Understanding the Problem: Petrol Price Increase and Expense Management

The question asks us to find the percentage by which Yogesh reduced his travel after the petrol price increased by 25%. The key condition is that his total monthly expenses on petrol remained the same as before the price hike.

Relating Price, Travel, and Expense

The total expense on petrol is calculated by multiplying the price per unit (like per litre) by the quantity consumed (which relates to travel distance or usage). We can express this as:

Expense = Price × Consumption

In this problem, 'Consumption' is directly related to 'Travel'. Let's use 'Travel' to represent the amount of petrol consumed or the distance covered.

Let's use variables to represent the initial situation:

• Original Price of petrol = \(P\)
• Original Travel (or Consumption) = \(T\)
• Original Expense = \(E = P \times T\)

Analyzing the Price Increase

The petrol price increased by 25%. So, the new price is:

New Price = Original Price + 25% of Original Price

New Price = \(P + 0.25P = 1.25P\)

Maintaining Constant Expenses

Yogesh reduced his travel so that his monthly expenses on petrol remained the same. This means the new expense is equal to the original expense.

Let the new travel (or consumption) be \(T_{new}\).

New Expense = New Price \(\times\) New Travel

New Expense = \(1.25P \times T_{new}\)

According to the problem, New Expense = Original Expense.

So, \(1.25P \times T_{new} = P \times T\)

Calculating the New Travel

We can solve the equation \(1.25P \times T_{new} = P \times T\) for \(T_{new}\):

\(T_{new} = \frac{P \times T}{1.25P}\)

Since \(P\) is a positive value (price of petrol), we can cancel \(P\) from the numerator and the denominator:

\(T_{new} = \frac{T}{1.25}\)

We know that \(1.25 = \frac{5}{4}\). So, \(\frac{1}{1.25} = \frac{1}{\frac{5}{4}} = \frac{4}{5} = 0.8\).

Thus, \(T_{new} = 0.8T\)

This means the new travel is 0.8 times the original travel, or 80% of the original travel.

Determining the Percentage Reduction in Travel

The percentage reduction in travel is calculated as:

Percentage Reduction = \(\frac{\text{Original Travel} - \text{New Travel}}{\text{Original Travel}} \times 100\%\)

Percentage Reduction = \(\frac{T - T_{new}}{T} \times 100\%\)

Substitute \(T_{new} = 0.8T\):

Percentage Reduction = \(\frac{T - 0.8T}{T} \times 100\%\)

Percentage Reduction = \(\frac{(1 - 0.8)T}{T} \times 100\%\)

Percentage Reduction = \(\frac{0.2T}{T} \times 100\%\)

Cancel \(T\) (assuming \(T\) is not zero):

Percentage Reduction = \(0.2 \times 100\%\)

Percentage Reduction = \(20\%\)

Alternative Approach: Using Formula for Constant Expense

When the price of a commodity increases by \(x\%\), and the consumer wants to keep the total expense constant, the consumption must be reduced by a certain percentage. The formula for this percentage reduction in consumption is:

Percentage Reduction = \(\frac{x}{100 + x} \times 100\%\)

In this problem, the petrol price increased by 25%. So, \(x = 25\).

Percentage Reduction in Travel = \(\frac{25}{100 + 25} \times 100\%\)

Percentage Reduction = \(\frac{25}{125} \times 100\%\)

Percentage Reduction = \(\frac{1}{5} \times 100\%\)

Percentage Reduction = \(20\%\)

Both methods yield the same result, confirming that Yogesh reduced his travel by 20%.

Since Original Expense = New Expense:

\(P \times T = 1.25P \times T_{new}\)

\(T_{new} = \frac{P \times T}{1.25P} = \frac{T}{1.25} = 0.8T\)

Reduction in travel = \(T - T_{new} = T - 0.8T = 0.2T\)

Percentage reduction = \(\frac{0.2T}{T} \times 100\% = 20\%\)

Revision Table: Percentage Calculations

Additional Information: Managing Expenses and Percentage Problems

This problem is a classic example of how changes in price affect consumption when the total expenditure is fixed. Understanding the relationship between price, consumption, and expense is crucial for solving such problems. It also highlights practical aspects of personal finance, like how consumers adjust their usage of goods or services when prices fluctuate.

• Inverse Relationship: When expense is kept constant, the price and consumption (travel in this case) have an inverse relationship. If the price goes up, consumption must go down proportionally, and vice versa.
• Base for Percentage Change: Always remember that the base for calculating percentage change is usually the original value, unless specified otherwise. In this problem, the price increase is based on the original price, and the travel reduction percentage is based on the original travel.
• Practical Application: This type of problem is relevant in various real-life scenarios involving budgeting, managing household expenses, and understanding economic concepts like elasticity of demand (though this problem simplifies it to fixed expenditure).

Solving percentage problems like this requires careful reading of the question to identify what values are changing and what value (expense) is staying constant. Setting up the initial and final scenarios with variables helps in formulating the correct equations.