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: 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: Introduction

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

[Solution Description]

The assertion states that the concept of introduction is fundamental in understanding any subject. This is true because an introduction sets the stage for learning by providing context and basic principles. The reason claims that introduction provides a foundational overview, which is essential for building deeper knowledge. This 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: 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: Proportional Relationship

d) 360 km

[Solution Description]

First, find the speed of the car by dividing the distance by time. So, $240 \div 4 = 60$ km/h. Now, multiply the speed by the time to find the distance covered in 6 hours. Therefore, $60 \times 6 = 360$ km.

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: Understanding Proportional Relationships

c) \$6.00

[Solution Description]

First, find the cost of one pencil by dividing the total cost by the number of pencils: $2.50 \div 5 = \$0.50$. Then, multiply the cost of one pencil by 12 to find the cost of 12 pencils: $0.50 \times 12 = \$6.00$.

Key Concept: Direct Variation

b) \$200

[Solution Description]

This is an example of direct variation, where the cost increases with the number of articles. Let the cost of 10 articles be $C$. Using the direct variation formula:

$C = \frac{100}{5} \times 10 = 200$

Therefore, the cost of 10 articles is \$200.

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: Inverse Variation

b) 8 minutes

[Solution Description]

This is an example of inverse variation, where more workers lead to less time taken. Let the time taken by five students be $t$ minutes. Using the inverse variation formula:

$2 \times 20 = 5 \times t$

Solving for $t$:

$t = \frac{40}{5} = 8 \text{ minutes}$

Therefore, five students will take 8 minutes to arrange the chairs.

Key Concept: Direct Variation

b) 750 mL

[Solution Description]

Mohan uses 300 mL of water for 2 persons. To find out how much water is needed for 5 persons, we can use the concept of direct variation. Since the amount of water increases proportionally with the number of persons, we calculate:

$\text{Water for 5 persons} = \frac{300 \text{ mL}}{2} \times 5 = 750 \text{ mL}$

Therefore, Mohan needs 750 mL of water for five persons.

Key Concept: Direct Proportion

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

[Solution Description]

Let’s verify whether the assertion is correct by using the concept of direct proportion. Given that the cost of 5 metres of cloth is \$210, we can find the cost of 10 metres. Since it is a case of direct proportion, we can use the relation:

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

Here, $x_1 = 5$, $y_1 = 210$, and $x_2 = 10$. We need to find $y_2$.

$\frac{5}{10} = \frac{210}{y_2}$

Simplifying this:

$5y_2 = 10 \times 210$

$5y_2 = 2100$

$y_2 = \frac{2100}{5} = 420$

So, the cost of 10 metres of cloth is \$420. Hence, the assertion is true. The reason states that the cost of cloth increases in direct proportion with its length, which is also true. Moreover, the reason correctly explains why the assertion holds true because it justifies the relationship between length and cost using direct proportion.

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

c) 21 metres

[Solution Description]

Let the height of the tree be $x$ metres. Since the height of an object is directly proportional to the length of its shadow, we can write:

$\frac{14}{10} = \frac{x}{15}$

Cross-multiplying gives:

$14 \times 15 = 10x$

Solving for $x$:

$x = \frac{210}{10} = 21$

Therefore, the height of the tree is 21 metres.

Key Concept: Inverse Proportion

c) 96 minutes

[Solution Description]

Let the time taken by 5 pipes to fill the tank be $x$ minutes. Since the number of pipes and the time taken are inversely proportional, we can write:

$6 \times 80 = 5 \times x$

Solving for $x$:

$x = \frac{480}{5} = 96$

Therefore, it will take 96 minutes for 5 pipes to fill the tank.

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: Total Cost Calculation

b) \$45

[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 \$15 and the number of articles is 3. Therefore, the total cost is $15 \times 3 = \$45$.

Key Concept: Direct Variation

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 articles purchased increases, the total cost also increases. This is a fundamental concept of direct variation where one quantity increases with the increase in another quantity. The reason states that the total cost is directly proportional to the number of articles purchased. This is a correct explanation because in direct variation, two quantities change in the same direction. Therefore, both the assertion and reason are true, and the reason correctly explains the assertion.

