Key Concept: Ratio and Percentage

b) 24

[Solution Description]

The ratio of apples to oranges is given as $5 : 3$. Let the number of oranges be $x$. According to the ratio:

$\frac{5}{3} = \frac{40}{x}$

Solving for $x$:

$5x = 120 \implies x = 24$

Therefore, the number of oranges is 24.

Key Concept: Ratios, Percentages

b) 60 : 1, 40%

[Solution Description]

To find the ratio and percentage:

1. Distance traveled by the train = 240 km.

2. Time taken = 4 hours.

3. Ratio of distance to time = $240 : 4$.

4. Simplifying the ratio by dividing both numbers by 4: $\frac{240}{4} : \frac{4}{4} = 60 : 1$.

5. To express the ratio as a percentage of the total distance of 600 km:

Percentage = $\frac{240}{600} \times 100 = 40\%$.

Key Concept: Ratios, Percentages

c) 27

[Solution Description]

The ratio of boys to girls is given as 3:2, which means for every 3 boys, there are 2 girls. Given that there are 18 girls, we can set up the proportion:

Let the number of boys be $x$. According to the ratio:

$\frac{3}{2} = \frac{x}{18}$

To find $x$, cross-multiply:

$3 \times 18 = 2 \times x$

$54 = 2x$

$x = \frac{54}{2} = 27$

Therefore, the number of boys in the class is 27.

Key Concept: Ratio and Percentage

c) 1 : 2

[Solution Description]

The total number of students is 24. Number of girls is 16, so number of boys = 24 - 16 = 8. Therefore, the ratio of boys to girls is $8 : 16$. Simplifying this ratio by dividing both numbers by 8, we get $\frac{8}{16} = \frac{1}{2}$. So the ratio is $1 : 2$.

Key Concept: Ratio to Percentage Conversion

d) 60%

[Solution Description]

To convert the ratio 3:5 into a percentage, we first express it as a fraction: $\frac{3}{5}$. Then, multiply by 100 to get the percentage: $\frac{3}{5} \times 100 = 60\%$.

Key Concept: Ratios, Percentages

a) 58.33%

[Solution Description]

Let the initial quantity of milk be $7x$ liters and water be $3x$ liters. After adding 20 liters of water, the amount of water becomes $3x + 20$. The new ratio is given as 7 : 5, so we have $7x / (3x + 20) = 7 / 5$. Solving this equation: $35x = 21x + 140$, which simplifies to $14x = 140$ and gives $x = 10$.

The final quantity of milk is $7x = 70$ liters, and the final quantity of water is $3x + 20 = 50$ liters. The total mixture is $70 + 50 = 120$ liters. The percentage of milk is $(70 / 120) \times 100 = 58.33\%$.

Key Concept: Finding Percentage

c) 80%

[Solution Description]

To find the percentage of students present, use the formula:

$\text{Percentage} = \left(\frac{\text{Number of students present}}{\text{Total number of students}}\right) \times 100$

Substitute the given values:

$\text{Percentage} = \left(\frac{24}{30}\right) \times 100 = 80\%$

Therefore, 80% of the class is present.

Key Concept: Ratio to Percentage Conversion

b) 37.5%

[Solution Description]

To convert the ratio 3 : 5 into a percentage, first add the parts of the ratio: $3 + 5 = 8$. Now, find the fraction of the first part: $\frac{3}{8}$. Then, multiply this fraction by 100 to get the percentage: $\frac{3}{8} \times 100 = 37.5\%$.

Key Concept: Sales Tax/Value Added Tax (VAT

d) Assertion is false, but Reason is true.

[Solution Description] To find the bill amount, calculate the GST on the original price. Let the original price be $Rs. x$. Given the selling price is $Rs. 500$, we can write: $Rs. x + 12\% \text{ of } Rs. x = Rs. 500$. Simplify this equation to find the original price. $1.12x = Rs. 500$ implies $x = Rs. \frac{500}{1.12} \approx Rs. 446.43$. Now, add the GST to the original price: $\$Rs. 446.43 + 12\% \text{ of } Rs. 446.43 = Rs. 446.43 + Rs. 53.57 = Rs. 500$. Therefore, the assertion that the total bill amount will be Rs. 560 is incorrect. However, the reason correctly states how GST is calculated.

Key Concept: Percentage of Girls

b) 40%

[Solution Description]

The total number of students is 20 and the number of girls is 8. To find the percentage of girls in the class, we use the formula:

$\text{Percentage of girls} = \left(\frac{8}{20}\right) \times 100 = 40\%$

Therefore, the percentage of girls in the class is 40%.

Key Concept: Finding Transportation Charge

b) 1200

[Solution Description]

To find the transportation charge, we need to calculate the total distance traveled both ways and multiply it by the rate per km.

The distance one way is 60 km, so the distance both ways is $60 \times 2 = 120$ km.

The transportation charge is given by:

$\text{Transportation Charge} = \text{Distance both ways} \times \text{Rate per km} = 120 \times 10 = 1200$

Therefore, the total transportation charge is \$1200.

Key Concept: Transportation charge, Total expenses, Cost per person

b) 178.95

[Solution Description]

First, calculate the transportation charge:

Transportation charge = Distance both ways × Rate

$= (60 \times 2) \times 15$

$= 120 \times 15 = 1800$

Next, calculate the total expenses:

Total expenses = Refreshment charge + Transportation charge

$= 5000 + 1800$

$= 6800$

Then, determine the total number of persons:

Total number of persons = Number of girls + Number of boys + Number of teachers

$= 20 + 15 + 3$

$= 38$

Finally, calculate the cost per person using the unitary method:

For 38 persons, the amount spent is $6800$.

The amount spent for 1 person = $\frac{6800}{38} = 178.95$.

Thus, the cost per person for the school trip is approximately \$178.95.

Key Concept: Finding the percentage of a journey remaining

c) 70%

[Solution Description]

To find the percentage of the journey remaining, we first calculate the percentage of the journey completed using the formula: $\frac{\text{Distance Covered}}{\text{Total Distance}} \times 100$. Here, the distance covered is 45 km and the total distance is 150 km. Substituting these values into the formula, we get: $\frac{45}{150} \times 100 = 30\%$. The percentage of the journey remaining is then calculated as $100\% - 30\% = 70\%$.

Key Concept: Percentage of journey completed, Percentage of journey remaining

b) 70% completed, 30% remaining

[Solution Description]

First, calculate the percentage of the journey completed. The formula for percentage completed is:

$\text{Percentage Completed} = \left(\frac{\text{Distance Completed}}{\text{Total Distance}}\right) \times 100$

Substituting the given values:

$\text{Percentage Completed} = \left(\frac{350}{500}\right) \times 100 = 70\%$

Next, calculate the percentage of the journey remaining. Since the total journey is 100%, the remaining percentage is:

$\text{Percentage Remaining} = 100\% - 70\% = 30\%$

Therefore, the train has completed 70% of the journey and 30% remains.

Key Concept: Sale Price Calculation

b) Rs. 200

[Solution Description]

First, find the discount amount using the formula:

$Discount = Discount \% \text{ of } Marked Price$

Given:

$Marked Price = Rs. 250, \quad Discount \% = 20\%$

Substituting the values:

$Discount = \frac{20}{100} \times 250 = Rs. 50$

Next, find the sale price by subtracting the discount from the marked price:

$Sale Price = Marked Price - Discount$

Substituting the values:

$Sale Price = 250 - 50 = Rs. 200$

The sale price of the book is Rs. 200.

Key Concept: Finding Discounts

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

[Solution Description]

The assertion states that the discount on an article can be calculated using the formula $Discount = Marked Price - Sale Price$. This is correct as per the syllabus, which clearly mentions that discount is the difference between the marked price and the sale price.

