Calculate Lamp Lighting Cost Based on Usage

This problem involves calculating the cost of lighting based on the number of lamps, the hours they are used per day, and the total number of days. The cost is directly proportional to the total amount of "lamp-hours" used. We can set up a relationship between the two scenarios provided.

Understanding the Relationship

The total usage can be calculated by multiplying the number of lamps, the hours per day, and the number of days. Let's call this total usage measure \(U\).

\[U = \text{Number of Lamps} \times \text{Hours per Day} \times \text{Number of Days}\]

The cost is directly proportional to this total usage. So, if \(C\) is the cost, then \(C \propto U\), or \(C = kU\) for some constant \(k\). This means the ratio of cost to usage is constant for both scenarios:

\[\frac{C_1}{U_1} = \frac{C_2}{U_2}\]

Applying the Formula to the Given Scenarios

We have two scenarios:

• Scenario 1:
• Number of Lamps (\(L_1\)): 20
• Hours per Day (\(H_1\)): 6 hr
• Number of Days (\(D_1\)): 20 days
• Cost (\(C_1\)): Rs. 100

• Scenario 2:
• Number of Lamps (\(L_2\)): 40
• Hours per Day (\(H_2\)): 8 hr
• Number of Days (\(D_2\)): 12 days
• Cost (\(C_2\)): ?

First, let's calculate the total usage (\(U\)) for each scenario.

For Scenario 1:

\[U_1 = L_1 \times H_1 \times D_1 = 20 \times 6 \times 20\]

\[U_1 = 120 \times 20 = 2400 \text{ lamp-hours}\]

For Scenario 2:

\[U_2 = L_2 \times H_2 \times D_2 = 40 \times 8 \times 12\]

\[U_2 = 320 \times 12\]

\[U_2 = 3840 \text{ lamp-hours}\]

Using the Proportionality Relation

Now we use the proportionality equation:

\[\frac{C_1}{U_1} = \frac{C_2}{U_2}\]

Substitute the known values:

\[\frac{100}{2400} = \frac{C_2}{3840}\]

To find \(C_2\), we rearrange the equation:

\[C_2 = \frac{100}{2400} \times 3840\]

Simplify the fraction \(\frac{100}{2400}\):

\[\frac{100}{2400} = \frac{1}{24}\]

So, the equation becomes:

\[C_2 = \frac{1}{24} \times 3840\]

Now, calculate \(C_2\):

\[C_2 = \frac{3840}{24}\]

Performing the division:

\[3840 \div 24\]

We can simplify this step-by-step or perform long division.

\[\frac{3840}{24} = \frac{1920}{12} = \frac{960}{6} = \frac{480}{3} = 160\]

So, \(C_2 = 160\).

Summary of Calculations

Using the ratio \(\frac{C_1}{U_1} = \frac{C_2}{U_2}\):

\[\frac{100}{2400} = \frac{C_2}{3840}\]

\[C_2 = 100 \times \frac{3840}{2400} = 100 \times \frac{384}{240}\]

Simplifying the fraction \(\frac{384}{240}\) by dividing both by 10, then by common factors like 12 or 24:

\[\frac{384}{240} = \frac{192}{120} = \frac{96}{60} = \frac{48}{30} = \frac{8}{5}\]

\[C_2 = 100 \times \frac{8}{5} = 20 \times 8 = 160\]

The cost of lighting 40 lamps for 8 hours a day for 12 days would be Rs. 160.