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 doubles, the time taken to cover a fixed distance reduces to half. This is true because, for a fixed distance, time taken is inversely proportional to speed. Mathematically, if speed $S$ doubles to $2S$, then time $T$ becomes $\frac{T}{2}$. The reason given is that speed and time are inversely proportional when covering a fixed distance. This is also true as per the definition of inverse proportion. Furthermore, the reason correctly explains the assertion since the reduction in time to half is directly due to the inverse proportionality between speed and time. Hence, both the assertion and reason are true, and the reason is the correct explanation of the assertion.

Key Concept: Increase in articles purchased increases total cost

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

[Solution Description]

Key Concept: Direct Proportion, Clock Hand Angle

b) 1 hour

[Solution Description]

The hour hand of a clock completes a full circle (360 degrees) in 12 hours. Therefore, the angle turned per hour is $\frac{360}{12} = 30$ degrees. If the hour hand has turned through 30 degrees, the time passed is 1 hour.

Key Concept: Understanding Proportional Relationships

c) \$3.50

[Solution Description]

First, find the cost of one orange by dividing the total cost by the number of oranges: $1.50 \div 3 = \$0.50$. Then, multiply the cost of one orange by 7 to find the cost of 7 oranges: $0.50 \times 7 = \$3.50$.

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: Interest, Principal, Time

a) \$11255.09

[Solution Description]

For semi-annual compounding, the formula is $A = P(1 + \frac{r}{n})^{nt}$, where $A$ is the final amount, $P$ is the principal amount, $r$ is the annual interest rate, $n$ is the number of times interest is compounded per year, and $t$ is the time in years.

Given:

- $P = \$10000$

- $r = 6\% = 0.06$

- $n = 2$ (semi-annual)

- $t = 2$ years

Plugging in the values:

$A = 10000(1 + \frac{0.06}{2})^{2 \times 2}$

Calculating step by step:

$\frac{0.06}{2} = 0.03$

$(1 + 0.03) = 1.03$

$(1.03)^4 = 1.1255088$

$A = 10000 \times 1.1255088 = 11255.088$

So, the total amount after 2 years is \$11255.09 (rounded to two decimal places).

Key Concept: Direct Proportion

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

[Solution Description]

Given that the cost of 1 kg of sugar is Rs.36. Since the weight of sugar and its cost are in direct proportion, the cost of 5 kg can be calculated as follows: Cost = Weight × Cost per kg. Substituting the values, Cost = 5 kg × Rs.36/kg = Rs.180. Therefore, the Assertion is true. The Reason states that weight and cost are in direct proportion, which is also true and correctly explains the Assertion.

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: Inverse Proportion

b) 1.5 hours

[Solution Description]

The initial speed of the car is 60 km/h (since 120 km / 2 hours = 60 km/h). We use the inverse proportion formula: $60 \times 2 = 80 \times t$. Solving for $t$, we get $t = \frac{60 \times 2}{80} = 1.5$ 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 is doubled, the time taken to cover a fixed distance is halved. This is true because speed and time are inversely proportional when the distance is constant. Mathematically, if $s$ represents speed and $t$ represents time, then $s \times t = k$, where $k$ is a constant. If speed doubles ($s_2 = 2s_1$), then time must halve ($t_2 = \frac{t_1}{2}$) to maintain the product constant ($s_1 t_1 = s_2 t_2$).

The reason states that speed and time are inversely proportional to each other when the distance is constant. This is also true because, for a fixed distance, increasing speed decreases time proportionally and vice versa, maintaining the inverse relationship.

Both the assertion and the reason are true, and the reason provides the correct explanation for the assertion.

Key Concept: Preparing tea

b) 10 teaspoons

[Solution Description]

To prepare tea for 15 people, where each person drinks 1 cup, you need a total of $15$ cups of tea. The recipe requires $2$ teaspoons of tea leaves for every $3$ cups of water. To find the number of teaspoons needed for $15$ cups, we use the proportion:

$\frac{2 \text{ teaspoons}}{3 \text{ cups}} = \frac{x \text{ teaspoons}}{15 \text{ cups}}$

Solving for $x$, we get:

$x = \frac{2 \times 15}{3} = 10 \text{ teaspoons}$

Therefore, $10$ teaspoons of tea leaves are needed.

