Chapter 8 Comparing Quantities

Here is the step-by-step solution:

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1. The ratio of the number of girls to the number of boys in the class?

Given:

Number of girls = 18

Percentage of girls = 60% of the total number of students.

Let the total number of students in the class be $x$.

According to the question,

$60\%$ of $x = 18$

$\implies \frac{60}{100} \times x = 18$

$\implies x = \frac{18 \times 100}{60}$

$\implies x = \frac{1800}{60} = 30$

So, the total number of students is 30.

Now, we can find the number of boys:

Number of boys = Total number of students - Number of girls

Number of boys = $30 - 18 = 12$

The ratio of the number of girls to the number of boys is:

Ratio = $\frac{\text{Number of girls}}{\text{Number of boys}} = \frac{18}{12}$

Simplifying the fraction:

Ratio = $\frac{\cancel{18}^{3}}{\cancel{12}_{2}} = \frac{3}{2}$

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

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2. The cost per head if two teachers are also going with the class?

First, we need to calculate the total cost of the picnic.

Total cost = Total cost of transport + Total cost of refreshments

Calculating the transport cost:

The picnic site is 55 km from the school. The transport will travel to the site and back.

Total distance to be covered = $55 \text{ km} + 55 \text{ km} = 110 \text{ km}$.

The rate of charging is $\textsf{₹} \, 12$ per km.

Total cost of transport = $110 \times 12 = \textsf{₹} \, 1320$.

Calculating the total cost:

Total cost of refreshments = $\textsf{₹} \, 4280$ (Given)

Total cost of the picnic = $\textsf{₹} \, 1320 + \textsf{₹} \, 4280 = \textsf{₹} \, 5600$.

Calculating the cost per head:

Total number of people going on the picnic:

Total number of students = 30

Number of teachers = 2

Total number of people = $30 + 2 = 32$.

Cost per head = $\frac{\text{Total cost of the picnic}}{\text{Total number of people}}$

Cost per head = $\frac{5600}{32} = \textsf{₹} \, 175$.

Therefore, the cost per head is $\textsf{₹} \, 175$.

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3. If their first stop is at a place 22 km from the school, what per cent of the total distance of 55 km is this? What per cent of the distance is left to be covered?

Total distance to the picnic site = 55 km.

Distance covered to the first stop = 22 km.

Percentage of the distance covered:

Percentage covered = $(\frac{\text{Distance covered}}{\text{Total distance}}) \times 100 \%$

Percentage covered = $(\frac{22}{55}) \times 100 \%$

Percentage covered = $(\frac{2}{5}) \times 100 \% = 40 \%$

So, 40% of the total distance is covered.

Percentage of the distance left:

Distance left to be covered = Total distance - Distance covered

Distance left = $55 - 22 = 33$ km.

Percentage left = $(\frac{\text{Distance left}}{\text{Total distance}}) \times 100 \%$

Percentage left = $(\frac{33}{55}) \times 100 \%$

Percentage left = $(\frac{3}{5}) \times 100 \% = 60 \%$

So, 60% of the total distance is left to be covered.

Alternate Method for percentage left:

Percentage left = $100\% - \text{Percentage covered}$

Percentage left = $100\% - 40\% = 60\%$

To find the ratio between two quantities, they must be in the same units. The ratio is expressed as a fraction in its simplest form.

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(a) Speed of a cycle 15 km per hour to the speed of scooter 30 km per hour.

The units are already the same (km per hour).

Ratio = Speed of cycle : Speed of scooter

Simplify the fraction by dividing the numerator and denominator by their greatest common divisor, 15.

The ratio can be written as 1:2.

The ratio is 1:2.

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(b) 5 m to 10 km

The units are different (meters and kilometers). We need to convert them to the same unit. Let's convert kilometers to meters, since 1 km = 1000 m.

10 km = $10 \times 1000 \text{ m} = 10000 \text{ m}$.

Now find the ratio of 5 m to 10000 m.

Ratio = 5 m : 10000 m

Simplify the fraction by dividing the numerator and denominator by 5.

The ratio can be written as 1:2000.

The ratio is 1:2000.

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(c) 50 paise to ₹ 5

The units are different (paise and rupees). We need to convert them to the same unit. Let's convert rupees to paise, since $\textsf{₹}$ 1 = 100 paise.

$\textsf{₹}$ 5 = $5 \times 100 \text{ paise} = 500 \text{ paise}$.

Now find the ratio of 50 paise to 500 paise.

Ratio = 50 paise : 500 paise

Simplify the fraction by dividing the numerator and denominator by 50.

The ratio can be written as 1:10.

The ratio is 1:10.

To convert a ratio to a percentage, first write the ratio as a fraction. Then, multiply the fraction by 100%.

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(a) 3 : 4

The ratio 3:4 can be written as the fraction $\frac{3}{4}$.

Percentage = $\frac{3}{4} \times 100\%$

Percentage = $3 \times \frac{\cancel{100}^{25}}{\cancel{4}_1}\%$

Percentage = $3 \times 25\% = 75\%$

The percentage is 75%.

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(b) 2 : 3

The ratio 2:3 can be written as the fraction $\frac{2}{3}$.

Percentage = $\frac{2}{3} \times 100\%$

Percentage = $\frac{200}{3}\%$

To express this as a mixed number or decimal:

$\frac{200}{3} = 66 \frac{2}{3}$

$\frac{200}{3} \approx 66.66...$

The percentage is $\frac{200}{3}\%$ or $66\frac{2}{3}\%$ or approximately 66.67%.

Here is the step-by-step solution:

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Given:

Total number of students = 25.

Percentage of students interested in mathematics = 72%.

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To Find:

The number of students who are not interested in mathematics.

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Solution:

First, we will find the number of students who are interested in mathematics.

Number of students interested in mathematics = 72% of 25

$\implies \frac{72}{100} \times 25$

$\implies \frac{72}{\cancel{100}_4} \times \cancel{25}^1$

$\implies \frac{72}{4}$

$\implies 18$

So, 18 students are interested in mathematics.

Now, we can find the number of students who are not interested in mathematics by subtracting the number of interested students from the total number of students.

Number of students not interested in mathematics = Total students - Number of students interested in mathematics

$\implies 25 - 18$

$\implies 7$

Therefore, 7 students are not interested in mathematics.

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Alternate Solution:

First, we find the percentage of students who are not interested in mathematics.

Percentage of students not interested = Total percentage - Percentage of students interested

$\implies 100\% - 72\%$

$\implies 28\%$

Now, we can find the number of students who are not interested in mathematics by calculating 28% of the total number of students.

Number of students not interested in mathematics = 28% of 25

$\implies \frac{28}{100} \times 25$

$\implies \frac{28}{\cancel{100}_4} \times \cancel{25}^1$

$\implies \frac{28}{4}$

$\implies 7$

Therefore, 7 students are not interested in mathematics.

Given:

Number of matches won = 10

Win percentage = 40%

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To Find:

Total number of matches played.

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Solution:

Let the total number of matches played by the team be $M$.

The win percentage is given by the formula:

Win Percentage = $\frac{\text{Number of matches won}}{\text{Total number of matches played}} \times 100\%$

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We are given that the win percentage is 40% and the number of matches won is 10.

We can write 40% as $\frac{40}{100}$.

Now, we can solve for $M$. Cross-multiply:

Divide both sides by 40:

The team played 25 matches in all.

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Alternate Solution:

Let the total number of matches played be $M$.

40% of the total matches played were won, and the number of matches won is 10.

So, 40% of $M$ is equal to 10.

Multiply both sides by $\frac{5}{2}$:

The team played 25 matches in all.

Here is the step-by-step solution:

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Given:

Amount of money Chameli had left = $\textsf{₹} \, 600$.

Percentage of money she spent = 75%.

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To Find:

The total amount of money she had in the beginning.

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Solution:

Let the total amount of money Chameli had in the beginning be $\textsf{₹} \, x$.

The percentage of money she spent is 75%.

So, the percentage of money she had left is:

$100\% - 75\% = 25\%$

This means that 25% of her initial money is equal to the $\textsf{₹} \, 600$ she had left.

According to the question,

$25\%$ of $x = 600$

$\implies \frac{25}{100} \times x = 600$

$\implies \frac{1}{4} \times x = 600$

$\implies x = 600 \times 4$

$\implies x = 2400$

Therefore, Chameli had $\textsf{₹} \, 2400$ in the beginning.

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Alternate Solution:

Let the total amount of money Chameli had in the beginning be $\textsf{₹} \, x$.

Amount of money spent = 75% of $x = \frac{75}{100}x = 0.75x$.

Amount of money left = Total money - Money spent

We are given that the money left is $\textsf{₹} \, 600$.

So, we can write the equation:

$x - 0.75x = 600$

$\implies 0.25x = 600$

$\implies x = \frac{600}{0.25}$

$\implies x = \frac{600}{\frac{25}{100}}$

$\implies x = \frac{600 \times 100}{25}$

$\implies x = 600 \times 4$

$\implies x = 2400$

Thus, Chameli had $\textsf{₹} \, 2400$ in the beginning.

Here is the step-by-step solution:

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Given:

Percentage of people who like cricket = 60%.

