Key Concept: Proportional Change in Dimensions

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

Key Concept: 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

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: Visual similarity through proportional change

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The Assertion states that Image A and Image D look visually similar due to proportional scaling of their dimensions. This is true because Image A has dimensions $60:40$ ($3:2$ ratio), and Image D has dimensions $90:60$ (also $3:2$ ratio). Both images are scaled proportionally by a factor of $\frac{3}{2}$ in width and height, preserving their aspect ratio. The Reason correctly explains that proportional scaling maintains the aspect ratio, which leads to visual similarity. Therefore, both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

Key Concept: Width-Height Ratio Comparison

c) Width: 75 mm, Height: 45 mm
[Solution Description]
To determine similarity, the ratios of width-to-height must be equal. The original ratio is $\frac{50}{30} = \frac{5}{3}$. We check each option:
1) For 25 mm width, height should be $30 \times \frac{25}{50} = 15$ mm (ratio $\frac{5}{3}$).
2) For 100 mm width, height should be $30 \times \frac{100}{50} = 60$ mm (ratio $\frac{5}{3}$).
3) For 75 mm width, height should be $45$ mm (ratio $\frac{5}{3}$).
4) Incorrect dimensions do not maintain the ratio.

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: Proportional changes, Multiplicative vs additive

a) Their heights must be in ratio 3:4
[Solution Description]
For two rectangles to be similar, their corresponding sides must be proportional.
Given width ratio = $\frac{w_1}{w_2} = \frac{3}{4}$, the height ratio must be equal:
$\frac{h_1}{h_2} = \frac{3}{4}$ ⇒ $4h_1 = 3h_2$
Option A states this exact proportion. Other options:
B) Additive difference doesn't guarantee proportionality
C) Unspecific relationship
D) Equal heights would only work if widths were equal

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: Proportional Ratios

c) $12 : 18$
[Solution Description]
A ratio proportional to $4 : 6$ must simplify to the same form or satisfy $ad = bc$. The simplest form of $4 : 6$ is $2 : 3$. Check each option:
- $8 : 12$ simplifies to $2 : 3$ (proportional).
- $6 : 9$ simplifies to $2 : 3$ (proportional).
- $10 : 15$ simplifies to $2 : 3$ (proportional).
- $12 : 18$ simplifies to $2 : 3$ (proportional).
Among the given options, $12 : 18$ is correct.

Key Concept: Ratio Simplification

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

Key Concept: 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: Simplifying Ratios

b) $3 : 2$
[Solution Description]
To simplify the ratio $54 : 36$, we find the HCF of 54 and 36, which is 18. Divide both terms by 18:
$\frac{54}{18} : \frac{36}{18} = 3 : 2$
The simplified ratio is $3 : 2$.

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: Simplest Form, Proportionality Check

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

Key Concept: Proportional Ratios

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: Proportionality, Simplifying Ratios

b) 25
[Solution Description]
To find the value of $x$, we use the property of proportional ratios: $a : b :: c : d$ implies $ad = bc$. Here, $16 : x :: 64 : 100$. So, $16 \times 100 = x \times 64$. Solving for $x$:
$1600 = 64x \\ \\ x = \frac{1600}{64} = 25$
Thus, $x = 25$.

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: 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: Problem Solving with Ratios

b) 25
[Solution Description]
Given the ratio of boys to girls is $3 : 5$. Let the number of boys be $3x$ and girls be $5x$. Since there are 15 boys, $3x = 15 \Rightarrow x = 5$. Number of girls $= 5x = 5 \times 5 = 25$.

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: Problem solving with proportional reasoning, Unit conversions

a) 304,920 sq ft
[Solution Description]
Given that 3 acres = 130,680 sq ft, then 1 acre = $\frac{130,680}{3} = 43,560$ sq ft. For 7 acres: $7 \times 43,560 = 304,920$ sq ft.

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: Dividing a quantity in a given ratio, Proportional reasoning

a) Rs.4,000 and Rs.5,600
[Solution Description]
Total parts = $5 + 7 = 12$. First part = $\frac{5}{12} \times 9600 = 4000$. Second part = $\frac{7}{12} \times 9600 = 5600$. Thus, the amounts are Rs.4,000 and Rs.5,600.

Key Concept: Age Ratio Problem

a) $11 : 15$
[Solution Description]
Currently, A is 12 years old and B is 20 years old, which matches the ratio $3 : 5$ (since $12 : 20$ simplifies to $3 : 5$). After 10 years, A will be $12 + 10 = 22$ years old and B will be $20 + 10 = 30$ years old. The new ratio is $22 : 30$. Simplifying this by dividing both terms by 2 gives $11 : 15$.

