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) 2 : 3

[Solution Description]

To find the ratio of boys to girls:

1. Total number of students = 40.

2. Number of girls = 60% of 40 = $\frac{60}{100} \times 40 = 24$.

3. Number of boys = Total students - Number of girls = $40 - 24 = 16$.

4. Ratio of boys to girls = $16 : 24$.

5. Simplifying the ratio by dividing both numbers by 8: $\frac{16}{8} : \frac{24}{8} = 2 : 3$.

Therefore, the ratio of boys to girls is $2 : 3$.

Key Concept: Recalling Ratios and Percentages

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.

1. Suppose the number of oranges = $x$ and the number of apples = $y$.

2. Given the ratio of oranges to apples is $1 : 4$, so $\frac{x}{y} = \frac{1}{4}$.

3. This implies $y = 4x$.

4. Now, calculate the percentage of apples: $\frac{y}{x + y} \times 100 = \frac{4x}{x + 4x} \times 100 = \frac{4x}{5x} \times 100 = 80\%$.

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

Key Concept: Expressing Ratios in Fraction Form

b) $\frac{1}{3}$

[Solution Description]

The ratio of bananas to mangoes can be expressed as $\frac{\text{Number of bananas}}{\text{Number of mangoes}} = \frac{4}{12}$. Simplifying this fraction by dividing both numerator and denominator by 4, we get $\frac{1}{3}$.

Key Concept: Concept of Percentages

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

[Solution Description]

The assertion states that if 40% of a number is 120, then the number is 300. To verify this, let's find the original number using the formula for calculating the original number from a percentage. The formula to find the original number when a percentage is given is:

$\text{Original Number} = \frac{\text{Given Value}}{\text{Percentage}} \times 100$

Here, the given value is 120 and the percentage is 40. Substituting these values into the formula:

$\text{Original Number} = \frac{120}{40} \times 100 = 3 \times 100 = 300$

Thus, the original number is indeed 300, which confirms that the assertion is true. The reason provides the correct method to calculate the original number from a given percentage, and it correctly explains the assertion. Therefore, both the assertion and the reason are true, and the reason is the correct explanation of the assertion.

Key Concept: Converting Ratios to Percentages

b) 40%

[Solution Description]

First, determine the number of boys by subtracting the number of girls from the total number of students: 200 - 120 = 80. Next, calculate the ratio of boys to the total number of students: $\frac{80}{200}$. Convert this ratio to a percentage by multiplying by 100: $\frac{80}{200} \times 100 = 40\%$.

Key Concept: Finding percentage of a quantity

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

[Solution Description]

To find the percentage of girls in the class, we use the formula for percentage:

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

Substituting the given values:

$\text{Percentage} = \left( \frac{30}{50} \right) \times 100 = 60\%$

The assertion states that the percentage of girls is 60%, which matches the calculation. The reason provides the correct formula used to calculate the percentage. Therefore, both the assertion and reason are true, and the reason correctly explains the assertion.

Key Concept: Ratio and Percentage

b) 40%

[Solution Description]

First, find the total number of fruits: $15 + 10 = 25$. Then, calculate the percentage of bananas: $\frac{10}{25} \times 100 = 40\%$.

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: Ratio of Boys to Girls

a) 2 : 3

[Solution Description]

The number of boys is 12 and the number of girls is 18. To find the ratio of boys to girls, we write the numbers as a fraction and simplify it:

$\frac{12}{18} = \frac{2}{3}$

Therefore, the ratio of boys to girls is 2 : 3.

Key Concept: Finding cost per person for a school trip

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

[Solution Description]

To find the cost per person, we first calculate the total expenses by adding the refreshment charge (\$4280) and the transportation charge (\$1320). This gives us \$5600. Next, we determine the total number of persons by adding the number of girls (18), boys (12), and teachers (2), which totals 32. Finally, we use the unitary method to calculate the cost per person by dividing the total expenses (\$5600) by the total number of persons (32). This results in \$175. Therefore, both the Assertion and Reason are true, and the Reason correctly explains the Assertion.

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: Percentage Calculation

c) 55%

[Solution Description]

To find the percentage of the journey completed, use the formula:

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

Substituting the given values:

$\text{Percentage Completed} = \left(\frac{33}{60}\right) \times 100 = 55\%$

Therefore, the percentage of the journey completed is 55%.

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

c) 30% completed, 70% 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:

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

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

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

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

Key Concept: Discount Calculation

a) Rs. 900[Solution Description]First, calculate the discount amount using the formula:

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

Here, 25% of the marked price (Rs. 1200) gives:

$Discount = \frac{25}{100} \times 1200 = Rs. 300$

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

$Sale\ Price = Marked\ Price - Discount$

Plugging in the values:

$Sale\ Price = 1200 - 300 = Rs. 900$

So, the sale price is Rs. 900.

Key Concept: Marked Price, Discount Percentage

c) Rs. 5000[Solution Description]Let the marked price be $MP$. Given that the sale price is Rs. 4750 and the discount is 5%, we can write:

$\text{Sale Price} = MP - (\text{Discount %} \times MP)$

Substituting the known values:

$4750 = MP - (0.05 \times MP)$

Simplify the equation:

$4750 = MP \times (1 - 0.05)$

$4750 = 0.95 \times MP$

Solve for $MP$:

$MP = \frac{4750}{0.95} = Rs. 5000$

Therefore, the marked price is Rs. 5000.

Key Concept: Discount Calculation

b) \$ 100

[Solution Description]

To find the discount, subtract the sale price from the marked price.

$Discount = Marked\ Price - Sale\ Price$

$Discount = \$ 500 - \$ 400 = \$ 100$

So, the discount offered is \$ 100.