Percentage of people who like football = 30%.

Total number of people in the city = 50 lakh = 50,00,000.

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To Find:

1. The percentage of people who like other games.

2. The exact number of people who like each type of game.

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Solution:

Part 1: Percentage of people who like other games

The total percentage of people must be 100%.

The percentage of people who like either cricket or football is the sum of their individual percentages:

$60\% + 30\% = 90\%$

The remaining percentage represents the people who like other games. This can be calculated by subtracting the percentage of people who like cricket and football from the total percentage.

Percentage of people who like other games = $100\% - 90\%$

$\implies 10\%$

So, 10% of the people like other games.

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Part 2: Exact number of people who like each type of game

The total number of people is 50 lakh (50,00,000).

Number of people who like cricket:

This is 60% of 50,00,000.

Number = $\frac{60}{100} \times 50,00,000$

$\implies 60 \times 50,000$

$\implies 30,00,000$ or 30 lakh.

Number of people who like football:

This is 30% of 50,00,000.

Number = $\frac{30}{100} \times 50,00,000$

$\implies 30 \times 50,000$

$\implies 15,00,000$ or 15 lakh.

Number of people who like other games:

This is 10% of 50,00,000.

Number = $\frac{10}{100} \times 50,00,000$

$\implies 10 \times 50,000$

$\implies 5,00,000$ or 5 lakh.

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Final Answer:

The percentage of people who like other games is 10%.

The number of people who like each type of game is as follows:

Cricket: 30 lakh

Football: 15 lakh

Other games: 5 lakh

Here is the step-by-step solution:

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Given:

The price of the scooter last year (original price) = $\textsf{₹} \, 34,000$.

Percentage increase in price = 20%.

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To Find:

The price of the scooter now (new price).

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Solution:

We can find the new price by first calculating the amount of the increase and then adding it to the original price.

Step 1: Calculate the increase in price.

Increase in price = 20% of $\textsf{₹} \, 34,000$

$\implies \frac{20}{100} \times 34,000$

$\implies \frac{1}{5} \times 34,000$

$\implies \textsf{₹} \, 6,800$

So, the price of the scooter has increased by $\textsf{₹} \, 6,800$.

Step 2: Calculate the new price.

New price = Original price + Increase in price

$\implies \textsf{₹} \, 34,000 + \textsf{₹} \, 6,800$

$\implies \textsf{₹} \, 40,800$

Therefore, the price of the scooter now is $\textsf{₹} \, 40,800$.

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Alternate Solution:

The original price represents 100%. A 20% increase means the new price is $100\% + 20\% = 120\%$ of the original price.

We can calculate the new price directly.

New price = 120% of the original price

$\implies 120\%$ of $\textsf{₹} \, 34,000$

$\implies \frac{120}{100} \times 34,000$

$\implies 1.2 \times 34,000$

$\implies \textsf{₹} \, 40,800$

Thus, the price of the scooter now is $\textsf{₹} \, 40,800$.

Here is the step-by-step solution:

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Given:

The marked price (MP) of the item = $\textsf{₹} \, 840$.

The selling price (SP) of the item = $\textsf{₹} \, 714$.

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To Find:

1. The discount amount.

2. The discount percentage (Discount %).

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Solution:

1. Calculating the Discount Amount

The discount is the reduction in price from the marked price to the selling price. The formula for the discount is:

Discount = Marked Price (MP) - Selling Price (SP)

Substituting the given values:

Discount = $\textsf{₹} \, 840 - \textsf{₹} \, 714$

Discount = $\textsf{₹} \, 126$

So, the discount on the item is $\textsf{₹} \, 126$.

2. Calculating the Discount Percentage

The discount percentage is always calculated on the marked price. The formula for the discount percentage is:

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

Substituting the values we have:

Discount % = $\left( \frac{126}{840} \right) \times 100$

Now, we simplify the expression:

Discount % = $\frac{12600}{840}$

Discount % = $\frac{1260}{84}$

Dividing both numerator and denominator by 12:

Discount % = $\frac{105}{7}$

Discount % = $15$

So, the discount percentage is 15%.

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Final Answer:

The discount on the item is $\textsf{₹} \, 126$ and the discount percentage is 15%.

Here is the step-by-step solution:

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Given:

The list price (or Marked Price, MP) of the frock = $\textsf{₹} \, 220$.

Discount percentage on sales = 20%.

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To Find:

1. The amount of discount on the frock.

2. The sale price (SP) of the frock.

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Solution:

1. Calculating the Amount of Discount

The discount is calculated as a percentage of the list price.

Discount = 20% of List Price

Discount = $20\% \times \textsf{₹} \, 220$

$\implies \frac{20}{100} \times 220$

$\implies \frac{1}{5} \times 220$

$\implies \frac{220}{5}$

$\implies \textsf{₹} \, 44$

So, the amount of discount on the frock is $\textsf{₹} \, 44$.

2. Calculating the Sale Price

The sale price is the price after the discount has been subtracted from the list price.

Sale Price (SP) = List Price (MP) - Discount

SP = $\textsf{₹} \, 220 - \textsf{₹} \, 44$

SP = $\textsf{₹} \, 176$

So, the sale price of the frock is $\textsf{₹} \, 176$.

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Alternate Solution:

If a discount of 20% is given, the customer pays $100\% - 20\% = 80\%$ of the list price.

So, we can directly calculate the sale price.

Sale Price (SP) = 80% of List Price

SP = $80\% \times \textsf{₹} \, 220$

$\implies \frac{80}{100} \times 220$

$\implies \frac{4}{5} \times 220$

$\implies 4 \times 44$

$\implies \textsf{₹} \, 176$

Now, we can find the discount amount:

Discount = List Price - Sale Price

Discount = $\textsf{₹} \, 220 - \textsf{₹} \, 176 = \textsf{₹} \, 44$.

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Final Answer:

The amount of discount is $\textsf{₹} \, 44$ and the sale price of the frock is $\textsf{₹} \, 176$.

Here is the step-by-step solution:

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Given:

Purchase price of the refrigerator = $\textsf{₹} \, 2,500$.

Amount spent on repairs (overhead expenses) = $\textsf{₹} \, 500$.

Selling price (SP) of the refrigerator = $\textsf{₹} \, 3,300$.

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To Find:

The loss or gain percentage.

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Solution:

First, we need to calculate the total Cost Price (CP) of the refrigerator. The total cost price includes the purchase price plus any overhead expenses like repairs.

Total Cost Price (CP) = Purchase Price + Repair Costs

CP = $\textsf{₹} \, 2,500 + \textsf{₹} \, 500$

CP = $\textsf{₹} \, 3,000$

Now, we compare the Selling Price (SP) with the Cost Price (CP) to determine if there was a gain or a loss.

SP = $\textsf{₹} \, 3,300$

CP = $\textsf{₹} \, 3,000$

Since the Selling Price (SP) is greater than the Cost Price (CP), Sohan made a gain (profit).

Next, we calculate the amount of gain.

Gain = Selling Price (SP) - Cost Price (CP)

Gain = $\textsf{₹} \, 3,300 - \textsf{₹} \, 3,000$

Gain = $\textsf{₹} \, 300$

Finally, we calculate the gain percentage. The gain percentage is always calculated on the Cost Price.

The formula for gain percentage is:

Gain % = $\left( \frac{\text{Gain}}{\text{Cost Price}} \right) \times 100$

Substituting the values:

Gain % = $\left( \frac{300}{3000} \right) \times 100$

$\implies \frac{1}{10} \times 100$

$\implies 10\%$

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Final Answer:

Sohan made a gain of 10% on the sale of the refrigerator.

Here is the step-by-step solution:

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Given:

Number of bulbs purchased = 200.

Cost price of each bulb = $\textsf{₹} \, 10$.

Number of fused bulbs = 5.

Selling price of each remaining bulb = $\textsf{₹} \, 12$.

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To Find:

The gain or loss percentage.

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Solution:

First, we need to calculate the total Cost Price (CP) of all the bulbs.

Total Cost Price (CP) = Number of bulbs purchased $\times$ Cost price of each bulb

CP = $200 \times \textsf{₹} \, 10$

CP = $\textsf{₹} \, 2000$

Next, we determine how many bulbs were actually sold.

Number of fused bulbs = 5.

Number of bulbs sold = Total bulbs - Fused bulbs

Number of bulbs sold = $200 - 5 = 195$

Now, we calculate the total Selling Price (SP) from the sale of the remaining bulbs.

Total Selling Price (SP) = Number of bulbs sold $\times$ Selling price of each bulb

SP = $195 \times \textsf{₹} \, 12$

SP = $\textsf{₹} \, 2340$

To determine if there was a gain or a loss, we compare the Selling Price (SP) with the Cost Price (CP).

SP = $\textsf{₹} \, 2340$

CP = $\textsf{₹} \, 2000$

Since SP > CP, the shopkeeper made a gain (profit).

Now, we calculate the amount of gain.

Gain = Selling Price (SP) - Cost Price (CP)

Gain = $\textsf{₹} \, 2340 - \textsf{₹} \, 2000$

Gain = $\textsf{₹} \, 340$

Finally, we calculate the gain percentage. The gain percentage is calculated on the Cost Price.

