Class 10 Mathematics Chapter 7 Ratio and Proportion ( Including Properties and Uses)

Key Concept: Coordinate Geometry

c) 5
[Solution Description]
The distance formula between two points $(x_1, y_1)$ and $(x_2, y_2)$ is:
$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$
Substituting the given points $(3, 4)$ and $(7, 1)$:
$d = \sqrt{(7 - 3)^2 + (1 - 4)^2} = \sqrt{16 + 9} = \sqrt{25} = 5$
Thus, the distance is $5$ units.

Key Concept: Alternendo, Invertendo

a) 27 : 28
[Solution Description]
Given: $\frac{7a + 3b}{7a - 3b} = \frac{13}{5}$
Apply Alternendo first:
$\frac{7a + 3b}{13} = \frac{7a - 3b}{5}$
Cross-multiplying:
$35a + 15b = 91a - 39b$
Bring like terms together:
$15b + 39b = 91a - 35a$
$54b = 56a$
Simplifying:
$\frac{a}{b} = \frac{54}{56} = \frac{27}{28}$
Thus, $a : b = 27 : 28$

Key Concept: Basic Distance Calculation

c) 200 km
[Solution Description]
To find the distance covered, we use the formula $Distance = Speed \times Time$. Here, Speed = 80 km/h and Time = 2.5 hours. Substituting these values into the formula, we get $Distance = 80\,km/h \times 2.5\,hours = 200\,km$.

Key Concept: Reciprocal Ratio

b) $11 : 9$
[Solution Description]
The reciprocal ratio of $a : b$ is $b : a$. Here, the given ratio is $9 : 11$. Swap the numerator and denominator to get the reciprocal ratio as $11 : 9$.

Key Concept: Continued Proportion

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
According to the definition of continued proportion, if $x$, $y$, and $z$ are in continued proportion, then $x : y = y : z$. Cross-multiplying gives $y^2 = x \times z$. This shows that the assertion is true. The reason states that the product of the first and third terms ($x \times z$) equals the square of the middle term ($y^2$), which directly explains why the assertion is true. Therefore, both the assertion and reason are true, and the reason correctly explains the assertion.

Key Concept: Speed, Time, and Distance

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that if a car travels at a constant speed of 60 km/h for 3 hours, the distance covered is 180 km. This can be verified using the formula:
$\text{Distance} = \text{Speed} \times \text{Time}$
Plugging in the values:
$60 \, \text{km/h} \times 3 \, \text{hours} = 180 \, \text{km}$
Hence, the assertion is true. The reason given is also correct since it correctly states the formula used to calculate distance when speed and time are known. Furthermore, the reason explains why the assertion is true.

Key Concept: Average Speed, Variable Speeds

b) 48 km/h
[Solution Description]
Let total distance be $D$.
Time for first half = $\frac{D/2}{40} = \frac{D}{80}$.
Time for second half = $\frac{D/2}{60} = \frac{D}{120}$.
Total time = $\frac{D}{80} + \frac{D}{120} = \frac{3D + 2D}{240} = \frac{5D}{240} = \frac{D}{48}$.
Average speed = $\frac{D}{\frac{D}{48}} = 48$ km/h.

Key Concept: Simplifying Ratios - Compound Ratio

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To find the compound ratio of $2 : 3$ and $4 : 5$, we multiply the antecedents ($2$ and $4$) to get $8$ and the consequents ($3$ and $5$) to get $15$. Thus, the compound ratio is indeed $8 : 15$. The reason correctly states the formula for the compound ratio. Therefore, both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

Key Concept: Continued Proportion, Mean Proportional

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that for $x = 16$ and $z = 25$ in continued proportion ($x : y = y : z$), the mean proportional $y$ must be 20. We can verify this by using the property of continued proportion: $\frac{x}{y} = \frac{y}{z}$. Cross-multiplying gives $y^2 = x \times z$.
Substituting the given values:
$ y^2 = 16 \times 25 $
$ y^2 = 400 $
$ y = \sqrt{400} $
$ y = 20 $
The reason states the general property of continued proportion: the square of the mean proportional equals the product of the first and third terms. This correctly explains why $y = 20$ when $x = 16$ and $z = 25$.
Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

Key Concept: Componendo, Dividendo

b) 4 : 1
[Solution Description]
Given: $\frac{x + y}{x - y} = \frac{5}{3}$
By Componendo-Dividendo:
$\frac{(x+y)+(x-y)}{(x+y)-(x-y)} = \frac{5+3}{5-3}$
Simplifying numerator and denominator:
$\frac{2x}{2y} = \frac{8}{2}$
$\Rightarrow \frac{x}{y} = 4$
Therefore, $x : y = 4 : 1$

Key Concept: Commensurable Quantities

a) $4\%$ and $6\%$
[Solution Description]
To determine if two quantities are commensurable, their ratio must be expressible as a fraction of integers.
Option 1: $4\%$ and $6\%$ can be written as $\frac{4}{6} = \frac{2}{3}$, which is a ratio of integers.
Option 2: $\sqrt{2}$ and $3$ have a ratio $\frac{\sqrt{2}}{3}$, which cannot be expressed as a ratio of integers.
Option 3: $5$ meters and $10$ meters have a ratio $\frac{5}{10} = \frac{1}{2}$, which is a ratio of integers.
Option 4: $\pi$ and $2\pi$ have a ratio $\frac{\pi}{2\pi} = \frac{1}{2}$, but since $\pi$ is irrational, this pair is incommensurable.
Therefore, the correct answer is the pair with a ratio that can be expressed as a fraction of integers.

Key Concept: Assertion-Reason

d) Assertion is false, but Reason is true.
[Solution Description]
Let the work done by $(x - 2)$ men in $(4x + 1)$ days be $W_1$ and the work done by $(4x + 1)$ men in $(2x - 3)$ days be $W_2$. According to the problem, $\frac{W_1}{W_2} = \frac{3}{8}$.
Since work is directly proportional to the number of men and the number of days, we can write:
$W_1 = k \cdot (x - 2)(4x + 1)$
$W_2 = k \cdot (4x + 1)(2x - 3)$
Thus, the ratio becomes:
$\frac{(x - 2)(4x + 1)}{(4x + 1)(2x - 3)} = \frac{3}{8}$
Simplifying:
$\frac{x - 2}{2x - 3} = \frac{3}{8}$
Solving for $x$:
$8(x - 2) = 3(2x - 3)$
$8x - 16 = 6x - 9$
$2x = 7 \Rightarrow x = 3.5$
However, the Assertion states $x = 5$, which is incorrect. The Reason correctly describes the proportionality principles but fails to explain Assertion as it leads to a different value of $x$.

Key Concept: Unit Conversion

c) 100 cm
[Solution Description]
By definition, 1 meter equals 100 centimeters.
$1 \text{ m} = 100 \text{ cm}$
So, 1 meter has 100 centimeters.

Key Concept: Mean Proportional, Continued Proportion

c) 12
[Solution Description]
For continued proportion, $a : b = b : c$. This implies $b^2 = a \times c$. Substituting the given values, $b^2 = 9 \times 16 = 144$. Taking the square root of both sides, $b = 12$ (since length cannot be negative).

Key Concept: Solving Equations Using Componendo and Dividendo

b) 11
[Solution Description]
Apply componendo and dividendo to the equation:
$\frac{(x + 1) + (x - 1)}{(x + 1) - (x - 1)} = \frac{6 + 5}{6 - 5}$
Simplify numerator and denominator:
$\frac{2x}{2} = \frac{11}{1} \implies x = 11$

Key Concept: Terms of Proportion

b) 7 and 15
[Solution Description]
In a proportion $a : b = c : d$, the middle terms $b$ and $c$ are called the means.
Here, $5 : 7 = 15 : 21$ implies $b = 7$ and $c = 15$ are the means.

Key Concept: Definition of Ratio

d) Assertion is false, but Reason is true.
[Solution Description]
To check if a ratio is in its simplest form, we need to verify whether the H.C.F. of its terms is 1. Here, the terms of the ratio are 4 and 16.
Step 1: Find the factors of 4 ($1, 2, 4$) and 16 ($1, 2, 4, 8, 16$).
Step 2: Identify common factors between 4 and 16, which are $1, 2, 4$.
Step 3: The highest common factor (H.C.F.) is 4.
Since the H.C.F. of the terms is not 1, the ratio $4:16$ is not in its simplest form. Thus, the Assertion (A) is false.
However, the Reason (R) claims that the H.C.F. is 1, which is incorrect as it is actually 4. Hence, the Reason (R) is also false.

Key Concept: Relative Speed, Uniform Motion

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
First, convert the speeds from km/h to m/s by multiplying with $\frac{1000}{3600}$ (or $\frac{5}{18}$). For Train 1: $60 \times \frac{5}{18} = \frac{50}{3}$ m/s. For Train 2: $90 \times \frac{5}{18} = 25$ m/s. Relative speed when moving in opposite directions is indeed the sum of their speeds: $\frac{50}{3} + 25 = \frac{125}{3}$ m/s. The time taken to cross each other is given as $10$ seconds. Total distance covered (combined length of trains) = relative speed $\times$ time = $\frac{125}{3} \times 10 = \frac{1250}{3}$ meters, which matches the assertion. The reason correctly explains the assertion since the relative speed formula for opposite directions is used to derive the result.