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: Marked Price Calculation

b) Rs. 2000

[Solution Description]

Let the marked price be $MP$. Given that the sale price after a 10% discount is Rs. 1800.

The sale price can be calculated using the formula:

$Sale Price = MP - \left( \frac{10}{100} \times MP \right).$

Simplifying the equation:

$Sale Price = MP \times \left(1 - \frac{10}{100}\right) = MP \times 0.9.$

Given that the sale price is Rs. 1800:

$0.9 \times MP = 1800.$

Solving for $MP$:

$MP = \frac{1800}{0.9} = Rs. 2000.$

Therefore, the marked price of the article was Rs. 2000.

Key Concept: Discount Calculation, Discount Percentage

c) 15%

[Solution Description]

To find the discount percentage, we first calculate the discount amount using the formula:

$Discount = Marked Price - Sale Price$.

Here, the Marked Price is \$800 and the Sale Price is \$680. So:

$Discount = 800 - 680 = \$120$.

Next, to find the discount percentage, we use the formula:

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

Substituting the values, we get:

$Discount \% = \frac{120}{800} \times 100\% = 15\%$.

Therefore, the discount percentage is 15%.

Key Concept: Sale Price Calculation

c) \$270

[Solution Description]

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

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

Given that the Discount % is 10% and the Marked Price (MP) is \$300:

$\text{Discount} = \frac{10}{100} \times 300 = 30$

Now, subtract the discount from the marked price to find the sale price:

$\text{Sale Price} = \text{Marked Price} - \text{Discount} = 300 - 30 = 270$

Therefore, the sale price of the bicycle is \$270.

Key Concept: Finding Discounts

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

[Solution Description]

To verify the assertion, we need to calculate the discount using the given formula. The marked price is \$500 and the sale price is \$450. The discount can be found by subtracting the sale price from the marked price:

$Discount = Marked Price - Sale Price = 500 - 450 = 50$

Thus, the discount is \$50, which matches the assertion. The reason provided is also true as it correctly states the formula used to calculate the discount. Therefore, both the assertion and reason are true, and the reason correctly explains the assertion.

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, first calculate the discount amount:

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

Now, using the formula for discount percentage:

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

Both the Assertion and Reason are true, and the Reason correctly explains how the discount percentage is calculated.

Key Concept: Discount Percentage

b) 10%

[Solution Description]

Applying the discount percentage formula:

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

Substituting the given values:

$\text{Discount %} = \left( \frac{50 - 45}{50} \right) \times 100 = \left( \frac{5}{50} \right) \times 100 = 10\%$

The discount percentage is 10%.

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: Discount, Marked Price

c) Rs. 9,600

[Solution Description]

To find the marked price, let us denote the marked price as $MP$. The sale price (SP) is Rs. 8,640 and the discount percentage is 10%.

First, calculate the discount amount using the formula: $Discount = \frac{10}{100} \times MP$.

The sale price can be expressed as: $Sale Price = MP - Discount$.

Substituting the discount value, $8640 = MP - \left(\frac{10}{100} \times MP\right)$.

Simplifying, $8640 = MP \times \left(1 - \frac{10}{100}\right) = MP \times \frac{90}{100}$.

Solving for $MP$, $MP = \frac{8640 \times 100}{90} = Rs. 9,600$.

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: Finding Discounts

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

[Solution Description]

Step 1: Calculate the discount using the formula $Discount = Marked Price - Sale Price$. Here, $Marked Price = Rs. 500$ and $Sale Price = Rs. 400$. So, $Discount = 500 - 400 = Rs. 100$.

Step 2: Calculate the discount percentage using the formula $\frac{\text{Discount}}{\text{Marked Price}} \times 100\%$. Substituting the values, $\frac{100}{500} \times 100\% = 20\%$.

Step 3: The Assertion states that the discount percentage is 20%, which matches our calculation.

Step 4: The Reason provides the correct formula for calculating the discount percentage, which directly explains the Assertion. Hence, both Assertion and Reason are true, and the Reason correctly explains the Assertion.

Key Concept: Finding Discount, Selling Price

b) \$81.60

[Solution Description]

First, calculate the first discount: 15% of \$120.

$\text{First Discount} = \frac{15}{100} \times 120 = \$18$

Now, subtract the first discount from the marked price to get the intermediate price:

$\text{Intermediate Price} = 120 - 18 = \$102$

Next, calculate the second discount: 20% of \$102.

$\text{Second Discount} = \frac{20}{100} \times 102 = \$20.40$

Finally, subtract the second discount from the intermediate price to get the final selling price:

$\text{Final Selling Price} = 102 - 20.40 = \$81.60$

Key Concept: Type of Question

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

[Solution Description]

To verify the assertion, we need to calculate the discount percentage using the given formula: $Discount Percentage = \left(\frac{Discount}{Marked Price}\right) \times 100$. The discount amount is obtained by subtracting the selling price from the marked price, which is Rs. 500 - Rs. 400 = Rs. 100. Now, applying the formula: $Discount Percentage = \left(\frac{100}{500}\right) \times 100 = 20\%$. Hence, both the assertion and reason are true, and the reason correctly explains the assertion.

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: Discount, Percentage Estimation

b) Rs. 810

[Solution Description]

First, calculate the initial discount of 25% on the marked price. The initial discount is $\frac{25}{100} \times 1200 = Rs. 300$. So, the price after the first discount is $1200 - 300 = Rs. 900$.

Next, apply the additional 10% coupon discount on the discounted price. The second discount is $\frac{10}{100} \times 900 = Rs. 90$. Therefore, the final amount the customer pays is $900 - 90 = Rs. 810$.