Key Concept: Direct Proportion

c) 6 days

[Solution Description]

Since the number of workers has doubled, the time taken should halve because more workers can complete the task faster. So, if 5 workers take 12 days, then 10 workers will take $\frac{12}{2} = 6$ days.

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, Principal

b) 5\%

[Solution Description]

The formula for simple interest is $I = P \times r \times t$, where $I$ is the interest earned, $P$ is the principal amount, $r$ is the annual interest rate, and $t$ is the time in years.

Given:

- $P = \$8000$

- $I = \$1200$

- $t = 3$ years

We need to find $r$. Rearranging the formula:

$r = \frac{I}{P \times t}$

Plugging in the values:

$r = \frac{1200}{8000 \times 3} = \frac{1200}{24000} = 0.05$

Converting to percentage:

$r = 0.05 \times 100 = 5\%$

So, the annual interest rate is 5\%.

Key Concept: Interest, Principal

a) \$5788.13

[Solution Description]

The formula for compound interest is $A = P(1 + r)^t$, where $A$ is the final amount, $P$ is the principal amount, $r$ is the annual interest rate, and $t$ is the time in years.

Given:

- $P = \$5000$

- $r = 5\% = 0.05$

- $t = 3$ years

Plugging in the values:

$A = 5000(1 + 0.05)^3$

Calculating step by step:

$1 + 0.05 = 1.05$

$1.05^3 = 1.157625$

$A = 5000 \times 1.157625 = 5788.125$

So, the total amount after 3 years is \$5788.13 (rounded to two decimal places).

Key Concept: Basic Concept

c) Interest doubles

[Solution Description] The interest earned is directly proportional to the principal amount. If the principal is doubled, the interest earned will also double. For example, if you deposit \$2000 instead of \$1000 at a 5\% annual interest rate, the interest earned will be $Interest = 2000 \times 0.05 \times 1 = \$100$, which is double the original \$50.

Key Concept: Inverse Proportion

c) Time decreases to $1/3$

[Solution Description]

Since cycling speed is 3 times the running speed, time taken while cycling will be $1/3$ of the time taken while running.

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: Travel Time Calculation

b) 1 hour

[Solution Description]

First, calculate the initial time taken at 60 km/h:

$t_1 = \frac{300}{60} = 5 \text{ hours}$

Next, calculate the new time taken at 75 km/h:

$t_2 = \frac{300}{75} = 4 \text{ hours}$

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

$\Delta t = t_1 - t_2 = 5 - 4 = 1 \text{ hour}$

Thus, the time saved is 1 hour.

Key Concept: Speed and Distance Relationship

c) 1.33 hours

[Solution Description]

First, calculate the distance traveled at 80 km/h:

$d = 80 \times 4 = 320 \text{ km}$

Now, calculate the time taken at 60 km/h to cover the same distance:

$t = \frac{320}{60} \approx 5.\overline{3} \text{ hours}$

The additional time taken is the difference between the new time and the original time:

$\Delta t = 5.\overline{3} - 4 = 1.\overline{3} \text{ hours}$

Thus, it takes approximately 1.33 hours longer.

Key Concept: Inverse Proportion

b) 6 hours

[Solution Description]

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

Given:

3 workers can build the wall in 12 hours.

To find:

Time taken by 6 workers to build the same wall.

Step 1: Understand the relationship between workers and time.

If 3 workers take 12 hours, then 1 worker would take $3 \times 12 = 36$ hours (since fewer workers take more time).

Step 2: Now, calculate the time taken by 6 workers.

Since 1 worker takes 36 hours, 6 workers will take $36 / 6 = 6$ hours.

Therefore, 6 workers will take 6 hours to build the wall.