Key Concept: 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

a) Yes
[Solution Description]
Simplify both ratios to their simplest forms. For $16 : 24$, the HCF is 8:
$\frac{16}{8} : \frac{24}{8} = 2 : 3$
For $20 : 30$, the HCF is 10:
$\frac{20}{10} : \frac{30}{10} = 2 : 3$
Both ratios simplify to $2 : 3$, so they are proportional.

Key Concept: Age Ratio Application

b) $1 : 3$
[Solution Description]
When Rohan is 6, his father’s age is $5 \times 6 = 30$ years. After 6 years, Rohan will be 12 and his father will be 36. The ratio is $12 : 36$, which simplifies to $1 : 3$ when divided by their HCF, 12.

Key Concept: Simplest form of ratios, Proportional ratios

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

Key Concept: Proportional ratios, Cross-multiplication

a) $10 : 16$
[Solution Description]
To check proportionality, we can cross-multiply or simplify the ratios. For two ratios to be proportional, their simplest forms must be equal. The simplest form of $5 : 8$ is already $5 : 8$. Now, let's check each option:
- $10 : 16$ simplifies to $5 : 8$ (HCF = 2).
Thus, $10 : 16$ is proportional to $5 : 8$.

Key Concept: Simplifying ratios using HCF, Proportionality of ratios

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

Key Concept: 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: 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: Assertion and Reason

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To check if the ratios $6 : 4$ and $9 : 6$ are proportional, we reduce them to their simplest forms.
1. For $6 : 4$, the HCF of 6 and 4 is 2. Dividing both terms by 2 gives $3 : 2$.
2. For $9 : 6$, the HCF of 9 and 6 is 3. Dividing both terms by 3 also gives $3 : 2$.
Since both ratios simplify to the same form $3 : 2$, they are proportional. Thus, both Assertion and Reason are true, and the Reason correctly explains the Assertion.

Key Concept: Equivalence of ratios in simplest form

c) 12
[Solution Description]
First, simplify the ratio $48 : 64$ to its simplest form. The HCF of 48 and 64 is 16. Dividing both terms by 16 gives $3 : 4$. Since $48 : 64 :: 9 : x$, their simplest forms must be equal. So, $3 : 4 = 9 : x$. Cross-multiplying gives $3x = 36$, so $x = 12$.

Key Concept: HCF Calculation

c) 16
[Solution Description]
The factors of 48 are: 1, 2, 3, 4, 6, 8, 12, 16, 24, 48. The factors of 64 are: 1, 2, 4, 8, 16, 32, 64. The highest common factor (HCF) of 48 and 64 is 16.

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: Simplest Form of a Ratio

c) $2 : 3$
[Solution Description]
To find the simplest form, divide both terms by their HCF. The HCF of 60 and 90 is 30. Dividing both terms by 30: $\frac{60}{30} = 2$, $\frac{90}{30} = 3$. 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: Simplifying Ratios, Proportionality

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To determine if the assertion is true or false:
1. Simplify $36:48$ by dividing both terms by their HCF. The HCF of 36 and 48 is 12. Simplifying gives $3:4$.
2. Simplify $27:36$ by dividing both terms by their HCF. The HCF of 27 and 36 is 9. Simplifying gives $3:4$.
3. Since both simplified ratios are equal ($3:4$), they are proportional, making the Assertion true.
4. The Reason states that two ratios are proportional if their simplest forms are equal, which is also true and correctly explains why the Assertion is true.

Key Concept: Problem Solving with 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 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 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: Proportional reasoning, unit conversion

c) 3 days
[Solution Description]
First find production rate: 1200 units/8 hours = 150 units/hour.
Daily production with 9 hours: 150 × 9 = 1350 units/day.
Days needed for 3150 units: 3150 ÷ 1350 = 2.33 days → 3 full days required (can't have partial day).

Key Concept: Compound ratios, proportional reasoning

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

Key Concept: Proportional Division

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

Key Concept: Ratio Application

b) 100 mL
[Solution Description]
The ratio $1 : 4$ means there is 1 part acid and 4 parts water, totaling 5 parts. The fraction of acid is $\frac{1}{5}$. For 500 mL, the amount of acid is $\frac{1}{5} \times 500 = 100$ mL.

Key Concept: Ratio Comparison

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: Assertion-Reason

c) Assertion is true, but Reason is false.
[Solution Description]
Let the initial quantity of juice be $3x$ litres and water be $5x$ litres. After adding 2 litres of juice, the new quantity of juice becomes $3x + 2$ litres. The new ratio is given as $1 : 1$, so we can write:
$\frac{3x + 2}{5x} = \frac{1}{1}$
Solving this equation:
$3x + 2 = 5x \\ \\ 2 = 2x \\ \\ x = 1$
Therefore, the initial quantities were 3 litres of juice and 5 litres of water. The assertion is true. However, the reason states that adding juice changes the ratio proportionally, which is incorrect because proportional change would require multiplying both terms by the same factor, not adding. Thus, the reason is false.

