Key Concept: Rule of Three, Proportional Change

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

Key Concept: Proportional Change in Dimensions

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

Key Concept: 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: Proportional Change

c) 30 mm
[Solution Description]
To maintain similarity, the scaling factor for width and height must be the same.
Scaling factor = $\frac{\text{New Width}}{\text{Original Width}} = \frac{45}{60} = \frac{3}{4}$.
New Height = Original Height × Scaling Factor = $40 \times \frac{3}{4} = 30$ mm.

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: Width–Height Comparison

c) 20 mm
[Solution Description]
The original ratio of width to height for Image A is $60 : 40$, which simplifies to $3 : 2$. To maintain the same ratio when the width is reduced to 30 mm, we set up the proportion $\frac{3}{2} = \frac{30}{h}$. Solving for $h$:
$3h = 60$
$h = 20$
The new height should be 20 mm.

Key Concept: Proportional Scaling

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

Key Concept: Multiplicative 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: 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 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: Proportional Ratios

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To determine if the ratios $6 : 9$ and $12 : 18$ are proportional, we will simplify both ratios to their simplest form.
Step 1: Simplify $6 : 9$ by dividing both terms by their HCF, which is 3. So, $\frac{6}{3} : \frac{9}{3} = 2 : 3$.
Step 2: Simplify $12 : 18$ by dividing both terms by their HCF, which is 6. So, $\frac{12}{6} : \frac{18}{6} = 2 : 3$.
Since both simplified forms are equal ($2 : 3$), the two ratios are proportional. Also, the Reason correctly explains why they are proportional—because simplifying both yields the same ratio.

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 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, Proportionality Check

c) $9 : 10$
[Solution Description]
Simplify $15 : 25$ to its simplest form by dividing both terms by their HCF (5): $3 : 5$. Now simplify the options:
Option a) $21 : 35$ simplifies to $3 : 5$ (HCF 7).
Option b) $30 : 50$ simplifies to $3 : 5$ (HCF 10).
Option c) $9 : 10$ remains $9 : 10$ (no common factor).
Option d) $45 : 75$ simplifies to $3 : 5$ (HCF 15).
Only option c) does not simplify to $3 : 5$, so it is not proportional.

Key Concept: Interpretation of Ratios

c) 15 cups
[Solution Description]
The given ratio of sugar to flour is $3 : 5$. For every 3 cups of sugar, 5 cups of flour are needed. If 9 cups of sugar are used, we need to find the corresponding amount of flour.
First, determine the scaling factor:
$\text{Scaling factor} = \frac{9}{3} = 3$
Multiply the flour part of the ratio by this scaling factor:
$5 \times 3 = 15$
Thus, 15 cups of flour are needed.

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) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
First, let's verify the assertion by simplifying both ratios:
- For $12:18$, dividing both terms by 6 (HCF) gives $2:3$.
- For $8:12$, dividing both terms by 4 (HCF) gives $2:3$.
Since both simplified ratios are equal ($2:3$), the assertion is true. The reason correctly explains why the assertion is true because it shows that both ratios simplify to the same simplest form.

Key Concept: Rule of Three

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

Key Concept: Proportional Ratios

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: 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: 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: Proportional Ratios

d) $5 : 7$
[Solution Description]
First, simplify $8 : 12$ by dividing both terms by their HCF (which is 4), giving $2 : 3$. Now check each option:
\begin{itemize}
- Option a) $4 : 6$ simplifies to $2 : 3$ (proportional).
- Option b) $10 : 15$ simplifies to $2 : 3$ (proportional).
- Option c) $6 : 9$ simplifies to $2 : 3$ (proportional).
- Option d) $5 : 7$ does not simplify to $2 : 3$ (not proportional).
\end{itemize}
All except option d) are proportional to $8 : 12$.

Key Concept: 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: 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: 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: 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: Simplest form of ratios, Proportional ratios

c) 48
[Solution Description]
First, find the simplest form of $48 : 36$. The HCF of 48 and 36 is 12. Dividing both terms by 12 gives $4 : 3$.
Since the ratios are proportional, $64 : x$ must reduce to $4 : 3$.
Set up the proportion $\frac{64}{x} = \frac{4}{3}$.
Cross-multiplying: $4x = 192$.
Solving for $x$: $x = 48$.

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: 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: Dividing quantities in a given ratio

c) Rs.4500$[Solution Description]
Total parts = $3 + 5 = 8$. The larger share corresponds to 5 parts. Value per part = Rs.7200 $\div$ 8 = Rs.900. Larger share = $5 \times 900 = Rs.4500$.

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, Proportionality of ratios

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

Key Concept: Simplifying Ratios

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: 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

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: Equivalence of ratios in simplest form

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

Key Concept: Assertion-Reason

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

Key Concept: 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: 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: 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 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, 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: 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: 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: Proportional Division

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

Key Concept: Ratio application, Cost calculation

d) \$880
[Solution Description]
Total parts = $2 + 7 = 9$.
Cement quantity = $(2/9) \times 180 = 40$ kg.
Sand quantity = $(7/9) \times 180 = 140$ kg.
Cement cost = $40 \times 15 = \$600$.
Sand cost = $140 \times 2 = \$280$.
Total cost = $600 + 280 = \$880$.

Key Concept: Ratio proportion, Age problems

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

Key Concept: Identifying proportional relationships, Applying factor of change

c) 25 liters
[Solution Description]
First, find the petrol consumption per kilometer: $\frac{15 \text{ liters}}{180 \text{ km}} = 0.0833$ liters/km. Then, multiply this rate by the new distance: $0.0833 \text{ liters/km} \times 300 \text{ km} = 25$ liters.

