Key Concept: Proportional Reasoning in Image Dimensions

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
For images to appear similar, their corresponding dimensions must change by the same multiplicative factor. In image A (60 mm × 40 mm), image C (30 mm × 20 mm) has both width and height scaled down by a factor of $\frac{1}{2}$. Similarly, image D (90 mm × 60 mm) is scaled up by a factor of $\frac{3}{2}$ from A. Since the scaling factors are equal for width and height in these cases, the shapes remain similar. However, image B (40 mm × 20 mm) has height scaled by $\frac{1}{2}$ but width scaled by $\frac{2}{3}$, making it distorted. The reason correctly explains why proportional scaling preserves similarity.

Key Concept: Proportional Change in Dimensions

b) 10 cm × 6 cm
[Solution Description]
Original dimensions: Width = 50 cm, Height = 30 cm.
Scaling factor = $\frac{1}{5}$.
New width = $50 \times \frac{1}{5} = 10$ cm.
New height = $30 \times \frac{1}{5} = 6$ cm.
Thus, the new dimensions after scaling are 10 cm (width) and 6 cm (height).

Key Concept: Simplified Ratio

c) $3:2$
[Solution Description]
To simplify the ratio $30:20$, divide both numbers by their HCF (10):
$30 / 10 : 20 / 10 = 3:2$
The simplified ratio is $3:2$.

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 proportionality, Similarity of images

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
First, we check the assertion:
- For Image A, width-to-height ratio is $\frac{60}{40} = \frac{3}{2}$.
- For Image D, width-to-height ratio is $\frac{90}{60} = \frac{3}{2}$.
Since both ratios simplify to the same proportion ($3 : 2$), the assertion is true.
Now, we check the reason:
- Scaling factor for width: $\frac{90}{60} = 1.5$.
- Scaling factor for height: $\frac{60}{40} = 1.5$.
Both dimensions are scaled by the same factor (1.5), so the reason is also true and correctly explains the assertion.

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, 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: Multiplicative vs additive changes, Proportional changes

b) Add 12 cm to both width and height
[Solution Description]
To preserve shape similarity, both dimensions must be scaled by the same multiplicative factor. Let's examine each case:
Option A: Multiply width by $\frac{1}{2}$ (24 cm) and height by $\frac{1}{2}$ (18 cm) → Same scaling factor → Similar
Option B: Add 12 cm to both (60 cm × 48 cm) → Different scaling factors: width scales by $\frac{60}{48} = 1.25$, height scales by $\frac{48}{36} \approx 1.33$ → Not similar
Option C: Scale both by 1.5 (72 cm × 54 cm) → Same scaling factor → Similar
Option D: Divide both by 4 (12 cm × 9 cm) → Same scaling factor → Similar
Only option B involves additive change which doesn't preserve similarity.

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: 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: 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: Proportionality, Simplifying Ratios

b) 25
[Solution Description]
To find the value of $x$, we use the property of proportional ratios: $a : b :: c : d$ implies $ad = bc$. Here, $16 : x :: 64 : 100$. So, $16 \times 100 = x \times 64$. Solving for $x$:
$1600 = 64x \\ \\ x = \frac{1600}{64} = 25$
Thus, $x = 25$.

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: Simplest Form of Ratio

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

Key Concept: 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: 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: 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: 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: Assertion-Reason

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

Key Concept: Problem solving with proportional reasoning, Unit conversions

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

Key Concept: Simplifying ratios

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To simplify the ratio $60 : 40$, we find the HCF of 60 and 40, which is 20. Dividing both terms by 20 gives $60 \div 20 = 3$ and $40 \div 20 = 2$. Therefore, the simplified ratio is $3 : 2$. Both the Assertion and Reason are true, and the Reason correctly explains the Assertion as it provides the correct HCF used for simplification.

Key Concept: Age Ratio Problem

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

Key Concept: Simplifying Ratios

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

Key Concept: Proportional ratios with fractions

c) 15 and 6
[Solution Description]
First simplify $15 : 25$ to $3 : 5$ by dividing by HCF 5.
For $9 : x$ to be proportional to $3 : 5$, set $\frac{9}{x} = \frac{3}{5}$.
Cross-multiply: $3x = 45 \Rightarrow x = 15$.
For $y : 10$ to be proportional to $3 : 5$, set $\frac{y}{10} = \frac{3}{5}$.
Cross-multiply: $5y = 30 \Rightarrow y = 6$.

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: Assertion and Reason

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

Key Concept: Ratios in Their Simplest Form

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

Key Concept: 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: Simplifying ratios using HCF, Proportionality of ratios

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To check proportionality, we simplify both ratios:
- For $45 : 30$, the HCF of 45 and 30 is 15. Dividing both terms by 15 gives $\frac{45}{15} : \frac{30}{15} = 3 : 2$.
- For $60 : 40$, the HCF of 60 and 40 is 20. Dividing both terms by 20 gives $\frac{60}{20} : \frac{40}{20} = 3 : 2$.
Both simplified forms are the same ($3 : 2$), so the assertion is true. The reason correctly explains why the two ratios are proportional, as it defines proportionality based on identical simplest forms obtained through HCF division.