Key Concept: Discount Calculation

c) Rs. 96

[Solution Description]

To find the discount amount, use the formula:

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

Given Discount % = 12% and Marked Price = Rs. 800,

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

Therefore, the discount amount is Rs. 96.

Key Concept: Discount Percentage Calculation

b) 10%

[Solution Description]

First, calculate the discount using the formula:

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

Given Marked Price = Rs. 500 and Sale Price = Rs. 450,

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

Now, calculate the discount percentage using the formula:

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

Substituting the values,

$\text{Discount %} = \left( \frac{50}{500} \right) \times 100 = 10\%$

Therefore, the discount percentage is 10%.

Key Concept: Sales Tax and GST

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

[Solution Description]

The assertion states that the bill amount of an item is always higher than its original price due to the inclusion of sales tax or GST. This is true because both sales tax and GST are additional charges levied on the original price. The reason explains that sales tax and GST are calculated as a percentage of the original price and added to it to determine the final bill amount. This is also true and correctly explains why the assertion is correct. Therefore, both the assertion and reason are true, and the reason is the correct explanation of the assertion.

Key Concept: VAT Calculation

b) Rs. 25,000

[Solution Description]

Let the original price of the refrigerator be Rs. 100.

VAT = 12%, so the price including VAT = Rs. 112

Given that the cost including VAT is Rs. 28,000

Original price = $\frac{100}{112} \times 28000$

Original price = Rs. 25,000

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: GST

a) Rs. 2119

[Solution Description]

Let the original price be Rs. 100. After adding 18% GST, the price becomes Rs. 118. So, if the price after GST is Rs. 2500, the original price = $\frac{100}{118} \times 2500$ = Rs. 2118.64 (approximately Rs. 2118.64, but rounded to Rs. 2119 for simplicity).

Key Concept: Sales Tax, GST

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

[Solution Description]

The assertion states that sales tax is always calculated on the selling price of an item and added to the bill amount. This is true as sales tax is indeed applied to the selling price and included in the final bill. The reason states that GST is a comprehensive tax levied on the supply of goods and services, replacing sales tax and VAT in India. This is also true as GST was introduced to replace multiple indirect taxes, including sales tax and VAT. However, the reason does not explain why sales tax is calculated on the selling price; it merely provides information about GST. Therefore, both statements are true, but the reason is not the correct explanation of the assertion.

Key Concept: VAT

c) Value Added Tax

[Solution Description]

Value Added Tax (VAT) is a type of tax that is included in the prices of items. This information is directly obtained from the syllabus.

Key Concept: Value Added Tax

c) Rs. 40,000

[Solution Description]

Let the original price of the laptop be $Rs. x$.

VAT is $12\%$ of Rs. x = $\frac{12}{100} \times x = 0.12x$.

The total price including VAT = Original price + VAT = \$x + 0.12x = 1.12x\$.

Given that the total price is Rs. 44,800, we have:

$1.12x = Rs. 44,800$.

To find the original price (\$x\$), divide both sides by 1.12:

$x = \frac{44,800}{1.12} = **Rs. 40,000**$.

Key Concept: Sales Tax/GST

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

[Solution Description]

Sales tax or GST is applied on the selling price of an item and is added to it to calculate the final bill amount. This means the bill amount is always greater than the selling price. Thus, the Assertion is true because the selling price is indeed less than the bill amount when taxes are applied. The Reason is also true because sales tax or GST is added to the selling price, increasing the bill amount. Moreover, the Reason correctly explains the Assertion, as the addition of tax directly causes the bill amount to be higher than the selling price.

Key Concept: Finding Sales Tax

c) \$53

[Solution Description]

Start by calculating the sales tax on the book's price.

Sales tax = $\frac{6}{100} \times 50 = 3$.

Then, add the sales tax to the original price.

Total amount payable = $50 + 3 = 53$.

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: Sales Tax Calculation

c) \$864

[Solution Description]

To find the total bill amount, we first calculate the sales tax using the formula:

$\text{Sales Tax} = \frac{\text{Tax Rate}}{100} \times \text{Selling Price}$

Here, the selling price is \$800 and the tax rate is 8%. Plugging in the values:

$\text{Sales Tax} = \frac{8}{100} \times 800 = \$64$

The total bill amount is the sum of the selling price and the sales tax:

$\text{Bill Amount} = 800 + 64 = \$864$

Key Concept: Sales Tax, GST

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: Original price = \$500, GST rate = 10%

GST = $\frac{10}{100} \times 500$ = \$50

Total bill amount = Original price + GST = \$500 + \$50 = \$550

Thus, the Assertion is true.

Reason states that GST is calculated on the original price and added to it, which is correct.

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

Key Concept: Value Added Tax, Finding Price Before Tax

b) \$2,000

[Solution Description]

Let the original price of the refrigerator be \$100. The VAT is 12%, so the price including VAT is \$112. When the price including VAT is \$2,240, the original price is calculated as follows: $\frac{100}{112} \times 2240 = \$2,000$.

Key Concept: Sales Tax

b) \$2400

[Solution Description]

Let the original price be \$100. With a 10% sales tax, the selling price becomes \$110. Now, when the selling price is \$2640, the original price can be calculated as:

$\text{Original Price} = 100 \times \frac{2640}{110} = 2400$

Key Concept: Reverse calculation to find price before tax

b) \$ 10,000

[Solution Description]

Let the original price of the laptop be \$ 100. With GST of 12%, the total price becomes \$ 112. Now, when the total price after GST is \$ 112, the original price is \$ 100. So, when the total price after GST is \$ 11,220, the original price can be calculated using the formula:

$\text{Original Price} = \frac{100 \times 11,220}{112} = \$ 10,000$

Therefore, the price of the laptop before GST was added is \$ 10,000.

