option:
1) For 25 mm width, height should be $30 \times \frac{25}{50} = 15$ mm (ratio $\frac{5}{3}$).
2) For 100 mm width, height should be $30 \times \frac{100}{50} = 60$ mm (ratio $\frac{5}{3}$).
3) For 75 mm width, height should be $45$ mm (ratio $\frac{5}{3}$).
4) Incorrect dimensions do not maintain the ratio.

Key Concept: Multiplicative vs. additive changes, Proportionality

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 remain similar because their dimensions change multiplicatively by the same factor. This is true because the ratio of width to height for these images is consistently $3:2$, indicating proportional scaling. The Reason states that multiplicative scaling preserves the ratio of width to height, while additive scaling does not. This is also true and correctly explains why the images remain similar under multiplicative scaling but not additive scaling. Therefore, both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

Key Concept: Proportional changes, Multiplicative vs additive

a) Their heights must be in ratio 3:4
[Solution Description]
For two rectangles to be similar, their corresponding sides must be proportional.
Given width ratio = $\frac{w_1}{w_2} = \frac{3}{4}$, the height ratio must be equal:
$\frac{h_1}{h_2} = \frac{3}{4}$ ⇒ $4h_1 = 3h_2$
Option A states this exact proportion. Other options:
B) Additive difference doesn't guarantee proportionality
C) Unspecific relationship
D) Equal heights would only work if widths were equal

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: 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: 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: 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: 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: 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: 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: 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: 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: 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: Rule of Three

b) 14 liters
[Solution Description]
We can solve this problem using the Rule of Three. The ratio of distance to petrol consumption is $120 : 8$.
Let the required petrol be $x$ liters for 210 km.
Setting up the proportion:
$120 : 8 :: 210 : x$
Cross-multiplying:
$120x = 8 \times 210$
$120x = 1680$
$x = \frac{1680}{120}$
$x = 14 \text{ liters}$

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: 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: Proportional reasoning

a) Yes, because their cross-products are equal
[Solution Description]
To check if two ratios are proportional, we can cross-multiply and see if the products are equal:
$8 \times 15 = 120 \\ \\ 10 \times 12 = 120$
Since both products are equal ($120 = 120$), the ratios are proportional.

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: 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: 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: Ratio Simplification

b) $2 : 3$
[Solution Description]
To simplify the ratio $60 : 90$, find the HCF of 60 and 90, which is 30. Divide both terms by 30: $60 \div 30 = 2$ and $90 \div 30 = 3$. Therefore, the simplest form is $2 : 3$.

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: 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: 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-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

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: 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: 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: Equivalence of ratios in simplest form

c) 12
[Solution Description]
First, simplify the ratio $48 : 64$ to its simplest form. The HCF of 48 and 64 is 16. Dividing both terms by 16 gives $3 : 4$. Since $48 : 64 :: 9 : x$, their simplest forms must be equal. So, $3 : 4 = 9 : x$. Cross-multiplying gives $3x = 36$, so $x = 12$.

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: Proportional Ratios

b) Yes, because both simplify to $4 : 5$
[Solution Description]
Check if the simplest forms of both ratios are equal. Simplifying $4 : 5$ gives $4 : 5$ (already in simplest form). For $20 : 25$, the HCF is 5, so $\frac{20}{5} = 4$, $\frac{25}{5} = 5$. The simplest form is $4 : 5$. Thus, they are proportional.

Key Concept: Simplifying Ratios, Proportionality

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To determine if the assertion is true or false:
1. Simplify $36:48$ by dividing both terms by their HCF. The HCF of 36 and 48 is 12. Simplifying gives $3:4$.
2. Simplify $27:36$ by dividing both terms by their HCF. The HCF of 27 and 36 is 9. Simplifying gives $3:4$.
3. Since both simplified ratios are equal ($3:4$), they are proportional, making the Assertion true.
4. The Reason states that two ratios are proportional if their simplest forms are equal, which is also true and correctly explains why the Assertion is true.