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: Inverse Proportion, Multiplicative Inverse

d) 6.67 days

[Solution Description]

Let the initial number of workers be $W_1 = 12$ and the time taken be $T_1 = 20$ days. The number of workers is increased by a factor of 3, so the new number of workers is $W_2 = 3 \times W_1 = 3 \times 12 = 36$. Since the number of workers and the time taken are inversely proportional, we have:

$W_1 \times T_1 = W_2 \times T_2$

Substituting the known values:

$12 \times 20 = 36 \times T_2$

Solving for $T_2$:

$240 = 36 \times T_2 \implies T_2 = \frac{240}{36} = \frac{20}{3} \approx 6.67 \text{ days}$

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: 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

c) 225 km

[Solution Description]

Let the distance traveled using 15 litres of petrol be y km. Since distance traveled is directly proportional to petrol consumption, we can use the relation $\frac{x1}{x2} = \frac{y1}{y2}$. Here, x1 = 4, y1 = 60, and x2 = 15. Substituting these values, we get $\frac{4}{15} = \frac{60}{y}$. Solving for y, we get $4y = 15 \times 60$ or $y = \frac{15 \times 60}{4} = 225$. Therefore, the car will travel 225 km using 15 litres of petrol.

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, 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

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, Ratio

b) 6 hours

[Solution Description]

First, find the constant of proportionality. Speed $S$ is directly proportional to distance $D$, so $S = k \cdot D$. Given that the car travels 240 km in 4 hours, the speed $S = \frac{240}{4} = 60$ km/h. Now, use this speed to find the time for 360 km. Time $T = \frac{D}{S} = \frac{360}{60} = 6$ hours.

Key Concept: Direct Proportion

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

[Solution Description] To determine if the cost of cloth is directly proportional to its length, we check if the ratio of the length of cloth to its cost remains constant. For 5 metres of cloth, the cost is \$210, so the ratio is $\frac{5}{210} = \frac{1}{42}$. For 10 metres of cloth, the cost is \$420, so the ratio is $\frac{10}{420} = \frac{1}{42}$. Since both ratios are equal, the cost of cloth is directly proportional to its length. The assertion is true. The reason correctly explains that two quantities are in direct proportion if their ratio remains constant. Therefore, both the assertion and reason are true, and the reason is the correct explanation of the assertion.

Key Concept: Direct Proportion, Height and Shadow Relationship

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

[Solution Description]

The problem involves the relationship between the height of an object and the length of its shadow under similar conditions. Given that an electric pole 14 metres high casts a shadow of 10 metres, we can use the concept of direct proportion to find the height of a tree that casts a shadow of 15 metres. Let the height of the tree be $x$ metres. Since it is a case of direct proportion, we can set up the relation $\frac{14}{10} = \frac{x}{15}$. Solving for $x$, we get $x = \frac{14 \times 15}{10} = 21$ metres. Thus, the assertion is true because the height of the tree is indeed directly proportional to the length of its shadow under similar conditions. The reason is also true because, in a direct proportion, as one quantity increases, the other also increases proportionally. Furthermore, the reason correctly explains the assertion by stating that the increase in height leads to a proportional increase in the length of the shadow.

Key Concept: Direct Proportion, Ratio

c) 180 km

[Solution Description]

Let x be the distance travelled in km and y be the petrol consumed in litres. The relationship between distance and petrol is directly proportional, so we can write:

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

Given $x_1 = 120 km$, $y_1 = 8 litres$, and $y_2 = 12 litres$, we need to find $x_2$.

Substituting the known values:

$\frac{120}{8} = \frac{x_2}{12}$

Solving for $x_2$:

$x_2 = \frac{120 \times 12}{8} = 180 km$

The distance travelled when 12 litres of petrol are consumed is 180 km.

Key Concept: Direct Proportion, Ratio

c) \$32

[Solution Description]

Let x be the cost in \$ and y be the weight in kg. The relationship between cost and weight is directly proportional, so we can write:

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

Given $x_1 = \$20$, $y_1 = 5 kg$, and $y_2 = 8 kg$, we need to find $x_2$.

Substituting the known values:

$\frac{20}{5} = \frac{x_2}{8}$

Solving for $x_2$:

$x_2 = \frac{20 \times 8}{5} = \$32$

The cost of 8 kg of sugar is \$32.

Key Concept: Direct Proportion

b) \$160

[Solution Description]

Given that the cost is directly proportional to the length of cloth, we use the relation $x1 / x2 = y1 / y2$, where $x1 = 7$, $y1 = 280$, and $x2 = 4$. Substituting the values into the equation, we get $7 / 4 = 280 / y2$ or $y2 = 4 \times 280 / 7 = 160$.