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 $x$ is directly proportional to $y$, then doubling $x$ will always double $y$. This is true because direct proportionality means that $x$ and $y$ are related by a constant ratio, i.e., $x = ky$, where $k$ is the constant of proportionality. If $x$ is doubled, then $2x = ky'$, where $y'$ is the new value of $y$. Solving for $y'$, we get $y' = \frac{2x}{k} = 2y$. Thus, $y$ is doubled when $x$ is doubled.

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, 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

a) 3 hours

3 hours

Key Concept: Inverse Proportion

c) 40 workers

[Solution Description]

We know that the number of workers and the time taken to build the wall are inversely proportional. Let $W_1 = 10$, $D_1 = 20$, and $D_2 = 5$. 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:

$10 \times 20 = W_2 \times 5$

Solving for $W_2$:

$200 = 5W_2$

$W_2 = \frac{200}{5} = 40$

Therefore, 40 workers are needed to build the wall in 5 days.

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: Inverse Proportion

c) 2 hours

[Solution Description]

We use the inverse proportion formula: $20 \times 3 = 30 \times t$. Solving for $t$, we get $t = \frac{20 \times 3}{30} = 2$ hours.

Key Concept: Direct Proportion, Missing Values

b) 4 days

[Solution Description]

Since the number of workers and the time taken to complete the task are inversely proportional, we use the formula $W_1 \times D_1 = W_2 \times D_2$. Here, $W_1 = 6$ workers, $D_1 = 10$ days, and $W_2 = 15$ workers. We need to find $D_2$. Rearranging the formula, $D_2 = \frac{W_1 \times D_1}{W_2} = \frac{6 \times 10}{15} = \frac{60}{15} = 4$ days.

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

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) 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

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, 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: 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 is doubled, the time taken to cover a fixed distance is halved. This is a direct consequence of the inverse proportionality relationship between speed and time, which is mathematically represented as $speed \times time = constant$. When speed is doubled, the time must be halved to keep the product constant. This confirms that the assertion is true.

The reason states that speed and time are inversely proportional quantities. This is also true because, according to the definition of inverse proportion, if two quantities are inversely proportional, an increase in one leads to a decrease in the other, and vice versa, while their product remains constant. In this case, speed and time satisfy this condition, making the reason true as well.

Furthermore, the reason correctly explains why the assertion is true because it provides the fundamental principle of inverse proportionality that underlies the assertion's statement.

Key Concept: Simple Interest

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

[Solution Description]

The assertion states that depositing more money leads to more interest earned. This is true because interest is calculated based on the principal amount. The reason explains that interest is directly proportional to the principal amount, which is also true. Moreover, the reason correctly explains why the assertion is true because it establishes the direct proportionality between the principal and the interest earned.

Key Concept: Inverse Proportion, Product Remains Constant

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

[Solution Description]

Let $d$ be the fixed distance, $s$ be the speed, and $t$ be the time taken to cover the distance $d$. We know that $d = s \times t$. If the speed is doubled ($2s$), then the new time $t'$ can be calculated as $d = 2s \times t'$. Solving for $t'$, we get $t' = \frac{d}{2s} = \frac{t}{2}$. This shows that the time taken is halved when the speed is doubled. The product of speed and time is $s \times t = d$, which is constant for a fixed distance. Therefore, both the Assertion and Reason are true, and the Reason correctly explains the Assertion.

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: 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: 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: 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: Inverse Proportion

b) 12 days

[Solution Description]

Let $W_1 = 15$ workers, $D_1 = 20$ days, and $W_2 = 25$ workers. We need to find $D_2$. 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,

$15 \times 20 = 25 \times D_极2$

Simplifying,

$300 = 25D_2$

Dividing both sides by 25,

$D_2 = \frac{300}{25} = 12$

So, it would take 12 days.

