Question

A man buys 100kg of sugar for Rs.2400. He sold \[{{\dfrac{1}{4}}^{th}}\] of the stock at a loss of 5 percent. At what percent profit should he sell the remaining stock so as to make an overall profit of 20% on the whole transaction?
(a) 25%
(b) \[27\dfrac{1}{3}%\]
(c) 28%
(d) \[28\dfrac{1}{3}%\]

Answer 1

Hint: Assume the cost price as C.P and selling price as S.P. First, find the C.P of 1kg of sugar by dividing the total cost of sugar with a total weight of sugar. Now, find the S.P for 25kg of sugar by using the formula: - Loss% = \[\dfrac{C.P-S.P}{100}\]. Then use the formula: - Profit% = \[\dfrac{S.P-C.P}{100}\] and find the S.P for 100kg of sugar, 25kg to determine the profit for 75kg sugar. Finally, apply the profit percent formula to determine the profit percent for 75kg of sugar which is the remaining stock.

Complete step-by-step solution
Here, let us assume the cost price as C.P. and selling price as S.P.
Now, it is given that the man buys 100kg of sugar for Rs.2400. Therefore, we have,
\[\Rightarrow \] Cost of 1 kg of sugar = (Total cost of sugar / Total weight of sugar)
 \[\Rightarrow \] Cost of 1 kg of sugar = \[\dfrac{2400}{100}\] = Rs.$24$
We have been provided with the data that he sold \[{{\dfrac{1}{4}}^{th}}\] of the stock at a loss of 5 percent. So, we have,
Total stock = 100kg
\[\Rightarrow \] \[{{\dfrac{1}{4}}^{th}}\] of the stock \[=\dfrac{1}{4}\times 100\] = 25kg
\[\Rightarrow \] Cost price of 25kg sugar = \[C.{{P}_{25}}=24\times 25=600\]
So, applying the formula for loss percent given as: - loss % = \[\left( \dfrac{C.P-S.P}{C.P} \right)\times 100\], we get,
For 25kg of sugar: -
\[\Rightarrow \] loss % = 5 %
\[\begin{align}
  & \Rightarrow \dfrac{C.{{P}_{25}}-S.{{P}_{25}}}{C.{{P}_{25}}}=\dfrac{5}{100} \\
 & \Rightarrow \dfrac{600-S.{{P}_{25}}}{600}=\dfrac{5}{100} \\
 & \Rightarrow 600-S.{{P}_{25}}=30 \\
 & \Rightarrow S.{{P}_{25}}=600-30 \\
 & \Rightarrow S.{{P}_{25}}=570 \\
\end{align}\]
Therefore, the selling price of 25kg sugar is Rs.570.
Now, it is given to us that the overall profit he wants to make is 20%. This is the profit on a total of 100kg of sugar. So, applying the formula for profit percent, we get,
\[\Rightarrow \] Profit % = \[\dfrac{S.{{P}_{100}}-C.{{P}_{100}}}{C.{{P}_{100}}}\times 100\]
\[\Rightarrow 20=\dfrac{S.{{P}_{100}}-2400}{2400}\times 100\]
\[\Rightarrow S.{{P}_{100}}=480+2400\]
\[\Rightarrow S.{{P}_{100}}=\]Rs.$2880$
Therefore, the selling price of 100kg of sugar is Rs.2880.
Now, we have to determine the profit percent on the remaining stock, i.e. 75kg. So, we have,
\[\begin{align}
  & \Rightarrow S.{{P}_{75}}=S.{{P}_{100}}-S.{{P}_{25}} \\
 & \Rightarrow S.{{P}_{75}}=2880-570 \\
\end{align}\]
\[\Rightarrow S.{{P}_{75}}=\] Rs.2310
So, applying the formula of profit percent for 75kg of sugar, we get,
\[\Rightarrow \] Profit % = \[\dfrac{S.{{P}_{75}}-C.{{P}_{75}}}{C.{{P}_{75}}}\times 100\]
Here, \[C.{{P}_{75}}=24\times 75\] = Rs.1800
\[\Rightarrow \] Profit % = \[\left( \dfrac{2310-1800}{1800} \right)\times 100\]
\[\Rightarrow \] Profit % = \[\dfrac{85}{3}\]
\[\Rightarrow \] Profit % = \[28\dfrac{1}{3}%\]
Hence, option (d) is the correct answer.

Note: One may note that we cannot take the sum or difference of percentage directly. One may think that the percentage of profit will be the sum of 20% and 5%. This will give the answer 25% but it will be a totally wrong approach. We have to proceed step – by – step just like we did in the above solution. You must remember the formula of profit percentage and loss percentage to solve the question.