Key Concept: Continued Proportion

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
Step 1: Check if 2, 6, and 18 are in continued proportion by verifying if the ratio $\frac{2}{6} = \frac{6}{18}$.
Step 2: Simplifying both ratios, we get $\frac{1}{3}$ on both sides, so they are in continued proportion.
Step 3: The mean proportional (middle term) is indeed 6.
Step 4: The reason states a fundamental property of continued proportion ($b^2 = a \times c$ for $a$, $b$, $c$ in continued proportion). Here, $6^2 = 2 \times 18$ or $36 = 36$, which is true.
Step 5: The reason correctly explains why 6 is the mean proportional.

Key Concept: Fourth Proportional, Units in Proportion

b) $28.125\,\text{kg}$
[Solution Description]
Given $A : B = B : C$.
Let $B = 8x$ and $A = 5x$ (since $A : B = 5 : 8$).
Then, $B : C = 8x : 72$.
Since $A : B = B : C$, we have:
$5x : 8x = 8x : 72 \implies \frac{5x}{8x} = \frac{8x}{72} \implies \frac{5}{8} = \frac{8x}{72}.$
Cross-multiplying gives:
$5 \times 72 = 8 \times 8x \implies 360 = 64x \implies x = \frac{360}{64} = \frac{45}{8}.$
Thus, weight of $A = 5x = 5 \times \frac{45}{8} = \frac{225}{8} = 28.125\,\text{kg}$.

Key Concept: Continued Proportion

c) 12
[Solution Description]
Given $a : b = b : c$, which implies:
$\frac{a}{b} = \frac{b}{c}$
Substituting the given values:
$\frac{9}{b} = \frac{b}{16}$
Cross-multiplying gives:
$b^2 = 9 \times 16 = 144$
Taking the square root of both sides:
$b = \sqrt{144} = 12$

Key Concept: Duplicate Ratio, Proportion

a) $\frac{270}{47}$
[Solution Description]
The duplicate ratio of $7 : 5$ is $7^2 : 5^2 = 49 : 25$. So, we have:
$\frac{4x + 1}{3x - 5} = \frac{49}{25}$
Cross-multiplying gives:
$25(4x + 1) = 49(3x - 5)$
Simplifying:
$100x + 25 = 147x - 245$
Bringing like terms together:
$47x = 270 \implies x = \frac{270}{47}$

Key Concept: Sub-triplicate Ratio

a) 4 : 5
[Solution Description]
The sub-triplicate ratio is the cube root of each term in the original ratio.
Original ratio: $64 : 125$
Sub-triplicate ratio: $\sqrt[3]{64} : \sqrt[3]{125} = 4 : 5$

Key Concept: Componendo

c) $4 + 7 : 7 = 8 + 14 : 14$
[Solution Description]
The componendo property states that if $a : b = c : d$, then $a + b : b = c + d : d$. For $4 : 7 = 8 : 14$, adding the numerator and denominator gives $\frac{4 + 7}{7} = \frac{8 + 14}{14}$, which simplifies to $\frac{11}{7} = \frac{22}{14}$ or $11 : 7 = 22 : 14$.

Key Concept: Reciprocal Ratio

b) $7 : 5$
[Solution Description]
The reciprocal ratio of $a : b$ is $b : a$. Therefore, the reciprocal ratio of $5 : 7$ is calculated as:
$7 : 5$

Key Concept: Ratio and Proportion

c) Assertion is true, but Reason is false.
[Solution Description]
Given the ratio $x : y = 3 : 4$, let $x = 3k$ and $y = 4k$ for some constant $k$.
Now, calculate $(3x + 4y) : (5x - 2y)$:
Substitute $x$ and $y$:
$3x + 4y = 3(3k) + 4(4k) = 9k + 16k = 25k$
$5x - 2y = 5(3k) - 2(4k) = 15k - 8k = 7k$
Thus, the ratio becomes $25k : 7k = 25 : 7$. Therefore, the Assertion is true.
The Reason states that adding the numerator and denominator of the first term to the numerator and denominator of the second term keeps the ratio the same. However, this is not always true and is incorrect. For example, take $a : b = 1 : 2$. Adding numerator and denominator gives $(1+1) : (2+2) = 2 : 4 = 1 : 2$, which holds here but fails for other ratios like $2 : 3$ where $(2+2) : (3+3) = 4 : 6 = 2 : 3$ holds again, but this is not a general rule for all ratios. Hence, the Reason is false.

Key Concept: Continued Proportion

a) $\frac{x + y}{y + z} = \frac{x}{y}$
[Solution Description]
In continued proportion, $y^2 = xz$. Check each option:
Option a: $\frac{x + y}{y + z} = \frac{x}{y}$. Substitute $y^2 = xz$:
$\frac{x + y}{y + z} = \frac{x}{y} \implies y(x + y) = x(y + z)$
Simplifies to $xy + y^2 = xy + xz$, which reduces to $y^2 = xz$ (True).

Key Concept: Basic Proportion Concept

a) 8 : 15
[Solution Description]
Given $a : b = 2 : 3$ and $b : c = 4 : 5$. To find $a : c$, multiply the two ratios such that $b$ cancels out. Multiply the first ratio by 4 and the second by 3 to make $b$ common: $(2 \times 4) : (3 \times 4) = 8 : 12$ and $(4 \times 3) : (5 \times 3) = 12 : 15$. Thus, $a : c = 8 : 15$.

Key Concept: Property of Proportion

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
From the given proportion $a : b = c : d$, we can apply the Componendo and Dividendo rule, which states that if $a : b = c : d$, then $(a + b) : (a - b) = (c + d) : (c - d)$. Therefore, both Assertion and Reason are true, and Reason correctly explains the Assertion.

Key Concept: Mixtures, Ratios

c) 24 kg
[Solution Description]
First, find the current quantities of milk and water in the mixture.
Total parts = $5 + 2 = 7$ parts.
Quantity of milk = $\frac{5}{7} \times 126 = 90$ kg.
Quantity of water = $\frac{2}{7} \times 126 = 36$ kg.
Let $x$ kg of water be added to achieve the new ratio $3 : 2$.
The new ratio of milk to water is $\frac{90}{36 + x} = \frac{3}{2}$.
Cross-multiplying: $180 = 108 + 3x$.
Solving for $x$: $3x = 72$ $\Rightarrow$ $x = 24$ kg.

Key Concept: Ratio and Saving

b) \$200
[Solution Description]
Let Ravi's pocket money = $5x$, Sanjeev's = $7x$.
Ravi's expenditure = $3y$, Sanjeev's = $5y$.
Savings for Ravi: $5x - 3y = 80$.
Savings for Sanjeev: $7x - 5y = 80$.
Solving these equations:
Multiply first by 5: $25x - 15y = 400$.
Multiply second by 3: $21x - 15y = 240$.
Subtract the second from the first: $4x = 160 \Rightarrow x = 40$.
Thus, Ravi's pocket money = $5 \times 40 = 200$.

Key Concept: Understanding the Problem

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that reading a problem carefully helps in understanding the given information and the requirement. This is true because without careful reading, one might miss key details. The reason states that understanding the problem involves recognizing variables and their relationships, which is also true. Furthermore, the reason correctly explains why careful reading is essential—it allows us to identify variables and relationships. Hence, both are true, and the reason explains the assertion.

Key Concept: Ratio Word Problem

a) $3 : 2$
[Solution Description]
The given ratio of girls to boys is $18 : 12$. To simplify, divide both terms by their HCF, which is 6. $18 \div 6 = 3$ and $12 \div 6 = 2$. The simplified ratio is $3 : 2$.

Key Concept: Ratio Adjustment

c) 6
[Solution Description]
Current ratio of boys to girls is $3 : 1$, so let boys = $3x$ and girls = $x$.
Total members = $3x + x = 4x = 36 \Rightarrow x = 9$.
Current boys = $27$, girls = $9$.
Let $y$ be the number of additional girls. Then, $\frac{27}{9 + y} = \frac{9}{5}$.
Cross-multiplying: $135 = 81 + 9y \Rightarrow 9y = 54 \Rightarrow y = 6$.

Key Concept: Alternendo

a) $3 : 6 = 5 : 10$
[Solution Description]
The alternendo property states that if $a : b = c : d$, then $a : c = b : d$. Applying this to $3 : 5 = 6 : 10$, we get $3 : 6 = 5 : 10$, which simplifies to $1 : 2 = 1 : 2$.

Key Concept: Assertion-Reason

d) Assertion is false, but Reason is true.
[Solution Description] To solve for $x$, we first note that the duplicate ratio of $\sqrt{5} : \sqrt{6}$ is $(\sqrt{5})^2 : (\sqrt{6})^2 = 5 : 6$. Thus, $(2x + 3) : (5x - 38) = 5 : 6$. Applying the property of proportion, we get $(2x + 3) \times 6 = (5x - 38) \times 5$. Simplifying, $12x + 18 = 25x - 190$. Rearranging terms: $12x - 25x = -190 - 18$ gives $-13x = -208$, so $x = 16$. However, the Assertion states $x = 12$, which is false. The Reason correctly defines the duplicate ratio as $a^2 : b^2$, which is true.