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]

From the given table in the syllabus, we observe that as time increases, the angle turned by the minute hand also increases. For example, when the time passed is 15 minutes, the angle turned is 90 degrees. Similarly, when the time passed is 30 minutes, the angle turned is 180 degrees. This shows that the angle turned and the time passed are directly proportional. The ratio of time passed to the angle turned can be calculated as follows: For 15 minutes and 90 degrees, the ratio is 15/90 = 1/6. For 30 minutes and 180 degrees, the ratio is also 30/180 = 1/6. This constant ratio confirms that the assertion is true. The reason states that the ratio of time passed to the angle turned remains constant, which is also true based on the calculations. Moreover, the reason correctly explains why the angle turned and time passed are directly proportional.

Key Concept: Work and Time

c) 16 days

[Solution Description]

The number of workers and the time taken are inversely proportional. 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 8 workers can build a wall in 12 days, we can find the number of workers needed to build the wall in 6 days as follows:

$8 \times 12 = W_2 \times 6$

Solving for $W_2$, we get:

$W_2 = \frac{8 \times 12}{6} = 16 \text{ workers}$

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, Proportional Relationships

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

[Solution Description]

The problem involves direct proportion, where the cost of cloth increases with its length in the same ratio. Given that 5 meters of cloth costs \$210, we can find the cost for 13 meters using the proportion formula:

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

Substituting the given values:

$\frac{5}{13} = \frac{210}{y_2}$

Solving for $y_2$:

$y_2 = \frac{13 \times 210}{5} = 546$

Therefore, the cost of 13 meters of cloth is indeed \$546. Both the assertion and the reason are true, and the reason correctly explains the assertion.

Key Concept: Direct Proportion

b) \$360

[Solution Description]

Let the cost of 12 metres of cloth be y. Since the cost is directly proportional to the length of the cloth, we can write:

$\frac{8}{12} = \frac{240}{y}$

Solving for y:

$8y = 12 \times 240$

$8y = 2880$

$y = \frac{2880}{8} = 360$

Therefore, the cost of 12 metres of cloth is \$360.

Key Concept: Direct Proportion

b) \$45

[Solution Description]

Let the cost be c and the number of notebooks be n. Since they are directly proportional, $\frac{c1}{n1} = \frac{c2}{n2}$. Here, c1 = \$25, n1 = 5, n2 = 9. We need to find c2. Substituting the values, $\frac{25}{5} = \frac{c2}{9}$. Simplifying, $5 = \frac{c2}{9}$. Multiplying both sides by 9, we get $c2 = 45$. Therefore, the cost of 9 notebooks is \$45.

Key Concept: Inverse Proportion

b) 12

8

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

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

[Solution Description]

According to the concept of direct proportion, when two quantities are in direct proportion, their ratio remains constant. Here, the cost of sugar and its weight are in direct proportion, so the ratio of cost to weight remains constant. Therefore, both the assertion and reason are true, and the reason correctly explains the assertion.

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

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: Basic Concept

c) \$50

[Solution Description] To find the interest earned, use the formula: $Interest = Principal \times Rate \times Time$. Here, the principal is \$500, the rate is 10\%, and the time is 1 year. First, convert the percentage rate to a decimal: $10\% = 0.10$. Now, multiply the principal by the rate and time: $Interest = 500 \times 0.10 \times 1 = \$50$.

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: Inverse Proportion

b) 4

[Solution Description]

Since $p$ and $q$ are inversely proportional, the product $pq$ remains constant. Given $p_1 = 12$, $q_1 = 2$, we have $p_1 q_1 = 12 \times 2 = 24$. Let $q_2 = 6$, then $p_2 q_2 = 24$. Solving for $p_2$, we get $p_2 = \frac{24}{6} = 4$. Therefore, the value of $p$ when $q = 6$ is $4$.

Key Concept: Direct Proportion

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

[Solution Description]

Let the length of cloth be $x$ and its cost be $y$. Given that the cost of 5 metres of cloth is \$210, we can write $\frac{x_1}{y_1} = \frac{x_2}{y_2}$. Here, $x_1 = 5$, $y_1 = 210$, and $x_2 = 10$. Substituting these values, we get:

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

Solving for $y_2$:

$5y_2 = 210 \times 10$

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

Therefore, the cost of 10 metres of cloth is \$420, which confirms the assertion. The reason is also true because the cost of cloth increases in direct proportion to its length.

