Key Concept: Proportional Reasoning, Similarity in Change

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that images A, C, and D appear similar due to proportional scaling of their dimensions, which is true as seen from the table (width and height scaled by $\frac{1}{2}$ for C and $1.5$ for D relative to A). The reason correctly explains this by stating that similarity requires equal scaling factors for both dimensions. Both statements are true, and the reason accurately justifies the assertion.

Key Concept: Proportional Reasoning, Similarity of Rectangles

d) Width = 48 cm, height = 32 cm
[Solution Description]
To find a similar rectangle, the ratio of width to height must be the same as the original rectangle. The original rectangle has a ratio of $\frac{24}{16} = \frac{3}{2}$. We need to check which option has the same ratio.
1. For width = 36 cm, height = 24 cm: $\frac{36}{24} = \frac{3}{2}$.
2. For width = 30 cm, height = 18 cm: $\frac{30}{18} = \frac{5}{3}$.
3. For width = 20 cm, height = 10 cm: $\frac{20}{10} = 2$.
4. For width = 48 cm, height = 32 cm: $\frac{48}{32} = \frac{3}{2}$.
Options 1 and 4 have the same ratio as the original rectangle.

Key Concept: Proportional Scaling and Visual Similarity

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
Let us analyze the assertion and reason step-by-step:
- **Assertion**: If Image X is scaled down such that both width and height are multiplied by the same factor (2/3), the resulting image will maintain its visual similarity because the proportions (width:height ratio) remain unchanged. This matches the key idea from the syllabus that proportional change preserves similarity.
- **Reason**: The reasoning correctly explains that keeping the scaling factor identical for both dimensions maintains the aspect ratio. This aligns with the conclusion in the syllabus that proportional changes prevent distortion.
Thus, both Assertion and Reason are true, and Reason correctly justifies the Assertion.

Key Concept: Non-Proportional Change

c) 30 mm width, 10 mm height
[Solution Description]
Distortion occurs when width and height do not change proportionally. First, simplify Image A's ratio: $60:40 = 3:2$ (divided by 20). Now analyze options:
- Option a) 120:80 → Simplified ratio $3:2$ (proportional).
- Option b) 45:30 → Simplified ratio $3:2$ (proportional).
- Option c) 30:10 → Simplified ratio $3:1$ (non-proportional).
- Option d) 90:60 → Simplified ratio $3:2$ (proportional).
Only option c) has a different simplified ratio ($3:1$) and will appear distorted.

Key Concept: Proportional Scaling

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

Key Concept: Ratios

d) 60 mm
[Solution Description]
Given the simplified ratio $3 : 2$ for Image D, let the height be $h$. We can write the proportion as $\frac{3}{2} = \frac{90}{h}$. Solving for $h$:
$3h = 180$
$h = 60$
The height of Image D is 60 mm.

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: Proportional Scaling

b) The image looks smaller or larger but maintains its shape.
[Solution Description]
Proportional scaling means both the width and height of the image are multiplied by the same factor. This preserves the shape of the image, making it look similar but different in size.

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: Ratio Application

b) 25
[Solution Description]
Given the ratio $3 : 5$ for apples to oranges, let the number of oranges be $x$. Using proportionality: $\frac{3}{5} = \frac{15}{x}$. Cross-multiply to get $3x = 75$. Solving, $x = 25$. Thus, there are 25 oranges.

Key Concept: Ratios, 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: Simplest Form of Ratio

c) $2 : 3$
[Solution Description]
To simplify the ratio $60 : 90$, divide both terms by their HCF. The HCF of 60 and 90 is 30. Dividing both by 30 gives $2 : 3$.

Key Concept: 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: Interpretation of Ratios

c) 15 cups
[Solution Description]
The given ratio of sugar to flour is $3 : 5$. For every 3 cups of sugar, 5 cups of flour are needed. If 9 cups of sugar are used, we need to find the corresponding amount of flour.
First, determine the scaling factor:
$\text{Scaling factor} = \frac{9}{3} = 3$
Multiply the flour part of the ratio by this scaling factor:
$5 \times 3 = 15$
Thus, 15 cups of flour are needed.

Key Concept: 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

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: 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: Rule of Three

b) 14 liters
[Solution Description]
We can solve this problem using the Rule of Three. The ratio of distance to petrol consumption is $120 : 8$.
Let the required petrol be $x$ liters for 210 km.
Setting up the proportion:
$120 : 8 :: 210 : x$
Cross-multiplying:
$120x = 8 \times 210$
$120x = 1680$
$x = \frac{1680}{120}$
$x = 14 \text{ liters}$

Key Concept: 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

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: Simplifying ratios

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To simplify the ratio $60 : 40$, we find the HCF of 60 and 40, which is 20. Dividing both terms by 20 gives $60 \div 20 = 3$ and $40 \div 20 = 2$. Therefore, the simplified ratio is $3 : 2$. Both the Assertion and Reason are true, and the Reason correctly explains the Assertion as it provides the correct HCF used for simplification.