Key Concept: Direct Proportion, Real-World Application

b) \$48

[Solution Description]

Since the cost of rice is directly proportional to its weight, we can find the cost per kilogram:

$\text{Cost per kg} = \frac{\$30}{5 \, \text{kg}} = \$6 \, \text{per kg}$

Now, calculate the cost for 8 kg:

$\text{Cost for 8 kg} = 8 \, \text{kg} \times \$6/\text{kg} = \$48$

The cost for 8 kg of rice is \$48.

Key Concept: Direct Proportion, Conceptual Complexity

c) Increases by 20\%

[Solution Description]

Since $x$ and $y$ are directly proportional, any increase in $x$ will result in the same percentage increase in $y$. Therefore, if $x$ increases by 20\%, $y$ will also increase by 20\%.

Key Concept: Direct Proportion

b) 300 km

[Solution Description]

Let the distance traveled in 5 hours be $x$ km. Since distance is directly proportional to time, we can write $\frac{120}{2} = \frac{x}{5}$. Cross-multiplying gives $2x = 600$. Dividing both sides by 2 gives $x = 300$. Therefore, the car will travel 300 km in 5 hours.

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) 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

d) 105 dollars

[Solution Description]

First, find the cost of 1 kg of sugar by dividing the total cost by the quantity:

$\text{Cost per kg} = \frac{45}{3} = 15 \text{ dollars}$

Now, multiply the cost per kg by the required quantity to find the total cost for 7 kg:

$\text{Total cost} = 15 \times 7 = 105 \text{ dollars}$

Key Concept: Direct Proportion

c) 60 dollars

[Solution Description]

First, find the cost of 1 notebook by dividing the total cost by the number of notebooks:

$\text{Cost per notebook} = \frac{40}{8} = 5 \text{ dollars}$

Now, multiply the cost per notebook by the required number to find the total cost for 12 notebooks:

$\text{Total cost} = 5 \times 12 = 60 \text{ dollars}$

Key Concept: Direct Proportion

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

[Solution Description]

The assertion states that if the distance travelled by a car increases, the consumption of petrol will also increase. This is true because more distance requires more petrol to cover it. The reason states that the consumption of petrol and the distance travelled by a car are in direct proportion. This is also true because as the distance increases, the petrol consumption increases in the same ratio, keeping their ratio constant. Therefore, both the assertion and reason are true, and the reason correctly explains the assertion.

Key Concept: Direct Proportion, Cost and Weight

c) Assertion is true, but Reason is false.

[Solution Description] Let the initial cost of sugar be \$C per kilogram and the weight of sugar purchased be \$W kilograms for a fixed amount of money \$M. Therefore, \$M = C \times W. If the cost increases by 20\%, the new cost becomes \$1.2C. The new weight of sugar purchased for the same amount \$M is \$W_{\text{new}} = \frac{M}{1.2C} = \frac{W}{1.2}. The percentage decrease in weight is calculated as $\frac{W - W_{\text{new}}}{W} \times 100 = \frac{W - \frac{W}{1.2}}{W} \times 100 = \frac{1 - \frac{1}{1.2}}{1} \times 100 = \frac{0.2}{1.2} \times 100 = 16.67\%$. The assertion is true. However, the reason is false because the cost of sugar and weight of sugar are directly proportional, not inversely proportional, when the amount of money is fixed.

Key Concept: Direct Proportion

d) \$ 216

[Solution Description]

Given, the cost of 4 kg sugar is \$ 144. The weight of sugar is directly proportional to its cost. Using the formula for direct proportion, $\frac{x_1}{x_2} = \frac{y_1}{y_2}$ , where $x_1$ is 4 kg, $y_1$ is \$ 144, and $x_2$ is 6 kg. We need to find $y_2$. Plugging in the values: $\frac{4}{6} = \frac{144}{y_2}$. Solving for $y_2$, we get $y_2 = \frac{144 \times 6}{4} = 216$. So, the cost of 6 kg of sugar is \$ 216.

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, Petrol Consumption and Distance Traveled

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

[Solution Description]

To verify the assertion, we first calculate the ratio of petrol consumption to distance traveled. Given that the car travels 240 km using 16 litres of petrol, the ratio is given by:

$\frac{16}{240} = \frac{1}{15}$

This matches the constant ratio mentioned in the syllabus. Thus, the assertion is true. The reason states that the ratio $\frac{x}{y}$ remains constant when x and y are in direct proportion, which is also true as per the syllabus. Furthermore, the reason correctly explains why the assertion holds true because the constant ratio is a fundamental property of direct proportion.