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: Proportional Reasoning

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that if 5 kg of sugar is needed for 25 liters of juice, then 10 kg of sugar is needed for 50 liters of juice. This is correct because the ratio $\frac{5\,\text{kg}}{25\,\text{liters}} = \frac{1}{5}$ and $\frac{10\,\text{kg}}{50\,\text{liters}} = \frac{1}{5}$. Since the ratios are equal, the assertion is true. The reason states that the ratio of sugar to juice remains constant in proportional reasoning, which is also true and correctly explains why the assertion holds.

Key Concept: Proportional Mixing

c) 15 mL blue and 25 mL yellow
[Solution Description]
The ratio of blue to yellow is $3 : 5$. Total parts = $3 + 5 = 8$. Quantity of blue = $\frac{3}{8} \times 40 = 15$ mL. Quantity of yellow = $\frac{5}{8} \times 40 = 25$ mL.

Key Concept: Proportional Division

a) \Rs.1,800 and \Rs.2,700
[Solution Description]
To divide \Rs.4,500 in the ratio $2 : 3$, we first find the total parts: $2 + 3 = 5$. Then, the share for the first part is $\frac{2}{5} \times 4500 = 1800$ and for the second part is $\frac{3}{5} \times 4500 = 2700$.

Key Concept: Proportional Reasoning

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To solve this problem, we need to check if the given assertion follows the principle of proportional reasoning.
First, calculate the ratio of rice to students in the first scenario:
$\frac{5\,\text{kg}}{20\,\text{students}} = 0.25\,\text{kg per student}$
Next, calculate the ratio for the second scenario:
$\frac{15\,\text{kg}}{60\,\text{students}} = 0.25\,\text{kg per student}$
Since both ratios are equal, the assertion is true because the amount of rice per student remains constant. The reason is also true and correctly explains the assertion, as proportional reasoning ensures that the ratio remains unchanged when scaling quantities proportionally.

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 Simplification

b) 100
[Solution Description]
The ratio $3 : 2$ means for every 3 boys, there are 2 girls. For 150 boys, the number of girls can be found by solving $\frac{3}{2} = \frac{150}{x}$. Cross-multiplying gives $3x = 300$, so $x = 100$.

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: 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: Proportional Relationships

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To verify proportionality, simplify both ratios:
1. $5 : 8$ is already in its simplest form.
2. Simplify $25 : 40$ by dividing both terms by their HCF (which is 5):
$\frac{25}{5} = 5$ and $\frac{40}{5} = 8$, giving the simplified ratio $5 : 8$.
Both ratios simplify to $5 : 8$, so they are proportional. The Reason correctly explains why these ratios are proportional.

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

a) Yes, because $\frac{4}{5} = \frac{16}{20}$
[Solution Description]
To determine if the ratios are proportional, simplify both ratios to their lowest terms. The ratio $4 : 5$ is already simplified. For the ratio $16 : 20$, divide both terms by their HCF, which is 4: $16 ÷ 4 = 4$ and $20 ÷ 4 = 5$. Thus, the simplified form of $16 : 20$ is $4 : 5$. Since both ratios are equal after simplification, they are proportional.

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: 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: 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: Dividing quantities in a given ratio

b) Rs.4000
[Solution Description]
Total parts = $3 + 2 = 5$. Value of one part:
$\frac{10000}{5} = 2000$
Smaller share = $2 \times 2000 = Rs.4000$.

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: 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: 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: 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: Cross Multiplication Method

b) 180
[Solution Description]
The problem can be modeled as $8:120 :: 12:x$. Cross multiplying gives:
$8 \times x = 120 \times 12$
Solving for $x$:
$x = \frac{120 \times 12}{8} = \frac{1440}{8} = 180$

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: 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, 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: 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

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
Given:
- $8$ bags $\equiv 40$ kg
For direct proportion, the ratio remains constant.
Let the weight for 5 bags be $x$ kg.
Using Rule of Three:
$8 : 40 :: 5 : x$
Cross-multiplying:
$$8x = 40 \times 5$$
$$x = \frac{200}{8} = 25$$
Hence, Assertion (A) is true.
Reason (R) states the correct concept of direct proportionality, and it explains why the calculation works.