Key Concept: Inverse Proportion

b) 60 books

[Solution Description]

First, calculate the number of books that can be bought at \$60 each. The total amount is \$4800. Using the inverse proportion formula $xy = k$, where $x$ is the price and $y$ is the number of books, we have $60 \times y = 4800$. Solving for $y$, we get $y = \frac{4800}{60} = 80$ books.

Next, calculate the number of books that can be bought at \$80 each. Using the same formula, $80 \times y = 4800$, solving for $y$, we get $y = \frac{4800}{80} = 60$ books.

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, 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: Direct Proportion

d) \$ 252

[Solution Description]

Given, the cost of 3 kg sugar is \$ 108. 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 3 kg, $y_1$ is \$ 108, and $x_2$ is 7 kg. We need to find $y_2$. Plugging in the values: $\frac{3}{7} = \frac{108}{y_2}$. Solving for $y_2$, we get $y_2 = \frac{108 \times 7}{3} = 252$. So, the cost of 7 kg of sugar is \$ 252.

Key Concept: Inverse Proportion Definition

b) Decreases by factor of 3

[Solution Description]

If $x$ and $y$ are in inverse proportion, then $xy = k$. If $x$ increases by a factor of 3, it becomes $3x$. Let the new value of $y$ be $y'$. Then:

$3x \times y' = k$

But $k = xy$, so:

$3x \times y' = xy$

Dividing both sides by $3x$ gives:

$y' = \frac{y}{3}$.

Thus, $y$ decreases by a factor of 3.

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, 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, Clock Hand Angle

b) 120 degrees

[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 moves for 20 minutes, the total angle turned is $20 \times 6 = 120$ degrees.

Key Concept: Inverse Proportion

b) 18

[Solution Description]

Let the number of workers needed to complete the task in 24 hours be $y$. Since the number of workers and time taken vary in inverse proportion, we have:

$12 \times 36 = 24 \times y$

Simplifying, we get:

$y = \frac{12 \times 36}{24}$

Performing the calculations:

$y = \frac{432}{24} = 18$

Therefore, 18 workers are needed to complete the task in 24 hours.

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

c) Time decreases to $1/5$

[Solution Description]

As per the concept of inverse proportion, when speed increases, time taken decreases. Here, speed increases to 5 times. Therefore, time taken will decrease to $1/5$ of the original time.

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: Inverse Proportion

b) 3

[Solution Description]

Since $x$ and $y$ are inversely proportional, $x \times y = k$, where $k$ is a constant. Given $x = 4$ and $y = 6$, we can find $k$ as follows:

$k = x \times y = 4 \times 6 = 24$

Now, to find $y$ when $x = 8$, we use the same equation:

$8 \times y = 24 \implies y = \frac{24}{8} = 3$

So, the value of $y$ is 3.

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: Direct Proportion

c) \$32

[Solution Description]

The cost of 5 notebooks is \$20. Let the cost of 8 notebooks be $x$. Using direct proportion, we have:

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

Solving for $x$:

$5x = 160 \quad \Rightarrow \quad x = 32$

So, the cost of 8 notebooks is \$32.

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: 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: Direct Proportion

b) 200 km

[Solution Description]

Let the distance be d and the amount of petrol be p. Since they are directly proportional, $\frac{d1}{p1} = \frac{d2}{p2}$. Here, d1 = 120 km, p1 = 15 liters, p2 = 25 liters. We need to find d2. Substituting the values, $\frac{120}{15} = \frac{d2}{25}$. Simplifying, $8 = \frac{d2}{25}$. Multiplying both sides by 25, we get $d2 = 200$. Therefore, the car can travel 200 km on 25 liters of petrol.

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: 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: 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 car increases, the time taken to cover a fixed distance decreases. This is true because speed and time are inversely proportional when the distance is constant. As speed increases, time decreases, and vice versa, maintaining the product $speed \times time = distance$ constant. The reason correctly explains why this relationship holds, as it directly links the inverse proportionality of speed and time under constant distance. Hence, both the assertion and reason are true, and the reason is the correct explanation of the assertion.

