Key Concept: Ratio Comparison

c) 30
[Solution Description]
Given the ratio of teachers to students is 1:34.
Let the number of teachers be $t$.
Therefore, $\frac{t}{1020} = \frac{1}{34}$.
Solving for $t$ gives $t = \frac{1020}{34} = 30$ teachers.

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 Reasoning, Comparing Ratios

a) Yes, both ratios simplify to $4 : 1$
[Solution Description]
Simplify Worker A's ratio: $24 : 6$ becomes $4 : 1$. Simplify Worker B's ratio: $16 : 4$ also becomes $4 : 1$. Since both ratios are equal, their building rates are proportional.

Key Concept: Non-Proportional Change

a) The image becomes wider and shorter
[Solution Description]
Non-proportional changes distort the image.
Let the original width $W$ and height $H$.
New width = $W + 0.5W = 1.5W$.
New height = $H - 0.25H = 0.75H$.
Since scaling factors differ ($1.5 \neq 0.75$), the image appears stretched horizontally and compressed vertically.

Key Concept: Similarity Verification

b) Yes, because their ratios are equal
[Solution Description]
Check if the width-to-height ratios are equal. For the first rectangle: $\frac{120}{90} = \frac{4}{3}$. For the second rectangle: $\frac{40}{30} = \frac{4}{3}$. Since both ratios are equal, the rectangles are similar.

Key Concept: 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 Scaling Factor

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

Key Concept: 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: 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: Ratios, Proportionality

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

Key Concept: Application of Proportional Reasoning

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

Key Concept: Simplest Form, 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: 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: Simplest Form of Ratios

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

Key Concept: 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: Assertion-Reason

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To check if the ratios are proportional, we simplify them:
1. Simplifying $60:40$ by dividing both terms by their HCF (20): $60 \div 20 : 40 \div 20 = 3:2$.
2. Simplifying $90:60$ by dividing both terms by their HCF (30): $90 \div 30 : 60 \div 30 = 3:2$.
Since both simplified forms are equal ($3:2$), the Assertion is true. The Reason correctly explains that they are proportional because their simplest forms match.

Key Concept: 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, Simplest Form

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

Key Concept: Comparing Ratios, Proportionality Check

b) Yes, because both simplify to $1:3$
[Solution Description] Simplify both ratios to their simplest form. For Project X, the ratio is $15:45$. The HCF of 15 and 45 is 15, so simplifying gives $\frac{15}{15} : \frac{45}{15} = 1:3$. For Project Y, the ratio is $10:30$. The HCF of 10 and 30 is 10, so simplifying gives $\frac{10}{10} : \frac{30}{10} = 1:3$. Since both simplified ratios are equal ($1:3$), the ratios are proportional.

Key Concept: 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: 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

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To simplify the ratio $12 : 18$, we find the HCF of 12 and 18. The factors of 12 are 1, 2, 3, 4, 6, and 12. The factors of 18 are 1, 2, 3, 6, 9, and 18. The highest common factor (HCF) of 12 and 18 is 6. Dividing both terms of the ratio $12 : 18$ by 6, we get $\frac{12}{6} : \frac{18}{6} = 2 : 3$. Therefore, both Assertion and Reason are true, and Reason correctly explains Assertion.

Key Concept: Ratios in Their Simplest Form, Proportional Ratios

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To determine whether the assertion and reason are correct:
1. Simplify the ratio $14 : 21$ by dividing both terms by their HCF, which is 7. This gives the simplified form as $2 : 3$.
2. Similarly, simplify the ratio $6 : 9$ by dividing both terms by their HCF, which is 3. This also gives the simplified form as $2 : 3$.
3. Since both simplified forms are equal ($2 : 3$), the ratios $14 : 21$ and $6 : 9$ are indeed proportional.
4. The Reason states that two ratios are proportional if their terms change by the same multiplicative factor. This is true, as seen here where both $14 : 21$ and $6 : 9$ reduce to the same simplest form. Hence, the Reason correctly explains the Assertion.
Therefore, both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

Key Concept: Ratios in Their Simplest Form

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The ratio $60 : 40$ simplifies to $3 : 2$ when divided by their HCF, which is 20. Similarly, the ratio $90 : 60$ simplifies to $3 : 2$ when divided by their HCF, which is 30. Since both ratios simplify to the same form ($3 : 2$), they are proportional. The Reason correctly explains why the ratios are proportional.

