Question

If a man were to sell his hand-cart for Rs.720, he would lose $25\%$. What must he sell it to gain $25\%$?

Answer 1

Hint: Let the Cost Price of the cart be C.P and selling price be S.P. The formula for a loss $L\%$ is given by $\dfrac{C.P-S.P}{C.P}=\dfrac{L}{100}$. In the question, it is given that the Selling Price S.P of the cart is Rs.720 which led to a loss of $25\%$. Substituting the values in the above equation, we get the value of Cost Price C.P. The formula for a profit of $P\%$ is given by $\dfrac{S.P-C.P}{C.P}=\dfrac{P}{100}$. Using this formula in the calculation of S.P for a profit of $25\%$, we get the required answer.

Complete step-by-step solution:
In the question, it is given that the Selling Price $S.P = Rs.720$.
Loss percentage $L\% = 25\%$.
Let the Cost Price for which the man has purchased the cart be C.P.
The formula for a loss $L\%$ is given by $\dfrac{C.P-S.P}{C.P}=\dfrac{L}{100}\to (1)$.
where C.P = Cost Price, S.P = Selling Price.
Substituting the values in equation-1, we get
$\dfrac{C.P-720}{C.P}=\dfrac{25}{100}$
Separating the L.H.S into 2 fractions and simplifying the R.H.S gives,
$\begin{align}
  & \dfrac{C.P}{C.P}-\dfrac{720}{C.P}=\dfrac{1}{4} \\
 & 1-\dfrac{720}{C.P}=\dfrac{1}{4} \\
 & 1-\dfrac{1}{4}=\dfrac{720}{C.P} \\
 & \dfrac{720}{C.P}=\dfrac{3}{4} \\
\end{align}$
Cross multiplying gives
$\begin{align}
  & 720\times 4=3\times C.P \\
 & C.P=\dfrac{720\times 4}{3}=240\times 4=960 \\
\end{align}$
$\therefore C.P= Rs.960$.
The required Selling Price S.P is to be calculated at a profit of $25\%$.
The formula for a profit of $P\%$ is given by $\dfrac{S.P-C.P}{C.P}=\dfrac{P}{100}\to \left( 2 \right)$.
Substituting $C.P = Rs.960$ and $P = 25$ in equation-2 we get,
\[\begin{align}
  & \dfrac{S.P-960}{960}=\dfrac{25}{100} \\
 & \dfrac{S.P}{960}-1=\dfrac{1}{4} \\
 & \dfrac{S.P}{960}=1+\dfrac{1}{4} \\
 & \dfrac{S.P}{960}=\dfrac{5}{4} \\
 & S.P=\dfrac{5\times 960}{4}=5\times 240=1200 \\
\end{align}\]
$\therefore $ The Selling price should be Rs.1200 to get a profit of $25\%$.

Note: The alternate way to do this question is to compare the given Selling Price with the
(1 + Profit Fraction) or (1- Loss Fraction) times the Cost Price. Mathematically,
S.P = $\left( 1+\dfrac{P}{100} \right)\times C.P$ or S.P =$\left( 1-\dfrac{L}{100} \right)\times C.P$. Using this concept, for the loss of $25\%$ S.P is Rs.720.
$\begin{align}
  & 720=\left( 1-\dfrac{25}{100} \right)C.P \\
 & C.P=\dfrac{720\times 4}{3}=Rs.960 \\
\end{align}$
For the profit of $25\%$
$\begin{align}
  & S.P=\left( 1+\dfrac{25}{100} \right)\times 960 \\
 & S.P=\dfrac{5\times 960}{4}=Rs.1200 \\
\end{align}$
This is another way to do the question.