Key Concept: Conceptual Complexity

c) The concept that every event is inevitable because of preceding causes

[Solution Description] Determinism is the philosophical belief that all events are determined completely by previously existing causes. It implies that every event is necessitated by antecedent events and conditions together with the laws of nature.

Key Concept: Introduction

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

[Solution Description]

The assertion is true because the introduction is indeed fundamental to understanding any subject as it sets the stage for further learning. The reason is also true because the introduction provides a foundation and context, making it easier to grasp the subject matter. Moreover, the reason correctly explains why the assertion is true.

Key Concept: Basic Concept

c) 24

[Solution Description]

To find the product of 4 and 6, we multiply the two numbers:

$4 \times 6 = 24$.

Key Concept: Real-World Application

c) Utilitarianism, which prioritizes the greatest good for the greatest number

[Solution Description] Utilitarianism is an ethical theory that focuses on the outcomes or consequences of actions, aiming to maximize overall happiness or utility.

Key Concept: Direct Proportion

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

[Solution Description]

Direct proportion is defined as a relationship between two quantities where their ratio remains constant. The assertion states that if two quantities x and y are in direct proportion, their ratio remains constant, which is true based on the definition of direct proportion. The reason given here is that the ratio $\frac{x}{y} = k$, where k is a positive constant, which directly explains the assertion because this equation defines the constant ratio for direct proportion.

Key Concept: Proportional Relationships, Real-World Application

c) 8 cups

[Solution Description] First, determine the ratio of cups of flour to cookies: 2 cups / 12 cookies = 1/6 cups per cookie. To find out how many cups are needed for 48 cookies, multiply the number of cookies by the ratio: 48 cookies * (1/6 cups per cookie) = 8 cups.

Key Concept: Direct Proportion

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

[Solution Description]

Let the cost of sugar be \$y and its weight be x kg. The cost of sugar is directly proportional to its weight, i.e., $y \propto x$. Given that 5 kg of sugar costs \$250, we can find the cost of 10 kg using direct proportion. Since the weight doubles from 5 kg to 10 kg, the cost should also double from \$250 to \$500. Therefore, the assertion is true. The reason is also true because it correctly states the concept of direct proportion. Moreover, the reason explains why the assertion is true.

Key Concept: Understanding Proportional Relationships

c) 420 miles

[Solution Description]

First, calculate the speed of the car by dividing the distance by the time: $240 \div 4 = 60$ miles per hour. Then, multiply the speed by the time to find the distance traveled in 7 hours: $60 \times 7 = 420$ miles.

Key Concept: Preparing tea for different numbers of people

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.

[Solution Description]

Key Concept: Scaling

c) 20 teaspoons

[Solution Description]

The recipe uses 8 teaspoons of tea leaves for 4 people. This ratio simplifies to 2 teaspoons per person. To scale this up to 10 people, multiply by the number of people: $10 \times 2 = 20$ teaspoons.

Key Concept: Proportionality in Preparing Tea

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

[Solution Description]

To determine the quantity of water needed for 5 people, we use the concept of direct proportionality. Mohan needs 300 mL of water for 2 people. For 5 people, we set up the proportion:

$\frac{300}{2} = \frac{x}{5}$

Solving for $x$:

$x = \frac{300 \times 5}{2} = 750 \, \text{mL}$

Thus, the assertion is true because it correctly calculates the required amount of water for 5 people using direct proportionality. The reason is also true because it states the correct principle that the quantity of water is directly proportional to the number of people.

Key Concept: Ratio and Proportion, Scaling

a) Option text

[Solution Description]

To prepare tea for one person, Mohan uses 300 mL of water, 2 spoons of sugar, 1 spoon of tea leaves, and 50 mL of milk. For five persons, the quantities are calculated by multiplying each item's amount by 5. Therefore, for five persons, the quantities required are:

$\text{Water} = 300 \, \text{mL} \times 5 = 1500 \, \text{mL}$

$\text{Sugar} = 2 \, \text{spoons} \times 5 = 10 \, \text{spoons}$

$\text{Tea Leaves} = 1 \, \text{spoon} \times 5 = 5 \, \text{spoons}$

$\text{Milk} = 50 \, \text{mL} \times 5 = 250 \, \text{mL}$

The Assertion is correct as it accurately calculates the quantities needed for five persons. The Reason is also correct because the quantity of each item is directly proportional to the number of persons.

Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

Key Concept: Direct Proportion

c) 360 km

[Solution Description]

This is a case of direct proportion. Let the distance traveled in 6 hours be $x$. According to direct proportion:

$\frac{240}{4} = \frac{x}{6}$

Solving for $x$:

$x = \frac{240 \times 6}{4} = 360$

Therefore, the car will travel 360 km in 6 hours.

Key Concept: Real-life scenarios involving proportions

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

[Solution Description]

The Assertion states that if the number of workers decreases, the time taken to complete a task increases proportionally. This is true because fewer workers would naturally take more time to complete the same task. The Reason states that the relationship between the number of workers and the time taken to complete a task is inversely proportional. This is also true because as one quantity increases, the other decreases, maintaining a constant product. Therefore, both Assertion and Reason are true, and the Reason correctly explains the Assertion.

Key Concept: Real-life scenarios involving proportions

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

[Solution Description]

To understand this, we need to recall the relationship between speed, time, and distance. The formula connecting these three quantities is given by:

$\text{Distance} = \text{Speed} \times \text{Time}$

If the distance remains constant, then speed and time are inversely proportional. This means that if the speed increases, the time decreases proportionally, and vice versa. In this case, if the speed of the car is doubled, the time taken to cover the same distance will indeed be halved. Therefore, both the Assertion and Reason are true, and the Reason correctly explains the Assertion.