Key Concept: Direct Proportion

b) 15 liters

[Solution Description]

Let the petrol consumed be $x$ liters for 180 km.

Since petrol consumption is directly proportional to distance traveled, we have:

$\frac{5}{60} = \frac{x}{180}$

Simplifying, $\frac{1}{12} = \frac{x}{180}$

Cross-multiplying, $x = \frac{180}{12}$

Therefore, $x = 15$ liters.

Key Concept: Direct Proportion

c) 15 liters

[Solution Description]

Here, the amount of petrol required is directly proportional to the distance traveled. Let $y$ be the amount of petrol required to travel 225 kilometers. The proportion is $\frac{10}{150} = \frac{y}{225}$. Cross-multiplying gives $150y = 2250$. Dividing both sides by 150, we get $y = 15$ liters.

Key Concept: Direct Proportion, Clock Hand Angle

c) 45 minutes

[Solution Description]

The minute hand of a clock completes a full circle (360 degrees) in 60 minutes. Therefore, the angle turned per minute is $\frac{360}{60} = 6$ degrees. If the minute hand has turned through 270 degrees, the time passed is $\frac{270}{6} = 45$ minutes.

Key Concept: Direct Proportion

c) 150 degrees

[Solution Description]

First, find the rate at which the minute hand turns. Since it turns 90 degrees in 15 minutes, the rate is:

$\text{Rate} = \frac{90\degree}{15\ \text{minutes}} = 6\degree/\text{minute}$

Now, calculate the angle turned in 25 minutes:

$\text{Angle} = 6\degree/\text{minute} \times 25\ \text{minutes} = 150\degree$

Key Concept: Direct Proportion

c) 60 minutes

[Solution Description]

The hour hand of a clock moves 360 degrees in 12 hours (720 minutes), so the rate of turning is $\frac{360}{720} = 0.5$ degrees per minute. To turn 30 degrees, the time passed is $\frac{30}{0.5} = 60$ minutes.

Key Concept: Direct Proportion

b) 12

[Solution Description]

Since x and y are in direct proportion, we have $\frac{x1}{y1} = \frac{x2}{y2}$. Here, x1 = 4, y1 = 8, and x2 = 6. We need to find y2. Substituting the values, we get $\frac{4}{8} = \frac{6}{y2}$. Simplifying, $\frac{1}{2} = \frac{6}{y2}$. Cross-multiplying gives $y2 = 12$. Therefore, the value of y is 12 when x is 6.

Key Concept: Direct Proportion

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

[Solution Description]

Let x be the weight of sugar in kg and y be the cost in \$. Given that the cost of 5 kg of sugar is \$250, we can find the constant ratio $k = \frac{y}{x}$. For 5 kg, $k = \frac{250}{5} = 50$. This means the cost per kg is \$50. Now, for 8 kg of sugar, the cost will be $y = k \times x = 50 \times 8 = 400$. Therefore, the cost of 8 kg of sugar is \$400. The reason correctly explains the assertion as the cost increases in direct proportion to the weight of sugar because the ratio remains constant.

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) 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

b) 5 hours

[Solution Description]

To find the time taken to travel 400 km, we first determine the speed of the car. The speed is calculated as distance divided by time. Here, the car travels 240 km in 3 hours, so the speed is $\frac{240}{3} = 80$ km/h. Now, to find the time taken to travel 400 km, we use the formula Time = Distance / Speed. Thus, Time = $\frac{400}{80} = 5$ hours.

Key Concept: Direct Proportion

c) 420 km

[Solution Description]

Let the distance traveled in 7 hours be $D$. Since the speed is constant, the distance is directly proportional to time. First, find the speed:

$\text{Speed} = \frac{\text{Distance}}{\text{Time}} = \frac{240}{4} = 60 \text{ km/h}$

Now, calculate the distance for 7 hours:

$D = 60 \times 7 = 420 \text{ km}$

Key Concept: Direct Proportion

b) \$30

[Solution Description]

Let the cost of 10 apples be y. Since cost is directly proportional to the number of apples, we can write:

$\frac{6}{10} = \frac{18}{y}$

Solving for y:

$6y = 10 \times 18$

$6y = 180$

$y = \frac{180}{6} = 30$

Therefore, the cost of 10 apples is \$30.