Key Concept: 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: Ratio Simplification

b) $2 : 3$
[Solution Description]
To simplify the ratio $60 : 90$, find the HCF of 60 and 90, which is 30. Divide both terms by 30: $60 \div 30 = 2$ and $90 \div 30 = 3$. Therefore, the simplest form is $2 : 3$.

Key Concept: Age ratios, Changing ratios

c) 3 : 4
[Solution Description]
At Rahul's age 6, sister's age is $6 \times 2 = 12$ years.
Age difference is always $12 - 6 = 6$ years.
When Rahul is 18, sister will be $18 + 6 = 24$ years.
The ratio will be $18 : 24$.
Simplifying by dividing both terms by 6 gives $3 : 4$.

Key Concept: 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: 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, 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: 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: 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: Ratios in Simplest Form

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 need to simplify both ratios:
1. For $18:12$, divide both terms by HCF of 18 and 12 which is 6. This gives $3:2$.
2. For $27:18$, divide both terms by HCF of 27 and 18 which is 9. This gives $3:2$.
Since both simplified ratios are equal ($3:2$), they are proportional. Also, the Reason correctly explains the condition for proportionality of ratios as it states that two ratios are proportional if their simplest forms are equal.

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) 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: Proportionality using simplest forms

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
First, simplify the ratio $4:6$ by dividing both terms by their HCF 2, giving $2:3$. Next, simplify $10:15$ by dividing both terms by their HCF 5, which also gives $2:3$. Since both simplified ratios are equal ($2:3$), they are proportional. Therefore, both Assertion and Reason are true, and 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: 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: Proportional Reasoning

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

Key Concept: Proportional reasoning, 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: Ratio Application

b) 6 cups
[Solution Description]
The ratio of flour to water is 4:6 or simplified as 2:3. To find the required flour for 9 cups of water:
$\frac{2}{3} = \frac{x}{9} \\ \\ 3x = 18 \\ \\ x = 6$
You need 6 cups of flour.

Key Concept: Ratio Comparison, Proportional Reasoning

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

Key Concept: Ratio application, Cost calculation

d) \$880
[Solution Description]
Total parts = $2 + 7 = 9$.
Cement quantity = $(2/9) \times 180 = 40$ kg.
Sand quantity = $(7/9) \times 180 = 140$ kg.
Cement cost = $40 \times 15 = \$600$.
Sand cost = $140 \times 2 = \$280$.
Total cost = $600 + 280 = \$880$.

Key Concept: Ratio Comparison in Real-life Application

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

Key Concept: Ratio comparison, Unit conversion

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

Key Concept: Proportional Reasoning

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

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

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

Key Concept: 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: 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 simplification, proportional reasoning

a) $9:12$
[Solution Description]
To find the proportional ratio, first simplify $45:60$ by dividing both terms by their HCF (15):
$\frac{45}{15} : \frac{60}{15} = 3 : 4$
Now check which option simplifies to $3:4$.

Key Concept: Problem Solving with Ratios

c) 10 cups
[Solution Description]
The ratio $3 : 5$ means for every 3 cups of sugar, 5 cups of flour are needed. To find the required flour for 6 cups of sugar, set up the proportion $3 : 5 :: 6 : x$.
The factor of change is $6 ÷ 3 = 2$. Multiply the second term by this factor:
$5 × 2 = 10$
So, 10 cups of flour are needed.

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: 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: 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: Cross Multiplication

b) 112.5 km
[Solution Description]
First, convert all time units to the same measure (minutes). 1 hour and 15 minutes is equal to 75 minutes.
Given the ratio of distance to time is proportional:
$30 : 45 :: 75 : x$.
Using cross multiplication:
$30 \times x = 45 \times 75$
$x = \frac{45 \times 75}{30}$
$x = \frac{3375}{30} = 112.5$
So, the car will cover 112.5 km in 1 hour and 15 minutes.

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) 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: 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: 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: 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: 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 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 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: Proportionality in Time and Work

c) 6 days
[Solution Description]
Here, more workers imply less time (inverse proportion). Let $d$ be the required days. By inverse proportion rule:
$3 \times 10 = 5 \times d \implies d = \frac{3 \times 10}{5} = 6$
5 workers will take 6 days.