Key Concept: Proportional ratios with fractions

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

Key Concept: 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: 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

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: 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-Reason

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To simplify the ratio $12 : 18$, we find the HCF of 12 and 18, which is 6. Dividing both terms by 6 gives $\frac{12}{6} : \frac{18}{6} = 2 : 3$. Therefore, the Assertion is true. The Reason states that the HCF of 12 and 18 is 6, which is also true and correctly explains why the ratio simplifies to $2 : 3$.

Key Concept: Equivalence of ratios in simplest form

d) 40 : 56
[Solution Description]
Simplify $25 : 35$ to its simplest form: divide both terms by 5 to get $5 : 7$. Now check each option:
- $10 : 14$ simplifies to $5 : 7$ (proportional).
- $30 : 42$ simplifies to $5 : 7$ (proportional).
- $50 : 70$ simplifies to $5 : 7$ (proportional).
- $40 : 56$ simplifies to $5 : 7$ (proportional).
However, $40 : 56$ is proportional as well, but since all options seem proportional, rechecking: The correct non-proportional ratio would be $15 : 21$ which simplifies to $5 : 7$ (all given options are proportional). Thus, the question may need adjustment, but assuming the intent was to test recognition of simplified forms.

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: 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: 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: 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: 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: 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 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 Division

c) \$450 and \$750
[Solution Description]
To divide \$1,200 in the ratio 3:5:
$\text{Total parts} = 3 + 5 = 8 \\ \\ \text{First person's share} = \frac{3}{8} \times 1200 = 450 \\ \\ \text{Second person's share} = \frac{5}{8} \times 1200 = 750$
The shares are \$450 and \$750.

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: 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: 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) 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 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: 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 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: Trairasika - Rule of Three

d) 24
[Solution Description]
Using the Rule of Three, identify the proportional relationship and solve for the unknown.
Given ratio: $8 : 12 :: 16 : x$.
Step 1: Find the factor of change between the known terms:
$16 ÷ 8 = 2$
Step 2: Apply the same factor to the second term to find $x$:
$12 × 2 = 24$
Therefore, $x = 24$.

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: 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: 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: Ratio simplification, proportional reasoning

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

Key Concept: Ratio Application

c) 27
[Solution Description]
Given the ratio of red to blue marbles is $5 : 3$.
Let the number of blue marbles be $x$.
The proportion can be written as $5 : 3 :: 45 : x$.
Cross-multiplying: $5x = 135$.
Solving for $x$, we get $x = 27$.

Key Concept: Proportional Ratios

d) 18
[Solution Description]
To solve for $x$, we set up the proportion $12 : 18 :: x : 27$.
Cross-multiplying gives: $12 \times 27 = 18 \times x$.
Simplifying, we get $324 = 18x$.
Dividing both sides by 18: $x = 18$.

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

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
Given the proportion $5 : 250 :: 8 : x$, cross multiplication gives $5x = 250 \times 8$. Solving for $x$, we get $x = \frac{2000}{5} = 400$. Here, Assertion correctly applies the Rule of Three and derives the cost of 8 kg rice as \$400. The Reason explains why the method works mathematically, stating that proportional ratios must satisfy $ad = bc$. Both statements are true, and the Reason justifies the Assertion.

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: 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: Trairasika — The Rule of Three

c) \$160
[Solution Description]
Using the Rule of Three, we can set up the proportion as $5 : 100 :: 8 : x$. Cross-multiplying gives $5x = 800$, so $x = \frac{800}{5} = 160$. Thus, 8 books cost \$160.

Key Concept: Cross Multiplication Method

b) 21
[Solution Description]
Given the proportion $5:7 :: 15:x$. Using cross multiplication, we get:
$5 \times x = 7 \times 15$
Solving for $x$:
$x = \frac{7 \times 15}{5} = \frac{105}{5} = 21$

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: Mixing ratios, Proportionality

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

Key Concept: Cross Multiplication Method

b) 15
[Solution Description]
Given the proportion $3:4 :: x:20$, cross multiplying yields:
$3 \times 20 = 4 \times x$
Solving for $x$:
$x = \frac{3 \times 20}{4} = \frac{60}{4} = 15$