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: Inverse Proportion

a) 3 hours

3 hours

Key Concept: Inverse Proportion, Identifying Inverse Proportion

b) 3 hours

[Solution Description]

Since distance is constant, speed and time are inversely proportional. Let the original speed be $s_1 = 60$ km/h and the original time be $t_1 = 4$ hours. The new speed is $s_2 = 80$ km/h. Using the inverse proportion formula $s_1 t_1 = s_2 t_2$, we substitute the known values: 60 * 4 = 80 * t_2. Solving for t_2 gives t_2 = (60 * 4) / 80 = 3 hours.

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 a direct consequence of the concept of inverse proportion in physics and mathematics. When speed increases, less time is required to cover the same distance, which means that speed and time are inversely proportional. The reason provided similarly affirms that speed and time change inversely in proportion. This directly explains the assertion because the relationship between speed and time for a fixed distance is indeed inverse proportionality. Therefore, both the assertion and reason are true, and the reason correctly explains the assertion.

Key Concept: Inverse Proportion

b) 4

[Solution Description]

Since x and y are inversely proportional, their product remains constant. Initially, $x_1 = 10$ and $y_1 = 6$, so the constant $k$ is calculated as $k = x_1 y_1 = 10 \times 6 = 60$. Now, when x changes to $x_2 = 15$, we can find the new y ($y_2$) using the equation $x_2 y_2 = k$. Substituting the known values: $15 \times y_2 = 60$. To find $y_2$, we divide both sides by 15: $y_2 = \frac{60}{15} = 4$.

Key Concept: Inverse Proportion

c) 4 hours

[Solution Description]

Let the speed be S and the time be T. Since they are inversely proportional, $S \times T$ is a constant. Initially, $S_1 = 60$ km/h and $T_1 = 5$ hours, so the constant is $S_1 T_1 = 60 \times 5 = 300$. Now, we need to find the time ($T_2$) when the speed is $S_2 = 75$ km/h. Using the inverse proportion formula: $S_2 T_2 = 300$. Substituting the known values: $75 \times T_2 = 300$. To find $T_2$, we divide both sides by 75: $T_2 = \frac{300}{75} = 4$ hours.

Key Concept: Inverse Proportion

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

[Solution Description]

According to the concept of inverse proportion, if two quantities x and y are inversely proportional, then the product of their corresponding values remains constant. This is mathematically represented by the equation $xy = k$, where k is a constant. When x increases, y decreases proportionally to maintain the product as k, and vice versa. Therefore, both the assertion and reason are true. Additionally, the reason correctly explains why the product of x and y remains constant in inverse proportion. Thus, option a) is correct.

Key Concept: Inverse Proportion

c) 200 notebooks

[Solution Description]

Let the price of each notebook be $p$ and the number of notebooks be $n$. Since the price and the number of notebooks are inversely proportional, we have:

$p_1 n_1 = p_2 n_2$

First, find the number of notebooks that can be purchased at Rs. 30 per notebook using the total amount:

$n_1 = \frac{9000}{30} = 300$

Now, given: $p_1 = 30$, $n_1 = 300$, and $p_2 = 45$. Substitute these values into the equation:

$30 \times 300 = 45 \times n_2$

Simplify the left side:

$9000 = 45n_2$

Solve for $n_2$:

$n_2 = \frac{9000}{45} = 200$

Therefore, 200 notebooks can be purchased at Rs. 45 per notebook.

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 are inversely proportional, their product is always constant. This is true because, by definition, inverse proportionality implies that one quantity increases while the other decreases in such a way that their product remains unchanged. The reason explains this concept accurately by stating that as one quantity increases, the other decreases to keep the product constant. Therefore, both the assertion and reason are true, and the reason correctly explains the assertion.

