There was a hike in petrol price by 12%. By how much percentage should a person decrease his petrol consumption such that there is no change in his expenditure on petrol ?
[amp_mcq option1=”8.6%” option2=”9.8%” option3=”10.7%” option4=”12%” correct=”option3″]
This question was previously asked in
UPSC CAPF – 2024
The original expenditure on petrol = P * C.
The petrol price hikes by 12%.
The new price P’ = P + 12% of P = P + 0.12P = 1.12P.
Let the new consumption be C’.
The new expenditure on petrol = P’ * C’ = 1.12P * C’.
The condition is that there is no change in expenditure.
Original expenditure = New expenditure
P * C = 1.12P * C’.
Divide both sides by P (assuming P > 0):
C = 1.12 * C’.
C’ = $\frac{C}{1.12}$.
The decrease in consumption = Original Consumption – New Consumption = C – C’.
Decrease = $C – \frac{C}{1.12} = C \left(1 – \frac{1}{1.12}\right) = C \left(\frac{1.12 – 1}{1.12}\right) = C \left(\frac{0.12}{1.12}\right)$.
The percentage decrease in consumption is $\frac{\text{Decrease in consumption}}{\text{Original consumption}} \times 100$.
Percentage decrease = $\frac{C \left(\frac{0.12}{1.12}\right)}{C} \times 100 = \frac{0.12}{1.12} \times 100$.
$\frac{0.12}{1.12} = \frac{12}{112}$.
Percentage decrease = $\frac{12}{112} \times 100 = \frac{3}{28} \times 100 = \frac{300}{28} = \frac{75}{7}$.
Now calculate the value of $\frac{75}{7}$ as a decimal:
$75 \div 7 \approx 10.714…$.
Rounding to one decimal place, this is approximately 10.7%.
Percentage decrease = $\frac{12}{100+12} \times 100 = \frac{12}{112} \times 100 = \frac{3}{28} \times 100 = \frac{300}{28} = \frac{75}{7}\%$.
This formula is a direct consequence of the relationship: Original Price * Original Consumption = New Price * New Consumption.