Key Concept: Ratio Comparison

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

Key Concept: Proportional Change in Images

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
From the given measurements, Image A has width 60 mm and height 40 mm, while Image D has width 90 mm and height 60 mm. The scaling factor for width is $\frac{90}{60} = \frac{3}{2}$. The scaling factor for height is $\frac{60}{40} = \frac{3}{2}$. Since both dimensions change by the same factor ($\frac{3}{2}$), the images look similar. This matches the conclusion that proportional scaling preserves similarity. Thus, both Assertion and Reason are true, and the Reason correctly explains the Assertion.

Key Concept: Proportional Reasoning, Simplifying Ratios

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

Key Concept: Non-Proportional Change

a) The image becomes wider and shorter
[Solution Description]
Non-proportional changes distort the image.
Let the original width $W$ and height $H$.
New width = $W + 0.5W = 1.5W$.
New height = $H - 0.25H = 0.75H$.
Since scaling factors differ ($1.5 \neq 0.75$), the image appears stretched horizontally and compressed vertically.

Key Concept: Proportional Scaling

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

Key Concept: Similarity Verification

b) Yes, because their ratios are equal
[Solution Description]
Check if the width-to-height ratios are equal. For the first rectangle: $\frac{120}{90} = \frac{4}{3}$. For the second rectangle: $\frac{40}{30} = \frac{4}{3}$. Since both ratios are equal, the rectangles are similar.

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: Multiplicative vs. Additive Changes

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that images look similar when both dimensions are scaled by the same multiplicative factor, which is true as per the syllabus. The reason explains that proportional change preserves the shape, which is also true and correctly explains why the assertion holds. Hence, both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

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, Problem-Solving

c) 15 kg
[Solution Description]
Given the ratio of sugar to flour is $3 : 5$ and 9 kg of sugar is used.
Let the amount of flour needed be $x$ kg.
According to the ratio:
$3 : 5 = 9 : x$
This can be written as:
$\frac{3}{5} = \frac{9}{x}$
Cross-multiplying:
$3x = 45$
Solving for $x$:
$x = \frac{45}{3} = 15$
Therefore, 15 kg of flour is needed.

Key Concept: Ratio Simplification

b) $1 : 3$
[Solution Description]
To simplify the ratio $15 : 45$, find the highest common factor (HCF) of 15 and 45, which is 15. Divide both terms by 15 to get $1 : 3$.

Key Concept: 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: 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: Assertion-Reason

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To simplify the ratio $6:4$, we find the HCF of 6 and 4, which is 2. Dividing both terms by 2 gives $3:2$. Thus, the Assertion is true. The Reason correctly explains that simplifying a ratio involves dividing both terms by their HCF. Therefore, both Assertion and Reason are true, and the Reason is the correct explanation of the Assertion.

Key Concept: Definition and Simplest Form of Ratios, Proportionality

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
First, simplify the ratio $12:18$ by dividing both terms by their HCF (6): $\frac{12}{6} : \frac{18}{6} = 2:3$. Similarly, simplify $20:30$ by dividing both terms by their HCF (10): $\frac{20}{10} : \frac{30}{10} = 2:3$. Since both simplified forms are identical, the ratios are proportional. The assertion is true. The reason states that two ratios are proportional if their simplest forms are identical, which is correct and directly explains the assertion.

Key Concept: Simplest Form, 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: Proportional Ratios

d) $25 : 18$
[Solution Description]
To determine if ratios are proportional, we simplify them to their simplest form. The given ratio is $5:3$. For another ratio to be proportional, it must simplify to the same ratio ($5:3$) when reduced.
Let's check each option:
\begin{itemize}

Key Concept: 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 them:
1. Simplifying $60:40$ by dividing both terms by their HCF (20): $60 \div 20 : 40 \div 20 = 3:2$.
2. Simplifying $90:60$ by dividing both terms by their HCF (30): $90 \div 30 : 60 \div 30 = 3:2$.
Since both simplified forms are equal ($3:2$), the Assertion is true. The Reason correctly explains that they are proportional because their simplest forms match.

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: 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: 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: 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: 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 quantities in a ratio

c) Rs.1,600 and Rs.2,000
[Solution Description]
To divide Rs.3,600 in the ratio $4 : 5$, first add the parts of the ratio: $4 + 5 = 9$. Then calculate each part:
$\text{First part} = \frac{4}{9} \times 3600 = Rs.1,600 \\ \\ \text{Second part} = \frac{5}{9} \times 3600 = Rs.2,000$
So, the amounts are Rs.1,600 and Rs.2,000.

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: Finding Missing Terms in Proportional Ratios

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

Key Concept: 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: 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: Using HCF for Simplification

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

Key Concept: Simplifying ratios using HCF, 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 Using HCF

a) $2 : 3$
[Solution Description]
To simplify the ratio $84 : 126$, first find the HCF of 84 and 126.
Step 1: Find the factors of 84. The factors are 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84.
Step 2: Find the factors of 126. The factors are 1, 2, 3, 6, 7, 9, 14, 18, 21, 42, 63, 126.
Step 3: Identify the highest common factor (HCF). The common factors are 1, 2, 3, 6, 7, 14, 21, 42. The HCF is 42.
Step 4: Divide both terms by the HCF.
$84 \div 42 = 2$
$126 \div 42 = 3$
So, the simplified ratio is $2 : 3$.