Key Concept: Duplicate Ratio, Solving for Variable

c) 6
[Solution Description]
The duplicate ratio of $3 : 5$ is $3^2 : 5^2 = 9 : 25$.
Given that $(x + 3) : (4x + 1) = 9 : 25$, we can write:
$\frac{x + 3}{4x + 1} = \frac{9}{25}$.
Cross-multiplying:
$25(x + 3) = 9(4x + 1)$.
Expanding both sides:
$25x + 75 = 36x + 9$.
Rearranging terms:
$75 - 9 = 36x - 25x$.
$66 = 11x$.
Solving for $x$:
$x = 6$.

Key Concept: Commensurable and Incommensurable Quantities

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To determine if two quantities are commensurable, their ratio must be expressible as the ratio of two integers.
1. Assertion states that the ratio between $4m$ and $6m$ is commensurable.
- The ratio is $\frac{4m}{6m} = \frac{2}{3}$, which is a ratio of two integers (2 and 3).
- Hence, the Assertion is true.
2. Reason states that the ratio $\frac{4}{6}$ can be expressed as the ratio of two integers.
- $\frac{4}{6}$ simplifies to $\frac{2}{3}$, which is indeed the ratio of two integers (2 and 3).
- Thus, the Reason is also true.
3. The Reason correctly explains why the Assertion is true because the commensurability of $4m$ and $6m$ depends on whether their ratio can be written as a fraction of integers, which it can ($\frac{2}{3}$).
Therefore, both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

Key Concept: Ratio simplification, Multiplicative property

b) 23 : 12
[Solution Description]
First, cross-multiply the given equation: $7(3x + 4y) = 9(5x - 2y)$.
Expand both sides: $21x + 28y = 45x - 18y$.
Bring all terms to one side: $21x - 45x + 28y + 18y = 0$.
Simplify: $-24x + 46y = 0$.
Rearrange to get $24x = 46y$.
Divide both sides by 2: $12x = 23y$.
The ratio $x : y = 23 : 12$, which is already in simplest form as 23 and 12 are co-prime.

Key Concept: Proportion Application

c) \$32
[Solution Description]
Since the cost is directly proportional to the number of books, the ratio of books to cost remains constant. Let the cost of 8 books be \$x. We set up the proportion: $\frac{5}{20} = \frac{8}{x}$. Cross-multiplying gives $5x = 160$, so $x = \frac{160}{5} = 32$. The cost of 8 books is \$32.

Key Concept: Proportion, Mean proportional, Third proportional

b) 7 and 28
[Solution Description]
Let the numbers be $a$ and $b$.
Mean proportional condition: $\sqrt{ab} = 14 \implies ab = 196$.
Third proportional condition: $\frac{b^2}{a} = 112 \implies b^2 = 112a$.
Substitute $a = \frac{196}{b}$ into $b^2 = 112a$:
$b^2 = 112 \left(\frac{196}{b}\right) \implies b^3 = 112 \times 196 \implies b^3 = 21952 \implies b = 28$
Then, $a = \frac{196}{28} = 7$.
The two numbers are 7 and 28.

Key Concept: Proving Identities, Componendo and Dividendo

c) $\frac{1}{4}$
[Solution Description]
Apply componendo and dividendo to the given equation:
$\frac{(\sqrt{5x + y} - \sqrt{5x - y}) + (\sqrt{5x + y} + \sqrt{5x - y})}{(\sqrt{5x + y} - \sqrt{5x - y}) - (\sqrt{5x + y} + \sqrt{5x - y})} = \frac{1 + 2}{1 - 2}$.
Simplify numerator and denominator:
$\frac{2\sqrt{5x + y}}{-2\sqrt{5x - y}} = -3$,
Or $\frac{\sqrt{5x + y}}{\sqrt{5x - y}} = 3$.
Square both sides:
$\frac{5x + y}{5x - y} = 9$.
Cross-multiply:
$5x + y = 45x - 9y$.
Rearrange terms:
$-40x + 10y = 0$,
Or $40x = 10y$.
Thus, $\frac{x}{y} = \frac{1}{4}$.

Key Concept: Compound Ratio, Proportion

b) 5
[Solution Description]
The compound ratio of $3 : 4$ and $5 : 6$ is calculated as $(3 \times 5) : (4 \times 6) = 15 : 24$. Simplify this ratio by dividing both terms by 3: $5 : 8$. Given that this is equal to $x : 8$, we can set up the proportion $\frac{5}{8} = \frac{x}{8}$. Solving for $x$, we get $x = 5$.

Key Concept: Basic Speed, Time, and Distance

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that if a car travels 60 km in 1 hour, its speed is 60 km/h. This is correct because speed is given by $\text{Speed} = \frac{\text{Distance}}{\text{Time}} = \frac{60\,\text{km}}{1\,\text{hour}} = 60\,\text{km/h}$. The reason explains that speed is calculated as distance traveled divided by the time taken, which is the correct formula for speed. Therefore, the reason correctly explains the assertion.

Key Concept: Duplicate Ratio

b) $9 : 16$
[Solution Description]
The duplicate ratio of $a : b$ is $a^2 : b^2$.
For $3 : 4$, the duplicate ratio is $3^2 : 4^2 = 9 : 16$.

Key Concept: Fundamental Property of Proportion

c) $ad = bc$
[Solution Description]
By the fundamental property of proportion, if $a : b = c : d$, then the product of the extremes equals the product of the means, i.e., $ad = bc$.

Key Concept: Componendo-Dividendo Property

c) $\frac{c - d}{c + d}$
[Solution Description]
Using the Componendo-Dividendo property, if $\frac{a}{b} = \frac{c}{d} = k$, then $\frac{a + b}{a - b} = \frac{c + d}{c - d}$. Taking the reciprocal, we get $\frac{a - b}{a + b} = \frac{c - d}{c + d}$.

Key Concept: Definition of Continued Proportion

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
First, let's check if the given numbers $2$, $6$, and $18$ are in continued proportion.
- The ratio of the first to the second term is $\frac{2}{6} = \frac{1}{3}$.
- The ratio of the second to the third term is $\frac{6}{18} = \frac{1}{3}$.
Since both ratios are equal, they are in continued proportion.
Now, verify Assertion (A): The middle term here is $6$, which is indeed the mean proportional between $2$ and $18$.
Next, Reason (R): For numbers in continued proportion, the property $b^2 = a \times c$ holds true, where $b$ is the middle term. Substituting the values: $6^2 = 2 \times 18 \implies 36 = 36$, which is true.
Hence, both Assertion (A) and Reason (R) are true, and Reason (R) correctly explains Assertion (A).

Key Concept: Proportion, Commensurable Quantities

a) $\sqrt{2} : \sqrt{3}$
[Solution Description]
First, find the combined ratio of $a : b : c$:
$a : b = 3 : 4, \quad b : c = 8 : 9$
To combine these, make the common term $b$ equal:
$a : b = 3 \times 2 : 4 \times 2 = 6 : 8$
Now, $a : b : c = 6 : 8 : 9$
Thus, $a : c = 6 : 9 = 2 : 3$
The sub-duplicate ratio of $a : c$ is:
$\sqrt{2} : \sqrt{3}$

Key Concept: Commensurable and Incommensurable Quantities

a) Commensurable
[Solution Description]
To determine if $4\frac{1}{3}$ and $6\frac{1}{2}$ are commensurable, we need to find their ratio expressed as a ratio of two integers. First, convert both mixed numbers to improper fractions.
$4\frac{1}{3} = \frac{13}{3}$ and $6\frac{1}{2} = \frac{13}{2}$.
Now, find the ratio $\frac{\frac{13}{3}}{\frac{13}{2}} = \frac{13}{3} \times \frac{2}{13} = \frac{2}{3}$.
Since $\frac{2}{3}$ is a ratio of two integers (2 and 3), the quantities are commensurable.

Key Concept: Ratio equivalence, Simplification

c) $29 : 9$
[Solution Description]
Cross-multiply the given equation:
$3(4x + 5y) = 7(3x - 2y)$
Expand both sides:
$12x + 15y = 21x - 14y$
Bring all terms to one side:
$12x - 21x + 15y + 14y = 0$
Combine like terms:
$-9x + 29y = 0$
Rearrange to solve for $x/y$:
$9x = 29y$
$x/y = 29/9$
So, the ratio $x : y$ is $29 : 9$.

Key Concept: Definition of Ratio

c) $2 : 3$
[Solution Description]
To find the simplest form of the ratio $8 : 12$, we identify the GCD of 8 and 12, which is 4. Dividing both terms by 4 gives:
$\frac{8}{4} : \frac{12}{4} = 2 : 3$
Therefore, the simplest form of the ratio $8 : 12$ is $2 : 3$.

Key Concept: Finding Unknowns in Proportions

c) 24
[Solution Description]
The duplicate ratio of $\sqrt{2} : \sqrt{3}$ is $(\sqrt{2})^2 : (\sqrt{3})^2 = 2 : 3$.
Given, $(3x + 4) : (5x - 6) = 2 : 3$.
Using proportion property: $3(3x + 4) = 2(5x - 6)$.
Expanding: $9x + 12 = 10x - 12$.
Simplifying: $-x + 24 = 0$ $\Rightarrow$ $x = 24$.
Thus, the value of $x$ is 24.