Gain % = $\left( \frac{\text{Gain}}{\text{Cost Price}} \right) \times 100$

Gain % = $\left( \frac{340}{2000} \right) \times 100$

Gain % = $\frac{34000}{2000}$

Gain % = $\frac{34}{2} = 17\%$

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Final Answer:

The shopkeeper made a gain of 17%.

Here is the step-by-step solution:

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Given:

Cost Price (CP) of each of the two fans = $\textsf{₹} \, 1200$.

Loss on the first fan = 5%.

Profit on the second fan = 10%.

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To Find:

1. The selling price (SP) of each fan.

2. The total profit or loss on the entire transaction.

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Solution:

We will calculate the selling price for each fan separately.

Case 1: First Fan (Sold at a 5% loss)

Cost Price (CP₁) = $\textsf{₹} \, 1200$.

Loss = 5% of CP₁

$\implies \text{Loss} = \frac{5}{100} \times 1200$

$\implies \text{Loss} = 5 \times 12 = \textsf{₹} \, 60$

The Selling Price (SP) is calculated as Cost Price - Loss.

SP₁ = CP₁ - Loss

SP₁ = $\textsf{₹} \, 1200 - \textsf{₹} \, 60 = \textsf{₹} \, 1140$

So, the selling price of the first fan is $\textsf{₹} \, 1140$.

Case 2: Second Fan (Sold at a 10% profit)

Cost Price (CP₂) = $\textsf{₹} \, 1200$.

Profit = 10% of CP₂

$\implies \text{Profit} = \frac{10}{100} \times 1200$

$\implies \text{Profit} = 10 \times 12 = \textsf{₹} \, 120$

The Selling Price (SP) is calculated as Cost Price + Profit.

SP₂ = CP₂ + Profit

SP₂ = $\textsf{₹} \, 1200 + \textsf{₹} \, 120 = \textsf{₹} \, 1320$

So, the selling price of the second fan is $\textsf{₹} \, 1320$.

Total Profit or Loss

To find the total profit or loss, we need to calculate the total cost price and the total selling price.

Total Cost Price (Total CP) = CP₁ + CP₂

Total CP = $\textsf{₹} \, 1200 + \textsf{₹} \, 1200 = \textsf{₹} \, 2400$

Total Selling Price (Total SP) = SP₁ + SP₂

Total SP = $\textsf{₹} \, 1140 + \textsf{₹} \, 1320 = \textsf{₹} \, 2460$

Now, we compare the Total SP and Total CP.

Since Total SP ($\textsf{₹} \, 2460$) > Total CP ($\textsf{₹} \, 2400$), there is a total profit.

Total Profit = Total SP - Total CP

Total Profit = $\textsf{₹} \, 2460 - \textsf{₹} \, 2400 = \textsf{₹} \, 60$

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Final Answer:

The selling price of the fan sold at a loss is $\textsf{₹} \, 1140$.

The selling price of the fan sold at a profit is $\textsf{₹} \, 1320$.

The total profit on the entire transaction is $\textsf{₹} \, 60$.

Here is the step-by-step solution:

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Given:

The cost of the pair of roller skates = $\textsf{₹} \, 450$.

The sales tax rate = 5%.

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To Find:

The final bill amount.

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Solution:

The bill amount is the original cost plus the sales tax applied to it.

First, we need to calculate the amount of sales tax.

Sales Tax Amount = 5% of the cost

$\implies \text{Sales Tax} = 5\% \times \textsf{₹} \, 450$

$\implies \text{Sales Tax} = \frac{5}{100} \times 450$

$\implies \text{Sales Tax} = \frac{1}{20} \times 450$

$\implies \text{Sales Tax} = \frac{450}{20} = \frac{45}{2}$

$\implies \text{Sales Tax} = \textsf{₹} \, 22.50$

Now, we find the final bill amount by adding the sales tax to the original cost.

Bill Amount = Cost + Sales Tax

Bill Amount = $\textsf{₹} \, 450 + \textsf{₹} \, 22.50$

Bill Amount = $\textsf{₹} \, 472.50$

Therefore, the bill amount is $\textsf{₹} \, 472.50$.

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Alternate Solution:

The original cost represents 100%. A sales tax of 5% is added to it.

So, the final bill amount will be $100\% + 5\% = 105\%$ of the original cost.

Bill Amount = 105% of Cost

$\implies \text{Bill Amount} = \frac{105}{100} \times 450$

$\implies \text{Bill Amount} = 1.05 \times 450$

$\implies \text{Bill Amount} = \textsf{₹} \, 472.50$

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Final Answer:

The bill amount for the pair of roller skates is $\textsf{₹} \, 472.50$.

Here is the step-by-step solution:

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Given:

The price of the air cooler including VAT = $\textsf{₹} \, 3300$.

The VAT (Value Added Tax) rate = 10%.

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To Find:

The price of the air cooler before VAT was added (the original price).

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Solution:

Let the original price of the air cooler (before VAT) be $\textsf{₹} \, x$.

The VAT is calculated on the original price. So, the VAT amount is 10% of $x$.

VAT Amount = $10\% \text{ of } x = \frac{10}{100} \times x = 0.1x$.

The final price including VAT is the sum of the original price and the VAT amount.

Price including VAT = Original Price + VAT Amount

According to the question, the price including VAT is $\textsf{₹} \, 3300$. We can set up the following equation:

$x + 0.1x = 3300$

$\implies 1.1x = 3300$

Now, we solve for $x$:

$x = \frac{3300}{1.1}$

To remove the decimal from the denominator, we multiply the numerator and the denominator by 10:

$x = \frac{3300 \times 10}{1.1 \times 10} = \frac{33000}{11}$

$x = 3000$

Therefore, the price of the air cooler before VAT was added is $\textsf{₹} \, 3000$.

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Alternate Solution:

The original price of an item is always considered as 100%.

A VAT of 10% is added to this price.

So, the final price ($\textsf{₹} \, 3300$) represents $100\% + 10\% = 110\%$ of the original price.

Let the original price be $\textsf{₹} \, x$.

We can write this as an equation:

$110\% \text{ of } x = 3300$

$\implies \frac{110}{100} \times x = 3300$

Now, we solve for $x$:

$x = 3300 \times \frac{100}{110}$

$x = \frac{330000}{110}$

$x = \frac{33000}{11}$

$x = 3000$

Thus, the price of the air cooler before VAT was $\textsf{₹} \, 3000$.

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Final Answer:

The price of the air cooler before VAT was added is $\textsf{₹} \, 3000$.

Here is the step-by-step solution:

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Given:

The final price of the article including GST = $\textsf{₹} \, 784$.

The GST (Goods and Services Tax) rate = 12%.

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To Find:

The price of the article before GST was added (the original price).

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Solution:

Let the original price of the article (before GST) be $\textsf{₹} \, x$.

The GST is calculated on the original price. So, the GST amount is 12% of $x$.

GST Amount = $12\% \text{ of } x = \frac{12}{100} \times x = 0.12x$.

The final price including GST is the sum of the original price and the GST amount.

Final Price = Original Price + GST Amount

We are given that the final price is $\textsf{₹} \, 784$. So, we can set up the following equation:

$x + 0.12x = 784$

$\implies 1.12x = 784$

Now, we solve for $x$:

$x = \frac{784}{1.12}$

To remove the decimal from the denominator, we can multiply the numerator and the denominator by 100:

$x = \frac{784 \times 100}{1.12 \times 100} = \frac{78400}{112}$

Now we perform the division:

$x = 700$

Therefore, the price of the article before GST was added is $\textsf{₹} \, 700$.

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Alternate Solution:

The original price of an item is considered as 100%.

A GST of 12% is added to this price.

So, the final price ($\textsf{₹} \, 784$) represents $100\% + 12\% = 112\%$ of the original price.

Let the original price be $\textsf{₹} \, x$.

We can write this as an equation:

$112\% \text{ of } x = 784$

$\implies \frac{112}{100} \times x = 784$

Now, we solve for $x$:

$x = 784 \times \frac{100}{112}$

We can simplify the fraction $\frac{784}{112}$. We know that $7 \times 112 = 784$.

$x = 7 \times 100$

$x = 700$

Thus, the price of the article before GST was $\textsf{₹} \, 700$.

---

Final Answer:

The price of the article before GST was added is $\textsf{₹} \, 700$.

Here is the step-by-step solution:

---

Given:

The new salary of the man after a 10% increase = $\textsf{₹} \, 1,54,000$.

Percentage increase in salary = 10%.

---

To Find:

The man's original salary.

---

Solution:

Let the original salary be $\textsf{₹} \, x$.

The original salary represents 100%. After a 10% increase, the new salary represents $100\% + 10\% = 110\%$ of the original salary.

According to the question, the new salary is $\textsf{₹} \, 1,54,000$.

So, we can write the equation:

$110\%$ of $x = 1,54,000$

$\implies \frac{110}{100} \times x = 1,54,000$

Now, we solve for $x$ to find the original salary:

$x = 1,54,000 \times \frac{100}{110}$

$x = \frac{154000 \times 100}{110}$

$x = \frac{1540000}{11}$

Since $154 \div 11 = 14$, we get:

$x = 140000$

---

Final Answer:

The man's original salary was $\textsf{₹} \, 1,40,000$.

