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 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: Scaling Factors

b) 1.5
[Solution Description]
To find the scaling factor:
Scaling factor = $\frac{\text{New Width}}{\text{Original Width}} = \frac{120}{80} = 1.5$.
Since the images are similar, the height also scales by the same factor.
New Height = $50 \times 1.5 = 75$ mm.

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 Ratio

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

Key Concept: 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: Identifying similarity in images, Multiplicative changes

a) 56.6 mm × 42.4 mm
[Solution Description]
Original area = $80 \times 60 = 4800$ mm². Half area needed = 2400 mm².
For similarity, scaling factors must be equal for both dimensions.
Let scaling factor be $k$. New dimensions = $80k \times 60k$.
Area condition: $(80k)(60k) = 2400$ ⇒ $4800k^2 = 2400$ ⇒ $k^2 = 0.5$ ⇒ $k = \sqrt{0.5} \approx 0.707$
Checking options:
A) $56.6 \times 42.4$ → $56.6/80 \approx 0.707$, $42.4/60 \approx 0.707$ → Correct scaling
B) $40 \times 30$ → Equal scaling but area is $\frac{1}{4}$th
C) $70 \times 50$ → Unequal scaling
D) $60 \times 40$ → Unequal scaling
Only option A satisfies both conditions.

Key Concept: Multiplicative vs. Additive Changes

a) Multiply both width and height by 1.5
[Solution Description]
For the image to remain similar, both width and height must be scaled by the same multiplicative factor. The original ratio of width to height is $50 : 30$, which simplifies to $5 : 3$.
Option a): Multiplies width by 1.5 to get 75 mm and height by 1.5 to get 45 mm. The new ratio is $75 : 45 = 5 : 3$ (similar).
Option b): Adds 10 mm to width (60 mm) and subtracts 5 mm from height (25 mm). The new ratio is $60 : 25 = 12 : 5$ (not similar).
Option c): Multiplies width by 2 (100 mm) but adds 10 mm to height (40 mm). The new ratio is $100 : 40 = 5 : 2$ (not similar).
Option d): Adds 20 mm to width (70 mm) and height (50 mm). The new ratio is $70 : 50 = 7 : 5$ (not similar).

Key Concept: Ratios, Proportionality

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

Key Concept: Proportional Ratios

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 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: Application of Proportional Reasoning

c) 20 spoons
[Solution Description]
Given the ratio $5 : 8$ (sugar to lemonade), let the required sugar be $x$ spoons for 32 glasses. The proportional relationship is:
$5 : 8 :: x : 32$
Cross-multiplying gives:
$8x = 5 \times 32$
Solving for $x$:
$x = \frac{160}{8} = 20$
She would need 20 spoons of sugar.

Key Concept: Simplest Form of Ratios

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

Key Concept: Proportional Ratios

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

Key Concept: 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: Dividing a Quantity in a Given Ratio

b) \$3,000 and \$2,000
[Solution Description]
To divide \$5,000 in the ratio $3 : 2$, first calculate the total parts: $3 + 2 = 5$.
First person's share: $3 \times \frac{5000}{5} = 3 \times 1000 = \$3,000$.
Second person's share: $2 \times \frac{5000}{5} = 2 \times 1000 = \$2,000$.

Key Concept: 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: 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, 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: Simplifying Ratios

b) $2 : 3$
[Solution Description]
To simplify the ratio $30 : 45$, find the Highest Common Factor (HCF) of 30 and 45, which is 15. Divide both terms by 15: $30 \div 15 = 2$ and $45 \div 15 = 3$. The simplified ratio is $2 : 3$.

Key Concept: 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 reasoning

a) Yes, because their cross-products are equal
[Solution Description]
To check if two ratios are proportional, we can cross-multiply and see if the products are equal:
$8 \times 15 = 120 \\ \\ 10 \times 12 = 120$
Since both products are equal ($120 = 120$), the ratios are proportional.