Key Concept: Sub-triplicate Ratio

c) $8 : 27$
[Solution Description]
The sub-triplicate ratio of $a : b$ is $\sqrt[3]{a} : \sqrt[3]{b}$. Given $\sqrt[3]{x} : \sqrt[3]{y} = 2 : 3$, cubing both sides gives:
$(\sqrt[3]{x})^3 : (\sqrt[3]{y})^3 = 2^3 : 3^3 \\ \\ \Rightarrow x : y = 8 : 27$

Key Concept: Componendo-Dividendo, Proportion

c) $\frac{c + 2d}{c - 2d}$
[Solution Description]
Given $\frac{a}{b} = \frac{c}{d} = k$ (let’s say), so $a = kb$ and $c = kd$. Now:
$\frac{a + 2b}{a - 2b} = \frac{kb + 2b}{kb - 2b} = \frac{b(k + 2)}{b(k - 2)} = \frac{k + 2}{k - 2}$
Similarly, $\frac{c + 2d}{c - 2d} = \frac{kd + 2d}{kd - 2d} = \frac{d(k + 2)}{d(k - 2)} = \frac{k + 2}{k - 2}$. Thus:
$\frac{a + 2b}{a - 2b} = \frac{c + 2d}{c - 2d}$

Key Concept: Continued Proportion, Adding a Number

b) $-2.5$
[Solution Description]
Let the number to be added be $k$. The numbers become $(7 + k)$, $(10 + k)$, and $(15 + k)$. For them to be in continued proportion:
$(7 + k)(15 + k) = (10 + k)^2$
Step 1: Expand both sides.
$105 + 22k + k^2 = 100 + 20k + k^2$
Step 2: Simplify by subtracting $k^2$ from both sides.
$105 + 22k = 100 + 20k$
Step 3: Bring like terms together.
$2k = -5$
Step 4: Solve for $k$.
$k = -2.5$
Therefore, $-2.5$ must be added to each number.

Key Concept: Third Proportional, Continued Proportion

c) $36$
[Solution Description]
To find the third proportional $x$ for $a = 4$ and $b = 12$, we use the relation $\frac{a}{b} = \frac{b}{x}$.
Step 1: Cross-multiply to solve for $x$.
$4 \times x = 12 \times 12$
Step 2: Simplify the equation.
$4x = 144$
Step 3: Solve for $x$.
$x = \frac{144}{4} = 36$
Therefore, the third proportional is $36$.

Key Concept: Proportion Problems

b) 5
[Solution Description]
Let the number to be subtracted be $x$. For the remainders to be in proportion:
$\frac{15 - x}{20 - x} = \frac{27 - x}{38 - x}.$
Cross-multiply:
$(15 - x)(38 - x) = (20 - x)(27 - x).$
Expand both sides:
$570 - 53x + x^2 = 540 - 47x + x^2.$
Simplify:
$570 - 53x = 540 - 47x.$
Rearrange:
$30 = 6x \implies x = 5.$
The required number is 5.

Key Concept: Componendo and Dividendo

c) Undefined
[Solution Description]
Given $\frac{a}{b} = \frac{5}{3}$, let $a = 5k$ and $b = 3k$ for some constant $k \neq 0$. Substitute these into the expression $\frac{3a + 5b}{3a - 5b}$:
$\frac{3(5k) + 5(3k)}{3(5k) - 5(3k)} = \frac{15k + 15k}{15k - 15k} = \frac{30k}{0}.$
Division by zero is undefined, so the expression does not exist. However, applying componendo and dividendo directly:
$\frac{a + b}{a - b} = \frac{5 + 3}{5 - 3} = \frac{8}{2} = 4.$
But the question requires $\frac{3a + 5b}{3a - 5b}$. Rechecking the solution, substitute $a = 5k$, $b = 3k$ again:
$\frac{3(5k) + 5(3k)}{3(5k) - 5(3k)} = \frac{30k}{0}.$
The correct answer is that the expression is undefined, but since none of the options reflect this, there may be an error in the question setup.

Key Concept: Simplification of Ratios

b) $2 : 3$
[Solution Description]
To simplify the ratio $16 : 24$, we need to divide both terms by their highest common factor (H.C.F.). The H.C.F. of 16 and 24 is 8. Dividing both terms by 8, we get:
$\frac{16}{8} : \frac{24}{8} = 2 : 3$
So, the simplified form of the ratio is $2 : 3$.

Key Concept: Basic Ratio Simplification

b) 2 : 3
[Solution Description]
To simplify the ratio $16 : 24$, we need to find the Highest Common Factor (H.C.F.) of both numbers. The H.C.F. of 16 and 24 is 8. Now, divide both terms of the ratio by 8:
$\frac{16}{8} = 2$
$\frac{24}{8} = 3$
So, the simplified ratio is $2 : 3$.

Key Concept: Adding Quantities to Change Ratios

b) -5
[Solution Description]
Let the quantity to be added be $k$. Then:
$\frac{9 + k}{17 + k} = \frac{1}{3}$
Cross-multiplying:
$3(9 + k) = 1(17 + k)$
Simplifying:
$27 + 3k = 17 + k$
Rearranging:
$2k = -10 \implies k = -5$
Therefore, $-5$ must be added (or equivalently, $5$ must be subtracted).

Key Concept: Commensurable and Incommensurable Quantities, Ratio Representation

b) Incommensurable, because $\sqrt{5}$ is irrational
[Solution Description]
To check if the quantities are commensurable, we need to find the ratio of A to B which is $\frac{\sqrt{5}}{10}$. This ratio cannot be expressed as a fraction of two integers since $\sqrt{5}$ is irrational. Therefore, the quantities are incommensurable.

Key Concept: Property of Proportion

c) $a \times d = b \times c$
[Solution Description]
By the property of proportion, if $\frac{a}{b} = \frac{c}{d}$, then the product of extremes equals the product of means: $a \times d = b \times c$.

Key Concept: Duplicate Ratio, Proportion

d) $-\frac{136}{3}$
[Solution Description]
The duplicate ratio of $5 : 6$ is $5^2 : 6^2 = 25 : 36$.
Given $(2x - 1) : (3x + 4) = 25 : 36$.
Using the property of proportion: $(2x - 1) \times 36 = (3x + 4) \times 25$.
Simplifying: $72x - 36 = 75x + 100$.
Rearranging: $-3x = 136$.
Thus, $x = -\frac{136}{3}$.

Key Concept: Definition of Ratio, Properties of Ratio

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To analyze the assertion and reason:
1. The assertion states that $\frac{15x + 10y}{3x + 2y}$ is always in its lowest terms. Let us simplify this ratio:
$\frac{15x + 10y}{3x + 2y} = \frac{5(3x + 2y)}{3x + 2y}$
If $3x + 2y \neq 0$, we can cancel $(3x + 2y)$ from numerator and denominator, giving $\frac{5}{1}$. This is indeed in its lowest terms. Thus, the assertion is true.
2. The reason claims that any ratio $\frac{a}{b}$ can be reduced to its lowest terms by dividing numerator and denominator by their H.C.F. This is a fundamental property of ratios and is always true.
3. The assertion is a specific case that aligns with the general principle stated in the reason, and the reason correctly explains why the assertion holds true.
Hence, both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

Key Concept: Duplicate Ratio

a) $16 : 49$
[Solution Description]
The duplicate ratio of $a : b$ is $a^2 : b^2$. For $4 : 7$:
$4^2 : 7^2 = 16 : 49$

Key Concept: Motion, Relative Speed

c) 240
[Solution Description]
1. **Time Until Trains Meet**: Combined speed = $60 + 40 = 100$ km/h. Distance = 300 km.
$$ \text{Time} = \frac{300}{100} = 3 \text{ hours}. $$
2. **Bird's Flight Distance**: The bird flies nonstop for 3 hours at 80 km/h.
$$ \text{Distance} = 80 \times 3 = 240 \text{ km}. $$

Key Concept: Proportion, Componendo and Dividendo

a) $\frac{\sqrt[3]{22}}{\sqrt[3]{3}}$
[Solution Description]
We are given $\frac{x^3 + y^3}{x^3 - y^3} = \frac{25}{19}$. Apply componendo and dividendo:
Step 1: Add 1 to both sides of the equation:
$\frac{x^3 + y^3 + x^3 - y^3}{x^3 - y^3} = \frac{25 + 19}{19} \implies \frac{2x^3}{x^3 - y^3} = \frac{44}{19}$
Step 2: Subtract 1 from both sides of the original equation:
$\frac{x^3 + y^3 - (x^3 - y^3)}{x^3 - y^3} = \frac{25 - 19}{19} \implies \frac{2y^3}{x^3 - y^3} = \frac{6}{19}$
Step 3: Divide the result of Step 1 by the result of Step 2:
$\frac{2x^3 / (x^3 - y^3)}{2y^3 / (x^3 - y^3)} = \frac{44/19}{6/19} \implies \frac{x^3}{y^3} = \frac{22}{3}$
Step 4: Take cube roots on both sides:
$\left(\frac{x}{y}\right)^3 = \frac{22}{3} \implies \frac{x}{y} = \sqrt[3]{\frac{22}{3}}$
Simplifying $\sqrt[3]{\frac{22}{3}}$, we get $\frac{\sqrt[3]{22}}{\sqrt[3]{3}}$.

Key Concept: Ratio and Proportion

c) Assertion is true, but Reason is false.
[Solution Description]
The ratio of boys to girls is $3 : 2$. Let the number of boys be $3x$ and girls be $2x$.
Total students = $3x + 2x = 5x = 30$.
Solving for $x$, we get $x = 6$.
Number of girls = $2x = 12$.
The Reason states that in a ratio $a : b$, the first term represents $\frac{a}{a + b}$ of the total. However, the first term represents the part relative to the second term, not necessarily $\frac{a}{a + b}$ directly. Thus, Assertion is true, but Reason is false.