Here is the step-by-step solution:

---

Given:

Number of people who went to the Zoo on Sunday (Original number) = 845.

Number of people who went to the Zoo on Monday (New number) = 169.

---

To Find:

The percentage decrease in the number of people visiting the Zoo on Monday.

---

Solution:

First, we find the decrease in the number of people.

Decrease in number = Number on Sunday - Number on Monday

Decrease = $845 - 169 = 676$

Now, we calculate the percentage decrease. The percentage decrease is always calculated based on the original value (number of people on Sunday).

The formula for percentage decrease is:

Percentage Decrease = $\left( \frac{\text{Decrease in number}}{\text{Original number}} \right) \times 100$

Substituting the values:

Percentage Decrease = $\left( \frac{676}{845} \right) \times 100$

To simplify the fraction, we can notice that $5 \times 169 = 845$ and $4 \times 169 = 676$. So, we can simplify the fraction by dividing the numerator and denominator by 169.

Percentage Decrease = $\left( \frac{\cancel{676}^4}{\cancel{845}_5} \right) \times 100$

Percentage Decrease = $\frac{4}{5} \times 100$

Percentage Decrease = $4 \times 20 = 80\%$

---

Final Answer:

The per cent decrease in the people visiting the Zoo on Monday is 80%.

Here is the step-by-step solution:

---

Given:

The total cost price (CP) of 80 articles = $\textsf{₹} \, 2,400$.

Profit percentage = 16%.

---

To Find:

The selling price (SP) of one article.

---

Solution:

First, we find the cost price (CP) of one article.

CP of one article = $\frac{\text{Total CP}}{\text{Number of articles}}$

CP of one article = $\frac{2400}{80} = \textsf{₹} \, 30$

Next, we calculate the profit on one article. The profit is 16% of the cost price.

Profit on one article = 16% of CP of one article

Profit = $\frac{16}{100} \times 30$

Profit = $\frac{480}{100} = \textsf{₹} \, 4.80$

The selling price (SP) is the cost price plus the profit.

SP of one article = CP of one article + Profit on one article

SP = $\textsf{₹} \, 30 + \textsf{₹} \, 4.80$

SP = $\textsf{₹} \, 34.80$

---

Alternate Solution:

First, calculate the total selling price (SP) for all 80 articles.

Total CP = $\textsf{₹} \, 2,400$.

Since the profit is 16%, the total SP will be $100\% + 16\% = 116\%$ of the total CP.

Total SP = $116\%$ of Total CP

Total SP = $\frac{116}{100} \times 2400$

Total SP = $116 \times 24 = \textsf{₹} \, 2,784$

Now, find the selling price of one article.

SP of one article = $\frac{\text{Total SP}}{\text{Number of articles}} = \frac{2784}{80}$

SP of one article = $\textsf{₹} \, 34.80$

---

Final Answer:

The selling price of one article is $\textsf{₹} \, 34.80$.

Here is the step-by-step solution:

---

Given:

Cost of the article = $\textsf{₹} \, 15,500$.

Amount spent on repairs = $\textsf{₹} \, 450$.

Profit percentage = 15%.

---

To Find:

The selling price (SP) of the article.

---

Solution:

First, we need to find the total Cost Price (CP) of the article. The total CP includes the initial cost and any overhead expenses like repairs.

Total CP = Cost of article + Amount spent on repairs

Total CP = $\textsf{₹} \, 15,500 + \textsf{₹} \, 450 = \textsf{₹} \, 15,950$

Next, we calculate the profit, which is 15% of the total CP.

Profit = 15% of Total CP

Profit = $\frac{15}{100} \times 15950$

Profit = $\frac{15 \times 1595}{10} = \frac{23925}{10} = \textsf{₹} \, 2392.50$

The selling price (SP) is the sum of the total cost price and the profit.

SP = Total CP + Profit

SP = $\textsf{₹} \, 15,950 + \textsf{₹} \, 2392.50$

SP = $\textsf{₹} \, 18,342.50$

---

Alternate Solution:

The total cost price (CP) is $\textsf{₹} \, 15,950$. This represents 100%.

A profit of 15% means the selling price is $100\% + 15\% = 115\%$ of the total CP.

SP = 115% of Total CP

SP = $\frac{115}{100} \times 15950$

SP = $1.15 \times 15950$

SP = $\textsf{₹} \, 18,342.50$

---

Final Answer:

The selling price of the article is $\textsf{₹} \, 18,342.50$.

Here is the step-by-step solution:

---

Given:

Cost Price (CP) of VCR = $\textsf{₹} \, 8,000$.

Cost Price (CP) of TV = $\textsf{₹} \, 8,000$.

Loss on VCR = 4%.

Profit on TV = 8%.

---

To Find:

The gain or loss percent on the whole transaction.

---

Solution:

First, we will find the Selling Price (SP) for both the VCR and the TV.

For the VCR:

CP = $\textsf{₹} \, 8,000$. Loss = 4%.

Loss amount = 4% of 8000 = $\frac{4}{100} \times 8000 = \textsf{₹} \, 320$.

SP of VCR = CP - Loss = $\textsf{₹} \, 8,000 - \textsf{₹} \, 320 = \textsf{₹} \, 7,680$.

For the TV:

CP = $\textsf{₹} \, 8,000$. Profit = 8%.

Profit amount = 8% of 8000 = $\frac{8}{100} \times 8000 = \textsf{₹} \, 640$.

SP of TV = CP + Profit = $\textsf{₹} \, 8,000 + \textsf{₹} \, 640 = \textsf{₹} \, 8,640$.

For the whole transaction:

Total Cost Price (Total CP) = CP of VCR + CP of TV

Total CP = $\textsf{₹} \, 8,000 + \textsf{₹} \, 8,000 = \textsf{₹} \, 16,000$.

Total Selling Price (Total SP) = SP of VCR + SP of TV

Total SP = $\textsf{₹} \, 7,680 + \textsf{₹} \, 8,640 = \textsf{₹} \, 16,320$.

Since Total SP ($\textsf{₹} \, 16,320$) > Total CP ($\textsf{₹} \, 16,000$), there is an overall gain.

Total Gain = Total SP - Total CP

Total Gain = $\textsf{₹} \, 16,320 - \textsf{₹} \, 16,000 = \textsf{₹} \, 320$.

Now, we find the gain percentage on the whole transaction.

Gain % = $\left( \frac{\text{Total Gain}}{\text{Total CP}} \right) \times 100$

Gain % = $\left( \frac{320}{16000} \right) \times 100$

Gain % = $\frac{32000}{16000} = 2\%$

---

Final Answer:

The gain percent on the whole transaction is 2%.

Here is the step-by-step solution:

---

Given:

Discount offered on all items = 10%.

Marked price (MP) of a pair of jeans = $\textsf{₹} \, 1450$.

Marked price (MP) of one shirt = $\textsf{₹} \, 850$.

Number of shirts purchased = 2.

---

To Find:

The total amount the customer has to pay.

---

Solution:

First, we need to calculate the total marked price of all the items purchased.

MP of one pair of jeans = $\textsf{₹} \, 1450$.

MP of two shirts = $2 \times \textsf{₹} \, 850 = \textsf{₹} \, 1700$.

Total Marked Price (Total MP) = MP of jeans + MP of two shirts

Total MP = $\textsf{₹} \, 1450 + \textsf{₹} \, 1700 = \textsf{₹} \, 3150$.

Next, we calculate the discount amount, which is 10% of the total marked price.

Discount = 10% of Total MP

Discount = $\frac{10}{100} \times 3150$

Discount = $\textsf{₹} \, 315$.

The final amount the customer has to pay is the total marked price minus the discount.

Amount to Pay = Total MP - Discount

Amount to Pay = $\textsf{₹} \, 3150 - \textsf{₹} \, 315$

Amount to Pay = $\textsf{₹} \, 2835$.

---

Alternate Solution:

Total Marked Price = $\textsf{₹} \, 1450 + (2 \times \textsf{₹} \, 850) = \textsf{₹} \, 1450 + \textsf{₹} \, 1700 = \textsf{₹} \, 3150$.

A discount of 10% means the customer pays $100\% - 10\% = 90\%$ of the marked price.

Amount to Pay = 90% of Total MP

Amount to Pay = $\frac{90}{100} \times 3150$

Amount to Pay = $90 \times 31.5$

Amount to Pay = $\textsf{₹} \, 2835$.

---

Final Answer:

The customer would have to pay $\textsf{₹} \, 2835$ for the pair of jeans and two shirts.

Here is the step-by-step solution:

---

Given:

Selling Price (SP) of each of the two buffaloes = $\textsf{₹} \, 20,000$.

Gain on the first buffalo = 5%.

Loss on the second buffalo = 10%.

---

To Find:

The overall gain or loss on the entire transaction.

---

Solution:

First, we need to find the Cost Price (CP) for each buffalo.

Case 1: First Buffalo (Sold at a 5% gain)

SP₁ = $\textsf{₹} \, 20,000$.