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 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: Inverse Proportion, Speed and Time

c) 2.67 hours

[Solution Description]

Let the initial speed be $S_1 = 60$ km/h and the time taken be $T_1 = 4$ hours. The speed is increased by 50\%, so the new speed is $S_2 = 1.5 \times S_1 = 1.5 \times 60 = 90$ km/h. Since speed and time are inversely proportional, we have:

$S_1 \times T_1 = S_2 \times T_2$

Substituting the known values:

$60 \times 4 = 90 \times T_2$

Solving for $T_2$:

$240 = 90 \times T_2 \implies T_2 = \frac{240}{90} = \frac{8}{3} \approx 2.67 \text{ hours}$

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: Proportional Relationships, Multi-step Solutions

b) 4 hours

[Solution Description] First, calculate the original speed of the car. Speed = Distance / Time = 240 miles / 4 hours = 60 mph. The speed is increased by 50\%, so the new speed is 60 mph + (0.5 * 60 mph) = 90 mph. Now, calculate the time required to travel 360 miles at this new speed: Time = Distance / Speed = 360 miles / 90 mph = 4 hours.

Key Concept: Direct Proportion

b) \$16

[Solution Description]

First, find the cost of one pen by dividing the total cost by the number of pens: \$10 \/ 5 = \$2 per pen. Now, multiply the cost of one pen by the number of pens to find the total cost for 8 pens: \$2 * 8 = \$16.

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

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

[Solution Description]

To verify the assertion, we start with the given information: 20 workers can build a wall in 60 hours. We need to find out if 30 workers can build the same wall in 40 hours. According to inverse proportion, the product of the number of workers and the time taken should remain constant. So, we calculate the constant for the initial scenario:

$\text{Constant} = 20 \text{ workers} \times 60 \text{ hours} = 1200 \text{ worker-hours}$

Now, we check the second scenario with 30 workers working for 40 hours:

$30 \text{ workers} \times 40 \text{ hours} = 1200 \text{ worker-hours}$

Since both scenarios yield the same constant, the assertion is true. The reason is also true because the number of workers and the time taken are indeed inversely proportional. Moreover, the reason can explain why the assertion is correct, as it directly relates to the concept of inverse proportion.

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: Direct Proportion

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

[Solution Description]

Direct proportion implies that as $x$ increases, $y$ also increases in such a way that the ratio $\frac{x}{y}$ remains constant. This is the definition of direct proportion, and the reason provided is a direct restatement of this condition. Therefore, both the assertion and the reason are true, and the reason correctly explains the assertion.

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: 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: Direct Proportion

c) \$32

[Solution Description]

Let the cost of 8 pens be $x$. Since the cost is directly proportional to the number of pens, we can write $\frac{5}{20} = \frac{8}{x}$. Cross-multiplying gives $5x = 160$. Dividing both sides by 5 gives $x = 32$. Therefore, the cost of 8 pens is \$32.

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: 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: 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: Direct Proportion

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

[Solution Description]

Key Concept: Direct Proportion

b) 21

[Solution Description]

Since $y$ is directly proportional to $x$, we can write $y = kx$, where $k$ is the constant of proportionality. Given that when $x = 4$, $y = 12$, we can find $k$ as follows:

$k = \frac{y}{x} = \frac{12}{4} = 3$

Now, to find the value of $y$ when $x = 7$, substitute $k = 3$ into the equation:

$y = 3 \times 7 = 21$

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) 4 hours

[Solution Description]

The initial speed of the train is 60 km/h (since 240 km / 4 hours = 60 km/h). We use the inverse proportion formula: $60 \times 4 = 60 \times t$. Solving for $t$, we get $t = \frac{60 \times 4}{60} = 4$ hours.

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: Proportional Relationship

d) \$36

[Solution Description]

First, find the cost of one apple by dividing the total cost by the number of apples. So, $24 \div 12 = \$2$ per apple. Now, multiply the cost of one apple by 18 to find the cost of 18 apples. Therefore, $2 \times 18 = \$36$.