The reason states that discount is a reduction given on the marked price to attract customers. This is also correct as per the syllabus, which explains that discounts are generally given to promote sales or attract customers.

Moreover, the reason correctly explains why the assertion is true: the formula for discount is directly derived from the concept of reducing the marked price to offer a discount. Therefore, both the assertion and reason are true, and the reason is the correct explanation of the assertion.

Key Concept: Understanding Discounts

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

[Solution Description]

The assertion states that the discount on an item is calculated by subtracting its sale price from its marked price. This is true according to the syllabus, as $Discount = Marked\ Price – Sale\ Price$. The reason states that discount is a reduction given on the marked price to attract customers or promote sales. This is also true and explains why discounts are given. Therefore, both the assertion and reason are true, and the reason correctly explains the assertion.

Key Concept: Sale Price Calculation

b) Rs. 704

[Solution Description]

First, calculate the discount amount using the formula $Discount = Discount \% \text{ of Marked Price}$.

Given, Discount % = 12% and Marked Price ($MP$) = Rs. 800.

So, $Discount = \frac{12}{100} \times 800 = Rs. 96$.

Now, calculate the sale price using the formula $Sale Price = Marked Price – Discount$.

Substituting the values:

$Sale Price = Rs. 800 - Rs. 96 = Rs. 704$.

Thus, the sale price of the shirt is Rs. 704.

Key Concept: Discount Calculation, Sale Price

c) \$300

[Solution Description]

To find the sale price after a discount, we first calculate the discount amount using the formula:

$Discount = \frac{Discount %}{100} \times Marked Price$.

Here, the Discount % is 25% and the Marked Price is \$400. So:

$Discount = \frac{25}{100} \times 400 = \$100$.

The sale price is then calculated by subtracting the discount from the marked price:

$Sale Price = Marked Price - Discount$.

Substituting the values, we get:

$Sale Price = 400 - 100 = \$300$.

Therefore, the sale price after the discount is \$300.

Key Concept: Sale Price Calculation

b) Rs. 1200

[Solution Description]

First, calculate the discount amount using the formula: $Discount = \left( \frac{Discount \%}{100} \right) \times Marked Price$.

Substituting the given values: $Discount = \left( \frac{20}{100} \right) \times Rs. 1500 = Rs. 300$.

Next, calculate the sale price using the formula: $Sale Price = Marked Price – Discount$.

Substituting the values: $Sale Price = Rs. 1500 – Rs. 300 = Rs. 1200$.

Therefore, the sale price is Rs. 1200.

Key Concept: Discount Amount

c) Rs. 60

[Solution Description]

To calculate the discount amount, use the formula:

$Discount = Discount\ \% \times Marked\ Price$

Given:

Marked Price (MP) = Rs. 500

Discount Percentage = 12%

Discount = $\frac{12}{100} \times 500$

$Discount = 0.12 \times 500 = Rs. 60$

Key Concept: Discount Calculation

b) \$50

[Solution Description]

To find the discount, we use the formula:

$\text{Discount} = \text{Marked Price} - \text{Sale Price}$

Given that the Marked Price (MP) is \$500 and the Sale Price (SP) is \$450, we can calculate the discount as follows:

$\text{Discount} = 500 - 450 = 50$

Therefore, the discount offered on the item is \$50.

Key Concept: Finding Discount Percentage

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

[Solution Description]

To find the discount percentage, we use the formula:

$Discount\ \% = \frac{Marked\ Price - Sale\ Price}{Marked\ Price} \times 100$

Given: Marked Price = Rs. 500, Sale Price = Rs. 400.

Step 1: Calculate the discount amount:

$Discount = Marked\ Price - Sale\ Price = 500 - 400 = Rs.\ 100$

Step 2: Calculate the discount percentage:

$Discount\ \% = \frac{100}{500} \times 100 = 20\%$

The assertion is true as the calculation correctly yields a 20% discount. The reason is also true because the discount percentage is indeed calculated on the marked price. Moreover, the reason correctly explains why the marked price is used in the formula.

Hence, both the assertion and reason are true, and the reason is the correct explanation of the assertion.

Key Concept: Discount, Sale Price

b) Rs. 3,000

[Solution Description]

To find the sale price, first calculate the discount amount using the formula: $\text{Discount} = \text{Discount %} \times \text{Marked Price}$. Substituting the given values, we get: $\text{Discount} = 0.25 \times Rs. 4,000 = Rs. 1,000$. Now, subtract the discount from the marked price to get the sale price: $\text{Sale Price} = \text{Marked Price} - \text{Discount} = Rs. 4,000 - Rs. 1,000 = Rs. 3,000$. Thus, the sale price is Rs. 3,000.

Key Concept: Finding Discount Amount

b) Rs. 8,100

[Solution Description]

To find the discount amount, use the formula $Discount = Discount \% \text{ of Marked Price}$. Here, $Discount \% = 18\%$ and $Marked Price = Rs. 45,000$. Thus, $Discount = \frac{18}{100} \times 45,000 = Rs. 8,100$.

Key Concept: Sale Price Calculation

b) Rs. 600

[Solution Description]

First, calculate the discount amount using the formula:

Discount = Discount % $\times$ Marked Price

Discount = 25% $\times$ Rs. 800

Convert 25% to decimal form:

25% = 0.25

Now, calculate the discount:

Discount = 0.25 $\times$ 800 = Rs. 200

Next, find the sale price by subtracting the discount from the marked price:

Sale Price = Marked Price - Discount

Sale Price = Rs. 800 - Rs. 200 = Rs. 600

The sale price is Rs. 600.

Key Concept: Calculating discount when percentage is given

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

[Solution Description]

1. Given: Marked Price ($MP$) = Rs. 500, Discount % = 20%.

2. The formula to calculate the sale price when a discount % is given is: Sale Price = MP - (Discount % $\times$ MP).

3. Substituting the values: Sale Price = 500 - (0.20 $\times$ 500).

4. Calculating the discount amount: $0.20 \times 500 = 100$.

5. Then, Sale Price = 500 - 100 = 400.

6. Hence, the Assertion is true.

7. The Reason provides the correct formula to calculate the sale price when a discount % is given, which directly explains the Assertion.

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

Key Concept: Calculating Discount Amount

c) Rs. 150

[Solution Description]

The marked price (MP) of the watch is Rs. 1500, and the discount percentage is 10%. To find the discount amount, we calculate 10% of Rs. 1500.

Step 1: Calculate the discount amount.

$Discount = \frac{10}{100} \times 1500$

$Discount = 150$

Therefore, the discount amount is Rs. 150.

Key Concept: Discount Calculation

c) 20%

[Solution Description]

Given: Marked Price ($MP$) = \$500, Sale Price ($SP$) = \$400.

Discount = $MP$ - $SP$ = $500$ - $400$ = $100$.

Discount % = $\left(\frac{\text{Discount}}{\text{Marked Price}} \times 100\right)\%$

Discount % = $\left(\frac{100}{500} \times 100\right)\%$

Discount % = $20%$

Key Concept: Selling Price Calculation

a) Rs. 840[Solution Description]The marked price (MP) of the shoes is Rs. 1200. The discount percentage is 30%. To find the discount amount, we use the formula:

$\text{Discount} = \text{Discount %} \times \text{Marked Price}$

Substituting the values:

$\text{Discount} = \frac{30}{100} \times 1200 = Rs. 360$

Now, the selling price (SP) can be calculated by subtracting the discount from the marked price:

$\text{Selling Price} = \text{Marked Price} - \text{Discount} = 1200 - 360 = Rs. 840$

Key Concept: Sale Price Estimation

b) \$337.5

[Solution Description]

To find the sale price, first calculate the discount amount using the formula:

$\text{Discount} = \left( \frac{\text{Discount %}}{100} \right) \times \text{Marked Price}$

Substituting the given values:

$\text{Discount} = \left( \frac{25}{100} \right) \times 450 = 112.5$

Next, subtract the discount from the marked price to get the sale price:

$\text{Sale Price} = \text{Marked Price} - \text{Discount} = 450 - 112.5 = 337.5$

Key Concept: Estimation in percentages, Discount calculation

d) Assertion is false, but Reason is true.

