Ahmed buys a plot of land for Rs. 480000. He sells \[\dfrac{2}{5}\] of it at a loss of 6 %. At what gain percent should he sell the remaining part of the plot to gain 10 % on the whole?

Hint:First of all, find the \[S{{P}_{1}}\] such that gain % would be 10 % on the whole. Now, find \[S{{P}_{2}}\] at which he sold \[\dfrac{2}{5}\] of the land at 6 % loss. Now subtract \[S{{P}_{2}}\text{ from }S{{P}_{1}}\] to get \[S{{P}_{3}}\]. Also, subtract \[C{{P}_{2}}\text{ of }\dfrac{2}{5}\] of the land from the total \[C{{P}_{1}}\text{ to get }C{{P}_{3}}\]. Now, find the gain % using \[\left( \dfrac{S{{P}_{3}}-C{{P}_{3}}}{C{{P}_{3}}} \right)\times 100\].

Complete step-by-step answer:
We are given that Ahmed buys a plot of land for Rs. 480000. He sell \[\dfrac{2}{5}\] of it at a loss of 6 %. We have to find the gain percent at which he should sell the remaining part of the plot to gain 10 % on the whole.
Now, let us consider our question. We are given that Ahmed buys a plot of Rs. 48000. So, we get,
\[\text{Total }C{{P}_{1}}=Rs.480000.....\left( i \right)\]
We are given that the total gain % should be 10 %. We know that,
Gain % = \[\left( \dfrac{S.P-C.P}{C.P} \right)\times 100\]
By substituting
\[CP=C{{P}_{1}}=Rs.480000\]
Gain % = 10 %
\[SP=S{{P}_{1}}\]
We get,
\[10=\dfrac{\left( S{{P}_{1}}-480000 \right)}{480000}\times 100\]
\[48000=S{{P}_{1}}-480000\]
\[S{{P}_{1}}=480000+48000\]
So, we get,
\[S{{P}_{1}}=Rs.528000....\left( ii \right)\]
He sells \[\dfrac{2}{5}\] of it at a loss of 6 %. So, we get, CP of \[\dfrac{2}{5}\] of the plot
\[=C{{P}_{2}}=\dfrac{2}{5}\times 480000\]
So, we get,
\[C{{P}_{2}}=2\times 96000=Rs.192000....\left( iii \right)\]
We know that,
Loss % \[=\left( \dfrac{CP-SP}{CP} \right)\times 100\]
By substituting
\[CP=C{{P}_{2}}=Rs.192000\]
Loss % = 6 %
\[SP=S{{P}_{2}}\]
We get,
\[6=\left( \dfrac{192000-S{{P}_{2}}}{192000} \right)\times 100\]
\[6\times 1920=192000-S{{P}_{2}}\]
So, we get,
\[S{{P}_{2}}=192000-11520\]
\[S{{P}_{2}}=Rs.180480....\left( iv \right)\]
So, we have got the total CP that is \[C{{P}_{1}}=Rs.480000\text{ and }C{{P}_{2}}=Rs.192000\]
So, we get CP of the remaining that is
\[C{{P}_{3}}=C{{P}_{1}}-C{{P}_{2}}\]
\[C{{P}_{3}}=Rs480000-Rs.192000\]
\[C{{P}_{3}}=Rs288000.....\left( v \right)\]
We have also got the total SP that is \[S{{P}_{1}}=Rs.528000,S{{P}_{2}}=Rs.180480\]. So, we get, SP of the remaining part, that is,
\[S{{P}_{3}}=S{{P}_{1}}-S{{P}_{2}}\]
\[S{{P}_{3}}=Rs.528000-Rs.180480\]
\[S{{P}_{3}}=Rs.347520...\left( vi \right)\]
We know that,
Gain % = \[\left( \dfrac{S.P-C.P}{C.P} \right)\times 100\]
So, we get, the gain % of the remaining part,
\[=\dfrac{\left( S{{P}_{3}}-C{{P}_{3}} \right)}{C{{P}_{3}}}\times 100\]
\[=\dfrac{\left( 347520-288000 \right)}{288000}\times 100\]
\[=\dfrac{59520}{288000}\times 100\]
= 20.6667 %
So, we get the gain percent of the remaining part as 20.67 %.

Note: In these types of questions, always take care of taking the SP and CP of the same part or the same quantity. For example, if the CP is of the \[\dfrac{2}{5}th\] part, that is, SP should also be of \[\dfrac{2}{5}th\] part only. Also, students must understand that the physical significance of the profit and loss is that when SP > CP, profit is increased while when CP > SP, there is a loss. Students should also remember that we see loss, loss %, profit %, etc. with respect to CP.