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: 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: 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: 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: Missing Term in Proportions

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

Key Concept: 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: Problem-Solving with Ratios

c) 6
[Solution Description]
The ratio of flour to eggs is $4:6$, which simplifies to $2:3$.
For 9 eggs, we set up the proportion $2:3 :: x:9$.
Cross-multiplying gives:
$3x = 18$
$x = 6$.
Hence, 6 cups of flour are needed.

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: Cross-Multiplication Method, Simplest Form

c) 21
[Solution Description]
Using the cross-multiplication method for proportion $a : b :: c : d$, we have $ad = bc$.
Here, $a = 5$, $b = 7$, $c = 15$.
So, $5 × x = 7 × 15$
$5x = 105$
$x = 105 / 5 = 21$.
Thus, the missing term $x$ is 21.

Key Concept: Proportional Reasoning

c) 600 km
[Solution Description]
To find the distance traveled in 10 hours at the same speed:
$\text{Speed} = \frac{360 \, \text{km}}{6 \, \text{hours}} = 60 \, \text{km/h} \\ \\ \text{Distance} = 60 \, \text{km/h} \times 10 \, \text{hours} = 600 \, \text{km}$
The car will travel 600 km.

Key Concept: Proportional 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: 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 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 in Real-life Application

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that the price per mL decreases with increasing volume, which is true because larger sizes generally have lower per-unit costs due to reduced packaging and handling expenses. The reason correctly explains this by mentioning economies of scale, which refers to cost advantages gained through increased production efficiency. Thus, both are true, and the reason correctly explains the assertion.

Key Concept: Ratio 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

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

Key Concept: Ratio Comparison

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

Key Concept: 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

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: Proportional Reasoning

c) 175 units
[Solution Description]
This is a proportional relationship problem where the ratio of units to hours must remain constant.
Given ratio: $50$ units : $2$ hours.
Step 1: Find the factor of change for hours:
$7 ÷ 2 = 3.5$
Step 2: Multiply the units by the same factor to maintain proportionality:
$50 × 3.5 = 175 \text{ units}$
The machine will produce 175 units in 7 hours.

Key Concept: Problem Solving with Proportional Reasoning, Identifying proportional relationships

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
First, check if Assertion is true: Simplify $4 : 5$ which is already in simplest form. Now simplify $36 : 45$ by dividing both terms by their HCF (9): $\frac{36}{9} : \frac{45}{9} = 4 : 5$. Since both ratios simplify to $4 : 5$, the Assertion is true. Next, check Reason: The condition for proportionality is $a \times d = b \times c$ for ratios $a : b$ and $c : d$. For $4 : 5$ and $36 : 45$, $4 \times 45 = 180$ and $5 \times 36 = 180$. Thus, the Reason correctly explains why the ratios are proportional. Hence, both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

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: Ratio Application

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

Key Concept: Proportional Ratios

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

Key Concept: Proportional Reasoning, Ratio Simplification

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To determine if the assertion and reason are correct:
1. Simplify the first ratio $24 : 36$ by dividing both terms by their HCF (12): $24 \div 12 = 2$, $36 \div 12 = 3$. Simplified form is $2 : 3$.
2. Simplify the second ratio $10 : 15$ by dividing both terms by their HCF (5): $10 \div 5 = 2$, $15 \div 5 = 3$. Simplified form is $2 : 3$.
3. Since both simplified ratios are equal ($2 : 3$), they are proportional. Thus, Assertion is true.
4. The Reason states that both ratios simplify to $2 : 3$, which matches our calculations. Hence, Reason is true and correctly explains the Assertion.
Therefore, the correct answer is option (a).

Key Concept: 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]
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: Proportional reasoning with unit conversion

c) 16 machines
[Solution Description]
First, find the rate per machine: $\frac{1200 \text{ pages}}{40 \text{ min} \times 8 \text{ machines}} = 3.75$ pages per machine per minute.
For 4500 pages in 50 minutes, total required machines = $\frac{4500}{50 \times 3.75} = 24$ machines.
Additional machines needed = $24 - 8 = 16$.

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: Compound Proportion

d) 11 pumps
[Solution Description]
This is a compound proportion problem with three variables. Let $x$ be the required pumps. The relationship is:
$\frac{4 \text{ pumps} \times 8 \text{ hrs} \times 2 \text{ days}}{1 \text{ tank}} = \frac{x \text{ pumps} \times 6 \text{ hrs} \times 1 \text{ day}}{1 \text{ tank}}$
Simplifying:
$4 \times 8 \times 2 = x \times 6 \times 1 \Rightarrow 64 = 6x \Rightarrow x = \frac{64}{6} \approx 10.\overline{6}$
Since we can't have a fraction of a pump, we round up to 11 pumps.