[Solution Description]

First, round off the bill amount Rs. 577.80 to the nearest tens, which gives Rs. 580. Next, calculate 10% of Rs. 580: $\frac{10}{100} \times 580 = Rs. 58$. Since the discount is 20%, we calculate 10% again: $\frac{10}{100} \times 580 = Rs. 58$. Adding these two amounts gives Rs. 116 as the estimated discount. However, this method introduces an error because the actual 20% discount on Rs. 577.80 is $\frac{20}{100} \times 577.80 = Rs. 115.56$. The assertion claims that estimating 20% by calculating 10% twice is accurate, but this is not true due to the rounding error. The reason states that calculating 10% twice is mathematically equivalent to finding 20% directly, which is correct. Therefore, the assertion is false, but the reason is true.

Key Concept: Discount, Discount Percentage

c) 20%

[Solution Description]

First, calculate the discount using the formula: $\text{Discount} = \text{Marked Price} - \text{Sale Price}$.

Here, Marked Price (\$1,200) - Sale Price (\$960) = \$240.

Now, to find the discount percentage, use the formula: $\text{Discount %} = \left(\frac{\text{Discount}}{\text{Marked Price}}\right) \times 100\%$.

Plugging in the values: $\text{Discount %} = \left(\frac{240}{1200}\right) \times 100\% = 20\%$.

Key Concept: Finding Discount Percentage

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

[Solution Description]

The assertion states that the discount percentage can be calculated using the formula $Discount\ \% = \left(\frac{Discount}{Marked\ Price}\right) \times 100$, which is correct as per the syllabus. The reason states that the discount is always calculated on the marked price of the item, which is also true. Furthermore, the reason correctly explains why the formula in the assertion uses the marked price as the base for calculating the discount percentage. Therefore, both the assertion and reason are true, and the reason is the correct explanation of the assertion.

Key Concept: Sales Tax, GST

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

[Solution Description]

To calculate the total bill amount, we first find the sales tax. The sales tax is 8% of the selling price, which is Rs. 1200.

Sales tax = $\frac{8}{100} \times 1200$ = Rs. 96.

Total bill amount = Selling price + Sales tax = Rs. 1200 + Rs. 96 = Rs. 1296.

Thus, the assertion is true. The reason correctly explains that sales tax is calculated as a percentage of the selling price and added to it to determine the total bill amount. Therefore, both the assertion and reason are true, and the reason correctly explains the assertion.

Key Concept: Goods and Services Tax

b) Rs.1071.43

[Solution Description]

Let the original price be Rs.x.

Price after GST = Original price + GST = x + 12% of x = 1.12x.

Given, 1.12x = Rs.1200.

Therefore, x = $\frac{1200}{1.12} = Rs.1071.43$.

Key Concept: Sales Tax

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

[Solution Description]

The assertion states that sales tax is always calculated on the selling price of an item. This is true because sales tax is a percentage of the selling price, as mentioned in the syllabus. The reason states that sales tax is a type of indirect tax collected by the shopkeeper and given to the government. This is also true, as per the syllabus. Moreover, the reason correctly explains why the assertion is true, as the nature of sales tax being an indirect tax necessitates its calculation on the selling price. Thus, both the assertion and reason are true, and the reason is the correct explanation of the assertion.

Key Concept: VAT

c) Rs. 1680

[Solution Description]

First, calculate the VAT. VAT = $\frac{12}{100} \times 1500$ = Rs. 180. Then, add the VAT to the original price: Total price = Rs. 1500 + Rs. 180 = Rs. 1680.

Key Concept: Sales Tax, GST

b) \$100

[Solution Description]

First, calculate the total amount paid by the customer under the sales tax system:

$\text{Sales Tax} = 10\% \text{ of } \$5000 = 0.10 \times 5000 = \$500$

$\text{Total Amount (Sales Tax)} = \$5000 + \$500 = \$5500$

Next, calculate the total amount paid by the customer under the GST system:

$\text{GST} = 12\% \text{ of } \$5000 = 0.12 \times 5000 = \$600$

$\text{Total Amount (GST)} = \$5000 + \$600 = \$5600$

Finally, find the difference in the total amounts:

$\text{Difference} = \$5600 - \$5500 = \$100$

Key Concept: Definition of Sales Tax

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

[Solution Description]

The assertion states that sales tax is always added to the selling price of an item, which is true as per the syllabus. The reason states that sales tax is collected by the shopkeeper from the customer and given to the government, which is also true. Furthermore, the reason correctly explains why the assertion is true because the process of collecting and remitting the tax ensures it is added to the selling price. Therefore, both the assertion and reason are true, and the reason is the correct explanation of the assertion.

Key Concept: Sales Tax

c) Rs. 864

[Solution Description]

First, calculate the sales tax amount.

Sales tax = $8\%$ of Rs. 800 = $\frac{8}{100} \times 800 = Rs. 64$.

Next, add the sales tax to the cost of the shirt to find the total bill amount.

Total bill amount = Cost of shirt + Sales tax = Rs. 800 + Rs. 64 = **Rs. 864**.

Key Concept: Sales Tax Calculation

b) Rs. 27,000

[Solution Description]

First, calculate the sales tax on the laptop.

Sales Tax = $\frac{8}{100} \times 25000 = Rs. 2000$.

Next, add the sales tax to the original price of the laptop.

Total Bill Amount = Original Price + Sales Tax = $Rs. 25000 + Rs. 2000 = Rs. 27000$.

Key Concept: Finding Sales Tax

c) Rs. 44,000

[Solution Description]

First, calculate the sales tax: $Rs. 40,000 \times \frac{10}{100} = Rs. 4,000$. Then, add the sales tax to the original cost: $Rs. 40,000 + Rs. 4,000 = Rs. 44,000$.

Key Concept: Sales Tax, GST

c) \$1440

[Solution Description]

First, calculate the sales tax on the laptop. Sales tax is 8% of \$1200.

$\text{Sales Tax} = \frac{8}{100} \times 1200 = 96$

Next, calculate the GST on the laptop. GST is 12% of \$1200.

$\text{GST} = \frac{12}{100} \times 1200 = 144$

Now, add the original price, sales tax, and GST to find the total amount to be paid.

$\text{Total Amount} = 1200 + 96 + 144 = 1440$

Key Concept: GST, Sales Tax

b) \$60

[Solution Description]

Let the original price be \$100. GST is 10%, so the selling price becomes \$110. When the selling price is \$660, the original price is:

$\text{Original Price} = \left( \frac{100}{110} \right) \times 660 = \$600$

The GST paid is the difference between the selling price and the original price:

$\text{GST} = 660 - 600 = \$60$

Key Concept: Sales Tax, GST

a) \$1016.95

[Solution Description]

Let the original price be \$100. GST is 18%, so the selling price becomes \$118. When the selling price is \$118, the original price is \$100. Therefore, when the selling price is \$1200, the original price is calculated as follows:

$\text{Original Price} = \left( \frac{100}{118} \right) \times 1200 = \$1016.95$

Key Concept: Finding Price Before Tax

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

[Solution Description] Let the original price be \$100. GST rate is 12%, so the total GST amount is 12% of \$100 = \$12. Therefore, the selling price including GST is \$100 + \$12 = \$112. To find the original price from the selling price, we use the formula: Original Price = Selling Price / $(1 + \frac{GST rate}{100}) = 112 / 1.12 = \$100$. Hence, both Assertion and Reason are true, and Reason correctly explains Assertion.

