Key Concept: Proportional Reasoning, Similarity in 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 appear similar due to proportional scaling of their dimensions, which is true as seen from the table (width and height scaled by $\frac{1}{2}$ for C and $1.5$ for D relative to A). The reason correctly explains this by stating that similarity requires equal scaling factors for both dimensions. Both statements are true, and the reason accurately justifies the assertion.

Key Concept: Ratio Comparison

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

Key Concept: Proportional Reasoning, Comparing Ratios

a) Yes, both ratios simplify to $4 : 1$
[Solution Description]
Simplify Worker A's ratio: $24 : 6$ becomes $4 : 1$. Simplify Worker B's ratio: $16 : 4$ also becomes $4 : 1$. Since both ratios are equal, their building rates are proportional.

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: Proportional Scaling

c) $\frac{1}{4}$
[Solution Description]
To preserve similarity, the scaling factor for height must match the width's scaling factor ($\frac{1}{4}$). Thus:
New height = Original height $\times$ Scaling factor = $80 \times \frac{1}{4} = 20$ mm.

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 Change Application

c) 36 inches
[Solution Description]
To preserve similarity, the height must be scaled by the same factor as the width. Scaling factor is $\frac{3}{4}$. Thus, the new height is calculated as $\frac{3}{4} \times 48 = 36$ inches.

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

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: 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: 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: 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: 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: 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: 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: 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, Simplest Form

a) Yes, because both simplify to $4:3$
[Solution Description] First, simplify both ratios to their simplest form. For the first ratio $48:36$, the HCF of 48 and 36 is 12. Dividing both terms by 12 gives $\frac{48}{12} : \frac{36}{12} = 4:3$. For the second ratio $64:48$, the HCF of 64 and 48 is 16. Dividing both terms by 16 gives $\frac{64}{16} : \frac{48}{16} = 4:3$. Since both simplified ratios are equal ($4:3$), the original ratios are proportional.

Key Concept: Simplifying Ratios

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

Key Concept: Proportional Ratios

d) $25 : 18$
[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: Simplifying ratios

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

Key Concept: 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: 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: 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: 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: 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 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: Ratios in Their Simplest Form, Proportional Ratios

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

Key Concept: Proportional ratios, Cross-multiplication

a) $10 : 16$
[Solution Description]
To check proportionality, we can cross-multiply or simplify the ratios. For two ratios to be proportional, their simplest forms must be equal. The simplest form of $5 : 8$ is already $5 : 8$. Now, let's check each option:
- $10 : 16$ simplifies to $5 : 8$ (HCF = 2).
Thus, $10 : 16$ is proportional to $5 : 8$.

Key Concept: Simplifying Ratios Using HCF

a)
[Solution Description]
To simplify the ratio $84:56$, we first find the HCF of 84 and 56.
Factors of 84: 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84
Factors of 56: 1, 2, 4, 7, 8, 14, 28, 56
The highest common factor is 28. Dividing both terms by 28 gives:
$84 \div 28 = 3$
$56 \div 28 = 2$
So the simplified form is $3:2$. Both Assertion and Reason are true, and Reason correctly explains how to simplify the ratio using HCF.

Key Concept: Proportional Ratios

a) $15 : 30$
[Solution Description]
First, simplify the given ratio $5 : 10$. The HCF of 5 and 10 is 5. Dividing both terms by 5:
$5 \div 5 = 1 \\ \\ 10 \div 5 = 2$
The simplified ratio is $1 : 2$. Now, check each option:
\begin{itemize}

Key Concept: Assertion-Reason

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To simplify the ratio $12 : 18$, we find the HCF of 12 and 18, which is 6. Dividing both terms by 6 gives $\frac{12}{6} : \frac{18}{6} = 2 : 3$. Therefore, the Assertion is true. The Reason states that the HCF of 12 and 18 is 6, which is also true and correctly explains why the ratio simplifies to $2 : 3$.

Key Concept: Proportional Ratios

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: 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: Assertion and Reason

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To check if the ratios $6 : 4$ and $9 : 6$ are proportional, we reduce them to their simplest forms.
1. For $6 : 4$, the HCF of 6 and 4 is 2. Dividing both terms by 2 gives $3 : 2$.
2. For $9 : 6$, the HCF of 9 and 6 is 3. Dividing both terms by 3 also gives $3 : 2$.
Since both ratios simplify to the same form $3 : 2$, they are proportional. Thus, both Assertion and Reason are true, and the Reason correctly explains the Assertion.