The selling price is $100\% + 5\% = 105\%$ of the cost price (CP₁).

So, SP₁ = $105\%$ of CP₁

$20,000 = \frac{105}{100} \times \text{CP}_1$

$\text{CP}_1 = 20,000 \times \frac{100}{105} = 20,000 \times \frac{20}{21} = \frac{400000}{21}$

$\text{CP}_1 = \textsf{₹} \, 19047.62$ (approx.)

Case 2: Second Buffalo (Sold at a 10% loss)

SP₂ = $\textsf{₹} \, 20,000$.

The selling price is $100\% - 10\% = 90\%$ of the cost price (CP₂).

So, SP₂ = $90\%$ of CP₂

$20,000 = \frac{90}{100} \times \text{CP}_2$

$\text{CP}_2 = 20,000 \times \frac{100}{90} = 20,000 \times \frac{10}{9} = \frac{200000}{9}$

$\text{CP}_2 = \textsf{₹} \, 22222.22$ (approx.)

Overall Gain or Loss

To find the overall result, we compare the total cost price and total selling price.

Total Selling Price (Total SP) = SP₁ + SP₂

Total SP = $\textsf{₹} \, 20,000 + \textsf{₹} \, 20,000 = \textsf{₹} \, 40,000$.

Total Cost Price (Total CP) = CP₁ + CP₂

Total CP = $\frac{400000}{21} + \frac{200000}{9}$

The LCM of 21 and 9 is 63.

Total CP = $\frac{400000 \times 3}{63} + \frac{200000 \times 7}{63} = \frac{1200000 + 1400000}{63} = \frac{2600000}{63}$

Total CP = $\textsf{₹} \, 41269.84$ (approx.)

Since Total CP ($\textsf{₹} \, 41269.84$) > Total SP ($\textsf{₹} \, 40,000$), there is an overall loss.

Overall Loss = Total CP - Total SP

Overall Loss = $\textsf{₹} \, 41269.84 - \textsf{₹} \, 40,000 = \textsf{₹} \, 1269.84$

Using the exact fraction: Loss = $\frac{2600000}{63} - 40000 = \frac{2600000 - (40000 \times 63)}{63} \ $$ = \frac{2600000 - 2520000}{63} = \frac{80000}{63}$.

---

Final Answer:

His overall loss is $\frac{80000}{63}$ or approximately $\textsf{₹} \, 1269.84$.

Here is the step-by-step solution:

---

Given:

The price of the TV = $\textsf{₹} \, 13,000$.

Sales tax rate = 12%.

---

To Find:

The total amount that Vinod will have to pay.

---

Solution:

First, we calculate the amount of sales tax charged on the TV. The tax is calculated on the price of the TV.

Sales Tax Amount = 12% of Price

Sales Tax = $\frac{12}{100} \times 13,000$

Sales Tax = $12 \times 130$

Sales Tax = $\textsf{₹} \, 1,560$

The total amount to be paid is the price of the TV plus the sales tax amount.

Total Amount = Price of TV + Sales Tax

Total Amount = $\textsf{₹} \, 13,000 + \textsf{₹} \, 1,560$

Total Amount = $\textsf{₹} \, 14,560$

---

Alternate Solution:

The price of the TV is 100%. With a 12% sales tax, the total amount to be paid is $100\% + 12\% = 112\%$ of the price.

Total Amount = 112% of Price

Total Amount = $\frac{112}{100} \times 13,000$

Total Amount = $112 \times 130$

Total Amount = $\textsf{₹} \, 14,560$

---

Final Answer:

The amount that Vinod will have to pay is $\textsf{₹} \, 14,560$.

Here is the step-by-step solution:

---

Given:

Discount given = 20%.

Amount paid (Selling Price, SP) = $\textsf{₹} \, 1,600$.

---

To Find:

The marked price (MP) of the skates.

---

Solution:

Let the marked price (MP) of the skates be $\textsf{₹} \, x$.

A discount of 20% means that the amount paid is $100\% - 20\% = 80\%$ of the marked price.

So, we can write the equation:

Selling Price = 80% of Marked Price

$1,600 = 80\% \text{ of } x$

$\implies 1,600 = \frac{80}{100} \times x$

Now, we solve for $x$ to find the marked price:

$x = 1,600 \times \frac{100}{80}$

$x = \frac{160000}{80}$

$x = \frac{16000}{8}$

$x = 2000$

---

Final Answer:

The marked price of the pair of skates was $\textsf{₹} \, 2,000$.

Here is the step-by-step solution:

---

Given:

Price of hair-dryer including VAT = $\textsf{₹} \, 5,400$.

VAT (Value Added Tax) rate = 8%.

---

To Find:

The price of the hair-dryer before VAT was added.

---

Solution:

Let the original price of the hair-dryer (before VAT) be $\textsf{₹} \, x$.

The original price represents 100%. With an 8% VAT, the final price is $100\% + 8\% = 108\%$ of the original price.

So, the price including VAT is 108% of the original price.

We can write the equation:

$108\% \text{ of } x = 5,400$

$\implies \frac{108}{100} \times x = 5,400$

Now, we solve for $x$:

$x = 5,400 \times \frac{100}{108}$

$x = \frac{540000}{108}$

Since $108 \times 5 = 540$, we can simplify:

$x = 50 \times 100$

$x = 5000$

---

Final Answer:

The price of the hair-dryer before VAT was added is $\textsf{₹} \, 5,000$.

Here is the step-by-step solution:

---

Given:

Price of the article including GST = $\textsf{₹} \, 1239$.

GST (Goods and Services Tax) rate = 18%.

---

To Find:

The price of the article before GST was added.

---

Solution:

Let the original price of the article (before GST) be $\textsf{₹} \, x$.

The original price represents 100%. With 18% GST, the final price becomes $100\% + 18\% = 118\%$ of the original price.

So, the price including GST is 118% of the original price.

We can set up the equation:

$118\% \text{ of } x = 1239$

$\implies \frac{118}{100} \times x = 1239$

Now, we solve for $x$:

$x = 1239 \times \frac{100}{118}$

$x = \frac{123900}{118}$

Performing the division:

$x = 1050$

---

Final Answer:

The price of the article before GST was added is $\textsf{₹} \, 1,050$.

Here is the step-by-step solution:

---

Given:

Principal (P) = $\textsf{₹} \, 10,000$.

Rate of Interest (R) = 15% per annum.

Time (T) = 2 years.

---

To Find:

1. The Simple Interest (SI).

2. The amount to be paid (A).

---

Solution:

The formula for Simple Interest (SI) is:

Substituting the given values into the formula:

$\text{SI} = \frac{10,000 \times 15 \times 2}{100}$

$\text{SI} = 100 \times 15 \times 2$

$\text{SI} = \textsf{₹} \, 3,000$

So, the simple interest is $\textsf{₹} \, 3,000$.

Now, we find the amount to be paid at the end of 2 years. The amount is the sum of the principal and the simple interest.

Amount (A) = Principal (P) + Simple Interest (SI)

A = $\textsf{₹} \, 10,000 + \textsf{₹} \, 3,000$

A = $\textsf{₹} \, 13,000$

---

Final Answer:

The simple interest on the sum is $\textsf{₹} \, 3,000$ and the amount to be paid at the end of 2 years is $\textsf{₹} \, 13,000$.

Here is the step-by-step solution:

---

Given:

Principal (P) = $\textsf{₹} \, 12,600$.

Time (n) = 2 years.

Rate of Interest (R) = 10% per annum, compounded annually.

---

To Find:

The Compound Interest (CI).

---

Solution:

We can find the Compound Interest using the formula for the Amount (A) first.

The formula for Amount when interest is compounded annually is:

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

Substituting the given values:

$A = 12600 \left( 1 + \frac{10}{100} \right)^2$

$A = 12600 \left( 1 + \frac{1}{10} \right)^2$

$A = 12600 \left( \frac{11}{10} \right)^2$

$A = 12600 \times \frac{11}{10} \times \frac{11}{10}$

$A = 126 \times 11 \times 11$

$A = 126 \times 121$

$A = \textsf{₹} \, 15,246$

Now, the Compound Interest (CI) is the difference between the Amount (A) and the Principal (P).

CI = A - P

CI = $\textsf{₹} \, 15,246 - \textsf{₹} \, 12,600$

CI = $\textsf{₹} \, 2,646$

---

Alternate Solution (Year-by-year calculation):

For the 1st year:

Simple Interest (SI₁) = $\frac{P \times R \times T}{100} = \frac{12600 \times 10 \times 1}{100} = \textsf{₹} \, 1,260$.

Amount at the end of 1st year = P + SI₁ = $\textsf{₹} \, 12,600 + \textsf{₹} \, 1,260 = \textsf{₹} \, 13,860$.

For the 2nd year:

The principal for the 2nd year is the amount from the 1st year, i.e., P' = $\textsf{₹} \, 13,860$.

Simple Interest (SI₂) = $\frac{P' \times R \times T}{100} = \frac{13860 \times 10 \times 1}{100} = \textsf{₹} \, 1,386$.

Total Compound Interest = SI₁ + SI₂ = $\textsf{₹} \, 1,260 + \textsf{₹} \, 1,386 = \textsf{₹} \, 2,646$.