Key Concept: Finding Price Before Tax

a) \$3813.56

[Solution Description]

Let the original price of the laptop be \$100. GST is 18%, so the total price including GST is \$118. Now, when the total price is \$4500, we can find the original price by using the formula:

$\text{Original Price} = \frac{\text{Total Price} \times 100}{118}$

Substituting the values:

$\text{Original Price} = \frac{4500 \times 100}{118} = \frac{450000}{118} = 3813.56$

Therefore, the price of the laptop before GST was added is \$3813.56.

Key Concept: Reverse calculation to find price before tax

b) \$ 8,000

[Solution Description]

Let the original price of the smartphone be \$ 100. With GST of 5%, the total price becomes \$ 105. Now, when the total price after GST is \$ 105, the original price is \$ 100. So, when the total price after GST is \$ 8,400, the original price can be calculated using the formula:

$\text{Original Price} = \frac{100 \times 8,400}{105} = \$ 8,000$

Therefore, the price of the smartphone before GST was added is \$ 8,000.

Key Concept: Reverse Calculation, GST

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

[Solution Description]

To verify the assertion, we use the formula for reverse calculation of GST. Given that the total price including GST is \$112 and the GST rate is $12\%$, we can calculate the original price as follows:

$Original Price = \frac{Total Price}{1 + \left(\frac{GST Rate}{100}\right)} = \frac{112}{1 + 0.12} = \frac{112}{1.12} = 100$

Thus, the original price is indeed \$100. The reason provides the correct formula for calculating the original price before GST, which is used to derive the assertion. Therefore, both the assertion and reason are true, and the reason correctly explains the assertion.

Key Concept: Sales Tax, GST

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

[Solution Description]

To find the total bill amount after adding GST:

Let the selling price (SP) be Rs. 1000.

GST rate = 18%.

GST amount = $\frac{18}{100} \times 1000 = Rs. 180$.

Total bill amount = SP + GST amount = $1000 + 180 = Rs. 1180$.

Thus, the assertion is true because the total bill amount is Rs. 1180. The reason is also true because GST is always calculated on the selling price and added to the bill amount. Furthermore, the reason correctly explains the assertion.

Key Concept: GST, Sales Tax

d) Rs. 30750

[Solution Description]

First, calculate the GST:

GST = $18\%$ of Rs. 25,000 = $0.18 \times 25000 = Rs. 4500$.

Next, calculate the sales tax:

Sales Tax = $5\%$ of Rs. 25,000 = $0.05 \times 25000 = Rs. 1250$.

Total Amount Payable = Cost + GST + Sales Tax = $Rs. 25000 + Rs. 4500 + Rs. 1250 = Rs. 30750$.

Key Concept: Finding Original Price before GST

b) \$ 42,000

[Solution Description]

Let the original price be \$ 100. GST is 12%, so the price including GST is \$ 112.

When the price including GST is \$ 112, the original price is \$ 100.

Therefore, when the price including GST is \$ 47,040, the original price is calculated as:

$\text{Original Price} = 100 \times \frac{47040}{112} = 42000$

So, the original price of the washing machine before GST was added is \$ 42,000.

Key Concept: Original Price before GST, Percentage Calculation

c) \$Rs. 30,000

[Solution Description]

Let the original price be $P$. GST is 18%, so the total price after GST is $1.18P$. Given that $1.18P = \$Rs. 35,400$, we can solve for $P$:

$P = \frac{\$Rs. 35,400}{1.18} = \$Rs. 30,000$

The original price of the washing machine before GST was added is \$Rs. 30,000.

Key Concept: Compound Interest Comparison

c) 96

[Solution Description]

First, calculate the compound interest:

$A_{\text{CI}} = P \left(1 + \frac{R}{100}\right)^n = 15000 \left(1 + \frac{8}{100}\right)^2 = 15000 \times \left(\frac{27}{25}\right)^2 = 15000 \times \frac{729}{625} = 17496$

$\text{CI} = A_{\text{CI}} - P = 17496 - 15000 = 2496$

Next, calculate the simple interest:

$\text{SI} = \frac{P \times R \times n}{100} = \frac{15000 \times 8 \times 2}{100} = 2400$

The difference between CI and SI is:

$2496 - 2400 = 96$

Key Concept: Compound Interest, Principal Calculation

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

[Solution Description]

In compound interest, the interest earned each year is added to the principal to form the new principal for the next year. Let’s assume an initial principal $P = \$100$ and an annual interest rate of 10%. For the first year, the interest earned is $I_1 = P \times \frac{10}{100} = \$10$. The amount at the end of the first year is $A_1 = P + I_1 = \$110$, which becomes the principal for the second year. Since $A_1 = \$110 > P = \$100$, the principal for the second year is greater than the principal for the first year. The reason correctly explains the assertion by stating that the interest earned in the first year is added to the principal to form the new principal for the second year. Thus, both the assertion and the reason are true, and the reason correctly explains the assertion.

Key Concept: Compound Interest Calculation

b) Rs. 1,576.25

[Solution Description]

To find the compound interest, we use the formula $A = P \left(1 + \frac{R}{100}\right)^n$, where $P$ is the principal, $R$ is the rate of interest, and $n$ is the number of years. Here, $P = Rs. 10,000$, $R = 5$, and $n = 3$.

Step 1: Calculate the amount after 3 years:

$A = Rs. 10,000 \left(1 + \frac{5}{100}\right)^3 = Rs. 10,000 \times \left(\frac{21}{20}\right)^3$

Step 2: Simplify the calculation:

$A = Rs. 10,000 \times \frac{9261}{8000} = Rs. 11,576.25$

Step 3: Calculate the compound interest:

$CI = A - P = Rs. 11,576.25 - Rs. 10,000 = Rs. 1,576.25$

The compound interest earned is Rs. 1,576.25.

Key Concept: Compound Interest Calculation

b) Rs. 5955

[Solution Description]

The formula to calculate the amount with compound interest is:

$A = P \left(1 + \frac{R}{100}\right)^n$

Here, Principal $P = Rs. 5000$, Rate $R = 6\%$, and Time $n = 3$ years.

Plugging in the values:

$A = Rs. 5000 \left(1 + \frac{6}{100}\right)^3 = Rs. 5000 \times \left(1.06\right)^3$

Calculating $1.06^3$:

$1.06^3 = 1.191016$

Therefore:

$A = Rs. 5000 \times 1.191016 = Rs. 5955.08$

So, the amount after 3 years is approximately Rs. 5955.08.

Key Concept: Compound Interest, Simple Interest, Principal Change

b) 1.00227[Solution Description]First, calculate the simple interest amount for 2 years:

Simple Interest = $P \times R \times T / 100$

Simple Interest = $2000 \times 5 \times 2 / 100 = \text{Rs.} 200$

Total Amount (Simple Interest) = $2000 + 200 = \text{Rs.} 2200$

Next, calculate the compound interest amount for 2 years:

Compound Interest = $P \times (1 + R/100)^T$

Compound Interest = $2000 \times (1 + 5/100)^2$

Compound Interest = $2000 \times (1.05)^2$

$(1.05)^2 = 1.1025$

Compound Interest = $2000 \times 1.1025 = \text{Rs.} 2205$

Ratio = Compound Interest Amount / Simple Interest Amount

Ratio = $2205 / 2200 = 1.00227$

Key Concept: Difference between SI and CI

b) \$10

[Solution Description]

Simple Interest formula: $SI = P \times r \times t$, where $P = \$1000$, $r = 10\% = 0.10$, and $t = 2$ years.

Substituting the values: $SI = 1000 \times 0.10 \times 2 = \$200$.

Compound Interest formula: $A = P \left(1 + \frac{r}{n}\right)^{nt}$, where $n = 1$ (compounded annually).