Key Concept: Reverse Calculation of GST

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

[Solution Description] To find the original price before GST was added, we use the formula:

$\text{Original Price} = \frac{\text{Price Including GST}}{1 + \frac{\text{GST Percentage}}{100}}$

Here, the price including GST is \$784 and the GST percentage is 12%. Substituting these values into the formula:

$\text{Original Price} = \frac{784}{1 + \frac{12}{100}} = \frac{784}{1.12} = 700$

Therefore, the original price before GST was added is \$700. The assertion is true, and the reason correctly explains how to derive the original price using the given formula.

Key Concept: Sales Tax Calculation

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

[Solution Description] To find the total bill amount, we first calculate the sales tax. The sales tax for an item priced at Rs. 800 with a tax rate of 8% is calculated as $\frac{8}{100} \times 800 = Rs. 64$. Then, the total bill amount is the sum of the original price and the sales tax: $Rs. 800 + Rs. 64 = Rs. 864$. Therefore, the assertion is true. The reason states that sales tax is calculated on the selling price and added to the bill, which is also correct and explains the assertion.

Key Concept: Finding Sales Tax

b) Rs. 12960

[Solution Description]

First, calculate the sales tax using the formula:

$\text{Sales Tax} = \frac{8}{100} \times 12000 = Rs. 960$

Now, add the sales tax to the original cost to get the total bill amount:

$\text{Total Bill Amount} = Rs. 12000 + Rs. 960 = Rs. 12960$

Key Concept: Finding Original Price before VAT

b) \$ 12,000

[Solution Description]

Let the original price be \$ 100. VAT is 10%, so the price including VAT is \$ 110.

When the price including VAT is \$ 110, the original price is \$ 100.

Therefore, when the price including VAT is \$ 13,200, the original price is calculated as:

$\text{Original Price} = 100 \times \frac{13200}{110} = 12000$

So, the original price of the laptop before VAT was added is \$ 12,000.

Key Concept: Finding original price before VAT/GST was added

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

[Solution Description] To find the original price before GST was added, we use the formula: $\text{Original Price} = \frac{\text{Price including GST}}{1 + \frac{GST}{100}}$ Here, the price including GST is \$540 and GST is 8%. Substituting these values into the formula, we get: $\text{Original Price} = \frac{540}{1 + \frac{8}{100}} = \frac{540}{1.08} = 500$ Thus, the original price before GST was added is indeed \$500. Therefore, both the Assertion and Reason are true, and the Reason correctly explains the Assertion.

Key Concept: Compound Interest Formula

b) 1025

[Solution Description]

To find the compound interest, we use the formula:

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

Substituting the given values:

$A = 10000 \left(1 + \frac{5}{100}\right)^2 = 10000 \times \left(\frac{21}{20}\right)^2 = 10000 \times \frac{441}{400} = 11025$

The compound interest is:

$CI = A - P = 11025 - 10000 = 1025$

Key Concept: Compound Interest, Multiple Rates

c) 11909

[Solution Description]

For each year, we calculate the amount separately using the compound interest formula:

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

After the first year:

$A_1 = 10000 \left(1 + \frac{5}{100}\right) = 10000 \times 1.05 = 10500$

After the second year:

$A_2 = 10500 \left(1 + \frac{6}{100}\right) = 10500 \times 1.06 = 11130$

After the third year:

$A_3 = 11130 \left(1 + \frac{7}{100}\right) = 11130 \times 1.07 = 11909.1$

Rounding to the nearest dollar:

$A_3 \approx 11909$

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: Simple Interest vs Compound Interest

b) Rs. 54

[Solution Description]

To find the difference between simple interest (SI) and compound interest (CI), we calculate both separately.

Step 1: Calculate the simple interest:

$SI = \frac{P \times R \times T}{100} = \frac{Rs. 15,000 \times 6 \times 2}{100} = Rs. 1,800$

Step 2: Calculate the compound interest using the formula $A = P \left(1 + \frac{R}{100}\right)^n$:

$A = Rs. 15,000 \left(1 + \frac{6}{100}\right)^2 = Rs. 15,000 \times \left(\frac{53}{50}\right)^2$

Step 3: Simplify the calculation:

$A = Rs. 15,000 \times \frac{2809}{2500} = Rs. 16,854$

Step 4: Calculate the compound interest:

$CI = A - P = Rs. 16,854 - Rs. 15,000 = Rs. 1,854$

Step 5: Find the difference between CI and SI:

$Difference = CI - SI = Rs. 1,854 - Rs. 1,800 = Rs. 54$

The difference between compound interest and simple interest is Rs. 54.

Key Concept: Difference between Simple and Compound Interest

c) Rs. 76.25

[Solution Description]

Step 1: Calculate Simple Interest (SI).

$SI = \frac{P \times R \times T}{100} = \frac{10000 \times 5 \times 3}{100} = Rs. 1500$

Step 2: Calculate Compound Interest (CI).

$CI = P \left(1 + \frac{R}{100}\right)^T - P = 10000 \left(1 + \frac{5}{100}\right)^3 - 10000$

$CI = 10000 \times 1.157625 - 10000 = Rs. 1576.25$

Step 3: Find the difference between CI and SI.

$\text{Difference} = CI - SI = 1576.25 - 1500 = Rs. 76.25$

Key Concept: Compound Interest

d) \$583.20

[Solution Description]

Compound Interest formula is given by: $A = P \left(1 + \frac{r}{n}\right)^{nt}$, where $P$ is the principal, $r$ is the rate, $n$ is the number of times interest is compounded per year, and $t$ is the time. Here, $P = \$500$, $r = 8\% = 0.08$, $n = 1$ (compounded annually), and $t = 2$ years.