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: 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: 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 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: 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: 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: 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 reasoning with unit conversion

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

Key Concept: 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: 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: Proportion Application

b) 7.5 days
[Solution Description]
First, let’s represent the given data proportionally. Here, the number of workers and the number of days are inversely proportional because more workers would take fewer days for the same amount of work. So, we model it as $5 : 12 :: 8 : x$ where $x$ is the unknown number of days.
Now, using cross multiplication:
$5 \times 12 = 8 \times x$
$60 = 8x$
$x = \frac{60}{8} = 7.5$
Thus, 8 workers will take 7.5 days to build the wall.

Key Concept: 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: 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: Proportionality Check

c) $4 \times 14 = 7 \times 8$
[Solution Description]
To verify proportionality, we use the cross multiplication rule:
$4 \times 14 = 7 \times 8$
Calculating both sides:
$56 = 56$
Since both sides are equal, the ratios are proportional.

Key Concept: Cross Multiplication 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) 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: Proportional Ratios

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

Key Concept: 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: Unit Conversion in Rule of Three

c) 90 m$^2$
[Solution Description]
Given: Pramāṇa = 2.5 L (a), Phala = 30 m$^2$ (b), Ichchhā = 7.5 L (c). Using the Rule of Three:
$d = \frac{b \times c}{a} = \frac{30 \times 7.5}{2.5} = 90$
7.5 liters will cover 90 m$^2$.

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, 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

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

Key Concept: 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: 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: 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: Dividing quantities in a given ratio

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

Key Concept: Dividing quantities in a given ratio

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) 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, real-world application

d) \$7,500
[Solution Description]
First, express the working hours as a ratio: 6:9:15.
Simplify the ratio by dividing each term by 3:
Simplified ratio = 2:3:5.
Total parts = 2 + 3 + 5 = 10 parts.
Value per part = $\frac{15000}{10} = 1500$.
Employee R's share = $5 \times 1500 = 7500$.
Therefore, employee R receives \$7,500.

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: 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: Combined ratios

d) No valid solution with given data
[Solution Description]
Let first class have $3x$ boys and $2x$ girls, total $5x$. Second class has $y$ boys and $y$ girls, total $2y$. Combined ratio $(3x + y):(2x + y) = 2:1$. Solving gives $3x + y = 4x + 2y \Rightarrow x = -y$. Negative values not possible, hence scenario invalid without additional constraints.

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: 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: 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: 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: 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 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 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:

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: Length, Area

b) 16146 square feet
[Solution Description]
First, calculate the area in square meters:
$Area = Length \times Width = 50 m \times 30 m = 1500 \text{ m}^2$.
Now, convert square meters to square feet using the conversion factor:
$1 \text{ m}^2 = 10.764 \text{ ft}^2$
So, $1500 \text{ m}^2 \times 10.764 \text{ ft}^2/\text{m}^2 = 16146 \text{ ft}^2$.

Key Concept:

c) 2,500 mL
[Solution Description]
Given, $1 \text{ litre} = 1,000 \text{ mL}$. Therefore, $2.5 \text{ litres} = 2.5 \times 1,000 \text{ mL} = 2,500 \text{ mL}$.

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: Unit Conversion - Area

b) 4.1322 acres
[Solution Description]
First, calculate the area of the plot in square feet:
$300 \text{ ft} \times 600 \text{ ft} = 180,000 \text{ sq ft}$
Next, convert square feet to acres using the conversion factor:
$\frac{180,000 \text{ sq ft}}{43,560 \text{ sq ft/acre}} \approx 4.1322 \text{ acres}$

Key Concept: 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: Age Ratio, Proportion Check

c) 8 years
[Solution Description]
Let the number of years after which the ratio becomes $3 : 5$ be $x$. At that time:
- Harmain's age will be $4 + x$.
- Brother's age will be $12 + x$.
According to the problem:
$\frac{4 + x}{12 + x} = \frac{3}{5}$
Cross-multiplying:
$5(4 + x) = 3(12 + x) \Rightarrow 20 + 5x = 36 + 3x \Rightarrow 2x = 16 \Rightarrow x = 8$
Therefore, after $8$ years, the ratio of their ages will be $3 : 5$.

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:
$\frac{500 \text{ s}}{60 \text{ s/min}} \approx 8.333 \text{ min}$