Question

A grocer bought sugar worth of RS. 4500. He sold one-third of it at a 10% gain. Then what percent of the remaining sugar be sold to get a 12% gain on the whole?

Answer 1

Hint: We solve this problem by using the formula of gain percentage.
The formula o gain percentage is given as
\[\text{Gain Percentage}=\dfrac{S.P-C.P}{C.P}\times 100\]
Where S.P is the selling price and C.P is the cost price.
By using this formula we find the selling price of one-third of sugar to get 10% gain and the selling piece of total sugar to get a 12% gain. Then we can find the selling price of the remaining two-third of sugar.

Complete step by step answer:
We are given that the cost price of sugar as RS. 4500
Let us assume that the cost price of whole sugar as
\[\Rightarrow C.P=4500\]
We are given that one-third of sugar is sold at a 10% gain.
Let us assume that the cost price of one third of sugar as \[{{C}_{1}}\]
We know that if cost price of whole product is \[x\] then the cost price of one third is given as \[\dfrac{x}{3}\]
By using the above condition we get
\[\begin{align}
  & \Rightarrow {{C}_{1}}=\dfrac{4500}{3} \\
 & \Rightarrow {{C}_{1}}=1500 \\
\end{align}\]
Let us assume that the selling price of one third of sugar as \[{{S}_{1}}\]
We know that the formula of gain percentage is given as
\[\text{Gain Percentage}=\dfrac{S.P-C.P}{C.P}\times 100\]
By using the above formula to one third sugar we get
\[\Rightarrow \text{Gain}=\dfrac{{{S}_{1}}-{{C}_{1}}}{{{C}_{1}}}\times 100\]
By substituting the required values in above equation we get
\[\begin{align}
  & \Rightarrow \dfrac{{{S}_{1}}-1500}{1500}\times 100=10 \\
 & \Rightarrow {{S}_{1}}-1500=150 \\
 & \Rightarrow {{S}_{1}}=1650 \\
\end{align}\]
We are given that the gain percentage of whole sugar as 12%
Let us assume that the selling price of whole sugar as \[S.P\]
Now, by using the gain percentage formula we get
\[\Rightarrow \text{Gain}=\dfrac{S.P-C.P}{C.P}\times 100\]
By substituting the required values in above equation we get
\[\begin{align}
  & \Rightarrow \dfrac{S.P-4500}{4500}\times 100=12 \\
 & \Rightarrow S.P-4500=540 \\
 & \Rightarrow S.P=5040 \\
\end{align}\]
Let us assume that the cost of remaining two third sugar as \[S\]
We know that cost of whole sugar is obtained by adding the costs of one third and the remaining two third.
By converting the above statement in to mathematical equation we get
\[\Rightarrow S.P=S+{{S}_{1}}\]
By substituting the required values in above equation we get
\[\begin{align}
  & \Rightarrow 5040=S+1650 \\
 & \Rightarrow S=3390 \\
\end{align}\]
Therefore we can conclude that the cost of remaining two third of sugar is RS. 3390 to get a total gain of 12% on the whole.

Note:
Students may do a mistake in taking the gain of one-third of the sugar.
We have the gain percentage of one-third of the sugar as 10%
By using the formula we got the equation as
\[\Rightarrow \text{Gain}=\dfrac{{{S}_{1}}-{{C}_{1}}}{{{C}_{1}}}\times 100\]
Here, the value \[{{C}_{1}}\] represents the cost price of one-third of the sugar.
But students may do mistake and take the formula as
\[\Rightarrow \text{Gain}=\dfrac{{{S}_{1}}-C.P}{C.P}\times 100\]
Here, the value C.P represents the total cost of sugar.
This formula gives the wrong answer because in the gain percentage formula the gain is calculated with respect to its cost price. So, the cost price we take in the formula should be the cost price of one-third of sugar.