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: 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: Proportional ratios with fractions

c) 15 and 6
[Solution Description]
First simplify $15 : 25$ to $3 : 5$ by dividing by HCF 5.
For $9 : x$ to be proportional to $3 : 5$, set $\frac{9}{x} = \frac{3}{5}$.
Cross-multiply: $3x = 45 \Rightarrow x = 15$.
For $y : 10$ to be proportional to $3 : 5$, set $\frac{y}{10} = \frac{3}{5}$.
Cross-multiply: $5y = 30 \Rightarrow y = 6$.

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: Proportional Ratios

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

Key Concept: Simplifying Ratios Using HCF

a) $2 : 3$
[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: 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: 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: 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: 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: 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, simplify both ratios to their simplest form:
\begin{itemize}

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: 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: 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: Problem Solving with Proportionality

c) 30 teachers
[Solution Description]
Given the ratio $1 : 34$, for every 1 teacher there are 34 students. To find the number of teachers for 1020 students, solve $1/34 = x/1020$. Cross-multiplying: $34x = 1020$. Dividing both sides by 34: $x = 30$. So, there are 30 teachers.

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 Division

a) \Rs.1,800 and \Rs.2,700
[Solution Description]
To divide \Rs.4,500 in the ratio $2 : 3$, we first find the total parts: $2 + 3 = 5$. Then, the share for the first part is $\frac{2}{5} \times 4500 = 1800$ and for the second part is $\frac{3}{5} \times 4500 = 2700$.

Key Concept: Proportional Mixing

c) 15 mL blue and 25 mL yellow
[Solution Description]
The ratio of blue to yellow is $3 : 5$. Total parts = $3 + 5 = 8$. Quantity of blue = $\frac{3}{8} \times 40 = 15$ mL. Quantity of yellow = $\frac{5}{8} \times 40 = 25$ mL.

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 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: Ratio Simplification

b) 100
[Solution Description]
The ratio $3 : 2$ means for every 3 boys, there are 2 girls. For 150 boys, the number of girls can be found by solving $\frac{3}{2} = \frac{150}{x}$. Cross-multiplying gives $3x = 300$, so $x = 100$.

Key Concept: 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: Proportional Reasoning

c) 20 kg
[Solution Description]
The ratio of fertilizer to land is $8 : 2$, which simplifies to $4 : 1$. For 5 acres, the required fertilizer is $4 \times 5 = 20$ kg.

Key Concept: Proportional Relationships

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To verify proportionality, simplify both ratios:
1. $5 : 8$ is already in its simplest form.
2. Simplify $25 : 40$ by dividing both terms by their HCF (which is 5):
$\frac{25}{5} = 5$ and $\frac{40}{5} = 8$, giving the simplified ratio $5 : 8$.
Both ratios simplify to $5 : 8$, so they are proportional. The Reason correctly explains why these ratios are proportional.

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, Rule of three

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

Key Concept: 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: 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: 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: 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, real-world application

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

Key Concept: 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) 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: Comparative Proportion

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

Key Concept: 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: Rule of Three, Cross Multiplication

d) Assertion is false, but Reason is true.
[Solution Description]
The cross multiplication method does not require ratios to be in their simplest form to be applied correctly. The proportionality condition $ad = bc$ holds true irrespective of whether the ratios $a:b$ and $c:d$ are simplified or not. For example, consider the unsimplified ratio $4:6 :: 6:9$. Cross multiplying gives $4 \times 9 = 6 \times 6$, which simplifies to $36 = 36$, proving proportionality. Thus, the Assertion is false. However, the Reason states that simplifying ratios eliminates common factors, which is a correct statement but not relevant to the applicability of the cross multiplication method. Therefore, the Reason is true independently.

Key Concept: Mixing ratios, Proportionality

b) 5 liters
[Solution Description]
Current mixture has $(5/8)\times40 = 25$ liters alcohol and $(3/8)\times40 = 15$ liters water. Let $x$ liters of water be added. New ratio is:
$\frac{25}{15 + x} = \frac{5}{4}$
Cross multiplying:
$25 \times 4 = 5 \times (15 + x)$
$100 = 75 + 5x \Rightarrow 5x = 25 \Rightarrow x = 5 \text{ liters}$

Key Concept: 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: 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: 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) 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: 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: 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: 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: Rule of Three, Inverse Proportion

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

Key Concept: Rule of Three, 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: 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: Sharing in given ratio

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

Key Concept: Ratio 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: 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 adjustment, mixture problem

d) 12 kg
[Solution Description]
Current ratio = 4:1, so sugar = (4/5)×60 = 48 kg, flour = (1/5)×60 = 12 kg.
Let $x$ be the sugar added. New ratio: (48 + $x$)/12 = 5/1.
Solving: 48 + $x$ = 60 → $x$ = 12 kg.

