Question

Due to the increase of $15\% $ in the price of milk, a family reduces its consumption of milk by $15\% $. What was the effect on the expenditure of that family on account of milk?
A. $2.50\% $ decrease
B. $2.25\% $ decrease
C. $3\% $ decrease
D. \[3.50\% \] decrease

Answer 1

Hint: We are given that the price is increasing at $15\% $ and the consumption decreases at the same rate $15\% $. Therefore apply the expenditure formula,
Price$ \times $ consumption=Expenditure
First, find the new increased price, since the price is increased so we add the increased amount to the original price i.e.
${P_n} = P + \dfrac{{15}}{{100}}P$
Where, ${P_n}$=New price, $P$ =Actual price
Find the new decreased quantity, since consumption is decreased so we subtract the decreased quantity to the original consumption i.e.
${C_n} = C - \dfrac{{15}}{{100}}C$
Where, ${C_n}$=New consumption, $C$ =Actual consumption
Apply the formula, Price$ \times $ consumption=Expenditure to find the effect on expenditure.
${E_n} = {P_n} \times {C_n}$ , where ${E_n}$ is the expenditure after $15\% $ increase in price and $15\% $ decrease in consumption.
Simplifying the above we get the quantity by which expenditure increases or decreases.

Complete step-by-step solution:
The price of milk is increased by $15\% $ and the consumption of the milk is decreased by $15\% $.
Suppose $P$ is the price of milk, then the price increment becomes $\dfrac{{15}}{{100}}P$.
Add the increment to the price to calculate the new increased price.
${P_n} = P + \dfrac{{15}}{{100}}P \ldots (1)$ , where ${P_n}$=New price, $P$ =Actual price
Suppose $C$ is the consumption of milk, then the decrease in consumption is $\dfrac{{15}}{{100}}C$.
Subtract the decrease in consumption to calculate the new consumption.
${C_n} = C - \dfrac{{15}}{{100}}C \ldots (2)$ , where ${C_n}$=New consumption, $C$ =Actual consumption
The expenditure is the product of the price and consumption.
Expenditure =Price$ \times $ consumption
$ \Rightarrow E = P \times C$
We have to calculate the expenditure after changing the price and consumption.
 $ \Rightarrow {E_n} = {P_n} \times {C_n} \ldots (3)$
, where ${E_n}$ is the expenditure after $15\% $increase in price and $15\% $decrease in consumption.
From equation $(1)$ , $(2)$ and $(3)$ we have,
${E_n} = {P_n} \times {C_n}$
$ \Rightarrow {E_n} = \left( {P + \dfrac{{15}}{{100}}P} \right) \times \left( {C - \dfrac{{15}}{{100}}C} \right)$
$ \Rightarrow {E_n} = P\left( {1 + \dfrac{{15}}{{100}}} \right) \times C\left( {1 - \dfrac{{15}}{{100}}} \right)$
$ \Rightarrow {E_n} = PC\left( {\dfrac{{100 + 15}}{{100}}} \right) \times \left( {\dfrac{{100 - 15}}{{100}}} \right)$
Simplify the expression,
$ \Rightarrow {E_n} = PC\left( {\dfrac{{115}}{{100}}} \right) \times \left( {\dfrac{{85}}{{100}}} \right)$
$ \Rightarrow {E_n} = 0.9775 \times PC$
Since expenditure is given by, $E = P \times C$we get,
$ \Rightarrow {E_n} = 0.9775 \times E$
We can write $0.9775 = 1 - 0.02225$,
$ \Rightarrow {E_n} = (1 - 0.0225) \times E$
$ \Rightarrow {E_n} = E - 0.0225E$
$ \Rightarrow {E_n} = E - \dfrac{{2.25}}{{100}}E$
By the definition of the percentage, we can write $\dfrac{{2.25}}{{100}} = 2.25\% $
${E_n} = E - 2.25\% $ of $E$
The decrease in expenditure is $2.25\% $.

Option B is the correct answer.

Note: The most important formula to solve this question is Price$ \times $ consumption=Expenditure.
The key points for this type of question are:
If the price is Increased by some rate then, add the increased amount to the actual amount.
If we have to find the decreased value then we have to subtract the decreased value from the actual value.