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: Proportional Reasoning, Similarity of Rectangles

d) Width = 48 cm, height = 32 cm
[Solution Description]
To find a similar rectangle, the ratio of width to height must be the same as the original rectangle. The original rectangle has a ratio of $\frac{24}{16} = \frac{3}{2}$. We need to check which option has the same ratio.
1. For width = 36 cm, height = 24 cm: $\frac{36}{24} = \frac{3}{2}$.
2. For width = 30 cm, height = 18 cm: $\frac{30}{18} = \frac{5}{3}$.
3. For width = 20 cm, height = 10 cm: $\frac{20}{10} = 2$.
4. For width = 48 cm, height = 32 cm: $\frac{48}{32} = \frac{3}{2}$.
Options 1 and 4 have the same ratio as the original rectangle.

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: 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: 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: Ratio comparison, Multiple images

a) Only A and B are similar
[Solution Description]
Let's simplify all ratios:
A: 84:56 = 3:2 (divided by 28)
B: 126:84 = 3:2 (divided by 42)
C: 35:25 = 7:5 (divided by 5)
Images A and B have identical simplified ratio (3:2), while C has different ratio (7:5).
Therefore, only A and B are similar.

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: Non-Proportional Scaling

c) The width is halved while the height remains unchanged.
[Solution Description]
Non-proportional scaling occurs when the width and height are scaled by different factors, causing distortion in the image. For example, if the width is doubled but the height remains the same, the image will appear stretched.

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: Ratio Application

b) 25
[Solution Description]
Given the ratio $3 : 5$ for apples to oranges, let the number of oranges be $x$. Using proportionality: $\frac{3}{5} = \frac{15}{x}$. Cross-multiply to get $3x = 75$. Solving, $x = 25$. Thus, there are 25 oranges.

Key Concept: Ratios, Proportionality

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The Assertion states that if two ratios $\frac{a}{b}$ and $\frac{c}{d}$ are proportional, then they are equal. This is true because proportionality of ratios means they represent the same relationship, hence their simplified forms are identical.
The Reason states that two ratios are proportional if and only if their cross-products are equal ($ad = bc$). This is also true as it is the fundamental condition for checking proportionality. Furthermore, the Reason correctly explains the Assertion because the equality of cross-products implies the equality of the ratios.
Therefore, both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

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: Definition and Simplest Form of Ratios

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To determine if the ratios $12:18$ and $20:30$ are proportional, we reduce them to their simplest forms.
For $12:18$, the HCF of 12 and 18 is 6. Dividing both terms by 6 gives $\frac{12}{6} : \frac{18}{6} = 2:3$.
For $20:30$, the HCF of 20 and 30 is 10. Dividing both terms by 10 gives $\frac{20}{10} : \frac{30}{10} = 2:3$.
Since both ratios simplify to $2:3$, they are proportional. The Reason correctly explains the Assertion as proportionality is determined by identical simplest forms.

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) $6 : 8$
[Solution Description]
Two ratios are proportional if their simplest forms are equal or if $ad = bc$. Simplifying $12 : 16$ gives $3 : 4$ (divided by HCF 4). Now check each option:
- Option a) $6 : 8$ simplifies to $3 : 4$ (proportional).
- Option b) $9 : 12$ simplifies to $3 : 4$ (proportional).
- Option c) $15 : 20$ simplifies to $3 : 4$ (proportional).
- Option d) $24 : 28$ simplifies to $6 : 7$ (not proportional).

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: 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: Proportional Ratios

a) $8 : 12$
[Solution Description]
Simplify $4 : 6$ by dividing both terms by their HCF, which is 2. This gives $2 : 3$. Now check which option simplifies to $2 : 3$. Option a) $8 : 12$ simplifies to $2 : 3$ (divided by 4). So, $4 : 6 :: 8 : 12$.