Key Concept: Inverse Proportion

b) 4 days

[Solution Description]

Let the number of workers be $w$ and the number of days be $d$. Since the number of workers and the number of days are inversely proportional, we have:

$w_1 d_1 = w_2 d_2$

Given: $w_1 = 10$, $d_1 = 6$, and $w_2 = 15$. Substitute these values into the equation:

$10 \times 6 = 15 \times d_2$

Simplify the left side:

$60 = 15d_2$

Solve for $d_2$:

$d_2 = \frac{60}{15} = 4$

Therefore, it will take 4 days for 15 workers to complete the job.

Key Concept: Inverse Proportion

a) 10 machines

[Solution Description]

Let the number of machines required be $m$. The production rate remains constant, so the product of the number of machines and the number of days is proportional to the number of units produced. Thus, we have:

$\frac{10 \times 5}{500} = \frac{m \times 10}{1000}$

Simplifying the left side:

$\frac{50}{500} = \frac{1}{10}$

Setting this equal to the right side:

$\frac{1}{10} = \frac{10m}{1000}$

Simplifying the right side:

$\frac{1}{10} = \frac{m}{100}$

Solving for $m$:

$m = \frac{1}{10} \times 100 = 10$

Therefore, 10 machines are needed to produce 1000 units in 10 days.

Key Concept: Inverse Proportion

b) 2 hours

[Solution Description]

Let the speed be $s$ and the time taken be $t$. Since speed and time are inversely proportional, we have $s_1 \times t_1 = s_2 \times t_2$. Given $s_1 = 60$ km/h, $t_1 = 3$ hours, and $s_2 = 90$ km/h, we can find $t_2$ as follows:

$60 \times 3 = 90 \times t_2$

$180 = 90 \times t_2$

$t_2 = \frac{180}{90} = 2$

So, it will take 2 hours to reach the destination at 90 km/h.

Key Concept: Inverse Proportion

b) 2 hours 24 minutes

[Solution Description]

First, find the original speed of the car. Speed is calculated as distance divided by time. The original speed is:

$\text{Speed} = \frac{240 \, \text{km}}{3 \, \text{h}} = 80 \, \text{km/h}$

If the speed increases by 20 km/h, the new speed is:

$80 + 20 = 100 \, \text{km/h}$

Time taken to cover the same distance at the new speed is distance divided by speed:

$\text{Time} = \frac{240 \, \text{km}}{100 \, \text{km/h}} = 2.4 \, \text{h}$

Converting 0.4 hours to minutes, we multiply by 60:

$0.4 \times 60 = 24 \, \text{minutes}$

Therefore, the total time taken is 2 hours and 24 minutes.

Key Concept: Inverse Proportion

b) y halves

[Solution Description]

Since $x$ and $y$ are inversely proportional, their product remains constant. Let the original product be $k$. Then $5y_1 = k$ and $10y_2 = k$. Thus, $y_2 = \frac{k}{10}$. Since $y_1 = \frac{k}{5}$, we find that $y_2 = \frac{y_1}{2}$. So y decreases by half.

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

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, then their product remains constant. This is true because, by definition, when two quantities are in inverse proportion, an increase in one causes a proportional decrease in the other such that the product $xy$ remains constant. The reason states that the product of two quantities in inverse proportion is equal to a constant value k. This is also true as it directly follows from the definition of inverse proportion. Moreover, the reason correctly explains the assertion because it provides the mathematical relationship ($xy = k$) that justifies why the product remains constant.

Key Concept: Inverse Proportion

b) 20 workers

[Solution Description]

The relationship between the number of workers and the number of days is inverse. Let $W_1 = 10$, $D_1 = 20$, and $D_2 = 10$. We need to find $W_2$.

Using the inverse proportion formula:

$W_1 \times D_1 = W_2 \times D_2$

Substituting the given values:

$10 \times 20 = W_2 \times 10$

Solving for $W_2$:

$200 = 10W_2 \implies W_2 = 20$

Therefore, 20 workers are needed to complete the task in 10 days.

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

c) 8

[Solution Description]

Since $a$ is inversely proportional to $b$, we can write $a = \frac{k}{b}$, where $k$ is a constant. Use the given values $a = 12$ and $b = 4$ to find $k$. Substitute these values into the equation: $12 = \frac{k}{4}$. Solving for $k$, we get $k = 48$. Now, use this value of $k$ to find $a$ when $b = 6$. The equation becomes $a = \frac{48}{6} = 8$.