Key Concept: 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: 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

b) 40
[Solution Description]
Model the problem as the proportion $5 : 25 :: 8 : x$.
Step 1: Write the proportion and use the Rule of Three:
$5 \times x = 25 \times 8$
Step 2: Multiply the known terms:
$5x = 200$
Step 3: Solve for $x$ by dividing both sides by 5:
$x = \frac{200}{5} = 40$
The machine can produce 40 toys in 8 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: Rule of Three, Mixed Numbers

c) 30 liters
[Solution Description]
Convert mixed numbers to improper fractions: Pramāṇa = $\frac{13}{4}$ kg, Phala = $\frac{15}{2}$ liters, Ichchhā = 13 kg. Using the Rule of Three:
$ichchhāphala = \frac{phala \times ichchhā}{pramāṇa} = \frac{\frac{15}{2} \times 13}{\frac{13}{4}} = \frac{195/2}{13/4} = \frac{195 \times 4}{2 \times 13} = \frac{780}{26} = 30 \text{ liters}$

Key Concept: Rule of Three - 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: Rule of Three, Unit Conversion

b) 112.5 palas
[Solution Description]
Given: Pramāṇa = 5 palas, Phala = $\frac{2}{3}$ niskas, Ichchhā = 15 niskas. Using Āryabhaṭa's Rule of Three:
$ichchhāphala = \frac{phala \times ichchhā}{pramāṇa} = \frac{\frac{2}{3} \times 15}{5} = \frac{10}{5} = 2$
However, the calculation is incorrect as we need to find the quantity of rice (palas) for 15 niskas. Let’s re-express the proportion correctly: $5 : \frac{2}{3} :: x : 15$. Cross-multiplying gives $\frac{2}{3}x = 5 \times 15$, so $x = \frac{75 \times 3}{2} = 112.5$ palas.

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: Sharing in a given ratio

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To divide 60 objects in the ratio of 5:1:
1. Total parts = $5 + 1 = 6$.
2. Value of each part = $\frac{60}{6} = 10$.
3. First person's share = $5 \times 10 = 50$ objects.
4. Second person's share = $1 \times 10 = 10$ objects.
Thus, Assertion is true because the shares are correctly calculated using the given ratio. Reason is also true and correctly explains how the shares are determined.

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: 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: 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

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

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

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: 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: 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: 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: 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: Finding parts from the whole using ratio

b) 16 and 20
[Solution Description]
The total ratio parts = 4 + 5 = 9. Size of each part = $36 ÷ 9 = 4$. First friend gets $4 × 4 = 16$ chocolates. Second friend gets $5 × 4 = 20$ chocolates.

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

Key Concept: Mixture Problem

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

Key Concept: 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: 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 cost calculation, Area conversion

c) \$1,721,649
[Solution Description]
First, find the price per acre: $\frac{\$250,000}{0.75} \approx \$333,333.33 \text{ per acre}$.
Calculate the area of the new plot in square feet: $500 \text{ ft} \times 450 \text{ ft} = 225,000 \text{ sq ft}$.
Convert this area to acres: $\frac{225,000}{43,560} \approx 5.165 \text{ acres}$.
Multiply by the cost per acre: $5.165 \times \$333,333.33 \approx \$1,721,649.48$.

Key Concept: Unit Conversion - 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: 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:

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

Key Concept: Volume Conversion

d) 3,000 mL
[Solution Description]
Given that $1 \text{ litre} = 1,000 \text{ mL}$, multiply the number of litres by 1,000 to get the volume in millilitres:
$3 \text{ litres} \times 1,000 \text{ mL/litre} = 3,000 \text{ mL}$
Hence, 3 litres is equal to 3,000 mL.

Key Concept: Temperature Conversion

c) $20^\circ C$
[Solution Description]
Given the Fahrenheit temperature is $68^\circ F$, substitute into the formula:
$Celsius = \frac{5}{9} \times (68 - 32) = \frac{5}{9} \times 36 = 20^\circ C$
Therefore, $68^\circ F$ is equal to $20^\circ C$.

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: 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: 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 Conversions, Proportional Reasoning

c) 27.55 litres
[Solution Description]
First, convert the area of the field from square feet to acres. The area is $800 \times 600 = 480,000$ square feet. Since $1$ acre = $43,560$ square feet, the area in acres is $\frac{480,000}{43,560} \approx 11.02$ acres.
Next, use the given ratio: $5$ litres for $2$ acres. The amount of diesel needed for $11.02$ acres can be calculated using proportionality:
$\text{Diesel} = \left( \frac{5 \text{ litres}}{2 \text{ acres}} \right) \times 11.02 \text{ acres} \approx 27.55 \text{ litres}$
Therefore, approximately $27.55$ litres of diesel are needed.

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.