Key Concept: Proportional Reasoning in Image Dimensions

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

Key Concept: Proportional Scaling

b) 10
[Solution Description]
The given ratio is 4 bags per 10 meters.
For 25 meters, let the required number of bags be $b$.
Therefore, $\frac{4}{10} = \frac{b}{25}$.
Solving for $b$, $b = \frac{4 \times 25}{10} = 10$ bags.

Key Concept: Non-Proportional Change

c) 30 mm width, 10 mm height
[Solution Description]
Distortion occurs when width and height do not change proportionally. First, simplify Image A's ratio: $60:40 = 3:2$ (divided by 20). Now analyze options:
- Option a) 120:80 → Simplified ratio $3:2$ (proportional).
- Option b) 45:30 → Simplified ratio $3:2$ (proportional).
- Option c) 30:10 → Simplified ratio $3:1$ (non-proportional).
- Option d) 90:60 → Simplified ratio $3:2$ (proportional).
Only option c) has a different simplified ratio ($3:1$) and will appear distorted.

Key Concept: Proportional Reasoning, Real-World Application

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

Key Concept: Width-height ratio, Simplest form

a) 72:48
[Solution Description]
First, we simplify the given ratio 48:32 by dividing both terms by their HCF 16 to get 3:2. Now we check which option has the same simplest form.
Option a) 72:48 simplifies to 3:2 (dividing by 24)
Option b) 64:32 simplifies to 2:1 (dividing by 32)
Option c) 90:45 simplifies to 2:1 (dividing by 45)
Option d) 120:100 simplifies to 6:5 (dividing by 20)
Only option a) has the same simplest form as the given image.

Key Concept: Width-Height Ratio

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

Key Concept: Proportional changes, Multiplicative vs additive

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

Key Concept: Proportional Reasoning

b) No, because the scaling factors for width and height are different
[Solution Description]
To check for similarity, compare the ratios of width to height for both images. For Image X: $80 : 60$ simplifies to $4 : 3$. For Image Y: $40 : 20$ simplifies to $2 : 1$.
Since $4 : 3 \neq 2 : 1$, the images are not similar because their proportional dimensions differ.

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: 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: 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: Ratios, Proportionality

b) 6 : 9
[Solution Description]
First, simplify the given ratio $8 : 12$ by dividing both terms by their HCF, which is 4:
$8 : 12 = 2 : 3$
Now, check each option to see if it simplifies to $2 : 3$:
\begin{itemize}

Key Concept: 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: Simplest Form of Ratios

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

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: 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 and Simplest Form

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To determine if the two ratios are proportional, we simplify them to their simplest form:
1) For $45:30$, the HCF of 45 and 30 is 15. Dividing both terms by 15 gives the simplified ratio $3:2$.
2) For $120:80$, the HCF of 120 and 80 is 40. Dividing both terms by 40 gives the simplified ratio $3:2$.
Since both ratios simplify to $3:2$, they are proportional. The Reason correctly explains the Assertion as it demonstrates that the simplification leads to the same ratio, confirming proportionality.

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: Maintaining Proportions, Problem Solving

b) 25 cups
[Solution Description] The given ratio is $5:7$ (flour to water). To find the required flour for 35 cups of water, set up the proportion $5:7 :: x:35$. The factor of change is $\frac{35}{7} = 5$. Multiply the flour quantity by this factor: $5 \times 5 = 25$ cups of flour.

Key Concept: Simplifying ratios, Proportional reasoning

b) $2 : 3$
[Solution Description]
To simplify the ratio $8 : 12$, find the HCF of 8 and 12, which is 4. Divide both terms by 4: $\frac{8}{4} = 2$ and $\frac{12}{4} = 3$. Thus, the simplest form is $2 : 3$.

Key Concept: 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 $60 : 40$, 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: Ratio Simplification

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

Key Concept: Age Ratio Application

b) $1 : 3$
[Solution Description]
When Rohan is 6, his father’s age is $5 \times 6 = 30$ years. After 6 years, Rohan will be 12 and his father will be 36. The ratio is $12 : 36$, which simplifies to $1 : 3$ when divided by their HCF, 12.

Key Concept: Assertion and Reason

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

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

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

Key Concept: 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: Proportional Ratios

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

Key Concept: Simplifying Ratios 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

c) 15 : 24
[Solution Description]
A ratio proportional to $5 : 8$ must simplify to the same value. Check each option by dividing both terms by their HCF.
For $15 : 24$, the HCF is 3:
$\frac{15}{3} = 5 \quad \text{and} \quad \frac{24}{3} = 8$
Hence, $15 : 24$ simplifies to $5 : 8$ and is proportional.