Key Concept: Proportion

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that if $\frac{a}{b} = \frac{c}{d}$, then $ad = bc$, which is correct as it follows the property of proportion. The reason explains that the product of the means equals the product of the extremes, which is also true and correctly justifies the assertion. Therefore, both statements are true, and the reason is the correct explanation.

Key Concept: Componendo and Dividendo, Continued Proportion

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion is a correct statement of the Componendo and Dividendo rule, which is a valid algebraic identity when the conditions $a \neq b$, $c \neq d$, and denominators are non-zero are met. The reason correctly states the conditions under which this rule is applicable, explaining why the assertion holds true.

Key Concept: Componendo and Dividendo

a) $5 : 3$
[Solution Description]
Given $\frac{a + b}{a - b} = \frac{4}{1}$. Applying componendo and dividendo, we get:
$\frac{(a + b) + (a - b)}{(a + b) - (a - b)} = \frac{4 + 1}{4 - 1}$
Simplifying numerator and denominator:
$\frac{2a}{2b} = \frac{5}{3} \implies \frac{a}{b} = \frac{5}{3}$
Therefore, $a : b = 5 : 3$.

Key Concept: Duplicate Ratio

b) $16 : 25$
[Solution Description]
The duplicate ratio of $a : b$ is $a^2 : b^2$. Here, the given ratio is $4 : 5$. So, square the terms: $4^2 = 16$ and $5^2 = 25$. The duplicate ratio is $16 : 25$.

Key Concept: Properties of Ratios

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To determine if the ratio $3 : 7$ is in its simplest form, we check the highest common factor (H.C.F.) of 3 and 7.
Step 1: Find the factors of 3. The only factors are 1 and 3.
Step 2: Find the factors of 7. The only factors are 1 and 7.
Step 3: The common factor is 1, which means the H.C.F. of 3 and 7 is 1.
Since the H.C.F. is 1, the ratio is in its simplest form. Therefore, both Assertion and Reason are true, and the Reason correctly explains why the ratio is in its simplest form.

Key Concept: Definition of Ratio

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that the ratio $5:10$ simplifies to $1:2$ when both terms are divided by 5. This is true because $\frac{5}{5} = 1$ and $\frac{10}{5} = 2$, resulting in $1:2$.
The reason states that if both terms of a ratio are divided by the same non-zero number, the ratio remains unchanged. This is also true as it is a fundamental property of ratios mentioned in the syllabus.
Moreover, the reason correctly explains the assertion because the simplification of $5:10$ to $1:2$ follows directly from this property.

Key Concept: Ratio and Proportion

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
Starting with the given equation:
$(7a + 8b)(7c - 8d) = (7a - 8b)(7c + 8d)$.
Expanding both sides:
$49ac - 56ad + 56bc - 64bd = 49ac + 56ad - 56bc - 64bd$.
Simplifying:
$-56ad + 56bc = 56ad - 56bc$.
Rearranging terms:
$-112ad = -112bc$, which reduces to $ad = bc$.
Thus, $\frac{a}{b} = \frac{c}{d}$. Both the Assertion and Reason are true, and the Reason correctly explains the Assertion.

Key Concept: Fourth Proportional

b) 27
[Solution Description]
Three quantities $a$, $b$, and $c$ are in continued proportion if $a : b = b : c$. Here, $a = 12$, $b = 18$, and $c = x$. Therefore, $\frac{12}{18} = \frac{18}{x}$. Cross-multiplying gives $12x = 324$. Solving for $x$, we get $x = 27$.

Key Concept: Average Speed

b) 58.33 km/h
[Solution Description]
Let the total distance be $D$.
Time for first half ($\frac{D}{2}$) = $\frac{D/2}{50}$ = $\frac{D}{100}$ hours
Time for second half ($\frac{D}{2}$) = $\frac{D/2}{70}$ = $\frac{D}{140}$ hours
Total time = $\frac{D}{100} + \frac{D}{140}$ = $\frac{7D + 5D}{700}$ = $\frac{12D}{700}$ = $\frac{3D}{175}$ hours
Average speed = $\frac{\text{Total distance}}{\text{Total time}}$ = $\frac{D}{3D/175}$ = $\frac{175}{3}$ ≈ $58.33$ km/h

Key Concept: Application of Ratio

b) 60 kg
[Solution Description]
Let the reduced weight be $x$.
Given the ratio of original to reduced weight is $8 : 5$, we have $\frac{96}{x} = \frac{8}{5}$.
Cross-multiplying: $8x = 5 \times 96$ $\Rightarrow$ $8x = 480$.
Solving for $x$: $x = \frac{480}{8} = 60$.
Hence, the reduced weight is $60\,\text{kg}$.

Key Concept: Componendo and Dividendo

c) Assertion is true, but Reason is false.
[Solution Description] The assertion states that if $\frac{a}{b} = \frac{c}{d}$, then applying componendo and dividendo gives $\frac{a + b}{a - b} = \frac{c + d}{c - d}$. This is a correct application of the rule. The reason claims that componendo and dividendo is applicable only when the ratio is in the form $\frac{a}{b} = \frac{c}{d}$. While it is true that the principle applies to such ratios, it is not strictly limited to this form; variations exist but are more complex. Therefore, the assertion is true, but the reason is false as it incorrectly restricts the applicability.

Key Concept: Sub-duplicate Ratio, Reciprocal Ratio

d) No real solution
[Solution Description]
The sub-duplicate ratio of $(x^2 - y^2) : (x^2 + y^2)$ is $\sqrt{x^2 - y^2} : \sqrt{x^2 + y^2}$.
The reciprocal ratio of $2 : 3$ is $3 : 2$.
Therefore, $\sqrt{x^2 - y^2} : \sqrt{x^2 + y^2} = 3 : 2$.
Squaring both sides: $\frac{x^2 - y^2}{x^2 + y^2} = \frac{9}{4}$.
Cross-multiplying: $4(x^2 - y^2) = 9(x^2 + y^2)$.
Expanding: $4x^2 - 4y^2 = 9x^2 + 9y^2$.
Simplifying: $-5x^2 = 13y^2$.
Taking ratios: $\left(\frac{x}{y}\right)^2 = -\frac{13}{5}$.
Since squares are non-negative, no real solution exists. However, assuming positive values for a valid ratio:
The problem implies $\frac{x}{y} = \sqrt{\frac{13}{5}}$ (but this contradicts the calculation).
Rechecking: There seems to be an inconsistency; thus, the correct interpretation requires further constraints.
(Note: User should verify problem statement.)
Assuming rephrased question with valid inputs, e.g., if sub-duplicate ratio matches another condition.
For example, if corrected to $\sqrt{x^2 + y^2} : \sqrt{x^2 - y^2} = 3 : 2$, then:
$\frac{x^2 + y^2}{x^2 - y^2} = \frac{9}{4}$.
Solving gives $x : y = \sqrt{13/5} : 1$, which simplifies to $\sqrt{65} : 5$.
From given options, select closest plausible answer:

Key Concept: Ratio Division

b) Rs.342.86
[Solution Description]
Given B : C = $4 : 5$, let B = $4k$ and C = $5k$. A = $\frac{2}{3}$ of B = $\frac{2}{3} \times 4k = \frac{8k}{3}$. Total amount = A + B + C = $\frac{8k}{3} + 4k + 5k = \frac{8k + 12k + 15k}{3} = \frac{35k}{3} = 1500$. Solving, $k = \frac{1500 \times 3}{35} = \frac{4500}{35} = \frac{900}{7}$. Thus, A = $\frac{8 \times 900}{7 \times 3} = \frac{7200}{21} \approx Rs.342.86$. However, simplifying further gives exact A = $\frac{2400}{7} ≈ Rs.342.86$.

Key Concept: Duplicate Ratio

b) 8.566
[Solution Description]
The duplicate ratio of $5 : \sqrt{6}$ is $5^2 : (\sqrt{6})^2 = 25 : 6$. Thus, we have:
$\frac{2x + 3}{5x - 38} = \frac{25}{6}$
Cross-multiplying gives:
$6(2x + 3) = 25(5x - 38)$
Simplifying further:
$12x + 18 = 125x - 950$
Rearranging terms:
$113x = 968 \implies x = \frac{968}{113} = 8.566$
However, none of the options match this result, but let's recheck calculations for any mistake.
Correctly simplifying:
$6(2x + 3) = 25(5x - 38)$
$12x + 18 = 125x - 950$
$-113x = -968 \implies x = \frac{968}{113} = 8.566$
But in solutions provided below, option b) is closest to this calculation. It seems there might be an error in input or options preparation.

Key Concept: Compound Ratio, Commensurable Quantities

c) Assertion is true, but Reason is false.
[Solution Description]
To verify the Assertion, first find the compound ratio of $\sqrt{2}:3$ and $5:\sqrt{8}$. The compound ratio is calculated as $(\sqrt{2} \times 5) : (3 \times \sqrt{8}) = 5\sqrt{2} : 3\sqrt{8}$. Simplify $\sqrt{8}$ to $2\sqrt{2}$, so the ratio becomes $5\sqrt{2} : 6\sqrt{2} = 5:6$. Since 5 and 6 are integers, this ratio is commensurable, making the Assertion true.
The Reason states that the product or ratio of two incommensurable quantities is always incommensurable. This is false, as seen in the solution above where the product of $\sqrt{2}$ (incommensurable) and $\sqrt{8}$ (incommensurable) resulted in a commensurable ratio of $5:6$.
Therefore, Assertion is true, but Reason is false.

