In a particular week the average number of people who visited the tajmahal is 40. If we exclude the holidays then the average is increased by 16. Further if we exclude also the day on which the maximum number of 112 people visited the tajmahal, then the average becomes 42. the number of holidays in the week is:

Options

A. 1
B. 2
C. 3
D. data insufficient

Answer

To solve the problem step by step, let's denote the following:

- Let \( N \) be the total number of visitors in the week.
- Let \( H \) be the number of holidays in the week.
- The week has 7 days.

### Step 1: Calculate the total number of visitors for the week

Given that the average number of visitors for the week is 40, we can express this as:
\[ \text{Total visitors} = \text{Average} \times \text{Number of days} = 40 \times 7 = 280 \]

### Step 2: Calculate the average excluding holidays

When we exclude the holidays, the average increases by 16, making it:
\[ 40 + 16 = 56 \]
Let \( D \) be the number of days excluding holidays. Thus, we have:
\[ D = 7 - H \]
The total number of visitors excluding holidays is:
\[ \text{Total visitors excluding holidays} = 56 \times D = 56 \times (7 - H) \]

### Step 3: Set up the equation for total visitors

From the above, we can set up the equation:
\[ 280 - \text{(visitors on holidays)} = 56 \times (7 - H) \]
Let \( V_H \) be the total number of visitors on holidays. Then:
\[ 280 - V_H = 56 \times (7 - H) \]

### Step 4: Calculate the total number of visitors on holidays

We know that:
\[ V_H = 280 - 56 \times (7 - H) \]

### Step 5: Exclude the day with maximum visitors

Next, when we exclude the day with the maximum visitors (112), the average becomes 42. The total number of days now is \( D - 1 = (7 - H) - 1 = 6 - H \). Thus:
\[ \text{Total visitors excluding holidays and the maximum day} = 42 \times (6 - H) \]

### Step 6: Set up the equation for total visitors excluding the maximum day

We can express this as:
\[ 280 - V_H - 112 = 42 \times (6 - H) \]

### Step 7: Solve the equations

We now have two equations:
1. \( 280 - V_H = 56 \times (7 - H) \)
2. \( 280 - V_H - 112 = 42 \times (6 - H) \)

Substituting \( V_H \) from the first equation into the second gives us:
\[ 280 - (280 - 56 \times (7 - H)) - 112 = 42 \times (6 - H) \]
This simplifies to:
\[ 56 \times (7 - H) - 112 = 42 \times (6 - H) \]

### Step 8: Solve for \( H \)

Now, we can solve for \( H \):
\[ 56 \times (7 - H) - 42 \times (6 - H) = 112 \]
Expanding both sides:
\[ 392 - 56H - 252 + 42H = 112 \]
Combining like terms:
\[ 140 - 14H = 112 \]
Rearranging gives:
\[ 14H = 28 \implies H = 2 \]

### Final Answer

The number of holidays in the week is \( H = 2 \).