Key Concept: Proportional Change

c) 60 mm
[Solution Description]
To find the height of the similar rectangle when the width changes proportionally:
First, calculate the ratio of original width to height: $\frac{50}{30} = \frac{5}{3}$.
For the new rectangle, the width-to-height ratio must remain the same for similarity.
Let the new height be $h$. Then, $\frac{100}{h} = \frac{5}{3}$.
Solving for $h$, we get $h = \frac{100 \times 3}{5} = 60$ mm.

Key Concept: Proportional Reasoning in Image Dimensions

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

Key Concept: 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

a) 90 mm width, 60 mm height
[Solution Description]
To find which image is proportional to Image A (60:40), simplify its ratio by dividing both sides by their HCF (20). Simplified ratio = $60/20 : 40/20 = 3:2$. Now check options:
- Option a) 90:60 → Divide by 30 → $3:2$ (matches).
- Option b) 30:15 → Divide by 15 → $2:1$ (does not match).
- Option c) 80:40 → Divide by 40 → $2:1$ (does not match).
- Option d) 60:60 → Divide by 60 → $1:1$ (does not match).
Thus, only option a) has the same simplified ratio as Image A.

Key Concept: Proportional change, Factor multiplication

a) Reducing both dimensions equally by a factor of 10
[Solution Description]
For similarity, both dimensions must be multiplied by the same factor. Original ratio is 150:100 = 3:2 (simplified).
We need to find an option where new dimensions maintain 3:2 ratio.
Option a) Dividing both by 10 → 15:10 = 3:2 ✔️
Option b) Subtracting 50 from both → 100:50 = 2:1 ✖️
Option c) Multiplying width by 0.5 and height by 0.4 → 75:40 = 15:8 ✖️
Option d) Adding 50 to width and 30 to height → 200:130 = 20:13 ✖️
Only option a) maintains the 3:2 proportion.

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: 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 scaling both width and height by the same multiplicative factor (0.8) preserves similarity. According to the syllabus, images remain similar only if their dimensions change proportionally (same multiplicative factor). Thus, the assertion is true. The reason correctly explains why multiplicative scaling preserves similarity, as it ensures proportional changes in dimensions. Therefore, both assertion and reason are true, and the reason correctly explains the assertion.

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: Ratios, Simplifying Ratios

c) 21
[Solution Description]
Given the ratio of boys to girls is $5 : 3$ and there are 35 boys.
Let the number of girls be $x$.
According to the ratio:
$5 : 3 = 35 : x$
This can be written as:
$\frac{5}{3} = \frac{35}{x}$
Cross-multiplying:
$5x = 105$
Solving for $x$:
$x = \frac{105}{5} = 21$
Therefore, there are 21 girls.

Key Concept: Ratios, Proportionality

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

Key Concept: 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: 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: 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: Proportional Ratios

a) $8 : 12$
[Solution Description]
To determine which ratio is proportional to $4 : 6$, first simplify $4 : 6$ to its simplest form. The HCF of 4 and 6 is 2. Dividing both terms by 2 gives:
$\frac{4}{2} : \frac{6}{2} = 2 : 3$
Now, check each option by simplifying it to see if it matches $2 : 3$.
Option a) simplifies to $8 : 12$ → $\frac{8}{4} : \frac{12}{4} = 2 : 3$. This matches the simplified form of $4 : 6$.
The other options do not simplify to $2 : 3$.

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: 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: 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: Simplification of Ratios

b) $2 : 3$
[Solution Description]
To simplify the ratio $72 : 108$, we divide both terms by their greatest common divisor (GCD). The GCD of 72 and 108 is 36.
$\frac{72}{36} : \frac{108}{36} = 2 : 3$
Therefore, the simplified ratio is $2 : 3$.

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 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: 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: Simplifying ratios, Proportional reasoning

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
Let the side lengths of the two cubes be $3k$ and $5k$ respectively. The surface area of a cube is given by $6 \times (\text{side length})^2$. So, the surface areas are:
Cube 1: $6 \times (3k)^2 = 6 \times 9k^2 = 54k^2$
Cube 2: $6 \times (5k)^2 = 6 \times 25k^2 = 150k^2$
The ratio of their surface areas is $54k^2 : 150k^2$, which simplifies to $9:25$ when divided by their HCF of 6$k^2$. The reason correctly explains the assertion since the surface area indeed depends on the square of the side length.