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

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that the cross multiplication method is used to find the fourth proportional in a proportion, which is true. The reason given is that in the proportion $a : b :: c : d$, the product of the extremes ($a$ and $d$) equals the product of the means ($b$ and $c$). This is also true, as it defines the cross multiplication rule $ad = bc$. Moreover, the reason correctly explains why the assertion is valid since the cross multiplication method is derived from this property. Therefore, both the Assertion and Reason are true, and the Reason is the correct explanation of the Assertion.

Key Concept: Cross Multiplication Method

b) 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: Proportional Ratios

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

Key Concept: Rule of Three Proportion

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

Key Concept: Proportional Ratios

b) 40
[Solution Description]
Model the problem as the proportion $5 : 25 :: 8 : x$.
Step 1: Write the proportion and use the Rule of Three:
$5 \times x = 25 \times 8$
Step 2: Multiply the known terms:
$5x = 200$
Step 3: Solve for $x$ by dividing both sides by 5:
$x = \frac{200}{5} = 40$
The machine can produce 40 toys in 8 hours.

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: Proportionality

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

Key Concept: 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: 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, 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: 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: Sharing in Given Ratio

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that when 20 sweets are shared in the ratio 3:2, one gets 12 and another gets 8. This is correct because:
1. Total parts = 3 + 2 = 5 parts
2. Value of each part = $20 \div 5 = 4$ sweets
3. First share = $3 \times 4 = 12$ sweets
4. Second share = $2 \times 4 = 8$ sweets
The reason correctly explains the method to find total parts (m+n) which is used to divide any quantity in given ratio. Therefore, both assertion and reason are true and reason correctly explains the assertion.

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

c) \$500 and \$750
[Solution Description]
To divide \$1,250 in the ratio 4:6:
1. Total parts = 4 + 6 = 10.
2. Value of each part = \$1,250 / 10 = \$125.
3. First friend's share = 4 $\times$ \$125 = \$500.
4. Second friend's share = 6 $\times$ \$125 = \$750.
So, the shares are \$500 and \$750.

Key Concept: Dividing quantities in a given ratio, 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) \$2,700
[Solution Description]
First, find the total number of parts in the ratio 2:3:4.
Total parts = 2 + 3 + 4 = 9 parts.
Next, calculate the value of each part:
Value per part = $\frac{8100}{9} = 900$.
Now, find B's share by multiplying the number of parts B gets by the value per part:
B's share = $3 \times 900 = 2700$.
Therefore, B's share is \$2,700.

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: 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: Finding parts from the whole using ratio

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

Key Concept: Dividing a Quantity in Given Ratio

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

Key Concept: 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: Business Investment

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

Key Concept: Profit Sharing in Business

d) Assertion is false, but Reason is true.
[Solution Description]
First, find the ratio of investments: Ram : Shyam = 20,000 : 30,000 = 2:3.
Total parts = 2 + 3 = 5.
Each part value = $\frac{5000}{5} = 1000$.
Ram's share = $2 \times 1000 = 2000$.
Thus, Assertion (A) is false as Ram's actual share is \$2,000, not \$3,000.
Reason (R) is true as it correctly describes the method of calculating profit share based on investment ratio.

Key Concept: Profit sharing and ratio application

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: 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: Temperature Conversion

c) $35^\circ C$
[Solution Description]
Use the formula for Fahrenheit to Celsius conversion:
$\text{Celsius} = \frac{5}{9} \times (\text{Fahrenheit} - 32)$
Substitute $95^\circ F$:
$\text{Celsius} = \frac{5}{9} \times (95 - 32) = \frac{5}{9} \times 63 = 35^\circ C$
The temperature is $35^\circ C$.

Key Concept: 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: Area Conversion

b) 1.67 hectares
[Solution Description]
First, calculate the area in square feet:
$300 \, \text{ft} \times 600 \, \text{ft} = 180{,}000 \, \text{sq ft}$
Convert square feet to square metres using $1 \, \text{sq m} = 10.764 \, \text{sq ft}$:
$\frac{180{,}000}{10.764} = 16{,}722.5 \, \text{sq m}$
Now, convert square metres to hectares using $1 \, \text{hectare} = 10{,}000 \, \text{sq m}$:
$\frac{16{,}722.5}{10{,}000} = 1.67225 \, \text{hectares}$
The area is approximately 1.67 hectares.

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: Temperature Conversion

c) $20^\circ C$
[Solution Description]
Given the Fahrenheit temperature is $68^\circ F$, substitute into the formula:
$Celsius = \frac{5}{9} \times (68 - 32) = \frac{5}{9} \times 36 = 20^\circ C$
Therefore, $68^\circ F$ is equal to $20^\circ C$.

Key Concept: 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: 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.

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 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:
$\text{Mass of gold} = \frac{37}{2} \times 1\,\text{kg} = 18.5\,\text{kg}$