Key Concept: Rule of Three

c) 4 hours
[Solution Description]
Since both workers have the same efficiency, working together means their rates add up. The first worker's rate is $\frac{1}{8}$ tasks/hour. Combined rate is $\frac{1}{8} + \frac{1}{8} = \frac{2}{8} = \frac{1}{4}$ tasks/hour. Thus, time taken together is $\frac{1}{\frac{1}{4}} = 4$ hours.

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: 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: 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: Ratio division, profit sharing

c) Rs.10,000
[Solution Description]
First, find the ratio of their investments:
Prashanti : Bhuvan = 1,20,000 : 80,000 = 3 : 2 (simplified).
Total parts = 3 + 2 = 5.
Each part's value = Rs.25,000 / 5 = Rs.5,000.
Bhuvan's share = 2 × Rs.5,000 = Rs.10,000.

Key Concept: Sharing in Given Ratio

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

Key Concept: Ratio Division

b) 50 and 10
[Solution Description]
First, find the total number of parts in the ratio: $5 + 1 = 6$. Next, divide the total chocolates by the total parts to find the size of each part: $\frac{60}{6} = 10$. Now, multiply the number of parts for each student by the size of each part. The first student gets $5 \times 10 = 50$ chocolates, and the second student gets $1 \times 10 = 10$ chocolates.

Key Concept: Dividing quantities in a given ratio

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

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

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

Key Concept: Dividing quantities in a given ratio

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: 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: 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 Profit Based on Investment Ratio

c) 2000 and 3000
[Solution Description]
The investment ratio is $8000 : 12000$, simplified to $2 : 3$.
Total parts = $2 + 3 = 5$.
Size of each part = $\frac{5000}{5} = 1000$ dollars.
First partner's share = $2 \times 1000 = 2000$ dollars.
Second partner's share = $3 \times 1000 = 3000$ dollars.

Key Concept: 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: Profit sharing, Ratio division

c) Rs.15,000 and Rs.9,000
[Solution Description]
The investment ratio is 5:3. Total parts = 5 + 3 = 8.
Each part of profit = $\frac{24000}{8} = Rs.3,000$.
Prashanti's share = $5 \times 3000 = Rs.15,000$.
Bhuvan's share = $3 \times 3000 = Rs.9,000$.

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 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: Profit Sharing in Business

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

Key Concept: Unit Conversions - Temperature

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
Step 1: Given the assertion that $68^\circ F$ is equal to $20^\circ C$.
Step 2: Verify the assertion by applying the reason formula $Celsius = \frac{5}{9} \times (Fahrenheit - 32)$.
Step 3: Substitute $68$ for Fahrenheit in the equation:
$\frac{5}{9} \times (68 - 32) = \frac{5}{9} \times 36 = 20$.
Step 4: The calculation confirms that $68^\circ F$ indeed equals $20^\circ C$, so both Assertion and Reason are true.
Step 5: Moreover, the Reason correctly explains how to derive Celsius from Fahrenheit, making it the correct explanation of the Assertion.

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 Conversion - Area

b) 12.355 acres
[Solution Description]
Given the conversion factor: $1 \text{ hectare} = 2.471 \text{ acres}$. To convert 5 hectares to acres:
$5 \text{ hectares} \times 2.471 \text{ acres/hectare} = 12.355 \text{ acres}$
Therefore, 5 hectares is equal to 12.355 acres.

Key Concept: Unit 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: 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: Temperature

b) 36 degrees Fahrenheit
[Solution Description]
The relationship between Celsius and Fahrenheit is given by:
$\Delta F = \frac{9}{5} \times \Delta C$.
Here, $\Delta C = 20^{\circ}C$.
So, $\Delta F = \frac{9}{5} \times 20 = 36^{\circ}F$.

Key Concept: Area Conversion

c) 217,800 square feet
[Solution Description]
To convert acres to square feet, multiply the number of acres by the conversion factor. Here, 1 acre = 43,560 square feet. Therefore:
$5 \text{ acres} \times 43,560 \text{ square feet/acre} = 217,800 \text{ square feet}$

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: 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:
$\text{Tractor's time} = \frac{120}{4} = 30\,\text{hours}$

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: Unit Conversion in Proportional Reasoning

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

Key Concept: Time and Volume Conversion

c) 300 seconds
[Solution Description]
First, convert 10 litres to millilitres: $10\,\text{L} = 10,000\,\text{mL}$. Using the given rate, the time taken is calculated as:
$\text{Time taken} = \frac{15 \times 10,000}{500} = 300\,\text{seconds}$