Key Concept: Equivalence of ratios in simplest form

d) 40 : 56
[Solution Description]
Simplify $25 : 35$ to its simplest form: divide both terms by 5 to get $5 : 7$. Now check each option:
- $10 : 14$ simplifies to $5 : 7$ (proportional).
- $30 : 42$ simplifies to $5 : 7$ (proportional).
- $50 : 70$ simplifies to $5 : 7$ (proportional).
- $40 : 56$ simplifies to $5 : 7$ (proportional).
However, $40 : 56$ is proportional as well, but since all options seem proportional, rechecking: The correct non-proportional ratio would be $15 : 21$ which simplifies to $5 : 7$ (all given options are proportional). Thus, the question may need adjustment, but assuming the intent was to test recognition of simplified forms.

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

c) Yes, because both speeds are equal
[Solution Description]
To check if the speed ratios are proportional, first find the speed of each car. Speed is calculated as distance divided by time.
First car's speed = $240 \text{ km} / 4 \text{ h} = 60 \text{ km/h}$.
Second car's speed = $360 \text{ km} / 6 \text{ h} = 60 \text{ km/h}$.
The ratio of speeds for both cars is $60 : 60$, which simplifies to $1 : 1$. Since both ratios are equal, they are proportional.

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: 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: Compound ratios, proportional reasoning

c) 1:1
[Solution Description]
Calculate worker-days for each wall:
Wall X: 4 workers × 15 days = 60 worker-days
Wall Y: 6 workers × 10 days = 60 worker-days
Since both walls require same worker-days, their efficiencies are equal.
Ratio is therefore 1:1.

Key Concept: Proportional reasoning, unit conversion

c) 3 days
[Solution Description]
First find production rate: 1200 units/8 hours = 150 units/hour.
Daily production with 9 hours: 150 × 9 = 1350 units/day.
Days needed for 3150 units: 3150 ÷ 1350 = 2.33 days → 3 full days required (can't have partial day).

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 Application

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

Key Concept: Ratio 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: 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: 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: Identifying proportional relationships

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To check if the ratios $4 : 5$ and $16 : 20$ are proportional, we simplify both ratios to their simplest form.
The ratio $4 : 5$ is already in its simplest form as the HCF of 4 and 5 is 1.
For the ratio $16 : 20$, we find the HCF of 16 and 20, which is 4. Dividing both terms by 4, we get $4 : 5$.
Since both ratios simplify to $4 : 5$, they are proportional.
The assertion is true, and the reason correctly explains why the assertion is true because it states the condition for proportionality (simplest forms being equal).

Key Concept: Proportional Relationships

a) Yes, both simplify to $3 : 4$
[Solution Description]
To determine if the ratios are proportional, simplify both to their simplest forms.
Step 1: Simplify $15 : 20$. Divide both terms by their HCF, which is 5:
$15 ÷ 5 = 3 \\ \\ 20 ÷ 5 = 4 \\ \\ \text{Simplified ratio} = 3 : 4$
Step 2: Simplify $18 : 24$. Divide both terms by their HCF, which is 6:
$18 ÷ 6 = 3 \\ \\ 24 ÷ 6 = 4 \\ \\ \text{Simplified ratio} = 3 : 4$
Since both ratios simplify to $3 : 4$, they are proportional.

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, Rule of three

c) 360 widgets
[Solution Description]
First, find the production rate per hour by dividing the number of widgets by the time taken: $\frac{150 \text{ widgets}}{5 \text{ hours}} = 30$ widgets/hour. Then, multiply this rate by the new time to find the total widgets produced: $30 \text{ widgets/hour} \times 12 \text{ hours} = 360$ widgets.

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: Proportional reasoning, real-world application

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

Key Concept: 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: 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: 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 proportional analysis

b) Factory A is more efficient
[Solution Description]
Factory A good units = $500 \times 0.75 = 375$ in 3 hours → 125/hour.
Factory B good units = $800 \times 0.70 = 560$ in 5 hours → 112/hour.
Factory A has higher efficiency (125 vs 112 good units/hour).

Key Concept: Trairasika — The Rule of Three, 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: 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: Proportionality Check

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

Key Concept: Cross Multiplication Basics

c) 15
[Solution Description]
We are given the proportion $3 : 5 :: 9 : x$. Using the cross multiplication rule, we have:
$3 \times x = 5 \times 9$
Solving for $x$:
$3x = 45 \implies x = \frac{45}{3} = 15$
Therefore, the value of $x$ is 15.