Key Concept: Commensurable and Incommensurable Quantities, Integer Ratio

b) Commensurable, because the ratio simplifies to $\frac{3}{2}$
[Solution Description]
The ratio of $T_1$ to $T_2$ is $\frac{\frac{4}{9}}{\frac{8}{27}} = \frac{4}{9} \times \frac{27}{8} = \frac{108}{72} = \frac{3}{2}$. This is a ratio of two integers, so $T_1$ and $T_2$ are commensurable.

Key Concept: Basic Ratio Concept

c) 15 and 20
[Solution Description]
Let the two numbers be $3x$ and $4x$. According to the problem, their sum is 35.
$3x + 4x = 35$
$7x = 35 \implies x = 5$
Therefore, the numbers are $3x = 15$ and $4x = 20$.

Key Concept: Compound Ratio

a) $6 : 35$
[Solution Description]
The compound ratio of two ratios $a : b$ and $c : d$ is given by $(a \times c) : (b \times d)$. Here, the given ratios are $2 : 5$ and $3 : 7$. So, multiply the antecedents: $2 \times 3 = 6$ and consequents: $5 \times 7 = 35$. The compound ratio is $6 : 35$.

Key Concept: Sub-duplicate Ratio, Continued Proportion

d) 16 : 36 : 81
[Solution Description]
Since $a$, $b$, $c$ are in continued proportion, $a : b = b : c$.
Let $a : b = b : c = k$. Then, $b = a k$ and $c = b k = a k^2$.
Sub-duplicate ratio of $a : c$ is $\sqrt{a} : \sqrt{c} = \sqrt{a} : \sqrt{a k^2} = 1 : k$.
Given $1 : k = 4 : 9$, so $k = \frac{9}{4}$.
Thus, $a : b : c = a : a k : a k^2 = 1 : \frac{9}{4} : \left(\frac{9}{4}\right)^2 = 16 : 36 : 81$.

Key Concept: Relative Speed, Distance Calculation

c) 240 km
[Solution Description]
Step 1: Calculate the time taken for the trains to meet.
Relative speed = $60 + 40 = 100$ km/h.
Distance = 300 km.
Time = $\frac{300}{100} = 3$ hours.
Step 2: Distance flown by the bird = Speed × Time = $80 \times 3 = 240$ km.
The bird keeps flying back and forth, but its total flight distance depends only on the time before the trains meet.

Key Concept: Triplicate Ratio

b) $8 : 1$
[Solution Description]
The triplicate ratio of $a : b$ is $a^3 : b^3$. Calculating for $2 : 1$, we get $2^3 : 1^3 = 8 : 1$.

Key Concept: Componendo and Dividendo, Square Roots

b) 20
[Solution Description]
Given the equation $\frac{\sqrt{x + 5} + \sqrt{x - 16}}{\sqrt{x + 5} - \sqrt{x - 16}} = \frac{7}{3}$, apply componendo and dividendo:
Step 1: Apply componendo and dividendo:
$\frac{(\sqrt{x + 5} + \sqrt{x - 16}) + (\sqrt{x + 5} - \sqrt{x - 16})}{(\sqrt{x + 5} + \sqrt{x - 16}) - (\sqrt{x + 5} - \sqrt{x - 16})} = \frac{7 + 3}{7 - 3}$
This simplifies to:
$\frac{2\sqrt{x + 5}}{2\sqrt{x - 16}} = \frac{10}{4} \implies \frac{\sqrt{x + 5}}{\sqrt{x - 16}} = \frac{5}{2}$
Step 2: Square both sides:
$\frac{x + 5}{x - 16} = \frac{25}{4}$
Step 3: Cross-multiply:
$4(x + 5) = 25(x - 16) \implies 4x + 20 = 25x - 400$
Step 4: Solve for $x$:
$20 + 400 = 25x - 4x \implies 420 = 21x \implies x = 20$

Key Concept: Ratio of Commensurable Quantities

b) $\frac{3}{5}$
[Solution Description]
The ratio of two quantities is calculated by dividing the first quantity by the second. Here, both quantities are in the same unit (cm), and their ratio should be simplified to its lowest terms.
Step 1: Write the ratio as $\frac{15}{25}$.
Step 2: Simplify the fraction by dividing numerator and denominator by 5: $\frac{15 \div 5}{25 \div 5} = \frac{3}{5}$.
The simplified ratio is $\frac{3}{5}$, which is a ratio of integers, confirming that the quantities are commensurable.

Key Concept: Ratio and Proportion

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that the ratio $4 : 5$ can be simplified to $8 : 10$ by multiplying both terms by 2. This is true because multiplying or dividing both terms of a ratio by the same non-zero number keeps the ratio unchanged as per the properties of ratios.
The reason given is also true, as it correctly explains the property of ratios that multiplying or dividing both terms by the same non-zero number does not change the ratio. Therefore, the reason is the correct explanation for the assertion.

Key Concept: Properties of Ratios: Duplicate Ratio

d) 25 : 36
[Solution Description]
The duplicate ratio of $a : b$ is $a^2 : b^2$.
$5^2 : 6^2 = 25 : 36$
Therefore, the duplicate ratio is $25 : 36$.

Key Concept: Compound Ratio

b) $8 : 105$
[Solution Description]
To find the compound ratio, multiply the antecedents together and the consequents together:
$(2 \times 3 \times 4) : (5 \times 7 \times 9) = 24 : 315$
Simplifying by dividing both terms by 3, we get: $8 : 105$.

Key Concept: Continued Proportion, Triplicate Ratio

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
Given that $a$, $b$, and $c$ are in continued proportion, so $a : b = b : c$. This implies $b^2 = a \cdot c$, which means $c = \frac{b^2}{a}$ (Reason is true).
Now, it's given that the triplicate ratio of $a : b$ is $64 : 27$, which means $\frac{a^3}{b^3} = \frac{64}{27}$. Taking cube root on both sides, we get $\frac{a}{b} = \frac{4}{3}$, or $b = \frac{3}{4}a$.
Substituting $b = \frac{3}{4}a$ into $c = \frac{b^2}{a}$, we get:
$c = \frac{\left(\frac{3}{4}a\right)^2}{a} = \frac{\frac{9}{16}a^2}{a} = \frac{9}{16}a$.
Therefore, the Assertion is also true, and the Reason correctly explains the Assertion.

Key Concept: Compound Ratios, Continued Proportion

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.
[Solution Description]
Given $(a^2 + c^2)$, $(ab + cd)$, and $(b^2 + d^2)$ are in continued proportion, we can write:
$\frac{a^2 + c^2}{ab + cd} = \frac{ab + cd}{b^2 + d^2}$
Cross-multiplying gives:
$(a^2 + c^2)(b^2 + d^2) = (ab + cd)^2$
Expanding both sides:
$a^2b^2 + a^2d^2 + c^2b^2 + c^2d^2 = a^2b^2 + 2abcd + c^2d^2$
Simplifying:
$a^2d^2 + b^2c^2 = 2abcd \implies (ad - bc)^2 = 0$
Therefore, $ad = bc$, which implies $\frac{a}{b} = \frac{c}{d}$. Hence, $a, b, c, d$ are in proportion. The Reason states the general condition for proportion, but it does not directly explain the Assertion, as the Assertion involves a specific case of continued proportion.

Key Concept: Mixture, Percentage

b) 4
[Solution Description]
1. **Define Variables**: Let $x$ = liters of Solution A (30% acid). Then, $(10 - x)$ = liters of Solution B (50% acid).
2. **Set Up Equation**: Total acid = Acid from A + Acid from B.
$$ 0.3x + 0.5(10 - x) = 0.42 \times 10 $$
3. **Solve**:
$ 0.3x + 5 - 0.5x = 4.2 $
$ -0.2x = -0.8 $
$ x = 4 $
4. **Answer**: 4 liters of Solution A.

Key Concept: Triplicate Ratio, Proportion

b) $-\frac{205}{68}$
[Solution Description]
The triplicate ratio of $2 : 3$ is $2^3 : 3^3 = 8 : 27$.
Given $(4y + 7) : (5y - 2) = 8 : 27$.
Using the property of proportion: $(4y + 7) \times 27 = (5y - 2) \times 8$.
Simplifying: $108y + 189 = 40y - 16$.
Rearranging: $68y = -205$.
Thus, $y = -\frac{205}{68}$.

Key Concept: Ratio in Lowest Terms

d) Assertion is false, but Reason is true.
[Solution Description]
To check if the ratio $12 : 36$ is in its lowest terms, we need to find the H.C.F. of 12 and 36.
Step 1: Find factors of 12 $\rightarrow$ 1, 2, 3, 4, 6, 12.
Step 2: Find factors of 36 $\rightarrow$ 1, 2, 3, 4, 6, 9, 12, 18, 36.
The highest common factor (H.C.F.) is 12.
Since the H.C.F. is 12 (not 1), the ratio is not in its lowest terms. Therefore, Assertion (A) is false.
Reason (R) correctly states that the H.C.F. is 12, which means the reason is true.
Thus, Assertion is false, but Reason is true.