Substituting the values: $A = 500 \left(1 + \frac{0.08}{1}\right)^{1 \times 2} = 500 \times (1.08)^2 = 500 \times 1.1664 = \$583.20$.

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:

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

Here, $P = \$10000$, $R = 10\%$, and $n = 2$ years. Substituting these values into the formula:

$A = \$10000 \left(1 + \frac{10}{100}\right)^2 = \$10000 \left(1.1\right)^2 = \$10000 \times 1.21 = \$12100$

Thus, the assertion is true. The reason correctly explains the assertion as it provides the formula used to derive the amount.

Key Concept: Compound Interest

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

[Solution Description]

In simple interest, the principal amount does not change over the investment period. The interest is calculated only on the initial principal amount every year. Therefore, both the assertion and the reason are true, and the reason correctly explains the assertion.

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 Formula

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

[Solution Description]

To find the compound interest, we first calculate the total amount using the formula $A = P \left(1 + \frac{R}{100}\right)^n$. Then, the compound interest is obtained by subtracting the principal from this amount, i.e., $CI = A - P$. The assertion correctly states that the compound interest is given by $CI = P \left(1 + \frac{R}{100}\right)^n - P$. The reason provides the correct formula for the amount after \$n\$ years and correctly relates it to the compound interest by stating $CI = A - P$. Therefore, both the assertion and reason are true, and the reason is the correct explanation of the assertion.

Key Concept: Compound vs Simple Interest

b) Rs. 48.64

[Solution Description]

Step 1: Calculate the Simple Interest (S.I.) for 3 years:

$S.I. = Rs. 10000 \times \frac{4}{100} \times 3 = Rs. 1200$

Step 2: Calculate the Compound Interest (C.I.) for 3 years.

For Year 1:

$SI_1 = Rs. 10000 \times \frac{4}{100} = Rs. 400$

$Amount_1 = Rs. 10000 + Rs. 400 = Rs. 10400$

For Year 2:

$SI_2 = Rs. 10400 \times \frac{4}{100} = Rs. 416$

$Amount_2 = Rs. 10400 + Rs. 416 = Rs. 10816$

For Year 3:

$SI_3 = Rs. 10816 \times \frac{4}{100} = Rs. 432.64$

$Amount_3 = Rs. 10816 + Rs. 432.64 = Rs. 11248.64$

Step 3: Calculate the Compound Interest (C.I.) earned over 3 years:

$C.I. = Rs. 11248.64 - Rs. 10000 = Rs. 1248.64$

Step 4: Calculate the difference between C.I. and S.I.:

$Difference = Rs. 1248.64 - Rs. 1200 = Rs. 48.64$

The difference between Compound Interest and Simple Interest at the end of 3 years is Rs. 48.64.

Key Concept: Compound Interest, Multi-step Calculation

b) \$2865.24

[Solution Description]

To calculate the compound interest, we need to find the interest for each year and then sum it up. Let’s break it down step by step.

Step 1: Find the interest for the first year.

Principal for the first year ($P_1$) = \$15,000

Interest for the first year ($SI_1$) = $15000 \times \frac{6}{100} = 900$

Step 2: Find the principal for the second year.

Principal for the second year ($P_2$) = $P_1 + SI_1 = 15000 + 900 = 15900$

Step 3: Find the interest for the second year.

Interest for the second year ($SI_2$) = $15900 \times \frac{6}{100} = 954$

Step 4: Find the principal for the third year.

Principal for the third year ($P_3$) = $P_2 + SI_2 = 15900 + 954 = 16854$

Step 5: Find the interest for the third year.

Interest for the third year ($SI_3$) = $16854 \times \frac{6}{100} = 1011.24$

Step 6: Calculate the total compound interest.

Total Compound Interest = $SI_1 + SI_2 + SI_3 = 900 + 954 + 1011.24 = 2865.24$

Therefore, the total compound interest earned over 3 years is \$2865.24.

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

b) \$59,550.80

[Solution Description]

Step 1: Calculate the amount after the first year. Principal for the first year ($P_1$) = \$50,000. Interest for the first year ($SI_1$) = \$50,000 $\times$ $\frac{6}{100}$ = \$3,000. Amount at the end of the first year = \$50,000 + \$3,000 = \$53,000.

Step 2: Calculate the amount after the second year. Principal for the second year ($P_2$) = \$53,000. Interest for the second year ($SI_2$) = \$53,000 $\times$ $\frac{6}{100}$ = \$3,180. Amount at the end of the second year = \$53,000 + \$3,180 = \$56,180.

Step 3: Calculate the amount after the third year. Principal for the third year ($P_3$) = \$56,180. Interest for the third year ($SI_3$) = \$56,180 $\times$ $\frac{6}{100}$ = \$3,370.80. Amount at the end of the third year = \$56,180 + \$3,370.80 = \$59,550.80.

Final amount after 3 years is \$59,550.80.

Key Concept: Compound Interest

b) Rs. 1248.64[Solution Description]To find the total interest earned, we calculate the amount year by year and then subtract the principal.

Step 1: Calculate the interest for the first year.

$SI1 = 10000 \times \frac{4}{100} = Rs. 400$

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

$Amount1 = 10000 + 400 = Rs. 10400$

Step 3: Calculate the interest for the second year.

$SI2 = 10400 \times \frac{4}{100} = Rs. 416$

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

$Amount2 = 10400 + 416 = Rs. 10816$

Step 5: Calculate the interest for the third year.

$SI3 = 10816 \times \frac{4}{100} = Rs. 432.64$

Step 6: Find the amount at the end of the third year.

$Amount3 = 10816 + 432.64 = Rs. 11248.64$

Step 7: Calculate the total interest earned.