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, Inverse Proportion

b) 4 days
[Solution Description]
This is an inverse proportion problem. The product of workers and days remains constant: $8 \times 6 = 12 \times d$. Solving for $d$:
$d = \frac{8 \times 6}{12} = 4 \text{ days}$

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: 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: 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: 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

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) Rs.720 and \Rs.480
[Solution Description]
To divide \Rs.1,200 in the ratio $3 : 2$:
$\text{Total parts} = 3 + 2 = 5$
$\text{Each part} = \frac{1200}{5} = Rs.240$
$\text{First share} = 3 \times 240 = Rs.720$
$\text{Second share} = 2 \times 240 = Rs.480$
The two shares are \Rs.720 and \Rs.480.

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: 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: Sharing in Ratio, Verification of Parts

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To verify the assertion:
- Total amount (\$1,800) is to be divided in the ratio 5:4.
- Sum of ratio terms = 5 + 4 = 9.
- First part = $5 \times \frac{1800}{9} = 5 \times 200 = \$1,000$.
- Second part = $4 \times \frac{1800}{9} = 4 \times 200 = \$800$.
- Both parts match the assertion.
The reason states the correct formula for dividing a quantity in a given ratio, which is used to derive the assertion. Hence, both Assertion and Reason are true, and Reason correctly explains the Assertion.

Key Concept: Combined ratios

d) No valid solution with given data
[Solution Description]
Let first class have $3x$ boys and $2x$ girls, total $5x$. Second class has $y$ boys and $y$ girls, total $2y$. Combined ratio $(3x + y):(2x + y) = 2:1$. Solving gives $3x + y = 4x + 2y \Rightarrow x = -y$. Negative values not possible, hence scenario invalid without additional constraints.

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: 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: 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

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 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: 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: Unit Conversions - Temperature

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
Step 1: Given the assertion that $68^\circ F$ is equal to $20^\circ C$.
Step 2: Verify the assertion by applying the reason formula $Celsius = \frac{5}{9} \times (Fahrenheit - 32)$.
Step 3: Substitute $68$ for Fahrenheit in the equation:
$\frac{5}{9} \times (68 - 32) = \frac{5}{9} \times 36 = 20$.
Step 4: The calculation confirms that $68^\circ F$ indeed equals $20^\circ C$, so both Assertion and Reason are true.
Step 5: Moreover, the Reason correctly explains how to derive Celsius from Fahrenheit, making it the correct explanation of the Assertion.

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: 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: 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:

c) 2,500 mL
[Solution Description]
Given, $1 \text{ litre} = 1,000 \text{ mL}$. Therefore, $2.5 \text{ litres} = 2.5 \times 1,000 \text{ mL} = 2,500 \text{ mL}$.

Key Concept: Unit Conversions - Temperature

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.
[Solution Description]
Assertion (A) states that $32^\circ F$ is equal to $0^\circ C$, which is true as per the given conversion in the syllabus. Reason (R) provides the correct formula to convert Celsius to Fahrenheit, which is also true. However, Reason (R) does not explain why $32^\circ F$ is equal to $0^\circ C$. Thus, both are true, but Reason is not the correct explanation of Assertion.

Key Concept: Area Unit Conversion

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To verify the assertion and reason:
1. The syllabus states that 1 hectare = 10,000 square metres.
2. Therefore, if a plot is 10,000 square metres, it must be equal to 1 hectare.
3. The reason correctly defines the conversion factor.
4. Thus, both the assertion and reason are true, and the reason correctly explains the assertion.

Key Concept: Unit Conversion in Proportional Reasoning

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that ensuring consistency in units is essential for accurate proportional reasoning. This is true because comparing or calculating with different units leads to incorrect results. The reason explains why this is necessary—different units cause wrong ratios and conclusions. Thus, both statements are true, and the reason correctly explains the assertion.

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: 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 Conversion - Volume and Time

b) 8.333 minutes
[Solution Description]
First, convert the tank volume from liters to milliliters:
$15 \text{ L} \times 1,000 \text{ mL/L} = 15,000 \text{ mL}$
Next, find the time taken using the given rate:
$\frac{25 \text{ s}}{750 \text{ mL}} \times 15,000 \text{ mL} = 500 \text{ s}$
Convert seconds to minutes:
$\frac{500 \text{ s}}{60 \text{ s/min}} \approx 8.333 \text{ min}$