Substituting the values: $A = 1000 \left(1 + \frac{0.10}{1}\right)^{1 \times 2} = 1000 \times (1.10)^2 = 1000 \times 1.21 = \$1210$.

Compound Interest = Total Amount - Principal = \$1210 - \$1000 = \$210.

Difference between CI and SI = \$210 - \$200 = \$10.

Key Concept: Compound Interest Calculation

c) Rs. 11025

[Solution Description]

To find the total amount with compound interest, we use the formula:

$A = P \left(1 + \frac{R}{100}\right)^n$

Here, $P = 10000$, $R = 5$, and $n = 2$.

Plugging in the values:

$A = 10000 \left(1 + \frac{5}{100}\right)^2 = 10000 \left(1.05\right)^2 = 10000 \times 1.1025 = Rs. 11025$

Key Concept: Compound Interest, Multi-step Calculation

c) \$6,038.28

[Solution Description]

To find the compound interest, we use the formula:

$CI = P \left(1 + \frac{R}{100}\right)^n - P$

Here, $P = 15000$, $R = 7$, and $n = 5$.

Step 1: Calculate the amount after 5 years.

$A = 15000 \left(1 + \frac{7}{100}\right)^5$

$A = 15000 \left(\frac{107}{100}\right)^5$

$A = 15000 \times 1.402552$

$A = 21038.28$

Step 2: Calculate the compound interest.

$CI = 21038.28 - 15000 = 6038.28$

The compound interest earned is \$6,038.28.

Key Concept: Compound Interest, Conceptual Complexity

b) 12 years

[Solution Description]

Let the principal be $P$ and the rate of interest be $R \%$.

Given that the principal doubles in 6 years:

$2P = P \left(1 + \frac{R}{100}\right)^6$

Divide both sides by $P$:

$2 = \left(1 + \frac{R}{100}\right)^6$

To find the time $n$ for the principal to become four times:

$4P = P \left(1 + \frac{R}{100}\right)^n$

Divide both sides by $P$:

$4 = \left(1 + \frac{R}{100}\right)^n$

Since $2 = \left(1 + \frac{R}{100}\right)^6$, we can write:

$\left(1 + \frac{R}{100}\right)^n = \left(\left(1 + \frac{R}{100}\right)^6\right)^2$

Therefore:

$n = 6 \times 2 = 12 \text{ years}$

So, the time required for the principal to become four times itself is 12 years.

Key Concept: Compound Interest vs Simple Interest

b) Compound interest is calculated on both the principal and accumulated interest.

[Solution Description]

Simple interest is calculated only on the principal amount, while compound interest is calculated on both the principal and the accumulated interest over previous periods. This results in compound interest growing faster than simple interest over time.

Key Concept: Compound Interest Calculation

b) 1576.25

[Solution Description]

To calculate the compound interest, we use the formula:

$A = P \left(1 + \frac{R}{100}\right)^n$.

Here, $P = 10000$, $R = 5\%$, and $n = 3$ years.

First, calculate the amount $A$:

$A = 10000 \left(1 + \frac{5}{100}\right)^3 = 10000 \times (1.05)^3$.

$(1.05)^3 = 1.157625$.

So, $A = 10000 \times 1.157625 = 11576.25$.

Now, calculate the compound interest:

$CI = A - P = 11576.25 - 10000 = 1576.25$.

Therefore, the compound interest is \$1576.25.

Key Concept: Compound Interest Calculation

b) Rs. 3090

[Solution Description]

Step 1: Calculate the principal for the first year, which is Rs. 25,000.

Step 2: Calculate the Simple Interest (S.I.) for the first year:

$SI_1 = Rs. 25000 \times \frac{6}{100} = Rs. 1500$

Step 3: Calculate the amount at the end of the first year:

$Amount_1 = Rs. 25000 + Rs. 1500 = Rs. 26500$

Step 4: Calculate the S.I. for the second year using Rs. 26500 as the principal:

$SI_2 = Rs. 26500 \times \frac{6}{100} = Rs. 1590$

Step 5: Calculate the amount at the end of the second year:

$Amount_2 = Rs. 26500 + Rs. 1590 = Rs. 28090$

Step 6: Calculate the Compound Interest (C.I.) earned over 2 years:

$C.I. = Rs. 28090 - Rs. 25000 = Rs. 3090$

The Compound Interest earned at the end of 2 years is Rs. 3090.

Key Concept: Compound Interest

b) Rs. 3090[Solution Description]To find the compound interest, we calculate the amount year by year and then subtract the principal.

Step 1: Calculate the interest for the first year.

$SI1 = 25000 \times \frac{6}{100} = Rs. 1500$

Step 2: Find the amount at the end of the first year.

$Amount1 = 25000 + 1500 = Rs. 26500$

Step 3: Calculate the interest for the second year.

$SI2 = 26500 \times \frac{6}{100} = Rs. 1590$

Step 4: Find the amount at the end of the second year.

$Amount2 = 26500 + 1590 = Rs. 28090$

Step 5: Calculate the compound interest.

$CI = 28090 - 25000 = Rs. 3090$

The compound interest earned is Rs. 3090.

Key Concept: Compound Interest, Year-by-Year Calculation

b) \$10,250

[Solution Description]

Step 1: Calculate the amount after the first year. Principal for the first year ($P_1$) = \$100,000. Interest for the first year ($SI_1$) = \$100,000 $\times$ $\frac{5}{100}$ = \$5,000. Amount at the end of the first year = \$100,000 + \$5,000 = \$105,000.

Step 2: Calculate the amount after the second year. Principal for the second year ($P_2$) = \$105,000. Interest for the second year ($SI_2$) = \$105,000 $\times$ $\frac{5}{100}$ = \$5,250. Amount at the end of the second year = \$105,000 + \$5,250 = \$110,250.

Total interest paid after 2 years is \$5,000 + \$5,250 = \$10,250.

Key Concept: Compound Interest, Changing Principal

c) \$11,576.25

[Solution Description]

To find the total amount after 3 years with compound interest, we use the formula:

$A = P \left(1 + \frac{R}{100}\right)^n$

where $P$ is the principal amount, $R$ is the annual interest rate, and $n$ is the number of years.

Plugging in the values:

$A = 10000 \left(1 + \frac{5}{100}\right)^3 = 10000 \times 1.157625 = 11576.25$

Therefore, the total amount after 3 years is \$11,576.25.

Key Concept: Compound Interest, Changing Principal

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

[Solution Description] Let’s calculate the interest for the first and second years. The principal for the first year (\$P1) is \$10,000. The interest for the first year ($SI1$) is calculated as: $SI1 = P1 \times \frac{R}{100} = 10000 \times \frac{5}{100} = \$500.$ The amount at the end of the first year (\$P2) is: $P2 = P1 + SI1 = 10000 + 500 = \$10500.$ The interest for the second year ($SI2$) is calculated on \$P2: $SI2 = P2 \times \frac{R}{100} = 10500 \times \frac{5}{100} = \$525.$ Since \$525 > \$500, the assertion is true. The reason is also true because in compound interest, the principal for the next year includes the interest from the previous year. Additionally, the reason correctly explains why the interest in the second year is higher than in the first year.

Key Concept: Simple Interest vs Compound Interest

c) 5832

[Solution Description]

The formula for compound interest is given by:

$\text{Amount} = P \times (1 + r)^t$

where $P = 5000$, $r = 0.08$, and $t = 2$.

$\text{Amount} = 5000 \times (1 + 0.08)^2 = 5000 \times 1.1664 = 5832$

Key Concept: Compound Interest, Simple Interest

b) \$74.16

[Solution Description]

Given: Simple Interest (SI) = \$72, Rate (r) = 6%, Time (t) = 2 years.

