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, 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 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: 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 Scaling

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

Key Concept: Multiplicative vs. additive changes, Proportionality

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The Assertion states that images A, C, and D remain similar because their dimensions change multiplicatively by the same factor. This is true because the ratio of width to height for these images is consistently $3:2$, indicating proportional scaling. The Reason states that multiplicative scaling preserves the ratio of width to height, while additive scaling does not. This is also true and correctly explains why the images remain similar under multiplicative scaling but not additive scaling. Therefore, both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

Key Concept: Proportional Scaling Factor

d) 2
[Solution Description]
To find the scaling factor, divide the dimensions of Image Y by those of Image X:
Width factor = $\frac{100}{50} = 2$,
Height factor = $\frac{60}{30} = 2$.
Since both factors are equal, the scaling factor is 2.

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: 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: 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: 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: 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: 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: 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: 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: 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: Proportional Ratios, Simplest Form

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

Key Concept: 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 $60 : 40$, 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: 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: 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: 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: 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: 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: Simplifying Ratios Using HCF

c) $3 : 4$
[Solution Description]
To simplify the ratio $36 : 48$, we find the HCF of 36 and 48, which is 12. Dividing both terms by 12:
$36 \div 12 = 3 \\ \\ 48 \div 12 = 4$
The simplified ratio is $3 : 4$.

Key Concept: Simplifying Ratios Using HCF

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

Key Concept: Proportional Ratios

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

Key Concept: Simplest Form Calculation

c) $2 : 3$
[Solution Description]
Find the HCF of 72 and 108, which is 36. Divide both terms by 36:
$72 \div 36 = 2 \\ \\ 108 \div 36 = 3$
The simplified ratio is $2 : 3$.

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: 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: Simplifying Ratios

c) 2 : 3
[Solution Description]
To find the simplest form, divide both terms by their HCF. The HCF of 84 and 126 is 42. Dividing both terms gives:
$\frac{84}{42} = 2 \quad \text{and} \quad \frac{126}{42} = 3$
Thus, the simplest form is $2 : 3$.

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: 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: 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 Ratios

b) Yes, because both simplify to $4 : 5$
[Solution Description]
Check if the simplest forms of both ratios are equal. Simplifying $4 : 5$ gives $4 : 5$ (already in simplest form). For $20 : 25$, the HCF is 5, so $\frac{20}{5} = 4$, $\frac{25}{5} = 5$. The simplest form is $4 : 5$. Thus, they are proportional.

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

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To solve this problem, we need to check if the given assertion follows the principle of proportional reasoning.
First, calculate the ratio of rice to students in the first scenario:
$\frac{5\,\text{kg}}{20\,\text{students}} = 0.25\,\text{kg per student}$
Next, calculate the ratio for the second scenario:
$\frac{15\,\text{kg}}{60\,\text{students}} = 0.25\,\text{kg per student}$
Since both ratios are equal, the assertion is true because the amount of rice per student remains constant. The reason is also true and correctly explains the assertion, as proportional reasoning ensures that the ratio remains unchanged when scaling quantities proportionally.

Key Concept: Proportional Reasoning

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: Ratio Calculation

b) 40 mL acid and 200 mL water
[Solution Description]
The ratio of acid to water is $1 : 5$. Total parts = $1 + 5 = 6$. Quantity of acid = $\frac{1}{6} \times 240 = 40$ mL. Quantity of water = $\frac{5}{6} \times 240 = 200$ mL.

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 comparison, Unit conversion

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

Key Concept: Ratio Comparison, Proportional Reasoning

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

Key Concept: Ratio 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: 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

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: 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: 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: 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: 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 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) 56
[Solution Description]
The proportion is $3 : 24 :: 7 : x$. Cross-multiplying gives $3x = 168$, so $x = \frac{168}{3} = 56$. Thus, the machine will produce 56 toys in 7 hours.

Key Concept: Trairasika — The Rule of Three

c) 18
[Solution Description]
The problem involves inverse proportion, so we set it as $6 \times 12 = x \times 4$. Solving for $x$ gives $x = \frac{72}{4} = 18$. Thus, 18 workers are needed to complete the task in 4 days.