Key Concept: Proportion Application

c) $16 : 11$
[Solution Description]
Let the number of boys be $3x$ and girls be $2x$. Total students = $3x + 2x = 5x = 450$. Solving, $x = 90$. Initially, boys = $3 \times 90 = 270$, girls = $2 \times 90 = 180$. After admission: boys = $270 + 50 = 320$, girls = $180 + 40 = 220$. New ratio = $\frac{320}{220} = \frac{16}{11}$.

Key Concept: Adding/Subtracting Quantities to Ratios

c) 30
[Solution Description]
Let the numbers be $5x$ and $9x$. After subtracting 6, we have:
$\frac{5x - 6}{9x - 6} = \frac{1}{2}.$
Cross-multiplying gives:
$2(5x - 6) = 1(9x - 6).$
Simplify:
$10x - 12 = 9x - 6.$
Solving for $x$:
$x = 6.$
The smaller number is $5x = 30$.

Key Concept: Continued Proportion, Mean Proportional

b) 8
[Solution Description]
Since $a, b, c$ are in continued proportion, we have $b^2 = ac$. Also, given:
$a + b + c = 28 \quad \text{and} \quad ab + bc = 160$
Let’s express $c$ as $\frac{b^2}{a}$ (from $b^2 = ac$). Substituting into the first equation:
$a + b + \frac{b^2}{a} = 28$
Multiply through by $a$ to eliminate denominators:
$a^2 + ab + b^2 = 28a$
Now, consider the second equation $ab + bc = 160$:
$ab + b\left(\frac{b^2}{a}\right) = 160 \implies ab + \frac{b^3}{a} = 160$
Multiply through by $a$:
$a^2b + b^3 = 160a$
From the earlier result ($a^2 + ab + b^2 = 28a$), let’s solve for $a$ in terms of $b$. However, solving these equations simultaneously is complex, but testing values shows $b = 8$ satisfies both when $a = 4$ and $c = 16$.

Key Concept: Definition of Ratio - Antecedent & Consequent

b) Antecedent is 5
[Solution Description]
In a ratio $a : b$, the first term ($a$) is called the antecedent, and the second term ($b$) is the consequent.
Here, for $5 : 7$, the antecedent is $5$.

Key Concept: Word Problem on Proportion

a) Rs.350
[Solution Description]
Let Rahul's pocket money = $5x$, Raj's pocket money = $7x$. Expenditure of Rahul = $3y$, Raj = $5y$. Savings relation: $5x - 3y = 100$ and $7x - 5y = 100$. Solving first equation for y: $3y = 5x - 100 ⇒ y = \frac{5x - 100}{3}$. Substitute into second equation: $7x - 5\left(\frac{5x - 100}{3}\right) = 100 ⇒ 21x - 25x + 500 = 300 ⇒ -4x = -200 ⇒ x = 50$. Thus, Raj's pocket money = $7 \times 50 = Rs.350$.

Key Concept: Concept of Continued Proportions, Mean Proportional

b) 10
[Solution Description]
To find the mean proportional $x$ between 5 and 20, we use the property of continued proportions:
$\frac{5}{x} = \frac{x}{20}$
Cross-multiplying gives:
$x^2 = 5 \times 20$
$x^2 = 100$
Taking the square root of both sides:
$x = 10$ (since length cannot be negative).

Key Concept: Componendo and Dividendo

a) 7 : 3
[Solution Description]
Using the Componendo and Dividendo rule: If $\frac{a}{b} = \frac{c}{d}$, then $\frac{a + b}{a - b} = \frac{c + d}{c - d}$. Applying this to $(x + y) : (x - y) = 5 : 2$, we have $\frac{x + y + x - y}{x + y - x + y} = \frac{5 + 2}{5 - 2}$. Simplifying, $\frac{2x}{2y} = \frac{7}{3}$, which gives $\frac{x}{y} = \frac{7}{3}$.

Key Concept: Linear Equations

c) 10
[Solution Description]
For both equations to represent the same line, their coefficients must be proportional. That means:
$\frac{3}{6} = \frac{5}{k} = \frac{15}{30}$
Simplifying $\frac{3}{6} = \frac{5}{k}$ gives:
$\frac{1}{2} = \frac{5}{k} \implies k = 10$
Checking with the third ratio: $\frac{1}{2} = \frac{15}{30}$, which holds true. Therefore, $k = 10$.

Key Concept: Mean Proportional

b) 20
[Solution Description]
The mean proportional ($b$) between two numbers $a$ and $c$ is given by $b = \sqrt{ac}$. Here, $a = 16$ and $c = 25$. Substituting these values:
$b = \sqrt{16 \times 25} = \sqrt{400} = 20$
The mean proportional is $20$.

Key Concept: Alternendo

c) $3 : 9 = 5 : 15$
[Solution Description]
The alternendo property implies that if $a : b = c : d$, then $a : c = b : d$. Here, $3 : 5 = 9 : 15$ transforms to $3 : 9 = 5 : 15$, which simplifies to $1 : 3 = 1 : 3$.

Key Concept: Mean Proportional, Algebraic Expression

a) $x(x - y)$
[Solution Description]
The mean proportional between two quantities $a$ and $b$ is $\sqrt{ab}$.
Here, $a = (x - y)$ and $b = (x^3 - x^2y) = x^2(x - y)$.
Mean proportional = $\sqrt{(x - y) \times x^2(x - y)} = \sqrt{x^2(x - y)^2} = x(x - y)$.

Key Concept: Componendo and Dividendo

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.
[Solution Description]
First, let's verify the Assertion (A): Subtract the two given equations:
$\frac{x + 3y}{y - x} - \frac{3x + y}{y - x} = \frac{(x + 3y) - (3x + y)}{y - x} = \frac{-2x + 2y}{y - x} = \frac{-2(x - y)}{-(x - y)} = 2.$
So, Assertion (A) is true.
Now, let's analyze Reason (R): Simplify $\frac{6xy + 2x^2}{2xy - 2x^2}$:
$\frac{6xy + 2x^2}{2xy - 2x^2} = \frac{2x(3y + x)}{2x(y - x)} = \frac{3y + x}{y - x}.$
Similarly, simplify $\frac{6xy + 2y^2}{2xy - 2y^2}$:
$\frac{6xy + 2y^2}{2xy - 2y^2} = \frac{2y(3x + y)}{2y(x - y)} = \frac{3x + y}{x - y}.$
Thus, $\frac{3y + x}{y - x} = \frac{3x + y}{x - y}$ is valid, so Reason (R) is also true. However, Reason (R) does not explain why Assertion (A) holds directly; it follows from a different algebraic manipulation. Hence, Reason (R) is not the correct explanation for Assertion (A).

Key Concept: Alternative Method for Proportions

a) $\frac{3}{4}$
[Solution Description]
Cross-multiply the given proportion:
$29(2x + 3y) = 18(3x + 5y)$
Expand both sides:
$58x + 87y = 54x + 90y$
Rearranging terms:
$58x - 54x = 90y - 87y \implies 4x = 3y$
Dividing both sides by $y$:
$\frac{x}{y} = \frac{3}{4}$

Key Concept: Ratio and Proportion: Application in Weight Reduction

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The problem involves reducing the weight in the given ratio $7 : 5$. The original weight is $84 \text{ kg}$. To find the reduced weight, multiply the original weight by the fraction of the reduced ratio over the original ratio.
Step 1: Identify the ratio components. Original ratio part = 7, Reduced ratio part = 5.
Step 2: Calculate the reduced weight using the formula:
$\text{Reduced weight} = \text{Original weight} \times \frac{\text{Reduced ratio part}}{\text{Original ratio part}}$
Substituting the values:
$\text{Reduced weight} = 84 \times \frac{5}{7} = 12 \times 5 = 60 \text{ kg}$
Therefore, the assertion that the weight reduces to $60 \text{ kg}$ is true, and the reason correctly explains how to calculate it using the ratio.

Key Concept: Ratio simplification, Order of terms in ratio

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To simplify the ratio $20 : 10$, we find the greatest common divisor (GCD) of 20 and 10. The GCD of 20 and 10 is 10. Dividing both terms of the ratio by 10 gives $20 \div 10 : 10 \div 10 = 2 : 1$. The simplified ratio $2 : 1$ is indeed in its lowest form because the GCD of 2 and 1 is 1, meaning the terms are co-prime. The reason correctly explains that simplifying a ratio by dividing both terms by their GCD ensures the ratio is in its simplest form. Thus, both the assertion and the reason are true, and the reason is the correct explanation of the assertion.

Key Concept: Substitution, Ratio Manipulation

c) 12
[Solution Description]
Let $\frac{p}{q} = k$, then $p = kq$. Substitute into the given equation:
$\frac{3k q + 5q}{5k q + 7q} = \frac{19}{31}$.
Factor out $q$ from numerator and denominator:
$\frac{q(3k + 5)}{q(5k + 7)} = \frac{19}{31}$,
Simplifying to $\frac{3k + 5}{5k + 7} = \frac{19}{31}$.
Cross-multiply:
$31(3k + 5) = 19(5k + 7)$.
Expand both sides:
$93k + 155 = 95k + 133$.
Rearrange terms:
$-2k = -22$,
Solving gives $k = 11$.
Therefore, $\frac{p}{q} = 11 = \frac{11}{1}$, with $m = 11$ and $n = 1$ being co-prime integers.
Hence, $m + n = 12$.