1. Find Principal (P):

$SI = P \times r \times t \implies 72 = P \times 0.06 \times 2 \implies P = \frac{72}{0.12} = 600$

2. Calculate Compound Interest (CI):

$CI = P \times (1 + r)^t - P = 600 \times (1 + 0.06)^2 - 600 = 674.16 - 600 = 74.16$

Key Concept: Compound Interest

c) Assertion is true, but Reason is false.

[Solution Description]

The assertion states that compound interest increases faster than simple interest over time. This is true because in compound interest, the interest earned each year is added to the principal for the next year, leading to exponential growth. The reason states that in compound interest, the principal remains the same every year. This is false because in compound interest, the principal changes every year as it includes the accumulated interest from previous years. Therefore, the assertion is true, but the reason is false.

Key Concept: Principal Calculation

b) Rs. 5,000

[Solution Description]

We know that:

$CI = P \left[\left(1 + \frac{R}{100}\right)^n - 1\right]$

Here, $CI = Rs. 1,050$, $R = 10$, and $n = 2$.

Plugging in the values:

$1,050 = P \left[\left(1 + \frac{10}{100}\right)^2 - 1\right]$

$1,050 = P \left[1.21 - 1\right]$

$1,050 = P \times 0.21$

$P = \frac{1,050}{0.21}$

$P = Rs. 5,000$

Key Concept: Simple Interest vs Compound Interest

b) \$1.53

[Solution Description]

For Simple Interest, the interest is calculated as $I = P \times r \times t$. Thus, $I = \$200 \times 0.05 \times 3 = \$30$.

For Compound Interest, the amount after 3 years is $A = P(1 + r)^t = \$200(1 + 0.05)^3 = \$200 \times 1.157625 = \$231.525$.

The interest earned is \$231.525 - \$200 = \$31.525. The difference in interest earned is \$31.525 - \$30 = \$1.525. Rounding to two decimal places, it is \$1.53.

Key Concept: Simple Interest Calculation

c) \$20

[Solution Description]

To calculate simple interest, use the formula $SI = \frac{P \times R \times T}{100}$, where $P$ is the principal, $R$ is the rate of interest, and $T$ is the time period. Here, $P = \$200$, $R = 5\%$, and $T = 2$ years.

$SI = \frac{200 \times 5 \times 2}{100} = \frac{2000}{100} = \$20$

Key Concept: Compound Interest, Real-world Application

b) Option B by 790[Solution Description]Calculate the amount under compound interest (Option A):

$A_A = P \left(1 + \frac{R}{100}\right)^n = \$20000 \left(1 + \frac{8}{100}\right)^4$

Simplify inside the parentheses:

$1 + \frac{8}{100} = 1.08$

Raise to the power of 4:

$1.08^4 = 1.3605$

Multiply by the principal:

$A_A = \$20000 \times 1.3605 = \$27210$

Calculate the amount under simple interest (Option B):

$A_B = P + P \times R \times n / 100 = \$20000 + \$20000 \times 10 \times 4 / 100$

Simplify:

$A_B = \$20000 + \$8000 = \$28000$

Compare the two amounts:

$\text{Difference} = \$28000 - \$27210 = \$790$

Option B yields a higher return by \$790.

Key Concept: Compound Interest Formula

b) Rs. 14600

[Solution Description]

To find the total amount, we use the formula $A = P \left(1 + \frac{R}{100}\right)^n$. Here, $P = \text{Rs.} 12000$, $R = 4$, and $n = 5$.

Step 1: Calculate $\left(1 + \frac{R}{100}\right)^n$:

$\left(1 + \frac{4}{100}\right)^5 = \left(\frac{104}{100}\right)^5 = 1.2166529$

Step 2: Multiply by the principal amount:

$12000 \times 1.2166529 = 14599.835$

So, the total amount after 5 years is approximately Rs. 14600.

Key Concept: Compound Interest, Real-World Application

b) 18937.15

[Solution Description]

Using the compound interest formula $A = P \left(1 + \frac{R}{100}\right)^n$, where $P = 15000$, $R = 6$, and $n = 4$, we calculate the total amount as follows:

$A = 15000 \left(1 + \frac{6}{100}\right)^4 = 15000 \times (1.06)^4$

Compute $(1.06)^4$:

$(1.06)^4 = 1.06 \times 1.06 \times 1.06 \times 1.06 = 1.26247696$

Multiply by the principal:

$A = 15000 \times 1.26247696 = 18937.1544$

Rounding to two decimal places, the total amount is \$18,937.15.

Key Concept: Compound Interest Formula

b) Rs. 1528

[Solution Description]

Given: Principal ($P$) = Rs. 8000, Rate ($R$) = 6%, Time ($n$) = 3 years.

The compound interest formula is:

$CI = P \left(1 + \frac{R}{100}\right)^n - P$

Substituting the values:

$CI = 8000 \left(1 + \frac{6}{100}\right)^3 - 8000$

Calculating step by step:

$1 + \frac{6}{100} = 1.06$

$1.06^3 = 1.191016$

$8000 \times 1.191016 = 9528.128$

$CI = 9528.128 - 8000 = 1528.128$

Therefore, the compound interest is approximately Rs. 1528.

Key Concept: Compound Interest, Multi-step Solution

b) \$1528

[Solution Description]

To find the compound interest, we first calculate the amount after 3 years using the formula $A = P \times \left(1 + \frac{R}{100}\right)^n$, where $P$ is the principal, $R$ is the rate, and $n$ is the time period. Here, $P = 8000$, $R = 6$, and $n = 3$. Plugging in the values, we get:

$A = 8000 \times \left(1 + \frac{6}{100}\right)^3 = 8000 \times \left(1.06\right)^3$

Calculating $(1.06)^3$:

$(1.06)^3 = 1.191016$

Now, multiply by 8000:

$A = 8000 \times 1.191016 = 9528.128$

The compound interest is then calculated as $CI = A - P = 9528.128 - 8000 = 1528.128$.

Rounding to the nearest whole number, the compound interest is \$1528.

Key Concept: Compound Interest, Real-World Application

b) 5%

[Solution Description]

We use the formula for compound interest: $A = P \times \left(1 + \frac{R}{100}\right)^n$, where $A$ is the final amount, $P$ is the principal, $R$ is the rate, and $n$ is the time period. Given $A = 14595.84$, $P = 12000$, and $n = 4$, we need to find $R$.

Rearranging the formula:

$14595.84 = 12000 \times \left(1 + \frac{R}{100}\right)^4$

Dividing both sides by 12000:

$\left(1 + \frac{R}{100}\right)^4 = \frac{14595.84}{12000} = 1.21632$

Taking the fourth root of both sides:

$1 + \frac{R}{100} = 1.21632^{1/4} \approx 1.05$

Subtracting 1:

$\frac{R}{100} = 0.05 \implies R = 5$

Therefore, the annual interest rate is 5%.

Key Concept: Compound Interest Formula

b) $A = P \left(1 + \frac{R}{100}\right)^n$

[Solution Description]

The formula for calculating the amount $A$ after $n$ years with principal $P$ and annual interest rate $R\%$ compounded annually is:

$A = P \left(1 + \frac{R}{100}\right)^n$

This formula is derived from the concept of compound interest where the interest is added to the principal each year.

Key Concept: Compound Interest, Amount Calculation

b) $P \left(1 + \frac{R}{100}\right)^5$

[Solution Description]

The formula for the amount after n years is $A = P \left(1 + \frac{R}{100}\right)^n$. To find the amount after 5 years, replace n with 5 in the formula:

$A_5 = P \left(1 + \frac{R}{100}\right)^5$

Key Concept: Application of Compound Interest Formula

a) 14,580

[Solution Description]

To find the value of the car after 3 years, we use the compound depreciation formula:

$A = P \left(1 - \frac{r}{100}\right)^n$

where $P = 20000$, $r = 10$, and $n = 3$. Plugging in the values:

$A = 20000 \left(1 - \frac{10}{100}\right)^3 = 20000 \times 0.729 = 14580$

The value of the car after 3 years is \$14,580.