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, Ratio

b) 30 litres

[Solution Description]

Let the amount of petrol required to travel 360 km be $x$ litres. Since distance and petrol consumption are in direct proportion, we have:

$\frac{240}{20} = \frac{360}{x}$

Simplifying this, we get:

$12 = \frac{360}{x}$

Cross-multiplying:

$12x = 360$

Solving for $x$:

$x = \frac{360}{12} = 30$

Therefore, 30 litres of petrol will be required to travel 360 km.

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: Direct Proportion

b) \$60

[Solution Description]

Cost is directly proportional to the number of pens. First, find the cost of one pen: $\frac{36}{12} = 3$ dollars per pen. Then, calculate the cost of 20 pens: $20 \times 3 = 60$ dollars. Therefore, 20 pens will cost 60 dollars.

Key Concept: Direct Proportion Constant

b) 0.5

[Solution Description]

The constant of proportionality k is calculated using the formula $\frac{x}{y} = k$. Substituting the given values: $\frac{10}{20} = k$. Simplifying the fraction, $k = 0.5$.

Key Concept: Inverse proportion, Multiplicative inverse

c) 4.5 hours

[Solution Description]

First, let's understand the relationship between speed and time. Speed and time are inversely proportional, meaning that if speed increases, time decreases proportionally, and vice versa. The product of speed and time remains constant. Mathematically, $S_1 \times T_1 = S_2 \times T_2$, where $S_1$ and $T_1$ are initial speed and time, and $S_2$ and $T_2$ are new speed and time.

Given:

$S_1 = 60$ km/h

$T_1 = 6$ hours

$S_2 = 80$ km/h

We need to find $T_2$.

Using the formula:

$60 \times 6 = 80 \times T_2$

Solving for $T_2$:

$360 = 80 \times T_2$

$T_2 = \frac{360}{80} = 4.5 \text{ hours}$

Therefore, the new time taken is 4.5 hours.

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

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: 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: Interest Comparison

b) Person B

[Solution Description]

Calculate the interest for Person A: $I_A = 1000 \times 0.04 \times 1 = \$40$.

Calculate the interest for Person B: $I_B = 2000 \times 0.03 \times 1 = \$60$.

Comparing the two interests, $\$60 > \$40$. Therefore, Person B earns more interest after one year.

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: 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 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

b) 7 kg

[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 3 kg, the cost is \$54: $3k = 54$.

Step 3: Solve for k: $k = \frac{54}{3} = 18$.

Step 4: Now, calculate the amount of sugar that can be bought for \$126: $\frac{126}{18} = 7$ kg.

The amount of sugar that can be bought for \$126 is 7 kg.

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

c) $xy = k$

[Solution Description]

According to the definition of inverse proportion, if x and y are in inverse proportion, then $xy = k$, where k is a constant. This means that as x increases, y decreases proportionally to keep the product xy constant.

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 a vehicle's speed increases, the time taken to travel a fixed distance decreases. This is true because speed and travel time are inversely proportional, meaning as one increases, the other decreases. The reason correctly explains the assertion by stating that speed and travel time are inversely proportional. Therefore, both the assertion and reason are true, and the reason correctly explains the assertion.

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

c) \$14

[Solution Description]

First, find the cost of one kilogram of apples by dividing the total cost by the quantity: \$8 \/ 4 = \$2 per kg. Now, multiply the cost of one kilogram by the quantity to find the total cost for 7 kg: \$2 * 7 = \$14.

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: 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) 2 hours

[Solution Description]

The original speed of the vehicle can be calculated as:

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

If the speed is doubled, the new speed becomes:

$60 \times 2 = 120 \text{ km/h}$

Since speed and time are inversely proportional, the time taken to cover the same distance at the new speed will be:

$\text{Time} = \frac{\text{Distance}}{\text{Speed}} = \frac{240}{120} = 2 \text{ hours}$

So, the time taken will be 2 hours.

Key Concept: Inverse Proportion

c) 9

[Solution Description]

Given that $y$ is inversely proportional to $x$, we have $y = \frac{k}{x}$. First, find the constant of proportionality $k$. Substituting $y = 12$ and $x = 3$, we get:

$12 = \frac{k}{3} \implies k = 12 \times 3 = 36.$

Now, use this $k$ to find $y$ when $x = 4$:

$y = \frac{36}{4} = 9.$

Therefore, the value of $y$ when $x = 4$ is 9.