Key Concept: Direct Proportion, Real-life Application

c) 40 metres

[Solution Description]

Let the height of the building be $x$ metres. Since the height of an object and the length of its shadow are in direct proportion, we can use the proportion $\frac{16}{12} = \frac{x}{30}$. Cross-multiplying gives $12x = 16 \times 30$. Solving for $x$, we get $x = \frac{16 \times 30}{12} = 40$ metres.

Key Concept: Total Cost Calculation

b) \$80

[Solution Description]

The total cost is calculated by multiplying the cost of one article by the number of articles. Here, the cost of one article is \$20 and the number of articles is 4. Therefore, the total cost is $20 \times 4 = \$80$.

Key Concept: Direct Variation

c) \$60

[Solution Description]

First, find the cost per article. If 5 articles cost \$25, then the cost per article is $25 \div 5 = \$5$. Now, to find the cost of 12 articles, multiply the cost per article by the number of articles: $5 \times 12 = \$60$.

Key Concept: Total Cost Calculation

b) \$50

[Solution Description]

The total cost is calculated by multiplying the cost of one article by the number of articles. Here, the cost of one article is \$10 and the number of articles is 5. Therefore, the total cost is $10 \times 5 = \$50$.

Key Concept: Direct Proportion

c) 420 km

[Solution Description]

First, find the speed of the car. The car travels 240 km in 4 hours, so the speed is $\frac{240}{4} = 60$ km/h. Now, calculate the distance traveled in 7 hours by multiplying the speed by the time: $60 \times 7 = 420$ km.

Key Concept: Interest Calculation

b) \$250

[Solution Description]

To calculate the interest earned, we use the formula: $I = P \times r \times t$, where $I$ is the interest, $P$ is the principal amount, $r$ is the rate of interest, and $t$ is the time in years.

Here, $P = \$5000$, $r = 5\% = 0.05$, and $t = 1$ year.

Plugging in the values: $I = 5000 \times 0.05 \times 1 = \$250$.

So, the interest earned after one year is \$250.

Key Concept: More money deposited leads to more interest earned

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

[Solution Description]

The assertion states that if the amount of money deposited in a bank increases, the interest earned on it will also increase. This is true because the interest earned depends on the principal amount deposited. The reason given is that the interest earned is directly proportional to the amount of money deposited. This is also true and correctly explains the assertion. There are no other factors like time or rate mentioned here which could affect the interest earned directly. Hence, the reason is the correct explanation of the assertion.

Key Concept: Basic Concept

b) \$50

[Solution Description] To find the interest earned, use the formula: $Interest = Principal \times Rate \times Time$. Here, the principal is \$1000, the rate is 5\%, and the time is 1 year. First, convert the percentage rate to a decimal: $5\% = 0.05$. Now, multiply the principal by the rate and time: $Interest = 1000 \times 0.05 \times 1 = \$50$.

Key Concept: Interest Variation

c) Doubles

[Solution Description]

Let the original principal be $P$. The original interest is given by $I = P \times r \times t$.

If the principal is doubled, the new principal is $2P$.

The new interest is $I_{\text{new}} = 2P \times r \times t = 2 \times (P \times r \times t) = 2I$.

This shows that the interest earned also doubles when the principal is doubled.

Key Concept: Time Saved Calculation

c) 2 hours

[Solution Description]

First, calculate the initial time taken at 40 km/h:

$t_1 = \frac{240}{40} = 6 \text{ hours}$

Next, calculate the new time taken at 60 km/h:

$t_2 = \frac{240}{60} = 4 \text{ hours}$

The time saved is the difference between $t_1$ and $t_2$:

$\Delta t = t_1 - t_2 = 6 - 4 = 2 \text{ hours}$

Thus, the time saved is 2 hours.

Key Concept: Higher vehicle speed decreases travel time

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

[Solution Description]

The assertion states that increasing the speed of a vehicle reduces the time taken to cover a fixed distance, which is true as per the formula $time = distance / speed$. The reason given is that speed is inversely proportional to time when distance is constant, which is also correct and directly explains the assertion. Hence, both the assertion and reason are true, and the reason correctly explains the assertion.

Key Concept: Inverse Proportion

c) 1 hour

[Solution Description]

Initially, the car takes 2 hours at 60 km/h. When speed is doubled (120 km/h), time taken will be halved. So, new time = $2/2 = 1$ hour.

Key Concept: Speed, Time, Acceleration

b) 2.78 m/s²

[Solution Description]

First, convert the speed from km/h to m/s: $100$ km/h $= 100 \times \frac{1000}{3600} = \frac{100000}{3600} \approx 27.78$ m/s. The acceleration $a$ is calculated using $a = \frac{v - u}{t}$, where $u$ is the initial velocity (0 m/s), $v$ is the final velocity (27.78 m/s), and $t$ is the time (10 seconds). Thus, $a = \frac{27.78 - 0}{10} \approx 2.78$ m/s².

