Question

A man sells two houses at the rate of 1.95 lacs rupees each on one. He gains $5\%$ and on others, he loses $5\%$ and finds his gain or loss percent in the whole transaction?

Answer 1

Hint: Let us assume the cost price of the first house be Rs. x and the second house be Rs. y. And the selling price of each of the two houses is the same (i.e. 1.95 lacs). It is given that in one house, he gains $5\%$ and in the other house, he lost $5\%$. So, we are going to use the formula when $5\%$ gain has occurred as follows: $S.P.=C.P.+\dfrac{5}{100}CP$. Now, substitute C.P. as Rs. x. And also we are going to use the formula for S.P. when loss of $5\%$ has happened is as follows: $S.P.=C.P.-\dfrac{5}{100}CP$. Now, substitute C.P. as Rs. y and S.P. as 1.95 lacs. From these equations, we are going to find the values of x and y or the cost prices of the two houses. Now, we are going to use the formulae for gain and loss percent as follows: $Gain\%=\dfrac{S.P.-C.P.}{C.P.}\times 100$ and the loss percent is equal to $Loss\%=\dfrac{C.P.-S.P.}{C.P.}\times 100$.

Complete step-by-step solution:
Let us assume the cost price of one of the houses is Rs. x and the other house is Rs. y.
Now, we have given the S.P. for each of the two houses are rupees 1.95 lacs.
In one of the houses, he gains $5\%$ so using the following formula, we can calculate the cost price of this house:
$S.P.=C.P.+\dfrac{5}{100}CP$
Substituting S.P. as Rs. 1.95 lacs and C.P. as Rs. x in the above we get,
$\begin{align}
  & 1.95lacs=x+\dfrac{5}{100}x \\
 & \Rightarrow 1.95lacs=x\left( 1+.05 \right) \\
 & \Rightarrow 1.95lacs=x\left( 1.05 \right) \\
\end{align}$
Dividing 1.05 on both the sides of the above equation we get,
$\begin{align}
  & \dfrac{1.95}{1.05}lacs=x \\
 & \Rightarrow 1.86lacs=x \\
\end{align}$
From the above, we got the C.P. of one of the houses as Rs 1.86 lacs.
Similarly, using the following formula we can find the cost price for the other house as follows:
$S.P.=C.P.-\dfrac{5}{100}CP$
Substituting S.P. as Rs. 1.95 lacs and C.P. as Rs. y in the above we get,
$\begin{align}
  & 1.95lacs=y-\dfrac{5}{100}y \\
 & \Rightarrow 1.95lacs=y\left( 1-0.05 \right) \\
 & \Rightarrow 1.95lacs=y\left( 0.95 \right) \\
\end{align}$
Dividing 0.95 on both the sides we get,
$\begin{align}
  & \dfrac{1.95lacs}{0.95}=y \\
 & 2.05lacs=y \\
\end{align}$
From the above, we got the C.P. of one of the houses as Rs 2.05 lacs.
The total cost price (C.P.) of the two houses is equal to the addition of x and y.
$\begin{align}
  & Rs\left( x+y \right) \\
 & =Rs\left( 1.86+2.05 \right)lacs \\
 & =Rs3.91lacs \\
\end{align}$
Now, we are going to find the total selling price (S.P.) of the two houses by adding 1.95 lacs twice.
$\begin{align}
  & \left( 1.95+1.95 \right)lacs \\
 & =3.90lacs \\
\end{align}$
Now, to find either gain or loss percent, we are subtracting the total selling price and cost price. As you can see the total cost price is greater than the total selling price so we are going to subtract the selling price from the cost price we get,
$\begin{align}
  & Rs\left( 3.91-3.90 \right)lacs \\
 & =Rs0.01lacs \\
\end{align}$
We know that when we subtract selling price from cost price then loss has occurred so to find the loss percent we are going to divide this loss by the total cost price and then multiplying the division by 100 we get,
$\begin{align}
  & \dfrac{0.01}{3.91}\times 100 \\
 & =0.25\% \\
\end{align}$
Hence, the loss percent is $0.25\%$.

Note: Make sure you know the required gain and loss formulae in percentage. Also, you should know how to write the selling price (S.P.) and cost price (C.P.) from gain and loss. Another thing, to know whether loss or gain has occurred in the total transaction of buying and selling two houses, you must see either the total cost price or selling price is higher in value then use the gain or loss percent formula.