$CI = 11248.64 - 10000 = Rs. 1248.64$

The total interest earned is Rs. 1248.64.

Key Concept: Compound Interest Calculation

c) Rs. 11,025

[Solution Description]

To find the total amount, we use the compound interest formula: $A = P \times \left(1 + \frac{R}{100}\right)^n$. Here, $P = Rs. 10,000$, $R = 5\%$, and $n = 2$ years. Plugging in the values, we get:

$A = 10000 \times \left(1 + \frac{5}{100}\right)^2 = 10000 \times \left(1.05\right)^2 = 10000 \times 1.1025 = Rs. 11,025$

Key Concept: Simple vs Compound Interest

c) Rs. 88.13

[Solution Description]

First, calculate the simple interest using the formula: $SI = P \times R \times T / 100$. Here, $P = Rs. 8,000$, $R = 6\%$, and $T = 3$ years.

$SI = 8000 \times 6 \times 3 / 100 = Rs. 1,440$

Next, calculate the compound interest using the formula: $A = P \times \left(1 + \frac{R}{100}\right)^n$. Here, $n = 3$ years.

$A = 8000 \times \left(1 + \frac{6}{100}\right)^3 = 8000 \times 1.191016 = Rs. 9,528.13$

The compound interest is $A - P = Rs. 9,528.13 - Rs. 8,000 = Rs. 1,528.13$. The difference between CI and SI is $Rs. 1,528.13 - Rs. 1,440 = Rs. 88.13$.

Key Concept: Compound Interest, Simple Interest

a) \$9.86

[Solution Description]

To find the difference between compound interest (CI) and simple interest (SI) over 3 years:

1. Simple Interest for 3 years = $P \times r \times t$ where $P = 500$, $r = 0.08$, $t = 3$.

$SI = 500 \times 0.08 \times 3 = 120$

2. Compound Interest for 3 years = $P \times (1 + r)^t - P$

$CI = 500 \times (1 + 0.08)^3 - 500 = 629.856 - 500 = 129.856$

3. Difference = $CI - SI = 129.856 - 120 = 9.856$

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, Principal Change

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

[Solution Description]

Under simple interest, the principal amount remains the same throughout the investment period. The interest is calculated only on the original principal for each year. However, in compound interest, the principal changes annually because the interest earned in one year is added to the principal for the next year. Therefore, the assertion that the principal remains constant under simple interest but changes annually under compound interest is true. The reason provided is also correct because compound interest is indeed calculated on the previous year's amount, which includes both the principal and the accrued interest. Hence, the reason correctly explains the assertion.

Key Concept: Compound Interest

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

[Solution Description]

In simple interest, the interest is calculated only on the original principal amount, and it remains constant every year. However, in compound interest, the interest is calculated on the principal plus the accumulated interest from previous periods. This means that each year, the principal for calculating interest increases, leading to a faster growth of the total amount. Therefore, both the assertion and the reason are true, and the reason correctly explains why the assertion is true.

Key Concept: Compound Interest, Simple Interest

c) \$15.25

[Solution Description]

Step 1: Calculate Simple Interest (SI).

SI = Principal × Rate × Time / 100

SI = \$2000 × 5 × 3 / 100 = \$300

Step 2: Calculate Compound Interest (CI).

CI = Principal × (1 + Rate/100)$^{Time}$ - Principal

CI = \$2000 × (1 + 5/100)$^3$ - \$2000

CI = \$2000 × (1.05)$^3$ - \$2000

CI = \$2000 × 1.157625 - \$2000

CI = \$2315.25 - \$2000 = \$315.25

Step 3: Find the difference between CI and SI.

Difference = CI - SI = \$315.25 - \$300 = \$15.25

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

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

[Solution Description]

The assertion states that the compound interest formula is derived from the simple interest formula, which is true because the compound interest formula $A = P \left(1 + \frac{R}{100}\right)^n$ can be seen as an extension of the simple interest formula where interest is added to the principal periodically. The reason given is that compound interest is calculated on both the initial principal and the accumulated interest of previous periods, which is also true and correctly explains why the compound interest formula is structured the way it is. Therefore, both the assertion and reason are true, and the reason correctly explains the assertion.

Key Concept: Compound Interest, Advanced Reasoning

b) 10210.20

[Solution Description]

First, calculate the total amount after 5 years using the compound interest formula:

$A = P \left(1 + \frac{R}{100}\right)^n = 20000 \left(1 + \frac{10}{100}\right)^5 = 20000 \times (1.10)^5$

Compute $(1.10)^5$:

$(1.10)^5 = 1.61051$

Multiply by the principal:

$A = 20000 \times 1.61051 = 32210.20$

The compound interest after 5 years is:

$CI = A - P = 32210.20 - 20000 = 12210.20$

Now, calculate the simple interest for the 5th year:

$SI = \frac{P \times R \times T}{100} = \frac{20000 \times 10 \times 1}{100} = 2000$

The difference between the compound interest in the 5th year and the simple interest in the 5th year is:

$\text{Difference} = CI - SI = 12210.20 - 2000 = 10210.20$

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, 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 vs Simple Interest

d) Compound interest is calculated on the initial principal and accumulated interest.

[Solution Description]

Compound interest is calculated on the initial principal as well as the accumulated interest of previous periods, whereas simple interest is calculated only on the initial principal. Therefore, compound interest typically results in a higher amount over time compared to simple interest, assuming the same principal, rate, and time period.

Key Concept: Compound Interest Formula

b) Rs. 1528.13

[Solution Description]

To find the compound interest, we use the formula:

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

Here, $P = 8000$, $R = 6$, and $n = 3$.

Step 1: Calculate the amount after 3 years.

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

Step 2: Simplify the expression inside the parentheses.

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

Step 3: Raise 1.06 to the power of 3.