---

Final Answer:

The Compound Interest (CI) is $\textsf{₹} \, 2,646$.

Here is the step-by-step solution:

---

Given:

Principal (P) = $\textsf{₹} \, 12,000$.

Time = $1\frac{1}{2}$ years = 1.5 years.

Rate of Interest = 10% per annum.

The interest is compounded half-yearly.

---

To Find:

The amount (A) to be repaid.

---

Solution:

Since the interest is compounded half-yearly, we need to adjust the rate and time period accordingly.

The time period is for half-years, so we multiply the years by 2.

Number of conversion periods (n) = $1.5 \text{ years} \times 2 = 3$ half-years.

The interest rate is for half-years, so we divide the annual rate by 2.

Rate of Interest (R) = $\frac{10\% \text{ per annum}}{2} = 5\%$ per half-year.

Now, we use the formula for the Amount (A):

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

Substituting the adjusted values:

$A = 12000 \left( 1 + \frac{5}{100} \right)^3$

$A = 12000 \left( 1 + \frac{1}{20} \right)^3$

$A = 12000 \left( \frac{21}{20} \right)^3$

$A = 12000 \times \frac{21}{20} \times \frac{21}{20} \times \frac{21}{20}$

$A = \frac{12000 \times 21 \times 21 \times 21}{8000}$

$A = \frac{12 \times 9261}{8} = \frac{3 \times 9261}{2}$

$A = \frac{27783}{2}$

$A = 13891.50$

---

Final Answer:

The amount to be repaid is $\textsf{₹} \, 13,891.50$.

Here is the step-by-step solution:

---

Given:

Principal (P) = $\textsf{₹} \, 10,000$.

Time = 1 year and 3 months.

Rate of Interest (R) = $8\frac{1}{2}\% = 8.5\%$ per annum, compounded annually.

---

To Find:

The Compound Interest (CI) paid.

---

Solution:

Since the time period is not a whole number of years, we first calculate the amount for the whole part (1 year) using the compound interest formula. Then, we calculate the simple interest for the fractional part (3 months) on the amount obtained.

Step 1: Calculate Amount for 1 year.

$A_1 = P \left( 1 + \frac{R}{100} \right)^1$

$A_1 = 10000 \left( 1 + \frac{8.5}{100} \right)$

$A_1 = 10000 \left( 1 + 0.085 \right)$

$A_1 = 10000 \times 1.085 = \textsf{₹} \, 10,850$

Step 2: Calculate Simple Interest for the next 3 months.

The principal for this calculation will be the amount after 1 year, which is $A_1 = \textsf{₹} \, 10,850$.

Time (T) = 3 months = $\frac{3}{12}$ years = $\frac{1}{4}$ years.

Simple Interest (SI) = $\frac{A_1 \times R \times T}{100}$

SI = $\frac{10850 \times 8.5 \times \frac{1}{4}}{100}$

SI = $\frac{10850 \times 8.5}{400} = \frac{92225}{400} = 230.5625$

SI $\approx \textsf{₹} \, 230.56$

Step 3: Calculate the total amount.

Total Amount (A) = Amount after 1 year + SI for next 3 months

A = $\textsf{₹} \, 10,850 + \textsf{₹} \, 230.5625 = \textsf{₹} \, 11,080.5625$

Step 4: Calculate the Compound Interest.

CI = Total Amount - Original Principal

CI = $\textsf{₹} \, 11,080.5625 - \textsf{₹} \, 10,000 = \textsf{₹} \, 1,080.5625$

---

Final Answer:

The Compound Interest paid is $\textsf{₹} \, 1,080.56$ (rounded to two decimal places).

Here is the step-by-step solution:

---

Given:

Initial Population (in 1997), $P_0 = 20,000$.

Rate of increase (R) = 5% per annum.

---

To Find:

The population at the end of the year 2000.

---

Solution:

This problem is an application of the compound interest formula where the principal is the initial population and the rate is the growth rate.

The time period (n) from the beginning of 1998 to the end of 2000 is 3 years (1998, 1999, 2000).

The formula for the final population ($P_n$) is:

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

Substituting the given values:

$P_3 = 20000 \left( 1 + \frac{5}{100} \right)^3$

$P_3 = 20000 \left( 1 + \frac{1}{20} \right)^3$

$P_3 = 20000 \left( \frac{21}{20} \right)^3$

$P_3 = 20000 \times \frac{21}{20} \times \frac{21}{20} \times \frac{21}{20}$

$P_3 = \frac{20000 \times 9261}{8000}$

$P_3 = \frac{20 \times 9261}{8} = \frac{5 \times 9261}{2}$

$P_3 = \frac{46305}{2} = 23152.5$

Since the population cannot be a fraction, we take the integer value. It's common in such problems to accept the calculated value, or round down.

So, the population at the end of the year 2000 would be 23,152.

---

Final Answer:

The population at the end of the year 2000 will be 23,152.

Here is the step-by-step solution:

---

Given:

Original Price (Principal, P) of the TV = $\textsf{₹} \, 21,000$.

Rate of depreciation (R) = 5% per annum.

Time (n) = 1 year.

---

To Find:

The value of the TV after one year.

---

Solution:

Depreciation means a reduction in value. We can find the depreciated value by first calculating the amount of depreciation and then subtracting it from the original price.

Step 1: Calculate the amount of depreciation.

Depreciation = 5% of the original price

Depreciation = $\frac{5}{100} \times 21,000$

Depreciation = $5 \times 210 = \textsf{₹} \, 1,050$

Step 2: Calculate the value after one year.

Value after 1 year = Original Price - Depreciation

Value after 1 year = $\textsf{₹} \, 21,000 - \textsf{₹} \, 1,050$

Value after 1 year = $\textsf{₹} \, 19,950$

---

Alternate Solution (using formula):

The formula for value after depreciation is similar to the compound interest formula, but with a minus sign.

Value after n years, $V_n = P \left( 1 - \frac{R}{100} \right)^n$

Substituting the given values:

$V_1 = 21000 \left( 1 - \frac{5}{100} \right)^1$

$V_1 = 21000 \left( \frac{95}{100} \right)$

$V_1 = 210 \times 95$

$V_1 = \textsf{₹} \, 19,950$

---

Final Answer:

The value of the TV after one year is $\textsf{₹} \, 19,950$.

(a) ₹ 10,800 for 3 years at $12\frac{1}{2}$% per annum compounded annually.

Given:

Principal (P) = $\textsf{₹} \, 10,800$

Rate (R) = $12\frac{1}{2}\% = \frac{25}{2}\%$ per annum

Time (n) = 3 years

The formula for the Amount (A) compounded annually is $A = P \left( 1 + \frac{R}{100} \right)^n$.

$A = 10800 \left( 1 + \frac{25/2}{100} \right)^3 = 10800 \left( 1 + \frac{25}{200} \right)^3$

$A = 10800 \left( 1 + \frac{1}{8} \right)^3 = 10800 \left( \frac{9}{8} \right)^3$

$A = 10800 \times \frac{9}{8} \times \frac{9}{8} \times \frac{9}{8} = 10800 \times \frac{729}{512}$

$A = \frac{7873200}{512} = 15377.34375 \approx \textsf{₹} \, 15,377.34$

Compound Interest (CI) = Amount - Principal

CI = $\textsf{₹} \, 15,377.34 - \textsf{₹} \, 10,800 = \textsf{₹} \, 4,577.34$

Amount = $\textsf{₹} \, 15,377.34$, Compound Interest = $\textsf{₹} \, 4,577.34$

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(b) ₹ 18,000 for $2\frac{1}{2}$ years at 10% per annum compounded annually.

Given:

Principal (P) = $\textsf{₹} \, 18,000$

Rate (R) = 10% per annum

Time = $2\frac{1}{2}$ years

First, calculate the amount for 2 years using the compound interest formula.

$A_2 = 18000 \left( 1 + \frac{10}{100} \right)^2 = 18000 \left( \frac{11}{10} \right)^2$

$A_2 = 18000 \times \frac{121}{100} = 180 \times 121 = \textsf{₹} \, 21,780$

Now, calculate the Simple Interest (SI) on this amount for the remaining $\frac{1}{2}$ year.

SI = $\frac{21780 \times 10 \times \frac{1}{2}}{100} = \frac{21780 \times 5}{100} = \textsf{₹} \, 1,089$

Total Amount = $A_2 + \text{SI} = \textsf{₹} \, 21,780 + \textsf{₹} \, 1,089 = \textsf{₹} \, 22,869$

Compound Interest (CI) = Total Amount - Principal

CI = $\textsf{₹} \, 22,869 - \textsf{₹} \, 18,000 = \textsf{₹} \, 4,869$

Amount = $\textsf{₹} \, 22,869$, Compound Interest = $\textsf{₹} \, 4,869$

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(c) ₹ 62,500 for $1\frac{1}{2}$ years at 8% per annum compounded half yearly.

Given:

Principal (P) = $\textsf{₹} \, 62,500$

Time = $1\frac{1}{2}$ years = 1.5 years

Rate = 8% per annum

Since interest is compounded half-yearly:

Rate (R) = $\frac{8}{2} = 4\%$ per half-year.