Key Concept: Work and Time

b) 24 workers

[Solution Description]

The number of workers and the time taken are inversely proportional. So, if $W_1$ workers can complete a task in $T_1$ days, then $W_2$ workers can complete a task in $T_2$ days where $W_1 \times T_1 = W_2 \times T_2$. Given that 20 workers can complete the project in 30 days, we can find the number of workers needed to complete the project in 25 days as follows:

$20 \times 30 = W_2 \times 25$

Solving for $W_2$, we get:

$W_2 = \frac{20 \times 30}{25} = 24 \text{ workers}$

Key Concept: Work and Time

d) 4 days

[Solution Description]

We know that the number of workers and the time taken are inversely proportional. So, if $W_1$ workers can complete a task in $T_1$ days, then $W_2$ workers can complete a task in $T_2$ days where $W_1 \times T_1 = W_2 \times T_2$. Given that 6 workers can complete the task in 10 days, we can find the time taken by 15 workers as follows:

$6 \times 10 = 15 \times T_2$

Solving for $T_2$, we get:

$T_2 = \frac{6 \times 10}{15} = 4 \text{ days}$

Key Concept: More workers reduce time taken to complete work

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

[Solution Description]

The assertion states that increasing the number of workers reduces the time taken to complete a task, which is a basic concept in work efficiency. The reason explains that more workers can divide the task among themselves, reducing the workload per worker. Both statements are true, and the reason correctly explains the assertion. Therefore, the correct answer is that both assertion and reason are true, and the reason is the correct explanation of the assertion.

Key Concept: Inverse Proportion

c) 8 days

[Solution Description]

Let’s solve the problem step by step. The number of workers and the time taken to complete the task are inversely proportional. This means that as the number of workers increases, the time taken decreases, and vice versa.

Given:

4 workers can complete the task in 10 days.

To find:

Time taken by 5 workers to complete the same task.

Step 1: Understand the relationship between workers and time.

If 4 workers take 10 days, then 1 worker would take $4 \times 10 = 40$ days (since fewer workers take more time).

Step 2: Now, calculate the time taken by 5 workers.

Since 1 worker takes 40 days, 5 workers will take $40 / 5 = 8$ days.

Therefore, 5 workers will take 8 days to complete the task.

Key Concept: Direct Proportion

c) 28 metres

[Solution Description]

Let the height of the building be x metres. Since height of the object is directly proportional to the length of its shadow, we can use the relation $\frac{x1}{x2} = \frac{y1}{y2}$. Here, x1 = 14, y1 = 10, and y2 = 20. Substituting these values, we get $\frac{14}{x} = \frac{10}{20}$. Solving for x, we get $10x = 14 \times 20$ or $x = \frac{14 \times 20}{10} = 28$. Therefore, the height of the building is 28 metres.

Key Concept: Direct Proportion, Mathematical Reasoning

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

[Solution Description]

The assertion states that if $x$ is directly proportional to $y$, then a 50\% increase in $x$ will result in a 50\% increase in $y$. This is true because in direct proportion, when one variable increases by a certain percentage, the other variable also increases by the same percentage to maintain the constant ratio $x/y$. The reason given is that in direct proportion, the ratio $x/y$ remains constant, which is also true. Moreover, the reason correctly explains why the assertion is true because the constant ratio ensures that both variables change proportionally. Therefore, both assertion and reason are true, and the reason is the correct explanation of the assertion.

Key Concept: Direct Proportion

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

[Solution Description]

Let the cost of sugar be \$C and its weight be \$w kg. According to the assertion, when \$w = 5, \$C = 250. For direct proportion, $\frac{C_1}{w_1} = \frac{C_2}{w_2}$. Substituting the values, $\frac{250}{5} = \frac{C_2}{8}$. Simplifying, $C_2 = 50 \times 8 = 400$. The assertion is true because the calculated cost matches the given value. The reason is also true because the cost of sugar increases with its weight, maintaining a constant ratio. Moreover, the reason correctly explains the assertion as it justifies the proportionality relationship used in the calculation.

Key Concept: Direct Proportion

c) Assertion is true, but Reason is false.

[Solution Description]

The assertion is true because in direct proportion, the ratio of the two quantities remains constant. However, the reason is false because in direct proportion, as one quantity increases, the other also increases, not decreases. Therefore, the correct answer is (c).

Key Concept: Direct Proportion

c) 5 days

[Solution Description]

This is a case of inverse proportion. More workers will take fewer days to complete the task. Since the number of workers has doubled, the time taken will be halved. Therefore, 8 workers will take 10 / 2 = 5 days.

Key Concept: Direct Proportion, Multiple Variables

b) \$900

[Solution Description]

Let $C$ be the cost, $N$ be the number of copies, and $P$ be the number of pages. Given $C \propto N \times P$, we can write $C = k \cdot N \cdot P$. For the first case, $400 = k \times 100 \times 200$, so $k = \frac{400}{100 \times 200} = 0.02$. Now, for the second case, $C = 0.02 \times 150 \times 300 = 900$.

Key Concept: Direct Proportion

b) \$60

[Solution Description]

The cost of 5 books is \$25. First, find the cost of one book by dividing the total cost by the number of books: \$25 / 5 = \$5 per book. Now, multiply the cost per book by the number of books to find the total cost for 12 books: \$5 * 12 = \$60.

Key Concept: Direct Proportion

c) \$16

[Solution Description]

Let the cost of 8 pencils be $x$. Since the number of pencils and their cost are in direct proportion, we have:

$\frac{5}{10} = \frac{8}{x}$

Cross-multiplying gives:

$5x = 80$

Solving for $x$:

$x = \frac{80}{5} = 16$

Therefore, the cost of 8 pencils is \$16.

Key Concept: Direct Proportion, Ratio

c) 270 degrees

[Solution Description]

Let x be the time in minutes and y be the angle turned in degrees. The relationship between time and angle turned is directly proportional, so we can write:

$\frac{y_1}{x_1} = \frac{y_2}{x_2}$

Given $y_1 = 180 degrees$, $x_1 = 30 minutes$, and $x_2 = 45 minutes$, we need to find $y_2$.