Key Concept: Simplification of Ratios

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

Key Concept: 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: 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: Ratio Application

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

Key Concept: Proportional Ratios

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To check if the ratios are proportional, we simplify them to their simplest forms. For $12 : 18$, the HCF of 12 and 18 is 6. Dividing both terms by 6 gives $2 : 3$. For $20 : 30$, the HCF of 20 and 30 is 10. Dividing both terms by 10 also gives $2 : 3$. Since both simplified ratios are equal, they are proportional. The reason correctly explains the assertion.

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: Simplifying Ratios

b) $2 : 3$
[Solution Description]
To simplify the ratio $24 : 36$, we find the HCF of 24 and 36, which is 12. Divide both terms by 12:
$24 ÷ 12 = 2$
$36 ÷ 12 = 3$
The simplified ratio is $2 : 3$.

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

c) \$160
[Solution Description]
Using the Rule of Three, we can set up the proportion as $5 : 100 :: 8 : x$. Cross-multiplying gives $5x = 800$, so $x = \frac{800}{5} = 160$. Thus, 8 books cost \$160.

Key Concept: Trairasika — The Rule of Three

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
Given the proportion $5 : 250 :: 8 : x$, cross multiplication gives $5x = 250 \times 8$. Solving for $x$, we get $x = \frac{2000}{5} = 400$. Here, Assertion correctly applies the Rule of Three and derives the cost of 8 kg rice as \$400. The Reason explains why the method works mathematically, stating that proportional ratios must satisfy $ad = bc$. Both statements are true, and the Reason justifies the Assertion.

Key Concept: 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: 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: 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: 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: 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

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 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: 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, 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 and Proportionality

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The Assertion claims that the Rule of Three can solve direct proportionality problems, which is true as it is designed for such cases. The Reason correctly describes the mathematical relationship in the Rule of Three, where the product of the first (pramāṇa) and fourth (ichchhāphala) terms equals the product of the second (phala) and third (ichchhā) terms. Since the Reason accurately explains why the Rule of Three works for proportional relationships, it validates the Assertion.

Key Concept: Rule of Three 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 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: 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: Ratio Division, Investment Sharing

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To verify Assertion (A), we use the formula for division in a given ratio $m:n$. Here, $x = Rs.12,000$, $m = 5$, and $n = 3$.
First, calculate the total parts: $5 + 3 = 8$.
Now, find the size of each part: $\frac{12,000}{8} = 1,500$.
Partner A's share: $5 \times 1,500 = 7,500$.
Thus, Assertion (A) is true.
Reason (R) states the correct formula for dividing a quantity into a given ratio, which was used to verify Assertion (A). Hence, both Assertion and Reason are true, and Reason correctly explains Assertion.

Key Concept: Ratio 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: 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: 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) 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: 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: Finding parts from the whole using ratio

c) \$80
[Solution Description]
Total ratio parts = 7 + 8 = 15. Size of each part = $150 ÷ 15 = 10$. Larger share is $8 × 10 = 80$.

Key Concept: Finding parts from the whole using ratio

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

Key Concept: Sharing 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: 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: Profit sharing and ratio application

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

Key Concept: Ratio Adjustment

b) 10 liters
[Solution Description]
Let the initial quantities be $4x$ (red) and $3x$ (blue). After adding 5 liters of blue, the new ratio is $4x : (3x + 5) = 4 : 5$. Solving: $\frac{4x}{3x + 5} = \frac{4}{5}$. Cross-multiplying gives $20x = 12x + 20$, so $8x = 20$ and $x = 2.5$. Initial red paint = $4 \times 2.5 = 10$ liters.

Key Concept: Mixture Problem

b) 20 kg
[Solution Description]
The ratio of sugar to flour is $2 : 5$. Total parts = $2 + 5 = 7$. Each part = $\frac{70}{7} = 10$ kg. Sugar quantity = $2 \times 10 = 20$ kg.

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: 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: Area conversion, Proportional reasoning

c) 20.66 tonnes
[Solution Description]
First, calculate the area of the plot in square feet: $300 \text{ ft} \times 600 \text{ ft} = 180,000 \text{ sq ft}$.
Convert this to acres using the given conversion factor ($1 \text{ acre} = 43,560 \text{ sq ft}$).
Divide the total area by the number of square feet in an acre: $\frac{180,000}{43,560} \approx 4.132 \text{ acres}$.
Multiply this by the manure rate per acre: $4.132 \times 5 = 20.66 \text{ tonnes}$.
The farmer needs approximately 20.66 tonnes of manure.

Key Concept: 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 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: 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: 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: 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: 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: Mass and Volume Conversion

a) 18.5 kg
[Solution Description]
Given the mass ratio of gold to water is $37 : 2$ for equal volumes. Since 1 litre of water has a mass of 1 kg, the mass of 1 litre of gold is calculated as:
$\text{Mass of gold} = \frac{37}{2} \times 1\,\text{kg} = 18.5\,\text{kg}$

Key Concept: Unit Conversion - Volume and Time

b) 8.333 minutes
[Solution Description]
First, convert the tank volume from liters to milliliters:
$15 \text{ L} \times 1,000 \text{ mL/L} = 15,000 \text{ mL}$
Next, find the time taken using the given rate:
$\frac{25 \text{ s}}{750 \text{ mL}} \times 15,000 \text{ mL} = 500 \text{ s}$
Convert seconds to minutes:
$\frac{500 \text{ s}}{60 \text{ s/min}} \approx 8.333 \text{ min}$

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.