Time (n) = $1.5 \times 2 = 3$ half-years.

$A = 62500 \left( 1 + \frac{4}{100} \right)^3 = 62500 \left( \frac{26}{25} \right)^3$

$A = 62500 \times \frac{17576}{15625} = 4 \times 17576 = \textsf{₹} \, 70,304$

Compound Interest (CI) = $\textsf{₹} \, 70,304 - \textsf{₹} \, 62,500 = \textsf{₹} \, 7,804$

Amount = $\textsf{₹} \, 70,304$, Compound Interest = $\textsf{₹} \, 7,804$

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(d) ₹ 8,000 for 1 year at 9% per annum compounded half yearly.

Given:

Principal (P) = $\textsf{₹} \, 8,000$

Time = 1 year

Rate = 9% per annum

Since interest is compounded half-yearly:

Rate (R) = $\frac{9}{2} = 4.5\%$ per half-year.

Time (n) = $1 \times 2 = 2$ half-years.

$A = 8000 \left( 1 + \frac{4.5}{100} \right)^2 = 8000 \left( 1 + \frac{9}{200} \right)^2$

$A = 8000 \left( \frac{209}{200} \right)^2 = 8000 \times \frac{43681}{40000} = \frac{43681}{5} = \textsf{₹} \, 8,736.20$

Compound Interest (CI) = $\textsf{₹} \, 8,736.20 - \textsf{₹} \, 8,000 = \textsf{₹} \, 736.20$

Amount = $\textsf{₹} \, 8,736.20$, Compound Interest = $\textsf{₹} \, 736.20$

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(e) ₹ 10,000 for 1 year at 8% per annum compounded half yearly.

Given:

Principal (P) = $\textsf{₹} \, 10,000$

Time = 1 year

Rate = 8% per annum

Since interest is compounded half-yearly:

Rate (R) = $\frac{8}{2} = 4\%$ per half-year.

Time (n) = $1 \times 2 = 2$ half-years.

$A = 10000 \left( 1 + \frac{4}{100} \right)^2 = 10000 \left( \frac{26}{25} \right)^2$

$A = 10000 \times \frac{676}{625} = 16 \times 676 = \textsf{₹} \, 10,816$

Compound Interest (CI) = $\textsf{₹} \, 10,816 - \textsf{₹} \, 10,000 = \textsf{₹} \, 816$

Amount = $\textsf{₹} \, 10,816$, Compound Interest = $\textsf{₹} \, 816$

Here is the step-by-step solution:

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Given:

Principal (P) = $\textsf{₹} \, 26,400$

Rate of Interest (R) = 15% per annum, compounded yearly.

Time = 2 years and 4 months.

---

To Find:

The total amount to be paid to clear the loan.

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Solution:

First, we will calculate the amount for the whole part of the time period, which is 2 years, using the compound interest formula.

Step 1: Calculate Amount for 2 years.

$A_2 = P \left( 1 + \frac{R}{100} \right)^n$

$A_2 = 26400 \left( 1 + \frac{15}{100} \right)^2 = 26400 \left( 1 + \frac{3}{20} \right)^2$

$A_2 = 26400 \left( \frac{23}{20} \right)^2 = 26400 \times \frac{529}{400}$

$A_2 = 66 \times 529 = \textsf{₹} \, 34,914$

Step 2: Calculate Simple Interest for the next 4 months.

The principal for this calculation is the amount after 2 years, which is $P' = \textsf{₹} \, 34,914$.

Time (T) = 4 months = $\frac{4}{12}$ years = $\frac{1}{3}$ years.

Simple Interest (SI) = $\frac{P' \times R \times T}{100}$

SI = $\frac{34914 \times 15 \times \frac{1}{3}}{100} = \frac{34914 \times 5}{100}$

SI = $\frac{174570}{100} = \textsf{₹} \, 1,745.70$

Step 3: Calculate the total amount to be paid.

Total Amount = Amount after 2 years + SI for next 4 months

Total Amount = $\textsf{₹} \, 34,914 + \textsf{₹} \, 1,745.70 = \textsf{₹} \, 36,659.70$

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Final Answer:

The amount Kamala will pay at the end of 2 years and 4 months is $\textsf{₹} \, 36,659.70$.

Here is the step-by-step solution:

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To Find:

Who pays more interest and by how much.

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Solution:

We need to calculate the interest paid by both Fabina and Radha and then compare them.

Case 1: Interest paid by Fabina (Simple Interest)

Principal (P) = $\textsf{₹} \, 12,500$

Rate (R) = 12% per annum

Time (T) = 3 years

Simple Interest (SI) = $\frac{P \times R \times T}{100}$

SI = $\frac{12500 \times 12 \times 3}{100} = 125 \times 36 = \textsf{₹} \, 4,500$

Case 2: Interest paid by Radha (Compound Interest)

Principal (P) = $\textsf{₹} \, 12,500$

Rate (R) = 10% per annum

Time (n) = 3 years

First, we find the amount (A).

$A = P \left( 1 + \frac{R}{100} \right)^n = 12500 \left( 1 + \frac{10}{100} \right)^3$

$A = 12500 \left( \frac{11}{10} \right)^3 = 12500 \times \frac{1331}{1000}$

$A = 12.5 \times 1331 = \textsf{₹} \, 16,637.50$

Compound Interest (CI) = A - P

CI = $\textsf{₹} \, 16,637.50 - \textsf{₹} \, 12,500 = \textsf{₹} \, 4,137.50$

Comparison

Interest paid by Fabina = $\textsf{₹} \, 4,500$

Interest paid by Radha = $\textsf{₹} \, 4,137.50$

Comparing the two amounts, $4,500 > 4,137.50$. Therefore, Fabina pays more interest.

Difference in interest = $\textsf{₹} \, 4,500 - \textsf{₹} \, 4,137.50 = \textsf{₹} \, 362.50$

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Final Answer:

Fabina pays more interest by $\textsf{₹} \, 362.50$.

Here is the step-by-step solution:

---

Given:

Principal (P) = $\textsf{₹} \, 12,000$

Rate of Interest (R) = 6% per annum

Time (T or n) = 2 years

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To Find:

The extra amount to be paid if the interest was compounded annually.

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Solution:

We need to find the difference between the compound interest and the simple interest for the given sum.

Step 1: Calculate the Simple Interest (SI)

SI = $\frac{P \times R \times T}{100}$

SI = $\frac{12000 \times 6 \times 2}{100} = 120 \times 12 = \textsf{₹} \, 1,440$

Step 2: Calculate the Compound Interest (CI)

First, find the amount (A) compounded annually.

$A = P \left( 1 + \frac{R}{100} \right)^n = 12000 \left( 1 + \frac{6}{100} \right)^2$

$A = 12000 \left( \frac{106}{100} \right)^2 = 12000 \times (1.06)^2$

$A = 12000 \times 1.1236 = \textsf{₹} \, 13,483.20$

CI = A - P = $\textsf{₹} \, 13,483.20 - \textsf{₹} \, 12,000 = \textsf{₹} \, 1,483.20$

Step 3: Find the extra amount

Extra Amount = Compound Interest - Simple Interest

Extra Amount = $\textsf{₹} \, 1,483.20 - \textsf{₹} \, 1,440 = \textsf{₹} \, 43.20$

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Final Answer:

The extra amount I would have to pay is $\textsf{₹} \, 43.20$.

Here is the step-by-step solution:

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Given:

Principal (P) = $\textsf{₹} \, 60,000$

Rate of Interest = 12% per annum, compounded half-yearly.

---

To Find:

(i) Amount after 6 months.

(ii) Amount after 1 year.

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Solution:

Since the interest is compounded half-yearly, we need to adjust the rate and time period for our calculations.

Rate (R) per half-year = $\frac{12\%}{2} = 6\%$.

(i) Amount after 6 months

Time period = 6 months = 1 half-year. So, n = 1.

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

$A = 60000 \left( 1 + \frac{6}{100} \right)^1 = 60000 \times \frac{106}{100}$

$A = 600 \times 106 = \textsf{₹} \, 63,600$

(ii) Amount after 1 year

Time period = 1 year = 2 half-years. So, n = 2.

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

$A = 60000 \left( 1 + \frac{6}{100} \right)^2 = 60000 \left( \frac{106}{100} \right)^2$

$A = 60000 \times (1.06)^2 = 60000 \times 1.1236$

$A = \textsf{₹} \, 67,416$

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Final Answer:

(i) After 6 months, he would get $\textsf{₹} \, 63,600$.

(ii) After 1 year, he would get $\textsf{₹} \, 67,416$.

Here is the step-by-step solution:

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Given:

Principal (P) = $\textsf{₹} \, 80,000$

Rate of Interest (R) = 10% per annum

Time = $1\frac{1}{2}$ years = 1.5 years

---

To Find:

The difference in amounts for interest compounded annually and half-yearly.

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Solution:

(i) Interest Compounded Annually

First, calculate the amount for 1 year using compound interest formula.

$A_1 = 80000 \left( 1 + \frac{10}{100} \right)^1 = 80000 \times 1.1 = \textsf{₹} \, 88,000$

Now, calculate Simple Interest on this amount for the next $\frac{1}{2}$ year.