Key Concept: Missing Term in Proportions

b) 28
[Solution Description]
Since the ratios are proportional, the product of means equals the product of extremes:
$7 \times 48 = 12 \times x \implies 336 = 12x \implies x = \frac{336}{12} = 28$
So, $x = 28$.

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: 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: 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: 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: 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 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

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]
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

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

Key Concept: Ratio Comparison

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

Key Concept: Ratio Comparison, Proportional Reasoning

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To check proportionality, we compare the given ratio $5 : 170$ with the new ratio $8 : 272$. Calculating the cross-products:
$5 \times 272 = 1360$ and $170 \times 8 = 1360$. Since both cross-products are equal, the ratios are proportional. Both Assertion and Reason are true, and Reason correctly explains Assertion.

Key Concept: Ratio comparison, Unit conversion

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

Key Concept: 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: 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

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: Trairasika - Rule of Three

d) 24
[Solution Description]
Using the Rule of Three, identify the proportional relationship and solve for the unknown.
Given ratio: $8 : 12 :: 16 : x$.
Step 1: Find the factor of change between the known terms:
$16 ÷ 8 = 2$
Step 2: Apply the same factor to the second term to find $x$:
$12 × 2 = 24$
Therefore, $x = 24$.

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: 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: 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: 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) 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

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

Key Concept: Cross Multiplication Method

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

Key Concept: 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: Time-Distance, Proportionality

c) 7.5 days
[Solution Description]
This is an inverse proportion problem since more workers will take less time. The product of workers and days remains constant. Let $x$ be the required number of days.
$5 \text{ workers} \times 12 \text{ days} = 8 \text{ workers} \times x \text{ days}$
$x = \frac{5 \times 12}{8} = \frac{60}{8} = 7.5 \text{ days}$

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 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: 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: 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 - 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: 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, Mixed Numbers

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

Key Concept: Rule of Three

c) 4 hours
[Solution Description]
Since both workers have the same efficiency, working together means their rates add up. The first worker's rate is $\frac{1}{8}$ tasks/hour. Combined rate is $\frac{1}{8} + \frac{1}{8} = \frac{2}{8} = \frac{1}{4}$ tasks/hour. Thus, time taken together is $\frac{1}{\frac{1}{4}} = 4$ hours.

Key Concept: Ratio division, profit sharing

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

Key Concept: 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: 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: 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

c) 10 liters
[Solution Description]
Given mixture: Milk = 7 parts, Water = 2 parts, total parts = 9.
1. Current milk quantity = $\frac{7}{9} \times 90 = 70$ liters.
2. Current water quantity = $\frac{2}{9} \times 90 = 20$ liters.
3. Let $x$ liters of water be added; new ratio = 7:3.
4. $\frac{70}{20 + x} = \frac{7}{3}$ ⇒ $210 = 140 + 7x$ ⇒ $7x = 70$ ⇒ $x = 10$.
Thus, 10 liters of water must be added.

Key Concept: Assertion-Reason

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

Key Concept: Dividing quantities in a given ratio

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

Key Concept: Dividing quantities in a given ratio

c) \$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: Finding parts from the whole using ratio

b) 16 and 20
[Solution Description]
The total ratio parts = 4 + 5 = 9. Size of each part = $36 ÷ 9 = 4$. First friend gets $4 × 4 = 16$ chocolates. Second friend gets $5 × 4 = 20$ chocolates.

Key Concept: Sharing 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: 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: Dividing a Quantity in Given Ratio

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

Key Concept: Business Investment

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

Key Concept: Profit sharing, 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 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: Profit Sharing

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

Key Concept: Unit Conversions - Area

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

Key Concept: 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: 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:

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: Length, Area

b) 16146 square feet
[Solution Description]
First, calculate the area in square meters:
$Area = Length \times Width = 50 m \times 30 m = 1500 \text{ m}^2$.
Now, convert square meters to square feet using the conversion factor:
$1 \text{ m}^2 = 10.764 \text{ ft}^2$
So, $1500 \text{ m}^2 \times 10.764 \text{ ft}^2/\text{m}^2 = 16146 \text{ ft}^2$.

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: Age Ratio, Proportion Check

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

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 - 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 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.