Key Concept: Simplifying Ratios

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

Key Concept: Proportional Ratios

c) 15
[Solution Description]
For ratios to be proportional, the cross-products must be equal. Thus, $5 \times 21 = 7 \times x$, which simplifies to $105 = 7x$. Solving for $x$ gives $x = 15$.

Key Concept: Simplifying Ratios

b) $3 : 5$
[Solution Description]
To simplify the ratio $45 : 75$, find the HCF of 45 and 75, which is 15. Divide both terms by 15 to get the simplest form:
$\frac{45}{15} : \frac{75}{15} = 3 : 5$
So, the simplest form of the ratio is $3 : 5$.

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: 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: 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: Proportional Ratios

a) Yes, both simplify to $3 : 2$
[Solution Description]
To check proportionality, simplify both ratios and compare their simplest forms.
For $54 : 36$:
Step 1: Find the HCF of 54 and 36. The HCF is 18.
Step 2: Divide both terms by 18.
$54 \div 18 = 3$
$36 \div 18 = 2$
Simplified ratio: $3 : 2$.
For $90 : 60$:
Step 1: Find the HCF of 90 and 60. The HCF is 30.
Step 2: Divide both terms by 30.
$90 \div 30 = 3$
$60 \div 30 = 2$
Simplified ratio: $3 : 2$.
Since both ratios simplify to $3 : 2$, they are proportional.

Key Concept: 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: Proportional Ratios

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

Key Concept: Assertion-Reason

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To check if the ratios are proportional, simplify both ratios to their simplest form:
\begin{itemize}

Key Concept: Equivalence of ratios in simplest form

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: 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: Simplification and Proportionality

c) The simplified form is $2:3$ and proportional to $2:3$
[Solution Description]
To simplify $24:36$, divide both terms by their HCF (12).
$24 ÷ 12 = 2$ and $36 ÷ 12 = 3$.
So, the simplified form is $2:3$.
Since the given ratio is also $2:3$, they are proportional.

Key Concept: Concept of proportionality using simplest forms, Rule of Three

b) 8 days
[Solution Description]
This problem uses inverse proportionality because more workers will take fewer days to complete the same work.
The relationship is given by $W_1 × D_1 = W_2 × D_2$.
Here, $W_1 = 8$ workers, $D_1 = 12$ days, $W_2 = 12$ workers.
Solve for $D_2$:
$8 × 12 = 12 × D_2$
$96 = 12D_2$
$D_2 = 96 / 12 = 8$ days.

Key Concept: 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 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: Ratio division, proportional scaling

c) 5:6:2
[Solution Description]
1kg = 1000g original mixture.
Total parts = 5+3+2 = 10 parts.
Original amounts: A = (5/10)×1000 = 500g, B = 300g, C = 200g.
After adding 300g B: A=500g, B=600g, C=200g.
New ratio = 500:600:200 = simplify by dividing by 100 → 5:6:2.

Key Concept: Compound ratios, proportional reasoning

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

Key Concept: Proportional Reasoning, Unit 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

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 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 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 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: 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: Identifying proportional relationships, Applying factor of change

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

Key Concept: 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 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: 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: Profit Sharing

b) \$8,000
[Solution Description]
The investment ratio is $4 : 5 : 6$.
Total parts = $4 + 5 + 6 = 15$.
Share of the friend who invested the least (4 parts):
$\text{Share} = \frac{4}{15} \times 30000 = 8000$.

Key Concept: Dividing quantities in a given ratio

b) Rs.4000
[Solution Description]
Total parts = $3 + 2 = 5$. Value of one part:
$\frac{10000}{5} = 2000$
Smaller share = $2 \times 2000 = Rs.4000$.