Key Concept: Inverse Proportion

c) 20 workers

[Solution Description]

Let $W_1 = 10$ workers, $D_1 = 18$ days, and $D_2 = 9$ days. We need to find $W_2$. Since the number of workers is inversely proportional to the time taken,

$W_1 \times D_1 = W_2 \times D_2$

Substituting the values,

$10 \times 18 = W_2 \times 9$

Simplifying,

$180 = 9W_2$

Dividing both sides by 9,

$W_2 = \frac{180}{9} = 20$

So, 20 workers are needed.

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) 50

[Solution Description]

Let the number of workers required to paint the wall in 10 hours be $x$. Since the number of workers and time taken vary in inverse proportion, we have:

$20 \times 25 = 10 \times x$

Simplifying, we get:

$x = \frac{20 \times 25}{10}$

Performing the calculations:

$x = \frac{500}{10} = 50$

Therefore, 50 workers are needed to paint the wall in 10 hours.

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

c) 4 hours

[Solution Description]

Let $t$ be the time it takes for Vehicle B to catch up with Vehicle A after Vehicle B starts moving. Since Vehicle A started 2 hours earlier, it would have already traveled $50 \times 2 = 100$ km before Vehicle B starts. The distance covered by Vehicle A in time $t$ is $50t$ km, and the distance covered by Vehicle B in the same time is $75t$ km. For Vehicle B to catch up with Vehicle A, the distances must be equal: $100 + 50t = 75t$. Subtracting $50t$ from both sides, we get $100 = 25t$. Dividing both sides by 25, we find $t = 4$ hours.

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, Price and Quantity

b) 4

[Solution Description]

The total budget is $16 \times 25 = \$400$. After the discount, the number of books that can be bought is $\frac{400}{20} = 20$. The additional number of books that can be purchased is $20 - 16 = 4$.

Key Concept: Inverse Proportion

b) Decreases to 100

[Solution Description]

Initially, at Rs. 40 per book, the number of books that can be bought is $\frac{6000}{40} = 150$. When the price increases to Rs. 60, the number of books decreases to $\frac{6000}{60} = 100$. Thus, the number of books decreases from 150 to 100 when the price increases from Rs. 40 to Rs. 60.

Key Concept: Inverse Proportion, Price and Quantity

b) 9

[Solution Description]

Let the total amount of money be $M$. Initially, the price per book is \$15, and the number of books that can be bought is 12. Therefore, $M = 12 \times 15 = \$180$. After the price increase, the new price per book is \$20. The number of books that can be bought now is $\frac{M}{20} = \frac{180}{20} = 9$.

Key Concept: Inverse Proportion

c) 1 hour

[Solution Description]

Let the distance be $D$. The time taken at 60 km/h is $t_1 = \frac{D}{60}$. The time taken at 40 km/h is $t_2 = \frac{D}{40}$. The difference in time is $t_2 - t_1 = \frac{D}{40} - \frac{D}{60} = D\left(\frac{1}{40} - \frac{1}{60}\right) = D\left(\frac{3 - 2}{120}\right) = D\left(\frac{1}{120}\right) = \frac{D}{120}$. Since the distance is the same, the time difference is $\frac{D}{120}$.

Key Concept: Inverse Proportion

a) 240, decreases

[Solution Description]

The number of notebooks that can be bought is inversely proportional to the price per notebook. First, calculate the number of notebooks that can be bought at Rs. 40 each:

Number of notebooks $= \frac{12000}{40} = 300$.

Next, calculate the number of notebooks that can be bought at Rs. 50 each:

Number of notebooks $= \frac{12000}{50} = 240$.

Thus, as the price increases from Rs. 40 to Rs. 50, the number of notebooks decreases from 300 to 240.

Key Concept: Inverse proportion, Product constant

b) Time taken halves

[Solution Description]

Let the initial speed be $x$ and the time taken be $y$. Since speed and time are inversely proportional, $xy = k$, where $k$ is a constant. If the speed is doubled, the new speed is $2x$. Let the new time taken be $y'$. Then, $2x \times y' = k$. Since $xy = k$, we can write $2x \times y' = xy$. Dividing both sides by $2x$, we get $y' = \frac{y}{2}$. Therefore, the time taken is halved.