Let's break down this problem about hostel expenses step-by-step to find the original expenditure.

Understanding the Hostel Expense Problem

We are given a situation where the number of students in a hostel changes, and this affects the total daily expenses and the average expenditure per student. We need to use the given information to find out what the total daily expenses were originally.

Setting Up the Equations

Let's use variables to represent the unknown values.

• Let the original average expenditure per head per day be Rs. x.
• The original number of students is 35.

The original total expenditure is the number of students multiplied by the average expenditure per head.

• Original Total Expenditure = Original Number of Students \(\times\) Original Average Expenditure per Head
• Original Total Expenditure = \(35 \times x = \text{Rs. } 35x\)

Now, consider the situation after the increase in students.

• The number of students increases by 7. So, the new number of students is \(35 + 7 = \text{42}\).
• The total expenses of the hostel increase by Rs. 42 per day. So, the new total expenditure is Original Total Expenditure + Rs. 42, which is \(\text{Rs. } (35x + 42)\).
• The average expenditure per head diminishes by Re. 1. So, the new average expenditure per head is Original Average Expenditure per Head - Re. 1, which is \(\text{Rs. } (x - 1)\).

The new total expenditure can also be calculated by multiplying the new number of students by the new average expenditure per head.

• New Total Expenditure = New Number of Students \(\times\) New Average Expenditure per Head
• New Total Expenditure = \(42 \times (x - 1)\)

We now have two expressions for the New Total Expenditure. We can set them equal to each other to form an equation.

\(42 \times (x - 1) = 35x + 42\)

Solving for the Average Expenditure

Let's solve the equation for \(x\):

\(42(x - 1) = 35x + 42\)

Distribute the 42 on the left side:

\(42x - 42 = 35x + 42\)

Now, collect the terms with \(x\) on one side and the constant terms on the other side. Subtract \(35x\) from both sides:

\(42x - 35x - 42 = 42\)

\(7x - 42 = 42\)

Add 42 to both sides:

\(7x = 42 + 42\)

\(7x = 84\)

Divide by 7 to find the value of \(x\):

\(x = \frac{84}{7}\)

\(x = 12\)

So, the original average expenditure per head was Rs. 12.

Calculating the Original Expenditure

The question asks for the original expenditure of the hostel. We defined the original total expenditure as \(35x\).

Original Total Expenditure = \(35 \times x\)

Substitute the value of \(x\) we found:

Original Total Expenditure = \(35 \times 12\)

Let's calculate \(35 \times 12\):

\(35 \times 10 = 350\)

\(35 \times 2 = 70\)

\(350 + 70 = 420\)

So, the original expenditure of the hostel was Rs. 420.

Let's quickly verify the new state:

• New number of students: 42
• New average expenditure: \(x - 1 = 12 - 1 = \text{Rs. } 11\)
• New total expenditure: \(42 \times 11 = \text{Rs. } 462\)
• Original total expenditure + 42 = \(420 + 42 = \text{Rs. } 462\)

The values match, confirming our calculation for \(x\) is correct.

Summary of Hostel Expenses

The original expenditure of the hostel is Rs. 420.

Revision Table: Hostel Expenditure Calculation

Reviewing the key steps involved in solving hostel expense problems like this one:

• Identify original and new conditions for students, average expense, and total expense.
• Use variables (like 'x' for original average expense) to represent unknowns.
• Formulate expressions for total expenses in both original and new states.
• Create an equation linking the original and new total expenses based on the given increase or decrease.
• Solve the equation for the unknown variable (average expense).
• Calculate the required value (original or new total expense) using the variable's value.
• Verify the answer by plugging the values back into the problem description.

Additional Information: Average and Total Values

In problems involving averages and totals, remember the basic relationship:

Total Value = Number of Items \(\times\) Average Value per Item

This relationship is fundamental to solving many problems related to averages, expenditures, scores, etc. When dealing with changes, always carefully note how the number of items, average value, and total value change. Sometimes the average changes, and sometimes the total changes, and these changes are interconnected.

For instance, if a group's average score increases, it could be because new members with higher scores joined, or the existing members' scores increased, or a combination of both, affecting the total score.