Understanding the Internet Download Bill Calculation

The problem asks us to calculate the internet download bill for a duration of 3 minutes 20 seconds, given that the bill for 2 minutes 30 seconds is 18 rupees. This is a problem involving proportional relationships, assuming the cost is directly proportional to the download time.

Step 1: Convert Time Durations to a Single Unit

To work with the durations easily, we should convert both given times into seconds. There are 60 seconds in one minute.

• First duration: 2 minutes 30 seconds
• Calculation: $2 \text{ minutes} \times 60 \text{ seconds/minute} + 30 \text{ seconds}$
• Total seconds for first duration: $120 + 30 = 150 \text{ seconds}$

• Second duration: 3 minutes 20 seconds
• Calculation: $3 \text{ minutes} \times 60 \text{ seconds/minute} + 20 \text{ seconds}$
• Total seconds for second duration: $180 + 20 = 200 \text{ seconds}$

Step 2: Determine the Cost Per Unit Time

We know the cost for 150 seconds is 18 rupees. We can find the cost per second by dividing the total cost by the total time in seconds.

• Cost for 150 seconds = 18 rupees
• Cost per second = $\frac{\text{Total Cost}}{\text{Total Time}}$
• Cost per second = $\frac{18 \text{ rupees}}{150 \text{ seconds}}$

Let's simplify this fraction:

Cost per second = $\frac{18}{150} = \frac{18 \div 6}{150 \div 6} = \frac{3}{25}$ rupees per second.

Alternatively, we can keep it as $\frac{18}{150}$ for the next step.

Step 3: Calculate the Bill for the New Duration

Now we need to find the bill for the second duration, which is 200 seconds. We multiply the cost per second by the new total time in seconds.

• New Total Time = 200 seconds
• Cost per second = $\frac{18}{150}$ rupees/second
• New Bill = Cost per second $\times$ New Total Time
• New Bill = $\frac{18}{150} \times 200$

Let's calculate the value:

New Bill = $\frac{18}{150} \times 200 = \frac{18 \times 200}{150}$

We can simplify by dividing both 200 and 150 by their greatest common divisor, which is 50:

New Bill = $\frac{18 \times (200 \div 50)}{(150 \div 50)} = \frac{18 \times 4}{3}$

Now, simplify further by dividing 18 by 3:

New Bill = $(18 \div 3) \times 4 = 6 \times 4$

New Bill = 24 rupees

Step 4: Round to One Decimal Place

The question asks for the answer up to one decimal place. The calculated bill is exactly 24.0 rupees.

Summary of Calculation Steps

Comparison with Options

Let's compare our calculated bill with the given options:

• A. 24
• B. 24.1
• C. 24.2
• D. 23.9

Our calculated bill of 24.0 rupees matches option A when considering the required precision of one decimal place.

Revision Table: Internet Bill Calculation

Additional Information: Understanding Proportional Relationships

A proportional relationship exists between two quantities when their ratio is constant. In this problem, the ratio of the total cost to the total time is constant. If $C$ is the cost and $T$ is the time, then $\frac{C}{T} = k$, where $k$ is the constant of proportionality (which is the cost per unit time). So, $C = kT$.

Given $C_1$ at $T_1$ and asked to find $C_2$ at $T_2$, we have:

• $C_1 = k T_1 \implies k = \frac{C_1}{T_1}$
• $C_2 = k T_2$

Substituting the value of $k$ from the first equation into the second equation, we get:

$C_2 = \frac{C_1}{T_1} \times T_2$

In this problem:

• $C_1 = 18$ rupees
• $T_1 = 2 \text{ minutes } 30 \text{ seconds} = 150 \text{ seconds}$
• $T_2 = 3 \text{ minutes } 20 \text{ seconds} = 200 \text{ seconds}$

Using the formula:

$C_2 = \frac{18 \text{ rupees}}{150 \text{ seconds}} \times 200 \text{ seconds}$

$C_2 = \frac{18}{150} \times 200 = \frac{18 \times 200}{150} = \frac{3600}{150} = \frac{360}{15} = 24$ rupees.

This confirms our earlier calculation and the principle of proportional relationships.