Substituting the known values:

$\frac{180}{30} = \frac{y_2}{45}$

Solving for $y_2$:

$y_2 = \frac{180 \times 45}{30} = 270 degrees$

The angle turned by the minute hand in 45 minutes is 270 degrees.

Key Concept: Direct Proportion

c) 135 km

[Solution Description]

Given that the distance travelled is directly proportional to the amount of petrol used, we can use the relation $x1 / x2 = y1 / y2$, where $x1 = 3$, $y1 = 45$, and $x2 = 9$. Substituting the values into the equation, we get $3 / 9 = 45 / y2$ or $y2 = 9 \times 45 / 3 = 135$.

Key Concept: Direct Proportion

b) 10

[Solution Description]

Let $W$ be the number of workers and $T$ be the time taken. Since work is directly proportional to the number of workers and inversely proportional to the time, we have the relationship $W \times T = \text{constant}$. Initially, $5$ workers take $8$ hours, so the constant is $5 \times 8 = 40$. To find the number of workers required to finish the task in $4$ hours, we set up the equation $W \times 4 = 40$. Solving for $W$, we get $W = \frac{40}{4} = 10$. Therefore, $10$ workers are needed.

Key Concept: Direct Proportion

c) 15 metres

[Solution Description]

Given that the height of the object is directly proportional to the length of its shadow, we use the relation $x1 / x2 = y1 / y2$, where $x1 = 10$, $y1 = 8$, and $x2 = h$, $y2 = 12$. Substituting the values into the equation, we get $10 / h = 8 / 12$ or $h = 10 \times 12 / 8 = 15$.

Key Concept: Inverse Proportion

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

[Solution Description]

The assertion states that if two quantities x and y are in inverse proportion, their product remains constant. This is true because the definition of inverse proportion states that $xy = k$, where k is a constant. The reason explains that an increase in x causes a proportional decrease in y, which is also true and directly supports the assertion. Therefore, both the assertion and reason are true, and the reason correctly explains the assertion.

Key Concept: Inverse Proportion

b) Speed and distance covered in a fixed time

[Solution Description]

In inverse proportion, as one quantity increases, the other decreases such that their product remains constant. Option b) does not follow this relationship because it suggests that both quantities increase together, which is not characteristic of inverse proportion.

Key Concept: Inverse Proportion

b) 3

[Solution Description]

Since $x$ and $y$ are inversely proportional, we can write $x \times y = k$, where $k$ is a constant. Given that $x = 4$ when $y = 6$, we can find the constant $k$ as $k = 4 \times 6 = 24$. Now, we need to find the value of $y$ when $x = 8$. Using the same relationship, we have $8 \times y = 24$. Solving for $y$, we get $y = \frac{24}{8} = 3$.

Key Concept: Inverse Proportion

a) 2

[Solution Description]

Given that x and y are in inverse proportion, we have $x_1 y_1 = x_2 y_2$. Substituting the given values: $(4)(5) = (10)(y_2)$. Solving for $y_2$: $20 = 10y_2$ => $y_2 = 2$.

Key Concept: Direct Proportion

b) 300 km

[Solution Description]

The distance traveled is directly proportional to the time taken at a constant speed. The speed of the car can be calculated as:

$\text{Speed} = \frac{\text{Distance}}{\text{Time}} = \frac{120}{2} = 60 \text{ km per hour}$

Now, the distance traveled in 5 hours can be calculated as:

$\text{Distance} = 60 \cdot 5 = 300 \text{ km}$

Key Concept: Direct Proportion

b) 4 days

[Solution Description]

The number of workers is inversely proportional to the number of days taken to complete the task. Let $D$ be the number of days taken by 6 workers. We can set up the proportion:

$3 \cdot 8 = 6 \cdot D \implies 24 = 6D \implies D = \frac{24}{6} = 4 \text{ days}$

Key Concept: Direct Proportion

c) 350 km

[Solution Description]

First, calculate the speed of the car by dividing the distance by time:

$\text{Speed} = \frac{150}{3} = 50 \text{ km/h}$

Now, multiply the speed by the total time to find the distance:

$\text{Distance} = 50 \times 7 = 350 \text{ km}$

Key Concept: Direct Proportion

d) Assertion is false, but Reason is true.

[Solution Description] The assertion states that increasing the number of workers decreases the time taken to complete a project. The reason given is that more workers allow for sharing the workload, which leads to faster completion. In the context of direct proportion, if the number of workers increases, the workload per worker decreases, and thus, the time taken to complete the project decreases. This is an example of inverse proportionality, not direct proportionality. Therefore, the assertion is false because it incorrectly describes the relationship as direct proportion when it is actually inverse. The reason, however, is true because more workers do indeed lead to faster completion of the project by allowing the workload to be shared.

Key Concept: Direct Proportion

c) \$180

[Solution Description]

Step 1: Let the cost of 1 kg of sugar be k. According to direct proportion, cost = weight $\times$ k.

Step 2: For 4 kg, the cost is \$72: $4k = 72$.

Step 3: Solve for k: $k = \frac{72}{4} = 18$.

Step 4: Now, calculate the cost of 10 kg: $10 \times 18 = 180$.

The cost of 10 kg of sugar is \$180.

Key Concept: Direct Proportion, Cost of sugar and weight of sugar

c) 540 Rs

[Solution Description]

First, find the cost per kg of sugar. Divide the total cost by the weight to get the cost per kg: $\text{Cost per kg} = \frac{180}{5} = 36$ Rs/kg. Now, multiply the cost per kg by the desired weight to find the total cost for 15 kg: $\text{Total cost} = 36 \times 15 = 540$ Rs.