SI = $\frac{88000 \times 10 \times \frac{1}{2}}{100} = \frac{88000 \times 5}{100} = \textsf{₹} \, 4,400$

Total Amount (compounded annually) = $A_1 + \text{SI} = \textsf{₹} \, 88,000 + \textsf{₹} \, 4,400 \ $$ = \textsf{₹} \, 92,400$

(ii) Interest Compounded Half-Yearly

Rate per half-year = $\frac{10}{2} = 5\%$

Time period (n) = $1.5 \times 2 = 3$ half-years

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

$A = 80000 \left( 1 + \frac{5}{100} \right)^3 = 80000 \left( \frac{21}{20} \right)^3$

$A = 80000 \times \frac{9261}{8000} = 10 \times 9261 = \textsf{₹} \, 92,610$

Difference in Amounts

Difference = Amount (half-yearly) - Amount (annually)

Difference = $\textsf{₹} \, 92,610 - \textsf{₹} \, 92,400 = \textsf{₹} \, 210$

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Final Answer:

The difference in the amounts he would be paying is $\textsf{₹} \, 210$.

Here is the step-by-step solution:

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Given:

Principal (P) = $\textsf{₹} \, 8,000$

Rate of Interest (R) = 5% per annum, compounded annually.

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Solution:

(i) The amount credited against her name at the end of the second year.

We need to find the Amount (A) after 2 years.

Time (n) = 2 years.

The formula for the amount is:

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

Substituting the values:

$A = 8000 \left( 1 + \frac{5}{100} \right)^2$

$A = 8000 \left( 1 + \frac{1}{20} \right)^2$

$A = 8000 \left( \frac{21}{20} \right)^2$

$A = 8000 \times \frac{21}{20} \times \frac{21}{20}$

$A = 20 \times 21 \times 21$

$A = 20 \times 441 = \textsf{₹} \, 8,820$

So, the amount at the end of the second year is $\textsf{₹} \, 8,820$.

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(ii) The interest for the 3rd year.

The interest for the 3rd year is the Simple Interest calculated on the amount at the end of the 2nd year.

Principal for the 3rd year (P') = Amount after 2 years = $\textsf{₹} \, 8,820$.

Time (T) = 1 year.

Rate (R) = 5%.

Interest for 3rd year (SI) = $\frac{P' \times R \times T}{100}$

SI = $\frac{8820 \times 5 \times 1}{100}$

SI = $\frac{44100}{100} = \textsf{₹} \, 441$

So, the interest for the 3rd year is $\textsf{₹} \, 441$.

Here is the step-by-step solution:

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Given:

Principal (P) = $\textsf{₹} \, 10,000$

Time = $1\frac{1}{2}$ years = 1.5 years

Rate = 10% per annum

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Solution:

Part 1: Interest Compounded Half-Yearly

Since interest is compounded half-yearly:

Rate (R) per half-year = $\frac{10\%}{2} = 5\%$

Time (n) in half-years = $1.5 \times 2 = 3$

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

$A = 10000 \left( 1 + \frac{5}{100} \right)^3 = 10000 \left( \frac{21}{20} \right)^3$

$A = 10000 \times \frac{9261}{8000} = \frac{10}{8} \times 9261 = 1.25 \times 9261 = \textsf{₹} \, 11,576.25$

Compound Interest (CI) = A - P = $\textsf{₹} \, 11,576.25 - \textsf{₹} \, 10,000 = \textsf{₹} \, 1,576.25$

Amount = $\textsf{₹} \, 11,576.25$, CI = $\textsf{₹} \, 1,576.25$

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Part 2: Interest Compounded Annually (for comparison)

First, find the amount for 1 year.

$A_1 = 10000 \left( 1 + \frac{10}{100} \right)^1 = \textsf{₹} \, 11,000$

Now, find Simple Interest on $A_1$ for the next $\frac{1}{2}$ year.

SI = $\frac{11000 \times 10 \times \frac{1}{2}}{100} = \frac{11000 \times 5}{100} = \textsf{₹} \, 550$

Total Amount = $A_1 + \text{SI} = \textsf{₹} \, 11,000 + \textsf{₹} \, 550 = \textsf{₹} \, 11,550$

Compound Interest (CI) = $\textsf{₹} \, 11,550 - \textsf{₹} \, 10,000 = \textsf{₹} \, 1,550$

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Comparison:

Interest when compounded half-yearly = $\textsf{₹} \, 1,576.25$

Interest when compounded annually = $\textsf{₹} \, 1,550$

Yes, the interest would be more if compounded half-yearly.

Difference = $\textsf{₹} \, 1,576.25 - \textsf{₹} \, 1,550 = \textsf{₹} \, 26.25$

Here is the step-by-step solution:

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Given:

Principal (P) = $\textsf{₹} \, 4,096$

Time = 18 months = 1.5 years

Annual Rate = $12\frac{1}{2}\% = \frac{25}{2}\%$

Interest is compounded half-yearly.

---

To Find:

The amount Ram will get.

---

Solution:

Since the interest is compounded half-yearly, we adjust the rate and time.

Rate per half-year (R) = $\frac{1}{2} \times \frac{25}{2}\% = \frac{25}{4}\%$

Time period (n) = 18 months = 3 half-years.

The formula for the Amount (A) is:

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

$A = 4096 \left( 1 + \frac{25/4}{100} \right)^3$

$A = 4096 \left( 1 + \frac{25}{400} \right)^3$

$A = 4096 \left( 1 + \frac{1}{16} \right)^3$

$A = 4096 \left( \frac{17}{16} \right)^3$

$A = 4096 \times \frac{17 \times 17 \times 17}{16 \times 16 \times 16}$

$A = 4096 \times \frac{4913}{4096}$

$A = \textsf{₹} \, 4,913$

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Final Answer:

The amount Ram will get is $\textsf{₹} \, 4,913$.

Here is the step-by-step solution:

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Given:

Population in 2003 = 54,000

Rate of increase = 5% per annum

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Solution:

(i) Find the population in 2001.

Let the population in 2001 be P.

The population in 2003 is the result of a 2-year increase from 2001.

Time (n) = 2003 - 2001 = 2 years.

We use the formula $A = P \left( 1 + \frac{R}{100} \right)^n$, where A is the population in 2003.

$54000 = P \left( 1 + \frac{5}{100} \right)^2$

$54000 = P \left( \frac{21}{20} \right)^2 = P \left( \frac{441}{400} \right)$

$P = 54000 \times \frac{400}{441}$

$P = \frac{21600000}{441} \approx 48979.59$

Since population must be a whole number, we round it to 48,980.

The population in 2001 was approximately 48,980.

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(ii) What would be its population in 2005?

Here, the population in 2003 acts as the initial population (P).

P = 54,000

Time (n) = 2005 - 2003 = 2 years.

Let the population in 2005 be A.

$A = 54000 \left( 1 + \frac{5}{100} \right)^2$

$A = 54000 \left( \frac{21}{20} \right)^2 = 54000 \times \frac{441}{400}$

$A = 135 \times 441 = 59,535$

The population in 2005 would be 59,535.

Here is the step-by-step solution:

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Given:

Initial count of bacteria (P) = 5,06,000

Rate of increase (R) = 2.5% per hour

Time (n) = 2 hours

---

To Find:

The count of bacteria after 2 hours.

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Solution:

This is a problem of compound growth. We use the formula:

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

Substituting the values:

$A = 506000 \left( 1 + \frac{2.5}{100} \right)^2$

$A = 506000 \left( 1 + \frac{1}{40} \right)^2$

$A = 506000 \left( \frac{41}{40} \right)^2$

$A = 506000 \times \frac{1681}{1600}$

$A = \frac{5060 \times 1681}{16} = \frac{8505860}{16} = 531616.25$

Since the count of bacteria must be a whole number, we can state the count is approximately 5,31,616.

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Final Answer:

The count of bacteria at the end of 2 hours will be approximately 5,31,616.

Here is the step-by-step solution:

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Given:

Original value of the scooter (P) = $\textsf{₹} \, 42,000$

Rate of depreciation (R) = 8% per annum

Time (n) = 1 year

---

To Find:

The value of the scooter after one year.

---

Solution:

First, calculate the amount of depreciation for one year.

Depreciation Amount = 8% of $\textsf{₹} \, 42,000$

Depreciation Amount = $\frac{8}{100} \times 42000 = 8 \times 420 = \textsf{₹} \, 3,360$

Now, subtract the depreciation amount from the original value to find the value after one year.

Value after 1 year = Original Value - Depreciation Amount

Value after 1 year = $\textsf{₹} \, 42,000 - \textsf{₹} \, 3,360 = \textsf{₹} \, 38,640$

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Alternate Solution (Using Formula):

The formula for depreciated value (V) is:

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

$V = 42000 \left( 1 - \frac{8}{100} \right)^1$

$V = 42000 \left( \frac{92}{100} \right)$

$V = 420 \times 92 = \textsf{₹} \, 38,640$

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Final Answer:

The value of the scooter after one year is $\textsf{₹} \, 38,640$.