Key Concept: 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

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: 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 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: 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

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: Rule of Three, Unit Conversion

b) 112.5 palas
[Solution Description]
Given: Pramāṇa = 5 palas, Phala = $\frac{2}{3}$ niskas, Ichchhā = 15 niskas. Using Āryabhaṭa's Rule of Three:
$ichchhāphala = \frac{phala \times ichchhā}{pramāṇa} = \frac{\frac{2}{3} \times 15}{5} = \frac{10}{5} = 2$
However, the calculation is incorrect as we need to find the quantity of rice (palas) for 15 niskas. Let’s re-express the proportion correctly: $5 : \frac{2}{3} :: x : 15$. Cross-multiplying gives $\frac{2}{3}x = 5 \times 15$, so $x = \frac{75 \times 3}{2} = 112.5$ palas.

Key Concept: Rule of Three, Inverse Proportion

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

Key Concept: Rule of Three 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, Proportionality

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that doubling the pramāṇa halves the ichchhāphala when the phala and ichchhā are unchanged. The formula provided in the reason confirms this relationship as $ichchhāphala$ is inversely proportional to $pramāṇa$. Hence, both the assertion and reason are true, and the reason correctly explains the assertion.

Key Concept: Sharing profit based on investment ratio

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

Key Concept: Combined ratio, multi-step problem

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

Key Concept: Ratio Division

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: 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: 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: Dividing quantities in a given ratio

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.
[Solution Description]
To solve this, let's analyze the assertion and reason step by step.
1. The assertion states that for a ratio $3 : 5$, the larger part is $\frac{5}{8}$ of the total quantity. This is correct because:
Total parts = $3 + 5 = 8$.
Larger part corresponds to $5$ parts, so its fraction is $\frac{5}{8}$.
2. The reason claims that for any ratio $m : n$ where $n > m$, the larger part is $\frac{n}{m + n}$. This is also correct mathematically. However, the reason does not correctly explain the assertion because it generalizes the concept but doesn't directly justify why the specific case ($3 : 5$) holds true.
Therefore, both assertion and reason are true, but the reason is not the correct explanation of the assertion.

Key Concept: 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) 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: 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: Adjusting Mixture Ratio

c) 45 liters
[Solution Description]
Current ratio is 4:5, total parts = $4 + 5 = 9$.
Water = $\frac{4}{9} \times 180 = 80$ liters.
Milk = $\frac{5}{9} \times 180 = 100$ liters.
Let $x$ be the extra water added for new ratio 5:4.
New equation: $\frac{80 + x}{100} = \frac{5}{4}$.
Solving, $4(80 + x) = 500 \implies 320 + 4x = 500 \implies 4x = 180 \implies x = 45$ liters.

Key Concept: Sharing in Ratio, Verification of Parts

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

Key Concept: Ratio division, Profit sharing

b) \$9,000
[Solution Description]
Total ratio parts = $5 + 3 = 8$. Each part corresponds to $\frac{24000}{8} = 3000$. Bina's share is $3 \times 3000 = 9000$.

Key Concept: Mixture problem, Ratio adjustment

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

Key Concept: Profit Sharing

c) Rs.3,000
[Solution Description]
The investment ratio of Prashanti to Bhuvan is $90000 : 30000 = 3 : 1$. Total parts = $3 + 1 = 4$. Each part = $\frac{12000}{4} = 3000$. Bhuvan's share = $1 \times 3000 = Rs.3000$.

Key Concept: Profit Sharing

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: Unit Conversion - Volume

b) 18.5 kg
[Solution Description]
The mass ratio of gold to water is $37 : 2$. Since 1 litre of water weighs 1 kg, we can find the mass of 1 litre of gold using the ratio:
$\text{Mass of gold} = \left( \frac{37}{2} \right) \times 1 \text{ kg} = 18.5 \text{ kg}$
Thus, the mass of 1 litre of gold is 18.5 kg.

Key Concept: Unit Conversions - Temperature

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

Key Concept: Unit 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 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:

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: 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: 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: 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: 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.

Key Concept: Unit Conversions, Proportional Reasoning

c) 27.55 litres
[Solution Description]
First, convert the area of the field from square feet to acres. The area is $800 \times 600 = 480,000$ square feet. Since $1$ acre = $43,560$ square feet, the area in acres is $\frac{480,000}{43,560} \approx 11.02$ acres.
Next, use the given ratio: $5$ litres for $2$ acres. The amount of diesel needed for $11.02$ acres can be calculated using proportionality:
$\text{Diesel} = \left( \frac{5 \text{ litres}}{2 \text{ acres}} \right) \times 11.02 \text{ acres} \approx 27.55 \text{ litres}$
Therefore, approximately $27.55$ litres of diesel are needed.

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: 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:
$\frac{180,000 \text{ sq ft}}{43,560 \text{ sq ft/acre}} \approx 4.1322 \text{ acres}$