Key Concept: Rule of Three, Proportional Change

c) 600 km
[Solution Description]
This is a problem of proportional reasoning where the distance is directly proportional to time when speed is constant.
1. Find the speed: $\frac{450 \text{ km}}{6 \text{ hours}} = 75 \text{ km/h}$.
2. Now, calculate the distance for 8 hours: $75 \text{ km/h} \times 8 \text{ hours} = 600 \text{ km}$.
Thus, the correct answer is 600 km.

Key Concept: Ratio Comparison

d) 1 : 20
[Solution Description]
Given: Teachers = 24, Students = 480.
The ratio of teachers to students is $24 : 480$.
To simplify this ratio, divide both terms by their greatest common divisor (GCD). Here, GCD of 24 and 480 is 24.
Simplifying: $\frac{24}{24} : \frac{480}{24} = 1 : 20$.
Therefore, the simplified ratio is $1 : 20$.

Key Concept: Proportional Reasoning, Simplifying Ratios

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

Key Concept: Proportional Change

a) 90 mm width, 60 mm height
[Solution Description]
To find which image is proportional to Image A (60:40), simplify its ratio by dividing both sides by their HCF (20). Simplified ratio = $60/20 : 40/20 = 3:2$. Now check options:
- Option a) 90:60 → Divide by 30 → $3:2$ (matches).
- Option b) 30:15 → Divide by 15 → $2:1$ (does not match).
- Option c) 80:40 → Divide by 40 → $2:1$ (does not match).
- Option d) 60:60 → Divide by 60 → $1:1$ (does not match).
Thus, only option a) has the same simplified ratio as Image A.

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: Verification of Similarity

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

Key Concept: Proportional Scaling

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

Key Concept: 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: Proportional Ratios

b) $16 : 24$
[Solution Description]
First, simplify the given ratio $12 : 18$ by dividing both terms by their HCF (6):
$\frac{12}{6} : \frac{18}{6} = 2 : 3$
Now check the options for ratios that simplify to $2 : 3$:
$16 : 24 \implies \frac{16}{8} : \frac{24}{8} = 2 : 3$
Thus, $16 : 24$ is proportional to $12 : 18$.

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: 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: 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: 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: 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: 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: 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: 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: 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

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: Dividing a quantity in a given ratio, Proportional reasoning

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

Key Concept: 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: 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: Finding Missing Terms in Proportional Ratios

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

Key Concept: 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 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

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: Using HCF for Simplification

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

Key Concept: Simplifying Ratios Using HCF

c) $3 : 4$
[Solution Description]
To simplify the ratio $36 : 48$, we find the HCF of 36 and 48, which is 12. Dividing both terms by 12:
$36 \div 12 = 3 \\ \\ 48 \div 12 = 4$
The simplified ratio is $3 : 4$.

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: 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: 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) 20 meters
[Solution Description]
Let the length of the shorter rope be $5k$ and the longer rope be $7k$. Given $7k = 28$, solving for $k$ gives $k = 4$. Therefore, the shorter rope's length is $5k = 5 \times 4 = 20$ meters.

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: Simplifying Ratios

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

Key Concept: Simplification and Proportionality

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

Key Concept: Proportional 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: Concept of proportionality using simplest forms, Rule of Three

b) 8 days
[Solution Description]
This problem uses inverse proportionality because more workers will take fewer days to complete the same work.
The relationship is given by $W_1 × D_1 = W_2 × D_2$.
Here, $W_1 = 8$ workers, $D_1 = 12$ days, $W_2 = 12$ workers.
Solve for $D_2$:
$8 × 12 = 12 × D_2$
$96 = 12D_2$
$D_2 = 96 / 12 = 8$ days.