Key Concept: Student Council Ratio Adjustment

c) 6
[Solution Description]
Current ratio of boys to girls is $3 : 1$. Total parts = 3 + 1 = 4. Number of boys = $\frac{3}{4} \times 36 = 27$. Number of girls = $\frac{1}{4} \times 36 = 9$. Let $x$ be the number of girls to be added. New ratio becomes:
$\frac{27}{9 + x} = \frac{9}{5}$
Cross-multiplying gives:
$135 = 81 + 9x$
Solving for $x$:
$9x = 54 \implies x = 6$
Therefore, 6 more girls should be added.

Key Concept: Compound Ratio

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To find the compound ratio of two ratios, multiply the numerators (antecedents) and denominators (consequents) separately. Here, the compound ratio of $2 : 3$ and $5 : 7$ is calculated as $(2 \times 5) : (3 \times 7) = 10 : 21$. Thus, the Assertion is correct. The Reason correctly states the method for calculating the compound ratio and explains the Assertion. Therefore, both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

Key Concept: Third Proportional

b) 20
[Solution Description]
The third proportional ($c$) to two numbers $a$ and $b$ is calculated using the formula $c = \frac{b^2}{a}$. Here, $a = 5$ and $b = 10$. Substituting these values:
$c = \frac{10^2}{5} = \frac{100}{5} = 20$
The third proportional is $20$.

Key Concept: Fundamental Property

b) $2 \times x = 3 \times 10$
[Solution Description]
The fundamental property states that for a proportion $a : b = c : d$, $a \times d = b \times c$.
Applying this to $2 : 3 = 10 : x$, we get $2 \times x = 3 \times 10$.
Simplifying: $2x = 30$.
Dividing by 2 gives $x = 15$.
Hence, $2 \times 15 = 3 \times 10$ must hold true.

Key Concept: Assertion-Reason: Continued Proportion

Assertion is false, but Reason is true.
[Solution Description]
The assertion states that if $\frac{a}{b} = \frac{b}{c}$, then $b$ is the mean proportional between $a$ and $c$. This is true because by definition, mean proportional implies $\frac{a}{b} = \frac{b}{c}$, which can be rewritten as $b^2 = a \cdot c$, meaning $b = \sqrt{a \cdot c}$.
The reason states that the mean proportional of two numbers $x$ and $y$ is given by $\sqrt{xy}$. This is also correct based on the definition of mean proportion.
However, the reason does not explain why $\frac{a}{b} = \frac{b}{c}$ implies $b$ is the mean proportional; it only provides an alternate definition of mean proportion.
Therefore, both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.

Key Concept: Duplicate Ratio

c) $25 : 49$
[Solution Description]
The duplicate ratio of $a : b$ is given by $a^2 : b^2$. For $5 : 7$, applying this formula gives $5^2 : 7^2 = 25 : 49$.

Key Concept: Mean Proportional

c) 45
[Solution Description]
Let the mean proportional be $x$. According to the definition of continued proportion:
$25 : x = x : 81$
Cross-multiplying gives:
$x^2 = 25 \times 81$
Taking the square root of both sides:
$x = \sqrt{2025} = 45$

Key Concept: Duplicate Ratio

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To find the duplicate ratio of a given ratio, we square both the antecedent and the consequent of the ratio. For the ratio $5 : 7$, the duplicate ratio is calculated as $(5)^2 : (7)^2 = 25 : 49$. This matches the assertion. The reason states the general rule for finding the duplicate ratio of any ratio $a : b$, which is indeed $a^2 : b^2$. Therefore, both the assertion and reason are true, and the reason correctly explains the assertion.

Key Concept: Ratio Basics

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To simplify the ratio 4:6, we find the H.C.F. of 4 and 6, which is 2. Then, we divide both terms by 2. So, $\frac{4}{2}:\frac{6}{2} = 2:3$. Therefore, the Assertion is true. The Reason states that both terms must be divided by their H.C.F. to simplify the ratio, which is also true and correctly explains why the Assertion holds. Hence, both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

Key Concept: Componendo and Dividendo, Proportion

c) 4
[Solution Description]
Given $\frac{a}{b} = \frac{c}{d}$, apply Componendo and Dividendo:
$\frac{a + b}{a - b} = \frac{c + d}{c - d}$
Let $k = \frac{a + b}{a - b} = \frac{c + d}{c - d}$. Then, the expression becomes:
$k + k = 2k$
Substitute $k = \frac{a + b}{a - b}$ back:
$2 \times \frac{a + b}{a - b}$
However, since $\frac{a + b}{a - b} = \frac{c + d}{c - d}$, the sum is $2k$. Without additional constraints, the answer remains in terms of $k$, but numerically, it simplifies to 2 if $k = 1$ (e.g., $a = 2b$ and $c = 2d$). Testing this case:
If $a = 2b$ and $c = 2d$, then $k = \frac{2b + b}{2b - b} = 3$ and $\frac{2d + d}{2d - d} = 3$, so $2k = 6$. None of the options fit, indicating a need for re-evaluation.
Alternatively, if $\frac{a}{b} = \frac{c}{d} = 1$ (i.e., $a = b$ and $c = d$), the expression is undefined (division by zero). Hence, the question may need adjustment or clarification. For illustrative purposes, assuming a non-trivial case where $a = 3b$ and $c = 3d$, then $k = 2$, and $2k = 4$.

Key Concept: Definition of Proportion

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To verify the assertion, calculate the product of extremes ($12 \times 12 = 144$) and the product of means ($18 \times 8 = 144$). Both products are equal, confirming the assertion is true. The reason states a fundamental property of proportions, which is also true. Moreover, the reason correctly explains why the assertion holds, as it describes the underlying principle used in the assertion.

Key Concept: Properties of Ratios: Invariance Property

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that $4 : 10$ simplifies to $2 : 5$, which is correct because dividing both terms by their greatest common divisor (GCD), which is 2, gives $2 : 5$. The reason correctly explains the invariance property of ratios, which states that dividing both terms of a ratio by the same non-zero number does not change its value. Therefore, both the assertion and reason are true, and the reason correctly explains the assertion.

Key Concept: Continued Proportion

d) Assertion is false, but Reason is true.
[Solution Description]
Let $a$, $b$, and $c$ be in continued proportion, so $a : b = b : c$. This implies $b^2 = a \times c$.
Now, consider $\frac{a + b}{b + c}$:
Since $b^2 = a c$, we can write $a = \frac{b^2}{c}$.
Substituting $a$ into $\frac{a + b}{b + c}$:
$\frac{\frac{b^2}{c} + b}{b + c} = \frac{b(b + c)}{c(b + c)} = \frac{b}{c}$
But $\frac{a}{c} = \frac{\frac{b^2}{c}}{c} = \frac{b^2}{c^2}$, which is not equal to
Therefore, the Assertion (A) is false since $\frac{a + b}{b + c} = \frac{b}{c} \neq \frac{a}{c}$.
The Reason (R) states that the ratio $\frac{a + b}{b + c} = \frac{a}{c}$ for quantities in continued proportion, which is incorrect as shown above. Hence, Reason (R) is false.
However, the given formula in Reason (R) is conceptually incorrect because it does not hold true for continued proportions.

Key Concept: Fourth Proportional

b) 9
[Solution Description]
Let the fourth proportional be $x$. Then, $2 : 3 = 6 : x$. Using the property of proportion, $2 \times x = 3 \times 6 \Rightarrow 2x = 18 \Rightarrow x = 9$.

Key Concept: Mean Proportional

b) 6
[Solution Description]
For the continued proportion $4 : x = x : 9$, the mean proportional $x$ satisfies $x^2 = 4 \times 9$. Solving for $x$, we get:
$x^2 = 36 \implies x = 6$
(Since $x$ is a length, we take the positive root.)

Key Concept: Product of Extremes and Means, Continued Proportion

b) $24\sqrt{2}$
[Solution Description]
Given $a : b = b : c$ and $b : c = c : d$. This implies $a, b, c, d$ are in continued proportion.
So, $b^2 = a \times c$ and $c^2 = b \times d$.
Let common ratio be $k$. Then, $b = a k$, $c = b k = a k^2$, $d = c k = a k^3$.
Given $a = 12$ and $d = 96$, so:
$96 = 12 \times k^3 \implies k^3 = 8 \implies k = 2.$
Now, the mean proportional between $a$ and $d$ is $\sqrt{a \times d} = \sqrt{12 \times 96} = \sqrt{1152} = 24\sqrt{2}$.

Key Concept: Time Calculation

c) 5 km
[Solution Description]
Let distance = $D$ km, correct time = $T$ hours.
At 4 km/h: $T + \frac{10}{60}$ = $\frac{D}{4}$ → $T = \frac{D}{4} - \frac{1}{6}$
At 5 km/h: $T - \frac{5}{60}$ = $\frac{D}{5}$ → $T = \frac{D}{5} + \frac{1}{12}$
Equating both expressions for $T$: $\frac{D}{4} - \frac{1}{6} = \frac{D}{5} + \frac{1}{12}$
Multiply by 60 (LCM): $15D - 10 = 12D + 5$ → $3D = 15$ → $D = 5$ km