$1.06^3 = 1.191016$

Step 4: Multiply by the principal.

$A = 8000 \times 1.191016 = 9528.128$

Step 5: Subtract the principal to find the compound interest.

$CI = 9528.128 - 8000 = 1528.128$

The compound interest is approximately Rs. 1528.13.

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 Calculation

c) Rs. 1800

[Solution Description]

We first calculate the amount using the formula $A = P \left(1 + \frac{R}{100}\right)^n$. Here, $P = Rs. 15000$, $R = 12\%$, and $n = 1$ year.

Step 1: Substitute the values into the formula: $A = 15000 \left(1 + \frac{12}{100}\right)^1$.

Step 2: Calculate $1 + \frac{12}{100} = 1.12$.

Step 3: Raise 1.12 to the power of 1: $1.12^1 = 1.12$.

Step 4: Multiply by the principal: $A = 15000 \times 1.12 = Rs. 16800$.

Step 5: Calculate the compound interest: $CI = A - P = 16800 - 15000 = Rs. 1800$.

Key Concept: Compound Interest, Application of Formula

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

[Solution Description] To verify the correctness of the assertion and reason, let's analyze the formulas provided. The compound interest ($CI$) is indeed calculated as the difference between the amount ($A$) and the principal ($P$). The amount $A$ is given by the formula $A = P \left(1 + \frac{R}{100}\right)^n$. Therefore, the compound interest can be expressed as $CI = A - P = P \left(1 + \frac{R}{100}\right)^n - P$. This confirms that both the assertion and the reason are true, and the reason correctly explains the assertion.

Key Concept: Compound Interest, Amount Deduction

d) Assertion is false, but Reason is true.

[Solution Description]

To solve this, we need to understand the relationship between simple interest and compound interest. Simple interest is calculated only on the principal amount using the formula $SI = P \times R \times T / 100$. On the other hand, compound interest is calculated on the principal plus any previously earned interest. This means that each year's interest is added to the principal before calculating the next year's interest. Therefore, the formula for compound interest cannot be directly derived from the simple interest formula because it involves compounding over multiple periods. The assertion is false because simple interest does not account for compounding, which is essential for compound interest. However, the reason is true as it correctly describes how compound interest is calculated.

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: Value of an Item

b) Rs.583,200

[Solution Description]

For depreciation, we use the formula:

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

Here, $P = 800000$, $R = 10$, and $n = 3$ years.

$A = 800000 \left(1 - \frac{10}{100}\right)^3 = 800000 \times \left(\frac{90}{100}\right)^3$

Calculating step by step:

$\left(\frac{90}{100}\right)^3 = \frac{90}{100} \times \frac{90}{100} \times \frac{90}{100} = \frac{729000}{1000000} = 0.729$

Now multiply by the principal:

$A = 800000 \times 0.729 = 583200$

The value of the car after 3 years is Rs.583,200.

Key Concept: Compound Interest Calculation

b) Rs.2,496

[Solution Description]

We use the compound interest formula:

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

Here, $P = 15000$, $R = 8$, and $n = 2$ years.

$A = 15000 \left(1 + \frac{8}{100}\right)^2 = 15000 \times \left(\frac{108}{100}\right)^2$

Calculating step by step:

$\left(\frac{108}{100}\right)^2 = \frac{108}{100} \times \frac{108}{100} = \frac{11664}{10000} = 1.1664$

Now multiply by the principal:

$A = 15000 \times 1.1664 = 17496$

The compound interest is calculated as:

$CI = A - P = 17496 - 15000 = 2496$

The compound interest is Rs.2,496.

Key Concept: Bacteria Growth

b) 5408

[Solution Description]

Using the compound interest formula for growth:

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

Here, $P = 5000$, $r = 4$, and $n = 2$. Substituting these values:

$A = 5000 \times \left(1 + \frac{4}{100}\right)^2 = 5000 \times \left(1.04\right)^2 = 5000 \times 1.0816 = 5408$

The number of bacteria after 2 hours is 5408.

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: 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, Compound Interest

b) 56,275

[Solution Description]

To find the population at the end of 2025, we use the compound interest formula:

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

Here, $P = 50000$, $r = 3$, and $n = 4$ (from 2021 to 2025). Plugging in the values:

$A = 50000 \left(1 + \frac{3}{100}\right)^4 = 50000 \times (1.03)^4$

Calculating $(1.03)^4$:

$(1.03)^4 = 1.12550881$

Therefore:

$A = 50000 \times 1.12550881 = 56275.44$

Rounding off, the population at the end of 2025 is approximately 56,275.

Key Concept: Population Growth

b) 53,045

[Solution Description]

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

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

Where $P_0 = 50000$, $r = 3$, and $n = 2$.

$P = 50000 \left(1 + \frac{3}{100}\right)^2 = 50000 \times \left(\frac{103}{100}\right)^2 = 50000 \times 1.0609 = 53045$

So, the population at the end of 2022 will be 53,045.

Key Concept: Compound Interest, Population Growth

b) 24952

[Solution Description]

First, calculate the population after the first three years with a 2% decrease each year:

$P_1 = 25000 \left(1 - \frac{2}{100}\right)^3 = 25000 \times 0.98^3 = 25000 \times 0.941192 = 23529.8$

Next, calculate the population for the next two years with a 3% increase each year:

$P_2 = 23529.8 \left(1 + \frac{3}{100}\right)^2 = 23529.8 \times 1.03^2 = 23529.8 \times 1.0609 = 24951.6$

The estimated population at the end of 2020 is approximately 24952.

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: Population Increase

b) 97,332

[Solution Description]

To find the population at the end of 2015, we use the compound interest formula:

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

where $P_0 = 80,000$, $r = 4\%$, and $n = 5$ years.