Key Concept: Direct Proportion

d) \$144

[Solution Description]

Step 1: Let the cost of 1 kg of sugar be k. According to direct proportion, cost = weight $\times$ k.

Step 2: For 5 kg, the cost is \$90: $5k = 90$.

Step 3: Solve for k: $k = \frac{90}{5} = 18$.

Step 4: Now, calculate the cost of 8 kg: $8 \times 18 = 144$.

The cost of 8 kg of sugar is \$144.

Key Concept: Direct Proportion, Ratio

c) 270 km

[Solution Description]

This problem involves direct proportion. Let the distance travelled with 18 litres of petrol be $y$ km. Using the formula $\frac{x_1}{x_2} = \frac{y_1}{y_2}$, where $x_1=5$, $x_2=18$, $y_1=75$, and $y_2=y$. Substituting these values:

$\frac{5}{18} = \frac{75}{y}$

Cross-multiplying:

$5y = 1350$

Dividing both sides by 5:

$y = 270$

Therefore, the vehicle can travel 270 km with 18 litres of petrol.

Key Concept: Direct Proportion

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

[Solution Description]

Given the car uses 8 litres of petrol to travel 120 km, we can determine the distance travelled with 16 litres of petrol by using the direct proportion formula $\frac{x_1}{y_1} = \frac{x_2}{y_2}$, where $x_1$ and $x_2$ are the amounts of petrol, and $y_1$ and $y_2$ are the distances travelled. Here, $x_1 = 8$ litres, $y_1 = 120$ km, and $x_2 = 16$ litres. We need to find $y_2$. Substituting the values into the formula: $\frac{8}{120} = \frac{16}{y_2}$. To find $y_2$, cross-multiply to get $8y_2 = 16 \times 120$, which simplifies to $8y_2 = 1920$. Dividing both sides by 8 gives $y_2 = \frac{1920}{8} = 240$ km. This shows that the distance travelled is 240 km when 16 litres of petrol is used, confirming the Assertion. The Reason states that the distance travelled by the car is directly proportional to the amount of petrol consumed, which is true as demonstrated by the calculation. Therefore, both Assertion and Reason are true, and Reason correctly explains the Assertion.

Key Concept: Direct Proportion

b) 10 liters

[Solution Description]

The consumption of petrol is directly proportional to the distance covered. Let $z$ be the amount of petrol consumed to cover 130 kilometers. The proportion is $\frac{7}{91} = \frac{z}{130}$. Cross-multiplying gives $91z = 910$. Dividing both sides by 91, we get $z = 10$ liters.

Key Concept: Direct Proportion

b) 120 degrees

[Solution Description]

Since the hour hand turns 30 degrees in 1 hour, the angle turned in 4 hours is:

$\text{Angle} = 30\degree/\text{hour} \times 4\ \text{hours} = 120\degree$

Key Concept: Direct Proportion

c) 15 minutes

[Solution Description]

The minute hand of a clock moves 360 degrees in 60 minutes, so the rate of turning is $\frac{360}{60} = 6$ degrees per minute. To turn 90 degrees, the time passed is $\frac{90}{6} = 15$ minutes.

Key Concept: Direct Proportion

b) 270 degrees

[Solution Description]

First, find the rate at which the second hand turns. Since it turns 360 degrees in 60 seconds, the rate is:

$\text{Rate} = \frac{360\degree}{60\ \text{seconds}} = 6\degree/\text{second}$

Now, calculate the angle turned in 45 seconds:

$\text{Angle} = 6\degree/\text{second} \times 45\ \text{seconds} = 270\degree$

Key Concept: Direct Proportion

d) 180 km

[Solution Description]

Given 4 litres of petrol travels 60 km. Since petrol consumption and distance travelled are in direct proportion, the ratio $\frac{x}{y}$ remains constant. Let $x_1 = 4$ litres, $y_1 = 60$ km, $x_2 = 12$ litres. Using the formula $\frac{x_1}{x_2} = \frac{y_1}{y_2}$, we substitute the values: $\frac{4}{12} = \frac{60}{y_2}$. Simplifying, we get $\frac{1}{3} = \frac{60}{y_2}$. Cross-multiplying gives $y_2 = 60 \times 3 = 180$ km.

Key Concept: Direct Proportion, Constant Ratio

d) 420 km

[Solution Description]

Distance is directly proportional to time when speed is constant. The ratio $\frac{240}{4} = 60$ km/h. So, distance in 7 hours = $60 \times 7 = 420$ km.

Key Concept: Direct Proportion

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

[Solution Description] To check if $x$ and $y$ are in direct proportion, we calculate the ratios $\frac{x_1}{y_1}$ and $\frac{x_2}{y_2}$. Here, $\frac{x_1}{y_1} = \frac{5}{10} = 0.5$ and $\frac{x_2}{y_2} = \frac{10}{20} = 0.5$. Since both ratios are equal, $x$ and $y$ are indeed in direct proportion. Therefore, the assertion is true. The reason correctly explains why the assertion is true because for direct proportion, the ratios of corresponding values must be equal.

Key Concept: Direct Proportion

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

[Solution Description]

Given that 3 metres of cloth costs \$ 90, we can find the cost per metre by dividing the total cost by the length of the cloth. So, cost per metre = \$ 90 / 3 = \$ 30 per metre. Now, to find the cost of 5 metres, we multiply the cost per metre by 5, which gives \$ 30 * 5 = \$ 150. Therefore, the assertion is true. The reason states that the cost of cloth is directly proportional to its length, which is a correct statement and explains why the cost increases as the length increases. Hence, both the assertion and reason are true, and the reason correctly explains the assertion.