Key Concept: Dividing quantities in a given ratio

c) 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: 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) 10 liters
[Solution Description]
Given mixture: Milk = 7 parts, Water = 2 parts, total parts = 9.
1. Current milk quantity = $\frac{7}{9} \times 90 = 70$ liters.
2. Current water quantity = $\frac{2}{9} \times 90 = 20$ liters.
3. Let $x$ liters of water be added; new ratio = 7:3.
4. $\frac{70}{20 + x} = \frac{7}{3}$ ⇒ $210 = 140 + 7x$ ⇒ $7x = 70$ ⇒ $x = 10$.
Thus, 10 liters of water must be added.

Key Concept: Dividing quantities in a given ratio

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: 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: 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: 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: Sharing in a given ratio

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.
[Solution Description]
To divide Rs.500 in the ratio 2:3:
1. Total parts = 2 + 3 = 5
2. Value of each part = Rs.500 / 5 = Rs.100
3. Larger share = 3 × Rs.100 = Rs.300
The Assertion is true as the larger share is indeed Rs.300. The Reason is also true because the total parts are 5. However, the Reason does not explain why the larger share is Rs.300; it only states the total parts. Hence, Reason is NOT the correct explanation of Assertion.

Key Concept: Profit Sharing

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

Key Concept: 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: Profit Sharing

c) \Rs.7,200
[Solution Description]
To divide \Rs.12,000 in the ratio 2:3:
Total parts = 2 + 3 = 5.
Each part = $\frac{12000}{5} = 2400$.
Second partner's share = $3 \times 2400 = \Rs.7,200$.

Key Concept: Profit sharing 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: Temperature conversion, Multi-step reasoning

c) $19^\circ \text{F}$
[Solution Description]
First, convert $-10^\circ \text{C}$ to Fahrenheit using the formula: $\frac{9}{5} \times (-10) + 32 = -18 + 32 = 14^\circ \text{F}$.
Then add the instrument error of $+5^\circ \text{F}$: $14 + 5 = 19^\circ \text{F}$.
The final temperature reading will be $19^\circ \text{F}$.

Key Concept: Unit 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: 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

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: 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:

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:

b) $20^\circ C$
[Solution Description]
Using the formula:
$Celsius = \frac{5}{9} \times (68 - 32) = \frac{5}{9} \times 36 = 20^\circ C$
So, $68^\circ F$ is equal to $20^\circ C$.

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: Unit Conversion Consistency

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The Assertion states that converting area measurements to acres is necessary for accurate cost calculation when fertilizing a field. This is true because using consistent units (e.g., acres) allows proper ratio application. The Reason correctly explains that proportional reasoning demands uniform units to prevent errors, which directly supports the Assertion. Thus, both are true, and the Reason justifies the Assertion.

Key Concept: 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: Unit Conversions, Proportional Reasoning

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
First, convert 7.5 hours to minutes: $7.5 \text{ hours} \times 60 \text{ minutes/hour} = 450 \text{ minutes}$.
Calculate the rate of filling: $50 \text{ gallons} / 10 \text{ minutes} = 5 \text{ gallons/minute}$.
Time to fill 2250 gallons: $2250 \text{ gallons} / 5 \text{ gallons/minute} = 450 \text{ minutes} = 7.5 \text{ hours}$.
Thus, both Assertion and Reason are true, and the Reason correctly explains the Assertion.

Key Concept: Work Rate Conversion

b) 30 hours
[Solution Description]
First, calculate the time taken by the oxen for 20 acres: $20 \times 6 = 120\,\text{hours}$. Since the tractor is 4 times faster, its time is:
$\text{Tractor's time} = \frac{120}{4} = 30\,\text{hours}$