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: 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 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: 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 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: Ratio Comparison

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

Key Concept: Ratio 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 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 Comparison

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

Key Concept: Simplification of Ratios

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

Key Concept: Proportional Reasoning Application

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

Key Concept: 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, 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: 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: 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: Ratio Application

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

Key Concept: Proportional Ratios

a) $15 : 21$
[Solution Description]
Two ratios are proportional if their simplest forms are equal. The ratio $5 : 7$ is already in its simplest form. Check each option by simplifying it:
Option a) $15 : 21$ simplifies to $5 : 7$ (HCF = 3).
Option b) $10 : 14$ simplifies to $5 : 7$ (HCF = 2).
Option c) $20 : 28$ simplifies to $5 : 7$ (HCF = 4).
Option d) $25 : 30$ simplifies to $5 : 6$ (HCF = 5).
Thus, options a), b), and c) are proportional to $5 : 7$, but only one is needed as the correct answer. Here, $15 : 21$ is chosen.

Key Concept: Trairasika — The Rule of Three

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

Key Concept: Trairasika — The Rule of Three

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

Key Concept: Proportion Application

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

Key Concept: 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: Cross Multiplication Method

b) 21
[Solution Description]
Given the proportion $5:7 :: 15:x$. Using cross multiplication, we get:
$5 \times x = 7 \times 15$
Solving for $x$:
$x = \frac{7 \times 15}{5} = \frac{105}{5} = 21$

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) 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: Mixing ratios, Proportionality

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

Key Concept: Cross Multiplication Method, Proportional Ratios

c) 35 kg
[Solution Description]
First, establish the given ratio: $7:5$ (cement:sand).
For the first case: $7/5 = 35/x$, solve for x using cross-multiplication:
$7x = 5 \times 35$
$7x = 175$
$x = 175 / 7 = 25$ kg sand.
Now, apply this proportion to the new amount of cement (49 kg):
$7/5 = 49/y$
Cross-multiplying: $7y = 5 \times 49$
$7y = 245$
$y = 245 / 7 = 35$ kg sand required.

Key Concept: Rule of Three Proportion

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

Key Concept: Proportional Ratios

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

Key Concept: 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: 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, 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: Proportionality

b) \$300
[Solution Description]
Using the Rule of Three:
$pramāṇa = 15$, $phala = 225$, $ichchhā = 20$. So,
$ichchhāphala = \frac{phala \times ichchhā}{pramāṇa} = \frac{225 \times 20}{15} = 300$
Hence, 20 notebooks cost \$300.

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 adjustment, mixture problem

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

Key Concept: Mixture Ratio

b) 70 kg
[Solution Description]
Total parts in the ratio = $7 + 1 = 8$. Size of each part = $\frac{80}{8} = 10$ kg. Sugar corresponds to 7 parts, so quantity of sugar = $7 \times 10 = 70$ kg.

Key Concept: Ratio Division

b) 50 and 10
[Solution Description]
First, find the total number of parts in the ratio: $5 + 1 = 6$. Next, divide the total chocolates by the total parts to find the size of each part: $\frac{60}{6} = 10$. Now, multiply the number of parts for each student by the size of each part. The first student gets $5 \times 10 = 50$ chocolates, and the second student gets $1 \times 10 = 10$ chocolates.

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: 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

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

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

Key Concept: Ratio division method

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

Key Concept: Finding parts from the whole using ratio

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: 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: Profit sharing and ratio application

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

Key Concept: Profit Sharing

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

Key Concept: 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: Profit Sharing and Ratio Application

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To verify the assertion, we calculate the shares of A and B using the given ratio 5:3. The total profit is \Rs.12,000. The total parts of the ratio are 5 + 3 = 8. Each part corresponds to \Rs.12,000 / 8 = \Rs.1,500. Partner A's share is 5 × \Rs.1,500 = \Rs.7,500, which matches the Assertion. The Reason correctly explains the method used to calculate the shares, as it describes the formula where the total profit is multiplied by the individual ratio divided by the sum of ratio parts.
Therefore, both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

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 Conversion - Area

b) 12.355 acres
[Solution Description]
Given the conversion factor: $1 \text{ hectare} = 2.471 \text{ acres}$. To convert 5 hectares to acres:
$5 \text{ hectares} \times 2.471 \text{ acres/hectare} = 12.355 \text{ acres}$
Therefore, 5 hectares is equal to 12.355 acres.

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: 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: 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: 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:

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:

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

Key Concept: 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: 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: 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 Consistency

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