Key Concept: Direct Proportion, Real-life Application

c) 7.5 hours

[Solution Description]

Let the time taken be $x$ hours. Since distance and time are in direct proportion when speed is constant, we can use the proportion $\frac{240}{4} = \frac{450}{x}$. Cross-multiplying gives $240x = 4 \times 450$. Solving for $x$, we get $x = \frac{4 \times 450}{240} = 7.5$ hours.

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: Direct Proportion

b) 20 liters

[Solution Description]

Let the petrol consumed be $x$ liters for 300 km.

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

$\frac{10}{150} = \frac{x}{300}$

Simplifying, $\frac{1}{15} = \frac{x}{300}$

Cross-multiplying, $x = \frac{300}{15}$

Therefore, $x = 20$ liters.

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, Price and Quantity

d) Assertion is false, but Reason is true.

[Solution Description]

Let the initial price of the book be \$100 and the initial number of books that can be purchased be 10. After a 20\% increase, the new price becomes \$120. The number of books that can be purchased now is calculated by dividing the fixed budget by the new price. Since the budget is \$1000 (100 * 10), the new number of books is $1000 / 120 = 8.33$. This means the number of books decreases by approximately 16.67\%, not 20\%. Therefore, the Assertion is false. However, the Reason is true because price and quantity are inversely proportional.

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, Ratio

b) 147 newtons

[Solution Description]

Let the weight of the 15 kg object be $W$ newtons. Since weight and mass are in direct proportion, we have:

$\frac{98}{10} = \frac{W}{15}$

Simplifying this, we get:

$9.8 = \frac{W}{15}$

Cross-multiplying:

$9.8 \times 15 = W$

Calculating $W$:

$W = 147$

Therefore, the weight of the 15 kg object is 147 newtons.

Key Concept: Direct Proportion Definition

c) Their ratio remains constant

[Solution Description]

Two quantities x and y are said to be in direct proportion if they increase (decrease) together in such a manner that the ratio of their corresponding values remains constant. That is, if $\frac{x}{y} = k$ (where k is a positive number), then x and y are said to vary directly.

Key Concept: Direct Proportion

c) 20

[Solution Description]

Let the number of workers needed be $x$. Since the number of workers is inversely proportional to the number of days, we have:

$12 \times 10 = x \times 6$

Solving for $x$:

$120 = 6x \quad \Rightarrow \quad x = 20$

So, 20 workers are needed to complete the task in 6 days.

Key Concept: Concept 3

b) 3.0 seconds

[Solution Description]

The time taken for an object to fall from a height can be calculated using the equation:

Height = $\frac{1}{2} g t^2$

Rearranging to solve for time $t$:

$t = \sqrt{\frac{2 \times \text{Height}}{g}}$

Substituting the values:

$t = \sqrt{\frac{2 \times 45 \, \text{m}}{9.8 \, \text{m/s}^2}}$

$t = \sqrt{\frac{90}{9.8}}$

$t = \sqrt{9.18}$

$t \approx 3.03$ seconds

Key Concept: Inverse Proportion, Multi-step Solution

c) 3 hours

[Solution Description]

First, find the time taken at 60 km/h:

$\text{Time}_1 = \frac{\text{Distance}}{\text{Speed}_1} = \frac{240}{60} = 4 \text{ hours}$

Since speed and time are inversely proportional:

$\text{Speed}_1 \times \text{Time}_1 = \text{Speed}_2 \times \text{Time}_2 \Rightarrow 60 \times 4 = 80 \times \text{Time}_2$

Solve for $\text{Time}_2$:

$\text{Time}_2 = \frac{60 \times 4}{80} = 3 \text{ hours}$

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: Direct Proportion

b) 360 km

[Solution Description]

Let the distance be $x$ km for 30 liters of petrol.

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

$\frac{240}{20} = \frac{x}{30}$

Simplifying, $12 = \frac{x}{30}$

Cross-multiplying, $x = 12 \times 30$

Therefore, $x = 360$ km.