Key Concept: Rule of Three

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

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

Key Concept: Simplifying ratios

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

Key Concept: Proportional Ratios

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

Key Concept: Proportional Ratios

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

Key Concept: 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: 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, 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: 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, 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

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: 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: HCF Calculation

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

Key Concept: Assertion-Reason

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To check if the ratios are proportional, simplify both ratios to their simplest form:
\begin{itemize}

Key Concept: Equivalence of ratios in simplest form

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

Key Concept: Proportionality using simplest forms

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

Key Concept: 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: Simplification and Proportionality

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

Key Concept: Concept of proportionality using simplest forms, 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: 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 Division

c) \$450 and \$750
[Solution Description]
To divide \$1,200 in the ratio 3:5:
$\text{Total parts} = 3 + 5 = 8 \\ \\ \text{First person's share} = \frac{3}{8} \times 1200 = 450 \\ \\ \text{Second person's share} = \frac{5}{8} \times 1200 = 750$
The shares are \$450 and \$750.

Key Concept: Proportional Reasoning

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 Reasoning, Unit Conversions

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

Key Concept: 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: 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 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: Assertion-Reason

c) Assertion is true, but Reason is false.
[Solution Description]
Let the initial quantity of juice be $3x$ litres and water be $5x$ litres. After adding 2 litres of juice, the new quantity of juice becomes $3x + 2$ litres. The new ratio is given as $1 : 1$, so we can write:
$\frac{3x + 2}{5x} = \frac{1}{1}$
Solving this equation:
$3x + 2 = 5x \\ \\ 2 = 2x \\ \\ x = 1$
Therefore, the initial quantities were 3 litres of juice and 5 litres of water. The assertion is true. However, the reason states that adding juice changes the ratio proportionally, which is incorrect because proportional change would require multiplying both terms by the same factor, not adding. Thus, the reason is false.

Key Concept: 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 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, 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 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: Profit Sharing

b) \$8,000
[Solution Description]
The investment ratio is $4 : 5 : 6$.
Total parts = $4 + 5 + 6 = 15$.
Share of the friend who invested the least (4 parts):
$\text{Share} = \frac{4}{15} \times 30000 = 8000$.

Key Concept: Proportional Ratios

d) 18
[Solution Description]
To solve for $x$, we set up the proportion $12 : 18 :: x : 27$.
Cross-multiplying gives: $12 \times 27 = 18 \times x$.
Simplifying, we get $324 = 18x$.
Dividing both sides by 18: $x = 18$.

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: 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]
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: Comparative Proportion

b) No, the second shop is cheaper
[Solution Description]
To check proportionality, compare the cost per notebook for both shops.
First shop: Cost per notebook = $\frac{\$120}{3} = \$40$ per notebook.
Second shop: Cost per notebook = $\frac{\$190}{5} = \$38$ per notebook.
Since $\$40 \neq \$38$, the two ratios are not proportional. The second shop offers notebooks at a cheaper rate (\$38 vs. \$40), so it is the better deal.

Key Concept: Cross Multiplication

b) 112.5 km
[Solution Description]
First, convert all time units to the same measure (minutes). 1 hour and 15 minutes is equal to 75 minutes.
Given the ratio of distance to time is proportional:
$30 : 45 :: 75 : x$.
Using cross multiplication:
$30 \times x = 45 \times 75$
$x = \frac{45 \times 75}{30}$
$x = \frac{3375}{30} = 112.5$
So, the car will cover 112.5 km in 1 hour and 15 minutes.

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: 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: Proportionality Check

c) $4 \times 14 = 7 \times 8$
[Solution Description]
To verify proportionality, we use the cross multiplication rule:
$4 \times 14 = 7 \times 8$
Calculating both sides:
$56 = 56$
Since both sides are equal, the ratios are proportional.

Key Concept: 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: Cross Multiplication

c) 8
[Solution Description]
Using the cross-multiplication method for proportions:
Given proportion is $4 : 9 :: x : 18$.
This can be written as $\frac{4}{9} = \frac{x}{18}$.
Cross-multiplying gives $4 \times 18 = 9 \times x$.
Calculating:
$72 = 9x$.
Solving for $x$:
$x = 72 / 9 = 8$.
Thus, the value of $x$ is 8.

Key Concept: Proportional 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: 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

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

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: 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: 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: Cross Multiplication

c) 8 liters
[Solution Description]
Given: $pramāṇa = 5$, $phala = 20$, $ichchhā = 32$.
Using Āryabhaṭa's Rule of Three:
$ichchhāphala = \frac{phala \times ichchhā}{pramāṇa} = \frac{20 \times 32}{5} = 128$
Therefore, 8 liters are needed.