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: 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: 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: Rule of Three Application

b) 300 km
[Solution Description]
The problem can be represented as $2 : 120 :: 5 : x$. Using the Rule of Three:
$2 \times x = 120 \times 5$
Solving for $x$:
$2x = 600 \implies x = \frac{600}{2} = 300$
Thus, the car will travel 300 km in 5 hours.

Key Concept: 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: Trairasika — The Rule of Three

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

Key Concept: 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: Cross Multiplication Method, Proportional Ratios

c) 35 kg
[Solution Description]
First, establish the given ratio: $7:5$ (cement:sand).
For the first case: $7/5 = 35/x$, solve for x using cross-multiplication:
$7x = 5 \times 35$
$7x = 175$
$x = 175 / 7 = 25$ kg sand.
Now, apply this proportion to the new amount of cement (49 kg):
$7/5 = 49/y$
Cross-multiplying: $7y = 5 \times 49$
$7y = 245$
$y = 245 / 7 = 35$ kg sand required.

Key Concept: Proportional Ratios

c) 28
[Solution Description]
Represent the problem as the proportion $3 : 12 :: 7 : x$.
Step 1: Apply cross multiplication:
$3 \times x = 12 \times 7$
Step 2: Multiply the known terms:
$3x = 84$
Step 3: Solve for $x$ by dividing both sides by 3:
$x = \frac{84}{3} = 28$
The farmer will need 28 kg of seeds for a 7-acre field.

Key Concept: Rule of Three 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, 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, 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: 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: Dividing a mixture in ratio

c) 28 kg
[Solution Description]
Total weight = 40 kg
Ratio = 7:3
Total parts = 7 + 3 = 10
Size of each part = $40 \div 10 = 4$ kg
Flour quantity = $7 \times 4 = 28$ kg.

Key Concept: Sharing in given ratio

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

Key Concept: Mixture Ratio

b) 70 kg
[Solution Description]
Total parts in the ratio = $7 + 1 = 8$. Size of each part = $\frac{80}{8} = 10$ kg. Sugar corresponds to 7 parts, so quantity of sugar = $7 \times 10 = 70$ kg.

Key Concept: 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

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

Key Concept: Dividing quantities in a given ratio, 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

c) 40
[Solution Description]
Let the number of \$1 coins = 5k, \$2 coins = 3k.
1. Total value = $(5k \times \$1) + (3k \times \$2) = 5k + 6k = 11k = \$88$.
2. Solving, $k = 8$.
3. Number of \$1 coins = $5k = 5 \times 8 = 40$.
Therefore, there are 40 \$1 coins.

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: 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: 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: 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: Ratio division method

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To verify the assertion:
1. Given quantity = 60 kg, ratio = 4:1.
2. Total parts = 4 + 1 = 5.
3. Size of each part = $\frac{60}{5} = 12$ kg.
4. Larger part = $4 \times 12 = 48$ kg (matches assertion).
The reason provides the correct formula for calculating the larger part, which explains how 48 kg was obtained. Therefore, both are true, and the reason correctly explains the assertion.

Key Concept: 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

d) 20 kg
[Solution Description]
Given ratio = 4:1, total parts = 4 + 1 = 5.
Sugar quantity = $\frac{4}{5} \times 25 = 20$ kg.

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: 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: 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: 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: Volume Conversion

c) 4{,}500 cc
[Solution Description]
Given $1 \, \text{litre} = 1{,}000 \, \text{mL}$ and $1 \, \text{mL} = 1 \, \text{cc}$, we can directly convert litres to cubic centimetres:
$4.5 \, \text{litres} = 4.5 \times 1{,}000 = 4{,}500 \, \text{cc}$
The tank contains 4{,}500 cubic centimetres of water.

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 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:

b) 2.471 acres
[Solution Description]
Given, $1 \text{ hectare} = 2.471 \text{ acres}$. So, the plot is equal to 2.471 acres.

Key Concept: Volume

b) 5,000,000 cubic centimeters
[Solution Description]
Convert liters to milliliters first:
$1 \text{ L} = 1000 \text{ mL}$.
So, $5000 \text{ L} = 5000 \times 1000 \text{ mL} = 5,000,000 \text{ mL}$.
Since $1 \text{ mL} = 1 \text{ cc}$, the volume in cubic centimeters is also $5,000,000 \text{ cc}$.

Key Concept: 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: 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 - 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: 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 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.