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: Trairasika — The Rule of Three, Cross Multiplication

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that if $a : b :: c : d$ is a proportional relationship, then $ad = bc$ must hold true. This is correct because, from the definition of proportionality, there exists a common factor $f$ such that $c = f \cdot a$ and $d = f \cdot b$. Substituting these into $ad = bc$, we get $a(f \cdot b) = b(f \cdot a)$, which simplifies to $abf = abf$, proving the assertion is true.
The reason states that the cross-multiplication rule ensures the proportionality factor $f$ is consistent for all terms. This is also true because the cross-multiplication rule directly follows from setting the two expressions for $f$ equal to each other: $\frac{c}{a} = \frac{d}{b}$. Therefore, the reason correctly explains the assertion.

Key Concept: Compound proportion with inverse relationship

c) 26 days
[Solution Description]
Total work = $15 \text{ workers} \times 28 \text{ days} \times 6 \text{ hours} = 2520$ worker-hours.
New scenario: Let $d$ be the required days. Then $20 \times d \times 5 = 2520$.
Solving: $100d = 2520 \Rightarrow d = \frac{2520}{100} = 25.2$ days ≈ 26 days.

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: 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: 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: 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: Proportional Reasoning

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
Given:
- $8$ bags $\equiv 40$ kg
For direct proportion, the ratio remains constant.
Let the weight for 5 bags be $x$ kg.
Using Rule of Three:
$8 : 40 :: 5 : x$
Cross-multiplying:
$$8x = 40 \times 5$$
$$x = \frac{200}{8} = 25$$
Hence, Assertion (A) is true.
Reason (R) states the correct concept of direct proportionality, and it explains why the calculation works.

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 Reasoning, Real-World Application

c) 6
[Solution Description]
First find the work rate per machine:
12,000 newspapers / (6 machines × 18 hours) = 12000/108 ≈ 111.11 newspapers per machine-hour
Now calculate required machines for new job:
Let x = required machines
$x \times 15 \times 111.11 = 20000$
$x = 20000 / (15 \times 111.11) ≈ 12$ machines
Additional machines needed = 12 - 6 = 6

Key Concept: Proportional Reasoning, Rule of Three

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To solve the problem using the Rule of Three, we start with the given proportional ratio:
$12 : 15 :: 48 : d$
By the definition of proportional ratios, the product of the means ($15 \times 48$) must equal the product of the extremes ($12 \times d$). This gives us the equation:
$15 \times 48 = 12 \times d$
Solving for $d$:
$d = \frac{15 \times 48}{12} = 60$
The Assertion correctly states the formula to find $d$ using the Rule of Three. The Reason explains the underlying principle of proportionality that justifies this formula. Therefore, both Assertion and Reason are true, and Reason correctly explains the Assertion.

Key Concept: Rule of Three, Proportionality

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

Key Concept: 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 - Basic Application

c) \$32
[Solution Description]
Given: Pramāṇa = 5 m (a), Phala = \$20 (b), Ichchhā = 8 m (c). Using Āryabhaṭa's Rule of Three:
$d = \frac{b \times c}{a} = \frac{20 \times 8}{5} = 32$
The cost of 8 meters is \$32.

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: Sharing profit based on investment ratio

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

Key Concept: Profit Sharing

a) Rs.4500 and Rs.1500
[Solution Description]
First, simplify the investment ratio: Rs.45,000 : Rs.15,000 simplifies to $3 : 1$. Total parts = $3 + 1 = 4$. Size of each part = $\frac{6000}{4} = 1500$. Ramesh's share = $3 \times 1500 = 4500$, Suresh's share = $1 \times 1500 = 1500$.

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: Combined ratio, multi-step problem

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

Key Concept: Dividing quantities in a given ratio

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To divide Rs.6,000 in the ratio 3:2, we calculate the total number of parts as $3 + 2 = 5$. Each part is worth $\frac{6000}{5} = Rs.1,200$. A's share is $3 \times 1200 = Rs.3,600$, which matches the Assertion. The Reason correctly states the formula for finding the first part of a division in a given ratio $m:n$, which is $m \times \frac{x}{m + n}$. Therefore, both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

Key Concept: Dividing quantities in a given ratio

c) Rs.720 and \Rs.480
[Solution Description]
To divide \Rs.1,200 in the ratio $3 : 2$:
$\text{Total parts} = 3 + 2 = 5$
$\text{Each part} = \frac{1200}{5} = Rs.240$
$\text{First share} = 3 \times 240 = Rs.720$
$\text{Second share} = 2 \times 240 = Rs.480$
The two shares are \Rs.720 and \Rs.480.