Key Concept: Direct Proportion

a) 2 days

[Solution Description]

Since the number of workers is inversely proportional to the number of days, we can use the formula: Workers$_{1}$ * Days$_{1}$ = Workers$_{2}$ * Days$_{2}$. Here, Workers$_{1}$ = 3, Days$_{1}$ = 6, and Workers$_{2}$ = 9. So, 3 * 6 = 9 * Days$_{2}$. Solving for Days$_{2}$, we get Days$_{2}$ = (3 * 6) / 9 = 2 days.

Key Concept: Direct Proportion, Missing Values

d) \$180

[Solution Description]

The cost of books is directly proportional to the number of books. First, find the cost of one book by dividing the total cost by the number of books. So, the cost of one book is $\frac{120}{8} = 15$ dollars. Now, to find the cost of 12 books, multiply the cost of one book by 12. Thus, Cost = $15 \times 12 = 180$ dollars.

Key Concept: Direct Proportion, Ratio

c) 50 kg

[Solution Description]

Let the weight of 25 liters of water be $x$. Since weight varies directly with volume, we can write $\frac{15}{25} = \frac{30}{x}$. Solving for $x$, we get $x = \frac{25 \times 30}{15} = 50$. So, the weight of 25 liters of water is 50 kg.

Key Concept: Direct Proportion

c) 5 hours

[Solution Description]

The car travels 240 km in 3 hours. Let the time taken to travel 400 km be $t$. Using direct proportion, we have:

$\frac{240}{3} = \frac{400}{t}$

Solving for $t$:

$240t = 1200 \quad \Rightarrow \quad t = 5$

So, it will take 5 hours to travel 400 km.

Key Concept: Direct Proportion

b) 150 km

[Solution Description]

Let the distance travelled in 5 hours be y. Since distance is directly proportional to time, we can write:

$\frac{3}{5} = \frac{90}{y}$

Solving for y:

$3y = 5 \times 90$

$3y = 450$

$y = \frac{450}{3} = 150$

Therefore, the car will travel 150 km in 5 hours.

Key Concept: Inverse Proportion

b) 3 hours

[Solution Description]

Let $S_1 = 60$ km/h, $T_1 = 4$ hours, and $S_2 = 80$ km/h. We need to find $T_2$. Since speed and time are inversely proportional, we use the formula:

$S_1 \times T_1 = S_2 \times T_2$

Substituting the values:

$60 \times 4 = 80 \times T_2$

Solving for $T_2$:

$T_2 = \frac{60 \times 4}{80} = 3$

So, it will take 3 hours to reach the destination at 80 km/h.

Key Concept: Inverse Proportion

b) 12

8

Key Concept: Inverse Proportion, Identifying Inverse Proportion

b) 4

[Solution Description]

Since x and y are in inverse proportion, their product remains constant. Let the constant be k, so $x y = k$. When x = 12 and y = 6, substitute these values into the equation to find k: 12 * 6 = k. Therefore, k = 72. Now, to find y when x = 18, use the equation 18 * y = 72. Solving for y gives y = 72 / 18 = 4.

Key Concept: Inverse Proportion

d) 16

[Solution Description]

Let the number of workers be W and the time taken be T. Since they are inversely proportional, $W \times T$ is a constant. Initially, $W_1 = 8$ and $T_1 = 12$, so the constant is $W_1 T_1 = 8 \times 12 = 96$. Now, we need to find the number of workers ($W_2$) when the time taken is $T_2 = 6$ days. Using the inverse proportion formula: $W_2 T_2 = 96$. Substituting the known values: $W_2 \times 6 = 96$. To find $W_2$, we divide both sides by 6: $W_2 = \frac{96}{6} = 16$.

Key Concept: Inverse Proportion, Determining missing values

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

[Solution Description]

To determine if the assertion and reason are correct, we first check if the number of workers and the time taken to build a wall are inversely proportional. According to the concept of inverse proportion, if two quantities are inversely proportional, their product remains constant.

Let’s denote the number of workers as $x$ and the time taken as $y$. Initially, $x_1 = 10$ workers and $y_1 = 20$ days. The product is $x_1 y_1 = 10 \times 20 = 200$.

Now, according to the assertion, if the number of workers increases to $x_2 = 20$, then the time taken should decrease to $y_2 = 10$ days. The product in this case is $x_2 y_2 = 20 \times 10 = 200$.

Since both products are equal ($200 = 200$), the assertion that the time taken decreases from 20 days to 10 days when the number of workers increases from 10 to 20 is correct.

Further, the reason states that the number of workers and the time taken are inversely proportional, which is also correct based on the constant product.

Therefore, both the assertion and reason are true, and the reason correctly explains the assertion.

Key Concept: Inverse Proportion

b) 10

[Solution Description]

Given that $a$ is inversely proportional to $b$, we have $a = \frac{k}{b}$. First, find the constant of proportionality $k$. Substituting $a = 15$ and $b = 2$, we get:

$15 = \frac{k}{2} \implies k = 15 \times 2 = 30.$

Now, use this $k$ to find $a$ when $b = 3$:

$a = \frac{30}{3} = 10.$

Therefore, the value of $a$ when $b = 3$ is 10.