Key Concept: Inverse Proportion

c) 2.5 days

[Solution Description]

Let the number of machines be $m$ and the number of days be $d$. Since the number of machines and the number of days are inversely proportional, we have $m_1 \times d_1 = m_2 \times d_2$. Given $m_1 = 8$, $d_1 = 5$, and $m_2 = 16$, we can find $d_2$ as follows:

$8 \times 5 = 16 \times d_2$

$40 = 16 \times d_2$

$d_2 = \frac{40}{16} = 2.5$

So, it will take 2.5 days for 16 machines to produce 1000 units.

Key Concept: Direct Proportion, Ratio

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 direct proportion, then the ratio of their corresponding values remains constant. This is true because, by definition, when $x$ and $y$ are in direct proportion, their ratio $\frac{x}{y}$ equals a positive constant $k$. The reason also states that $\frac{x}{y} = k$, which is the correct explanation for the assertion. Therefore, both the 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]

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: 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: Direct Proportion

c) 96 kilometers

[Solution Description]

To solve this, we use the concept of direct proportion. The distance traveled is directly proportional to the amount of petrol consumed. Let $x$ be the distance traveled with 8 liters of petrol. We can set up the proportion as follows: $\frac{5}{60} = \frac{8}{x}$. Cross-multiplying gives $5x = 480$. Dividing both sides by 5, we get $x = 96$ kilometers.

Key Concept: Direct Proportion

c) 120 degrees

[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. In 20 minutes, the minute hand will turn $6 \times 20 = 120$ degrees.

Key Concept: Direct Proportion

b) \Rs. 420

[Solution Description]

Let the cost of 10 metres of cloth be $y$. Since the cost of cloth is directly proportional to its length, we can write:

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

Cross-multiplying gives:

$5y = 210 \times 10$

Solving for $y$:

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

Therefore, the cost of 10 metres of cloth is \Rs. 420.

Key Concept: Direct Proportion

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

[Solution Description]

The cost of 5 pens is \$10. To find the cost of 10 pens, we use the concept of direct proportion. Since the cost increases proportionally with the number of pens, the cost of 10 pens will be twice the cost of 5 pens. Therefore, the cost of 10 pens is \$20. Both the Assertion and Reason are true, and the Reason correctly explains the Assertion.

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: Direct 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 articles purchased increases, the total cost also increases. This is true because if the cost per article remains constant, the total cost will increase in direct proportion to the number of articles purchased. The reason explains that the cost of each article remains constant, which is why the total cost increases proportionally with the number of articles. Therefore, both the assertion and the 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: Work and Time, Efficiency

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

[Solution Description]

The assertion states that increasing the number of workers by a factor of 3 reduces the time taken by a factor of 3. This is based on the assumption that the work done is directly proportional to the number of workers and inversely proportional to the time taken. However, this is only true if all workers are equally efficient and there are no other constraints like coordination or resource allocation. The reason states that efficiency is inversely proportional to the time taken, which is generally true in the context of work and time problems. However, the reason does not correctly explain the assertion because it does not account for the specific relationship between the number of workers and the time taken. Therefore, both the assertion and the reason are true, but the reason is not the correct explanation of the assertion.

Key Concept: Direct Proportion

c) \$72

[Solution Description]

Let the cost of 8 books be $C$. Since the cost is directly proportional to the number of books, we can set up the proportion:

$\frac{5}{45} = \frac{8}{C}$

Cross-multiplying gives:

$5C = 45 \times 8$

Solving for $C$:

$C = \frac{45 \times 8}{5} = \frac{360}{5} = 72$

So, the cost of 8 books is \$72.

Key Concept: Inverse Proportion, Identifying Inverse Proportion

d) 40

[Solution Description]

Given the relationships $x \propto \frac{1}{y}$ and $y \propto \frac{1}{z}$, we can combine these to get $x \propto z$. Let the constant of proportionality be k, so $x = kz$. When x = 10 and z = 5, substitute these values into the equation to find k: 10 = k * 5. Solving for k gives k = 2. Now, to find x when z = 20, use the equation x = 2 * 20. Therefore, x = 40.