Calculating Percentage Decrease in Consumption When Petrol Price Increases

This problem involves the relationship between the price of a commodity (petrol), its consumption, and the total expenditure. The key information is that the expenditure on petrol remains unchanged despite the price increase.

The fundamental relationship is:

\(\text{Expenditure} = \text{Price} \times \text{Consumption}\)

If the expenditure is to remain constant, and the price increases, the consumption must decrease proportionally. We need to find the percentage decrease in consumption required.

Step-by-Step Solution for Petrol Consumption Adjustment

Let's assume initial values to make the calculation easier. Suppose:

• Initial Price of petrol = \( \text{Rs. } 100 \) per unit
• Initial Consumption = \( 100 \) units

Using these assumptions, the initial expenditure on petrol is:

\( \text{Initial Expenditure} = \text{Initial Price} \times \text{Initial Consumption} \)

\( \text{Initial Expenditure} = 100 \times 100 = \text{Rs. } 10000 \)

Now, the problem states that the price of petrol increased by \(7\%\). So, the new price will be:

\( \text{New Price} = \text{Initial Price} + 7\% \text{ of Initial Price} \)

\( \text{New Price} = 100 + \frac{7}{100} \times 100 \)

\( \text{New Price} = 100 + 7 = \text{Rs. } 107 \) per unit

The problem also states that the expenditure on petrol remains unchanged. This means the new expenditure is the same as the initial expenditure:

\( \text{New Expenditure} = \text{Initial Expenditure} = \text{Rs. } 10000 \)

Let the new consumption be \( C_{\text{new}} \). We can use the relationship \( \text{Expenditure} = \text{Price} \times \text{Consumption} \) again for the new values:

\( \text{New Expenditure} = \text{New Price} \times \text{New Consumption} \)

\( 10000 = 107 \times C_{\text{new}} \)

Now, we can find the value of the new consumption:

\( C_{\text{new}} = \frac{10000}{107} \) units

The decrease in consumption is the difference between the initial consumption and the new consumption:

\( \text{Decrease in Consumption} = \text{Initial Consumption} - \text{New Consumption} \)

\( \text{Decrease in Consumption} = 100 - \frac{10000}{107} \)

To subtract, we find a common denominator:

\( \text{Decrease in Consumption} = \frac{100 \times 107}{107} - \frac{10000}{107} \)

\( \text{Decrease in Consumption} = \frac{10700 - 10000}{107} = \frac{700}{107} \) units

Finally, we need to calculate the percentage decrease in consumption. This is the decrease in consumption divided by the original consumption, multiplied by \(100\%\):

\( \text{Percentage Decrease in Consumption} = \frac{\text{Decrease in Consumption}}{\text{Initial Consumption}} \times 100\% \)

\( \text{Percentage Decrease in Consumption} = \frac{\frac{700}{107}}{100} \times 100\% \)

\( \text{Percentage Decrease in Consumption} = \frac{700}{107} \times \frac{1}{100} \times 100\% \)

\( \text{Percentage Decrease in Consumption} = \frac{700}{107} \% \)

To express this as a mixed fraction, we divide 700 by 107:

\( 700 \div 107 \)

\( 107 \times 6 = 642 \)

\( 700 - 642 = 58 \)

So, \( \frac{700}{107} \) is \( 6 \) with a remainder of \( 58 \). This means \( \frac{700}{107} = 6 \frac{58}{107} \).

Therefore, the percentage decrease in consumption should be \( 6 \frac{58}{107} \% \).

Alternatively, a direct formula can be used for such problems where expenditure remains constant: If the price increases by R%, the consumption must be reduced by \( \frac{R}{100+R} \times 100\% \). Here, R = 7%. So, the percentage decrease in consumption is \( \frac{7}{100+7} \times 100\% = \frac{7}{107} \times 100\% = \frac{700}{107}\% = 6 \frac{58}{107} \% \).

Summary of Calculation

The consumer should decrease the consumption of petrol by \( 6 \frac{58}{107} \% \) to keep the expenditure unchanged when the price increases by \(7\%\).

Revision Table: Price, Consumption, and Expenditure

Additional Information: Percentage Calculations

Understanding percentage changes is crucial for many real-world problems, including those related to finance, economics, and daily expenses like petrol. When dealing with percentage changes and a constant product (like Expenditure = Price x Consumption), a percentage increase in one factor requires a specific percentage decrease in the other, and these percentages are not simply opposites.

• If a value increases by x%, the new value is Original Value \( \times (1 + \frac{x}{100}) \).
• If a value decreases by y%, the new value is Original Value \( \times (1 - \frac{y}{100}) \).
• In this problem, Price\(_{new}\) = Price\(_{old}\) \( \times (1 + \frac{7}{100}) = \) Price\(_{old}\) \( \times \frac{107}{100} \).
• Since Expenditure is constant, Price\(_{old}\) \( \times \) Consumption\(_{old}\) = Price\(_{new}\) \( \times \) Consumption\(_{new}\).
• Price\(_{old}\) \( \times \) Consumption\(_{old}\) = Price\(_{old}\) \( \times \frac{107}{100} \times \) Consumption\(_{new}\).
• This simplifies to Consumption\(_{old}\) = \( \frac{107}{100} \times \) Consumption\(_{new}\), or Consumption\(_{new}\) = Consumption\(_{old}\) \( \times \frac{100}{107} \).
• The decrease in consumption is Consumption\(_{old}\) - Consumption\(_{new}\) = Consumption\(_{old}\) - Consumption\(_{old}\) \( \times \frac{100}{107} = \) Consumption\(_{old}\) \( (1 - \frac{100}{107}) = \) Consumption\(_{old}\) \( (\frac{107-100}{107}) = \) Consumption\(_{old}\) \( \times \frac{7}{107} \).
• The percentage decrease is \( \frac{\text{Decrease}}{\text{Original}} \times 100\% = \frac{\text{Consumption}_{old} \times \frac{7}{107}}{\text{Consumption}_{old}} \times 100\% = \frac{7}{107} \times 100\% = \frac{700}{107}\% \).
• This confirms the result \( 6 \frac{58}{107} \% \).