Key Concept: Compound Interest, Real-World Application

b) \$9869.57

[Solution Description]

The formula for the amount after $n$ years with compound interest is $A = P \left(1 + \frac{R}{100}\right)^n$. Here, $A = \$12000$, $R = 5\%$, and $n = 4$.

Rearrange the formula to solve for $P$:

$P = \frac{A}{\left(1 + \frac{R}{100}\right)^n} = \frac{12000}{\left(1 + \frac{5}{100}\right)^4}$

Calculate $\left(1 + \frac{5}{100}\right)^4$:

$\left(1 + \frac{5}{100}\right)^4 = \left(\frac{105}{100}\right)^4 = \frac{105}{100} \times \frac{105}{100} \times \frac{105}{100} \times \frac{105}{100} = \frac{121550625}{100000000} = 1.21550625$

Now, calculate $P$:

$P = \frac{12000}{1.21550625} = \$9869.57$

Key Concept: Compound Interest Calculation

b) Rs.2597.12

[Solution Description]

Using the formula for compound interest:

$CI = P \left[\left(1 + \frac{R}{100}\right)^n - 1\right]$

Substituting the given values:

$CI = 10000 \left[\left(1 + \frac{8}{100}\right)^3 - 1\right]$

$CI = 10000 \left[(1.08)^3 - 1\right]$

$CI = 10000 [1.259712 - 1]$

$CI = 10000 \times 0.259712 = Rs.2597.12$

The compound interest is Rs.2597.12.

Key Concept: Compound Interest Calculation

b) Rs.1996

[Solution Description]

Given the amount $A = Rs.13996.80$, principal $P = Rs.12000$, and time $n = 2$ years. The compound interest can be calculated using the formula $CI = A - P$.

Step 1: Substitute the given values into the formula:

$CI = Rs.13996.80 - Rs.12000 = Rs.1996.80$

Therefore, the compound interest earned is Rs.1996.80.

Key Concept: Compound Interest

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

[Solution Description]

To verify the assertion, we use the compound interest formula. Given: Principal $P = \$10,000$, Rate $R = 5\%$, Time $n = 2$ years.

Step 1: Calculate the amount $A$ using the formula $A = P \left(1 + \frac{R}{100}\right)^n$.

Substituting the values: $A = \$10,000 \left(1 + \frac{5}{100}\right)^2 = \$10,000 \times (1.05)^2$.

Step 2: Compute $(1.05)^2 = 1.1025$.

Step 3: Multiply to find $A = \$10,000 \times 1.1025 = \$11,025$.

Step 4: Calculate Compound Interest $CI = A - P = \$11,025 - \$10,000 = \$1,025$.

Thus, the assertion is true.

The reason provides the correct formula for compound interest, which was used in the solution. Therefore, both the assertion and reason are true, and the reason correctly explains the assertion.

Key Concept: Compound Interest, Depreciation

d) 162000

[Solution Description]

To find the value of the machine after 2 years with a 10% annual depreciation, we use the compound interest formula for depreciation:

$A = P \left(1 - \frac{R}{100}\right)^n$

Here, $P = 200000$, $R = 10$, and $n = 2$.

$A = 200000 \left(1 - \frac{10}{100}\right)^2 = 200000 \times \left(0.9\right)^2$

Calculating step by step:

$0.9^2 = 0.9 \times 0.9 = 0.81$

$A = 200000 \times 0.81 = 162000$

Therefore, the value of the machine after 2 years will be \$162,000.

Key Concept: Population Growth

c) 10927

[Solution Description]

To find the population at the end of 2023, we use the compound interest formula:

$A = P \left(1 + \frac{r}{100}\right)^n$

where $P = 10000$, $r = 3$, and $n = 3$. Substituting these values into the formula:

$A = 10000 \times \left(1 + \frac{3}{100}\right)^3 = 10000 \times \left(1.03\right)^3 = 10000 \times 1.092727 = 10927.27$

Since population cannot be in decimals, it is rounded to 10927.

Key Concept: Bacteria Growth

b) 12,167

[Solution Description]

The bacterial growth follows the compound interest formula:

$A = P \left(1 + \frac{r}{100}\right)^n$

where $P = 10000$, $r = 4$, and $n = 5$ hours.

Substituting the values:

$A = 10000 \left(1 + \frac{4}{100}\right)^5$

Simplify inside the bracket:

$1 + \frac{4}{100} = 1.04$

Now calculate $1.04^5$:

$1.04^5 = 1.2166529$

Multiply by $P$:

$A = 10000 \times 1.2166529 = 12166.529$

After 5 hours, the number of bacteria will be approximately 12,167.

Key Concept: Depreciation of Value

b) 20,250

[Solution Description]

To find the value of the car at the end of 2020, we use the compound interest formula for depreciation:

$A = P \left(1 - \frac{r}{100}\right)^n$

where $P = 25000$, $r = 10\%$, and $n = 2$ years.

Substituting the values, we get:

$A = 25000 \left(1 - \frac{10}{100}\right)^2 = 25000 \times (0.9)^2 = 25000 \times 0.81 = 20250$

Therefore, the value of the car at the end of 2020 is \$20,250.

Key Concept: Population Growth

b) 10612

[Solution Description]

To find the population in 2018, we use the compound interest formula:

$A = P \left(1 + \frac{r}{100}\right)^n$

Here, $P = 10000$, $r = 2$, and $n = 3$ (from 2015 to 2018).

Substituting the values:

$A = 10000 \left(1 + \frac{2}{100}\right)^3 = 10000 \times (1.02)^3$

Calculating step by step:

$1.02^3 = 1.061208$

$A = 10000 \times 1.061208 = 10612.08$

The population in 2018 is approximately 10612.

Key Concept: Population Growth using Compound Interest

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

[Solution Description]

The assertion states that the population of a town increases at a rate of 3% per annum, and the population after 2 years can be calculated using the formula $P \left(1 + \frac{r}{100}\right)^n$. This is true because the population grows at a compounded rate each year. The reason states that the population growth is an application of the compound interest formula because the increase in population is compounded annually. This is also true and directly explains why the assertion is correct. Therefore, both the assertion and reason are true, and the reason correctly explains the assertion.

Key Concept: Population Growth

b) 76,832

[Solution Description]

To find the population at the end of 2017, we use the compound interest formula for population decrease:

$P = P_0 \left(1 - \frac{r}{100}\right)^n$

Where $P_0 = 80000$, $r = 2$, and $n = 2$.

$P = 80000 \left(1 - \frac{2}{100}\right)^2 = 80000 \times \left(\frac{98}{100}\right)^2 = 80000 \times 0.9604 = 76832$

So, the population at the end of 2017 will be 76,832.

Key Concept: Compound Interest, Population Increase

b) 12,124

[Solution Description]

To find the population at the end of 2024, we need to apply the compound interest formula step by step.

Step 1: Calculate the population for the first four years (2018-2022) with a 6% increase.

$P_{2022} = 10000 \left(1 + \frac{6}{100}\right)^4 = 10000 \times 1.06^4$

$10000 \times 1.262477 = 12624.77 \approx 12625$

Step 2: Calculate the population for the next two years (2022-2024) with a 2% decrease.

$P_{2024} = 12625 \left(1 - \frac{2}{100}\right)^2 = 12625 \times 0.98^2$

$12625 \times 0.9604 = 12123.55 \approx 12124$

Therefore, the population at the end of 2024 is approximately 12,124.

Key Concept: Population Increase

c) 53,045

[Solution Description]

To find the population at the end of 2022, we use the compound interest formula for population increase:

$P = P_0 \left(1 + \frac{r}{100}\right)^n$

where $P_0$ is the initial population, $r$ is the rate of increase, and $n$ is the number of years.