Key Concept: Sharing in given ratio

b) 12 and 6
[Solution Description]
Total chocolates = 18
Ratio = 2:1
Total parts = 2 + 1 = 3
Size of each part = $18 \div 3 = 6$
First friend's share = $2 \times 6 = 12$ chocolates
Second friend's share = $1 \times 6 = 6$ chocolates.

Key Concept: Ratio 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: 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: 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: Dividing quantities in a given ratio, adjusting ratios

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

Key Concept: Dividing quantities in a given ratio

b) 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) 20 liters
[Solution Description]
Given ratio $2 : 3$, total parts = $2 + 3 = 5$.
$\text{Quantity of blue paint} = \frac{2}{5} \times 50 = 20 \text{ liters}$
The quantity of blue paint needed is 20 liters.

Key Concept: 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: Ratio mixture adjustment

c) 80 mL
[Solution Description]
Initially, alcohol = $\frac{4}{9} \times 360 = 160$ mL, water = $\frac{5}{9} \times 360 = 200$ mL. For new ratio 4:7, required water = $\frac{7}{4} \times 160 = 280$ mL. Water to add = $280 - 200 = 80$ mL.

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: 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: 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: Combined ratio application

a) 3:2
[Solution Description]
Assume 5 units from each mixture.
From A: Blue = $\frac{2}{5} \times 5 = 2$ units, White = $\frac{3}{5} \times 5 = 3$ units.
From B: Blue = $\frac{4}{5} \times 5 = 4$ units, White = $\frac{1}{5} \times 5 = 1$ unit.
Combined Blue = 2 + 4 = 6 units.
Combined White = 3 + 1 = 4 units.
New ratio = 6:4 = 3:2 after simplifying by 2.

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, 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: 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: 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: Unit Conversion - Length

c) 32.81 feet
[Solution Description]
Using the conversion factor: $1 \text{ metre} = 3.281 \text{ feet}$. To convert 10 metres to feet:
$10 \text{ metres} \times 3.281 \text{ feet/metre} = 32.81 \text{ feet}$
Hence, 10 metres is equal to 32.81 feet.

Key Concept: Unit cost calculation, Area conversion

c) \$1,721,649
[Solution Description]
First, find the price per acre: $\frac{\$250,000}{0.75} \approx \$333,333.33 \text{ per acre}$.
Calculate the area of the new plot in square feet: $500 \text{ ft} \times 450 \text{ ft} = 225,000 \text{ sq ft}$.
Convert this area to acres: $\frac{225,000}{43,560} \approx 5.165 \text{ acres}$.
Multiply by the cost per acre: $5.165 \times \$333,333.33 \approx \$1,721,649.48$.

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: 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: Area conversion, Temperature conversion

d) Assertion is false, but Reason is true.
[Solution Description]
To verify the assertion, we need to convert 1 hectare to square feet.
Step 1: We know that 1 hectare = 10,000 square meters (from syllabus).
Step 2: Using the conversion factor, 1 square meter = 10.764 square feet.
Step 3: So, 1 hectare = 10,000 square meters × 10.764 square feet/square meter = 107,640 square feet.
The assertion states 107,639 square feet, which is not equal to 107,640 square feet (difference of 1 square foot). So, the Assertion is false.
However, the Reason statement about the conversion factor is correct as per the syllabus.

Key Concept: Volume Conversion

d) 3,000 mL
[Solution Description]
Given that $1 \text{ litre} = 1,000 \text{ mL}$, multiply the number of litres by 1,000 to get the volume in millilitres:
$3 \text{ litres} \times 1,000 \text{ mL/litre} = 3,000 \text{ mL}$
Hence, 3 litres is equal to 3,000 mL.

Key Concept: Unit Conversion - Proportionality

b) \$40,173
[Solution Description]
First, find the cost per square foot:
$\frac{\$50,000}{43,560 \text{ sq ft}} \approx \$1.1478 \text{ per sq ft}$
Multiply by the area to get the total cost:
$35,000 \text{ sq ft} \times \$1.1478/\text{sq ft} \approx \$40,173$
Rounding to the nearest dollar gives \$40,173.

Key Concept: 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 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.