Question

In the new budget, the price of petrol rose by $10\%$ . By how much percent must one reduce the consumption so that the expenditure does not increase?

Answer 1

Hint: First we will assume the cost of petrol as $x$ . Now we will calculate the increase in the price of petrol by using the petrol rose in budget, from this value we can calculate the new price of petrol after releasing the budget. Now we will assume the initial consumption as $y$ and final consumption as $z$. To balance the expenditure, we need to equate the initial expenditures to final expenditure so that we will get the reduction in the consumption.

Complete step-by-step answer:
Let the initial price of the petrol is $x$.
Given that the petrol price is rose by $10\%$ in the new budget, then the increase in petrol price is given by
$\begin{align}
  & I=10\%x \\
 & \Rightarrow I=\dfrac{10}{100}\times x \\
 & \Rightarrow I=0.1x \\
\end{align}$
Now the final price of the petrol after new budget is given by
$\begin{align}
  & {{F}_{p}}=x+0.1x \\
 & \Rightarrow {{F}_{p}}=x\left( 1+0.1 \right) \\
 & \Rightarrow {{F}_{p}}=1.1x \\
\end{align}$
Let the initial and final consumption are $y$ and $z$ respectively.
Now the initial expenditures are $x\times y$.
Final expenditures are ${{F}_{p}}\times z$
If the initial expenditures are equal to final expenditures, then only there is no loss. So
$\begin{align}
  & x\times y={{F}_{p}}\times z \\
 & \Rightarrow x\times y=1.1x\times z \\
 & \Rightarrow z=\dfrac{y}{1.1} \\
\end{align}$
So, the final consumption should be equal to $\dfrac{1}{1.1}$ of the initial consumption.
So, the required percentage is $\left( 1-\dfrac{1}{1.1} \right)\times 100\approx 9\%$
$\therefore $ The person should decrease his consumption by $9\%$.

Note: For the problem that are dealing with the percentages we can assume the count as $100$ and calculate the values of the given variables according to given conditions. For this problem we can assume the initial price of petrol as Rs.$100$ from this we can calculate the increase in petrol price as $10\%\text{ of }100=10$ and increased price of petrol as Rs.$110$ and percentage can be calculated by
$\dfrac{110-100}{110}\times 100=\dfrac{10}{110}\times 100\approx 9\%$
From both the methods we got the same result.