Key Concept: Assertion-Reason

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.
[Solution Description]
The Assertion states that to divide \Rs.5,000 in the ratio $3 : 2$, we use the formula $\frac{x}{m + n}$.
The Reason explains that the formula $\frac{x}{m + n}$ helps find the size of each part when dividing a quantity in a given ratio.
Here, both Assertion and Reason are true. However, while the formula $\frac{x}{m + n}$ is indeed used to find the size of one part, it alone does not fully explain how the division is done in the ratio $3 : 2$. The complete process involves multiplying this value by the respective ratio terms ($3$ and $2$). Hence, the Reason does not correctly explain the Assertion.

Key Concept: Dividing quantities in a given ratio

c) 20 liters
[Solution Description]
Given ratio $2 : 3$, total parts = $2 + 3 = 5$.
$\text{Quantity of blue paint} = \frac{2}{5} \times 50 = 20 \text{ liters}$
The quantity of blue paint needed is 20 liters.

Key Concept: 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: 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 mixture adjustment

c) 80 mL
[Solution Description]
Initially, alcohol = $\frac{4}{9} \times 360 = 160$ mL, water = $\frac{5}{9} \times 360 = 200$ mL. For new ratio 4:7, required water = $\frac{7}{4} \times 160 = 280$ mL. Water to add = $280 - 200 = 80$ mL.

Key Concept: 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: 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: Combined ratio application

a) 3:2
[Solution Description]
Assume 5 units from each mixture.
From A: Blue = $\frac{2}{5} \times 5 = 2$ units, White = $\frac{3}{5} \times 5 = 3$ units.
From B: Blue = $\frac{4}{5} \times 5 = 4$ units, White = $\frac{1}{5} \times 5 = 1$ unit.
Combined Blue = 2 + 4 = 6 units.
Combined White = 3 + 1 = 4 units.
New ratio = 6:4 = 3:2 after simplifying by 2.

Key Concept: Profit Sharing and Ratio Application

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To verify the assertion, we calculate the shares of A and B using the given ratio 5:3. The total profit is \Rs.12,000. The total parts of the ratio are 5 + 3 = 8. Each part corresponds to \Rs.12,000 / 8 = \Rs.1,500. Partner A's share is 5 × \Rs.1,500 = \Rs.7,500, which matches the Assertion. The Reason correctly explains the method used to calculate the shares, as it describes the formula where the total profit is multiplied by the individual ratio divided by the sum of ratio parts.
Therefore, both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

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: 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: Area conversion, Proportional reasoning

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

Key Concept: Unit 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: 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:

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: 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: 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:

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: Mass Ratio, Unit Consistency

c) 17.5 kg
[Solution Description]
Given the mass ratio of gold to water as $37 : 2$, and the mass of $1$ litre of water is $1$ kg. Let the mass of $1$ litre of gold be $m$.
From the ratio:
$\frac{m}{1 \text{ kg}} = \frac{37}{2} \Rightarrow m = \frac{37}{2} \text{ kg} = 18.5 \text{ kg}$
The difference in mass between gold and water is:
$18.5 \text{ kg} - 1 \text{ kg} = 17.5 \text{ kg}$
So, the difference is $17.5$ kg.

Key Concept: Mass and Volume Conversion

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

Key Concept: Unit Conversion - Volume and Time

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

Key Concept: Unit Conversion in Proportional Reasoning

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that ensuring consistency in units is essential for accurate proportional reasoning. This is true because comparing or calculating with different units leads to incorrect results. The reason explains why this is necessary—different units cause wrong ratios and conclusions. Thus, both statements are true, and the reason correctly explains the assertion.