Given:

$P_0 = 50000$,

$r = 3\%$,

$n = 2$ years.

Plugging in the values:

$P = 50000 \left(1 + \frac{3}{100}\right)^2$

$P = 50000 \times 1.03^2$

$P = 50000 \times 1.0609$

$P = 53045$

So, the population at the end of 2022 is 53,045.

Key Concept: Depreciation of Value

b) \$23,500

[Solution Description]

Principal = \$25,000

Depreciation rate = 6%

Depreciation amount = 6% of \$25,000 = $\frac{6}{100} \times 25000 = 1500$

Value after one year = $25000 - 1500 = 23500$

So, the value of the car after one year is \$23,500.

Key Concept: Depreciation of Value

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

[Solution Description] The assertion states that the value of a machinery depreciates by 5% each year, which is true as depreciation is a reduction in value over time. The reason provides the correct formula for calculating the value after depreciation, $V = P \left(1 - \frac{r}{100}\right)$. Since the reason correctly explains how the depreciation is calculated, both the assertion and reason are true, and the reason is the correct explanation of the assertion.

Key Concept: Depreciation and Compound Interest Formula

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.[Solution Description]Let's verify the assertion and reason step by step.

Given:

Principal ($P$) =  \$20,000

Rate of depreciation ($r$) = 10% = 0.10

Time ($t$) = 1 year

Using the formula for depreciation:

$V = P(1 - r)$

Substituting the values:

$V = 20000(1 - 0.10) = 20000 \times 0.90 =  \$18,000$

The calculated value matches the assertion (\$18,000), so the Assertion is true.

The Reason provides the correct formula and explains how the value is calculated, so the Reason is also true and correctly explains the Assertion.

Key Concept: Decrease in value over time

a) \$26,100

[Solution Description]

The machine's value decreases at a compound rate of 15% per year. To find its value after 4 years, we use the formula:

$V = P \left(1 - \frac{r}{100}\right)^n$

where $P = 50000$, $r = 15$, and $n = 4$.

Substituting the values:

$V = 50000 \left(1 - \frac{15}{100}\right)^4 = 50000 \times (0.85)^4$

Calculating $(0.85)^4$:

$0.85 \times 0.85 = 0.7225 \\ \\ 0.7225 \times 0.85 = 0.614125 \\ \\ 0.614125 \times 0.85 = 0.52200625$

Now multiply by the principal:

$V = 50000 \times 0.52200625 = 26100.3125$

So, the machine's value after 4 years is approximately \$26,100.

Key Concept: Compound Interest Formula, Growth of Bacteria

a) 335,024

[Solution Description]

Given:

Initial count of bacteria (P) = 2,50,000

Rate of increase (R) = 5% per hour

Time (T) = 6 hours

The formula for the amount after compound interest is:

$A = P \left(1 + \frac{R}{100}\right)^T$

Substituting the values:

$A = 250000 \left(1 + \frac{5}{100}\right)^6$

$A = 250000 \times (1.05)^6$

Calculate $(1.05)^6$:

$1.05^2 = 1.1025$

$1.05^4 = 1.1025 \times 1.1025 = 1.21550625$

$1.05^6 = 1.21550625 \times 1.1025 = 1.34009564$

Now, calculate A:

$A = 250000 \times 1.34009564 = 335023.91$

Since the count of bacteria must be a whole number, we round it to the nearest whole number: **335,024**.

Key Concept: Applications of Compound Interest Formula

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

[Solution Description] The assertion states that the count of bacteria increases exponentially over time when the growth rate is constant, which is true because exponential growth occurs when the growth rate remains consistent. The reason given is that the compound interest formula $A = P \left(1 + \frac{r}{100}\right)^t$ can be used to calculate the bacterial count after a certain period. This is also true, as the compound interest formula is applicable for calculating exponential growth, such as bacterial growth. Moreover, the reason correctly explains the assertion because the formula mathematically represents exponential growth. Therefore, both the assertion and reason are true, and the reason is the correct explanation of the assertion.

Key Concept: Population Growth

c) 56,243

[Solution Description]

To find the population at the end of 2013, we use the compound interest formula:

$A = P \left(1 + \frac{r}{100}\right)^n$

Here, $P = 50000$, $r = 4\%$, and $n = 3$ years. Substituting these values into the formula, we get:

$A = 50000 \left(1 + \frac{4}{100}\right)^3 = 50000 \times \left(1.04\right)^3$

First, calculate $\left(1.04\right)^3$:

$1.04 \times 1.04 \times 1.04 = 1.124864$

Now, multiply this by the principal amount:

$50000 \times 1.124864 = 56243.2$

Therefore, the estimated population at the end of 2013 is 56,243.

Key Concept: Bacteria Growth

c) 16,000

[Solution Description]

Since the bacteria double every hour, the number of bacteria follows exponential growth with a doubling time of 1 hour. The formula for exponential growth is:

$N = N_0 \times 2^t$

where $N_0 = 500$ and $t = 5$ hours. Substituting these values into the formula, we get:

$N = 500 \times 2^5$

Calculate $2^5$:

$2^5 = 32$

Now, multiply this by the initial number of bacteria:

$500 \times 32 = 16000$

Therefore, the number of bacteria after 5 hours is 16,000.

Key Concept: Future Population Calculation

b) 75295

[Solution Description]

To find the future population, we use the compound interest formula for decrease:

$P = P_0 \left(1 - \frac{r}{100}\right)^n$

where $P$ is the future population, $P_0$ is the initial population, $r$ is the annual decrease rate, and $n$ is the number of years. Here, $P_0 = 80000$, $r = 2$, and $n = 3$. Plugging in the values:

$P = 80000 \left(1 - \frac{2}{100}\right)^3 = 80000 \times (0.98)^3$

Calculating $(0.98)^3$:

$(0.98)^3 = 0.941192$

Then,

$P = 80000 \times 0.941192 = 75295.36$

Since population cannot be in decimals, we round it to the nearest whole number, which is 75295.

Key Concept: Future Population Calculation

b) 12167

[Solution Description]

To find the future population, we use the compound interest formula:

$P = P_0 \left(1 + \frac{r}{100}\right)^n$

where $P$ is the future population, $P_0$ is the initial population, $r$ is the annual growth rate, and $n$ is the number of years. Here, $P_0 = 10000$, $r = 4$, and $n = 5$. Plugging in the values:

$P = 10000 \left(1 + \frac{4}{100}\right)^5 = 10000 \times (1.04)^5$

Calculating $(1.04)^5$:

$(1.04)^5 = 1.2166529$

Then,

$P = 10000 \times 1.2166529 = 12166.529$

Since population cannot be in decimals, we round it to the nearest whole number, which is 12167.

Key Concept: Depreciation

b) \$1,020

[Solution Description]

To find the value of the laptop after one year, we use the depreciation formula:

$\text{Final Value} = P \times \left(1 - \frac{r}{100}\right)$

Here, $P$ is the principal amount (\$1,200), and $r$ is the rate of depreciation (15%).

Substituting the values:

$\text{Final Value} = 1200 \times \left(1 - \frac{15}{100}\right) = 1200 \times 0.85 = \$1,020$

So, the value of the laptop after one year is \$1,020.

Key Concept: Depreciation

b) \$972

[Solution Description]

The initial value of the laptop is \$1,200. The depreciation rate is 10% per year. To find the value after two years, we use the formula:

$\text{Final Value} = \text{Initial Value} \times \left(1 - \frac{\text{Depreciation Rate}}{100}\right)^{\text{Number of Years}}$

Substituting the values:

$\text{Final Value} = 1200 \times \left(1 - \frac{10}{100}\right)^2 = 1200 \times (0.9)^2 = 1200 \times 0.81 = 972$

So, the value of the laptop after two years is \$972.