Key Concept: Inverse Proportion

b) 4

[Solution Description]

Since $p$ and $q$ are inversely proportional, $p \times q = k$, where $k$ is a constant. Given $p = 12$ and $q = 3$, we can find $k$ as follows:

$k = p \times q = 12 \times 3 = 36$

Now, to find $q$ when $p = 9$, we use the same equation:

$9 \times q = 36 \implies q = \frac{36}{9} = 4$

So, the value of $q$ is 4.

Key Concept: Inverse Proportion

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

[Solution Description]

The assertion states that as the number of workers increases, the time taken to complete the task decreases. This is true because, in inverse proportion, an increase in one quantity causes a decrease in the other. The reason states that the product of the number of workers and the time taken to complete the task remains constant. This is also true because, in inverse proportion, the product of the two quantities remains constant. The reason correctly explains the assertion, as the constant product is what defines inverse proportion.

Key Concept: Inverse Proportion

b) 2

[Solution Description]

Since $a$ and $b$ are inversely proportional, $a \times b = k$, where $k$ is a constant. Given $a = 10$ and $b = 5$, we can find $k$ as follows:

$k = a \times b = 10 \times 5 = 50$

Now, to find $b$ when $a = 25$, we use the same equation:

$25 \times b = 50 \implies b = \frac{50}{25} = 2$

So, the value of $b$ is 2.

Key Concept: Inverse Proportion

c) 6 days

[Solution Description]

Let the number of days required by 16 workers be $d$. Since the number of workers and the number of days are inversely proportional, the product of the two quantities remains constant. Thus, we have:

$12 \times 8 = 16 \times d$

Solving for $d$, we get:

$96 = 16d$

Dividing both sides by 16:

$d = \frac{96}{16} = 6$

Therefore, 16 workers will take 6 days to complete the task.

Key Concept: Inverse Proportion

c) Time becomes one-third

[Solution Description]

In inverse proportion, as one quantity increases, the other decreases in such a way that their product remains constant. Here, speed and time are inversely proportional. If speed is tripled (i.e., increased by a factor of 3), the time taken will decrease by the same factor. Thus, the time taken will become one-third of the original time.

Key Concept: Inverse Proportion

d) Number of books decreases to $\frac{2}{3}$

[Solution Description]

Since the price of a book increases from \$40 to \$60, the number of books that can be bought decreases. The ratio of the increase in price is $60 \div 40 = 1.5$. Since the price has increased by a factor of 1.5, the number of books that can be bought decreases by the same factor. Thus, the number of books that can be bought becomes $\frac{1}{1.5}$ or $\frac{2}{3}$ of the original number.

Key Concept: Inverse Proportion, Multi-step Reasoning

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

[Solution Description]

Let us analyze the assertion and reason step by step.

First, consider the relationship between speed ($v$), time ($t$), and distance ($d$). The formula connecting these quantities is $d = v \times t$. For a fixed distance $d$, if the speed $v$ is doubled, the time $t$ must be halved to maintain the equation $d = v \times t$. This confirms that the assertion is true.

Now, let us examine the reason. Two quantities are said to be inversely proportional if their product remains constant. In this case, since $d$ is constant, $v \times t = d$ implies that $v$ and $t$ are inversely proportional. Thus, the reason is also true.

Furthermore, the reason correctly explains the assertion because it provides the fundamental principle of inverse proportionality that justifies why doubling the speed results in halving the time for a fixed distance.

Therefore, both the assertion and reason are true, and the reason correctly explains the assertion.

Key Concept: Inverse Proportion

b) 2

[Solution Description]

Let the time taken be $t$. Since speed and time are inversely proportional, we use the inverse proportion formula:

$60 \times 3 = 90 \times t$

Solving for $t$:

$180 = 90t$

$t = \frac{180}{90} = 2$

So, the car will take 2 hours to travel the same distance at 90 km/h.

Key Concept: Inverse Proportion

b) 6 hours

[Solution Description]

Speed is constant, so distance and time are directly proportional. First find the speed: speed = distance / time = 240 km / 4 hours = 60 km/h. Now find the time to travel 360 km: time = distance / speed = 360 km / 60 km/h = 6 hours.

Key Concept: Inverse Proportion

c) 2.5

[Solution Description]

Given that $x$ is inversely proportional to $y$, we can write $x = \frac{k}{y}$, where $k$ is a constant. First, find the constant $k$ using the values $x = 10$ and $y = 5$. Substitute these values into the equation: $10 = \frac{k}{5}$. Solving for $k$, we get $k = 50$. Now, use this value of $k$ to find $x$ when $y = 20$. The equation becomes $x = \frac{50}{20} = 2.5$.

Key Concept: Inverse Proportion

b) 5

[Solution Description]

Given that $p$ is inversely proportional to $q$, we can write $p = \frac{k}{q}$, where $k$ is a constant. Use the given values $p = 15$ and $q = 3$ to find $k$. Substitute these values into the equation: $15 = \frac{k}{3}$. Solving for $k$, we get $k = 45$. Now, use this value of $k$ to find $p$ when $q = 9$. The equation becomes $p = \frac{45}{9} = 5$.

Key Concept: Inverse Proportion

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

[Solution Description] The assertion states that if the speed of a vehicle increases, the time taken to cover a fixed distance decreases. This is true because as speed increases, the time decreases for a fixed distance. The reason states that speed and time are inversely proportional to each other, which is also true. Moreover, the reason correctly explains the assertion, as the inverse proportionality between speed and time directly leads to the conclusion that increasing speed decreases the time taken.