Key Concept: Simplifying ratios, Proportionality

b) $2 : 3$
[Solution Description]
To simplify the ratio $28 : 42$, find the HCF of 28 and 42. The HCF is 14. Divide both terms by 14: $28 \div 14 = 2$ and $42 \div 14 = 3$. 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: 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: 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: 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: 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: Assertion-Reason

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To check if the ratios are proportional, we simplify both ratios to their simplest form.
Step 1: Simplify $6:4$. The HCF of 6 and 4 is 2. Dividing both terms by 2 gives $3:2$.
Step 2: Simplify $9:6$. The HCF of 9 and 6 is 3. Dividing both terms by 3 also gives $3:2$.
Since both simplified ratios are equal ($3:2$), they are proportional. Additionally, the Reason correctly explains why the Assertion is true.

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: Cross-Multiplication Method, Simplest Form

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

Key Concept: 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: Ratio Application

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

Key Concept: Proportional 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

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: Proportional Division

d) Rs.4,500
[Solution Description]
The total parts in the ratio $3 : 5$ are $3 + 5 = 8$. The larger share corresponds to $\frac{5}{8}$ of the total amount. Thus, $\frac{5}{8} \times 7200 = 4500$. The larger share is Rs.4,500.

Key Concept: Ratio proportion, Age problems

b) 12
[Solution Description]
Let present ages be $3x$ and $5x$.
After 6 years: $(3x + 6)/(5x + 6) = 9/13$.
Cross-multiplying: $39x + 78 = 45x + 54$.
Solving gives $6x = 24$, so $x = 4$.
Younger sibling's age is $3x = 12$ years.

Key Concept: Ratio Application

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

Key Concept: Ratio 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: Proportional Reasoning Application

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

Key Concept: Proportional 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: 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: Problem Solving with Ratios

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

Key Concept: 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: Proportional reasoning, real-world application

b) 12 cups
[Solution Description]
Given ratio of flour to sugar is $4:3$. Let the required flour be $x$ cups for 9 cups of sugar.
$4 : 3 :: x : 9$
Cross-multiplying:
$3x = 36 \implies x = 12$
So, 12 cups of flour are needed.

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: Compound proportion with inverse relationship

c) 26 days
[Solution Description]
Total work = $15 \text{ workers} \times 28 \text{ days} \times 6 \text{ hours} = 2520$ worker-hours.
New scenario: Let $d$ be the required days. Then $20 \times d \times 5 = 2520$.
Solving: $100d = 2520 \Rightarrow d = \frac{2520}{100} = 25.2$ days ≈ 26 days.

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

Key Concept: 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: Rule of Three, Cross Multiplication

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

Key Concept: 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: 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: 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) 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: Proportional Ratios

c) 28
[Solution Description]
Represent the problem as the proportion $3 : 12 :: 7 : x$.
Step 1: Apply cross multiplication:
$3 \times x = 12 \times 7$
Step 2: Multiply the known terms:
$3x = 84$
Step 3: Solve for $x$ by dividing both sides by 3:
$x = \frac{84}{3} = 28$
The farmer will need 28 kg of seeds for a 7-acre field.

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: 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 - 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, 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: 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: Sharing profit based on investment ratio

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

Key Concept: 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: Combined ratio, multi-step problem

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

Key Concept: 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: 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

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

Key Concept: 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: 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 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: Sharing Profit Based on Investment Ratio

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

Key Concept: Finding parts from the whole using ratio

c) 36
[Solution Description]
Total ratio parts = 3 + 2 = 5. Size of each part = $60 ÷ 5 = 12$. First child gets $3 × 12 = 36$ marbles.

Key Concept: Profit sharing, Ratio division

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

Key Concept: Profit Sharing 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: Mixture problem, Ratio adjustment

b) 21:4
[Solution Description]
Initial sugar = $\frac{4}{5} \times 60 = 48$ kg.
Initial salt = $\frac{1}{5} \times 60 = 12$ kg.
After adding 15 kg sugar, new sugar = 48 + 15 = 63 kg.
New ratio = 63:12 = 21:4 when simplified by dividing by 3.

Key Concept: Mixture Problem

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

Key Concept: 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: Temperature Conversion

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

Key Concept: 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, Real-world application

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To verify the assertion, we need to calculate the amount of manure required for 1 hectare of land given that 10 tonnes of manure are needed for 1 acre.
Step 1: From the syllabus, we know that 1 hectare = 2.471 acres.
Step 2: The manure requirement is 10 tonnes per acre. Therefore, for 1 hectare (which is 2.471 acres), the total manure required is:
$10 \text{ tonnes/acre} \times 2.471 \text{ acres} = 24.71 \text{ tonnes}$
The assertion states that 1 hectare requires exactly 24.71 tonnes of manure, which matches our calculation. The reason correctly states that 1 hectare is equal to 2.471 acres, and this is indeed the correct explanation for the assertion.

Key Concept: Temperature Conversion

c) $20^\circ C$
[Solution Description]
Given the Fahrenheit temperature is $68^\circ F$, substitute into the formula:
$Celsius = \frac{5}{9} \times (68 - 32) = \frac{5}{9} \times 36 = 20^\circ C$
Therefore, $68^\circ F$ is equal to $20^\circ C$.

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: 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: 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: Time and Volume Conversion

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

Key Concept: Mass Ratio, Unit Consistency

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

Key Concept: Unit Conversion - Area

b) 4.1322 acres
[Solution Description]
First, calculate the area of the plot in square feet:
$300 \text{ ft} \times 600 \text{ ft} = 180,000 \text{ sq ft}$
Next, convert square feet to acres using the conversion factor:
$\frac{180,000 \text{ sq ft}}{43,560 \text{ sq ft/acre}} \approx 4.1322 \text{ acres}$

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.