Substituting the values:

$P = 80,000 \left(1 + \frac{4}{100}\right)^5 = 80,000 \left(1.04\right)^5$

Calculating $1.04^5$:

$1.04^5 \approx 1.2166529$

Therefore,

$P \approx 80,000 \times 1.2166529 = 97,332.23$

The estimated population at the end of 2015 is approximately 97,332.

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, Compound Interest

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

[Solution Description]

The given problem involves calculating the depreciated value of machinery using the compound interest formula for depreciation. The assertion states that the machinery's value after one year will be $\$14,100$, and the reason provides the formula used for this calculation.

Step 1: Identify the principal amount ($P$) and the depreciation rate ($r$). Here, $P = \$15,000$ and $r = 6\%$.

Step 2: Apply the formula $V = P \left(1 - \frac{r}{100}\right)$. Substitute the values:

$V = 15000 \left(1 - \frac{6}{100}\right)$

Step 3: Simplify the expression inside the parentheses:

$1 - \frac{6}{100} = 1 - 0.06 = 0.94$

Step 4: Multiply to find the final value:

$V = 15000 \times 0.94 = 14100$

Thus, the machinery's value after one year is indeed $\$14,100$. Both the assertion and reason are correct, and the reason correctly explains the assertion.

Key Concept: Decrease in value over time

b) \$21,109

[Solution Description]

The equipment's value decreases at a compound rate of 12% per year. To find its value after 5 years, we use the formula:

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

where $P = 40000$, $r = 12$, and $n = 5$.

Substituting the values:

$V = 40000 \left(1 - \frac{12}{100}\right)^5 = 40000 \times (0.88)^5$

Calculating $(0.88)^5$:

$0.88 \times 0.88 = 0.7744 \\ \\ 0.7744 \times 0.88 = 0.681472 \\ \\ 0.681472 \times 0.88$

$= 0.59969536 \\ \\ 0.59969536 \times 0.88 = 0.5277319168$

Now multiply by the principal:

$V = 40000 \times 0.5277319168 = 21109.27667$

So, the equipment's value after 5 years is approximately \$21,109.

Key Concept: Bacterial Growth, Depreciation

a) 64,000

[Solution Description]

The initial count of bacteria is 1,00,000. The decrease rate is 20% per hour. To find the count after 2 hours, we use the formula for compound decrease:

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

Here, $P = 1,00,000$, $r = 20$, and $n = 2$. Substituting the values:

$\text{Count} = 1,00,000 \times \left(1 - \frac{20}{100}\right)^2 = 1,00,000 \times \left(0.8\right)^2$

Calculating $\left(0.8\right)^2$:

$0.8 \times 0.8 = 0.64$

Now multiply by the initial count:

$1,00,000 \times 0.64 = 64,000$

Hence, the count of bacteria after 2 hours will be 64,000.

Key Concept: Compound Interest Application

b) 2,92,465

[Solution Description]

Given:

Initial count $P = 2,50,000$

Rate of increase $r = 4\%$ per half hour

Time $t = 2$ hours (which is 4 half-hour intervals)

The formula for the final amount $A$ is:

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

Substituting the values:

$A = 2,50,000 \left(1 + \frac{4}{100}\right)^4$

$A = 2,50,000 \times (1.04)^4$

$A = 2,50,000 \times 1.16985856$

$A = 2,92,464.64$

Thus, the bacterial count after 2 hours is approximately 2,92,465.

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: Bacterial Growth, Compound Interest Formula

b) 15,869

[Solution Description]

The initial number of bacteria is 10,000, and the rate of growth is 8% per hour. The time period is 6 hours. Using the compound interest formula:

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

where:

- $P = 10,000$ (initial number of bacteria)

- $r = 8$ (rate of growth)

- $n = 6$ (number of hours)

Plugging in the values:

$A = 10000 \left(1 + \frac{8}{100}\right)^6 = 10000 \left(1.08\right)^6$

Now, calculate $(1.08)^6$:

$(1.08)^6 = 1.586874322$

Therefore:

$A = 10000 \times 1.586874322 = 15868.74322$

The estimated number of bacteria after 6 hours is approximately 15,869.

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) 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: Finding future population

c) 11,041

[Solution Description]

Using the compound interest formula:

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

Here, $P = 10000$, $r = 2$, and $n = 5$. Substituting these values:

$\text{Future Population} = 10000 \left(1 + \frac{2}{100}\right)^5 = 10000 \times \left(\frac{102}{100}\right)^5$

Calculating step by step:

$\frac{102}{100} = 1.02$

$1.02^5 = 1.10408$

$10000 \times 1.10408 = 11040.8$

Hence, the population after 5 years is approximately 11,041.

Key Concept: Depreciation

b) \$704

[Solution Description]

To find the value of the bicycle after one year, we apply the depreciation formula:

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

Here, $P$ is the principal amount (\$800), and $r$ is the rate of depreciation (12%).

Substituting the values:

$\text{Final Value} = 800 \times \left(1 - \frac{12}{100}\right) = 800 \times 0.88 = \$704$

So, the value of the bicycle after one year is \$704.

Key Concept: Depreciation, Compound Interest

b) \$35,820

[Solution Description]

To calculate the value of the machine after 4 years with an annual depreciation rate of 8%, we apply the compound depreciation formula:

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

Substituting the given values:

$\text{Final Value} = \$50,000 \times \left(1 - \frac{8}{100}\right)^4$

$\text{Final Value} = \$50,000 \times \left(\frac{92}{100}\right)^4$

$\text{Final Value} = \$50,000 \times 0.71639296$

$\text{Final Value} = \$35,819.65 \approx \$35,820$