Key Concept: Inverse Proportion

b) 16

[Solution Description]

Let the number of workers required to complete the task in 15 days be $x$. Since the number of workers and the time taken are inversely proportional, we have:

$12 \times 20 = x \times 15$

Solving for $x$, we get:

$x = \frac{12 \times 20}{15} = \frac{240}{15} = 16$

Therefore, 16 workers are required.

Key Concept: Inverse Proportion

b) 10

[Solution Description]

Let the number of workers required to build the house in 24 days be $y$. Since the number of workers and the time taken are inversely proportional, we have:

$8 \times 30 = y \times 24$

Solving for $y$, we get:

$y = \frac{8 \times 30}{24} = \frac{240}{24} = 10$

Therefore, 10 workers are required.

Key Concept: Inverse Proportion

c) 16

[Solution Description]

Let the number of workers required to complete the job in 30 days be $z$. Since the number of workers and time taken vary in inverse proportion, we have:

$8 \times 60 = 30 \times z$

Simplifying, we get:

$z = \frac{8 \times 60}{30}$

Performing the calculations:

$z = \frac{480}{30} = 16$

Therefore, 16 workers are needed to complete the job in 30 days.

Key Concept: Vehicle speed and travel time

b) 40 km/h

[Solution Description]

Let the original speed be $S$ km/h. The time taken to cover 300 km at this speed is $T = \frac{300}{S}$ hours. When the speed increases by 20 km/h, the new speed becomes $S + 20$ km/h, and the time taken is 5 hours. So, we have the equation $\frac{300}{S + 20} = 5$. Solving for $S$, we multiply both sides by $S + 20$ to get $300 = 5(S + 20)$. Simplifying, we get $300 = 5S + 100$. Subtracting 100 from both sides gives $200 = 5S$. Finally, dividing both sides by 5, we find $S = 40$ km/h.

Key Concept: Inverse Proportion, Vehicle speed and travel time

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

[Solution Description]

Let the initial speed of the vehicle be $s$ and the initial time taken to cover a fixed distance $d$ be $t$. We know that $d = s \times t$.

If the speed is tripled, the new speed becomes $3s$. To cover the same distance $d$, the new time $t'$ can be calculated as:

$d = 3s \times t'$

Solving for $t'$, we get:

$t' = \frac{d}{3s} = \frac{t}{3}$

Thus, the time is reduced to one-third of the original time. This shows that the Assertion is true.

The Reason states that speed and time are inversely proportional when the distance remains constant. This is also true because the relationship $d = s \times t$ implies $s \times t = \text{constant}$, which is the definition of inverse proportionality.

Therefore, both Assertion and Reason are true, and the Reason correctly explains the Assertion.

Key Concept: Vehicle speed and travel time

b) 3 hours

[Solution Description]

First, calculate the distance using the formula $Distance = Speed \times Time$. Here, the speed is 60 km/h and the time is 4 hours. So, the distance is $60 \times 4 = 240$ km. Now, we need to find the time taken when the speed is 80 km/h. Using the same formula, we can rearrange it to find time: $Time = \frac{Distance}{Speed}$. Substituting the values, we get $Time = \frac{240}{80} = 3$ hours.

Key Concept: Inverse Proportion

c) 120

[Solution Description]

Let the number of books that can be bought be y. Given the total amount is Rs. 6000 and the price of each book is Rs. 50. Since the price and the number of books are in inverse proportion, we have $xy = k$, where $k$ is the constant. Here, $x = 50$. So, $50y = 6000$. Solving for y, $y = \frac{6000}{50} = 120$. Therefore, 120 books can be bought.

Key Concept: Inverse Proportion

b) 75

[Solution Description]

Let the number of books be y. Given the total amount is Rs. 6000 and the price of each book is Rs. 80. Since the price and the number of books are in inverse proportion, we have $xy = k$, where $k$ is the constant. Here, $x = 80$. So, $80y = 6000$. Solving for y, $y = \frac{6000}{80} = 75$. Therefore, 75 books can be bought.

Key Concept: Inverse Proportion, Price and Quantity

b) 2

[Solution Description]

Initially, the number of books that can be bought is $\frac{240}{30} = 8$. After the price decrease, the number of books that can be bought is $\frac{240}{24} = 10$. The additional number of books that can be bought is $10 - 8 = 2$.

Key Concept: Inverse proportion, Product constant

c) 60 days

[Solution Description]

Let the number of cows be $x$ and the number of days be $y$. Since the number of cows and the number of days are inversely proportional, $xy = k$, where $k$ is a constant. Initially, $x_1 = 50$ and $y_1 = 30$, so $k = 50 \times 30 = 1500$. When the number of cows is reduced to 25, let the new number of days be $y'$. Then, $25 \times y' = 1500$. Solving for $y'$, we get $y' = \frac{1500}{25} = 60$ days.

Key Concept: Inverse Proportion

b) 3 hours

[Solution Description]

Since speed and time are inversely proportional for a fixed distance, we can use the relationship $s_1 t_1 = s_2 t_2$, where $s_1 = 60$ km/h, $t_1 = 4$ hours, and $s_2 = 80$ km/h.

Substitute the known values:

$60 \times 4 = 80 \times t_2$.

Calculate the left side:

$240 = 80 \times t_2$.

Solve for $t_2$:

$t_2 = \frac{240}{80} = 3$ hours.

Key Concept: Inverse Proportion

c) Assertion is true, but Reason is false.

[Solution Description]

The assertion is true because in inverse proportion, the product of the two quantities remains constant. However, the reason is false because the product of inversely proportional quantities is equal to the constant of proportionality, not its square. Therefore, the correct answer is that the Assertion is true, but the Reason is false.