Question

Pure Ghee costs Rs.100 per kg. After adulterating it with vegetable oil costing Rs.50 per kg. A shopkeeper sells the mixture at Rs.96 per kg. Thereby making a profit of 20%. In what ratio does he mix the two?
(a) 1:2
(b) 3:2
(c) 3:1
(d) 2:3

Answer 1

Hint: In this question, we first need to find the cost of pure ghee by equating the cost of mixture to Rs.96. Then we get the actual price of the ghee now as 1 kg ghee is Rs.100 and 1 kg vegetable oil is Rs.50 then by calculating how much pure ghee and vegetable oil both together give that cost gives the ratio of the ghee and the vegetable oil.

Complete step-by-step answer:

Let us assume that the cost of pure ghee and vegetable oil together as x.

Now, the cost of mixture can be written as the sum of the cost of them together and the profit.

\[\Rightarrow \text{cost of mixture}=x+\text{20 percent of }x\text{ }\]

Given, in the question that the cost of the mixture is Rs.96

According to the percentage formula x% of y can be written as

\[\dfrac{x}{100}\times y\]

Now, we can write the equation as follows.

\[\Rightarrow x+\dfrac{20}{100}\times x=96\]

This can be further simplified as.

\[\Rightarrow \dfrac{100x+20x}{100}=96\]

Now, this can also be written as.


\[\Rightarrow \dfrac{120x}{100}=96\]

Let us now multiply with 100 on both the sides.

\[\Rightarrow 120x=96\times 100\]

Now, on dividing with 120 on both the sides we get,

\[\begin{align}

  & \Rightarrow x=\dfrac{9600}{120} \\

 & \therefore x=Rs.80 \\

\end{align}\]

Now, this is the original cost of the pure ghee and vegetable oil together.

Given, that

Cost of pure ghee per kg is Rs.100

Cost of vegetable oil per kg is Rs.50

As we already know that Rs.80 is the original cost of the mixture per kg so if we consider the amount of ghee as y then the amount of vegetable oil will be the remaining in the 1 kg mixture.

Now, we can write this as:

\[\Rightarrow y\times 100+\left( 1-y \right)\times 50=80\]

Now, on further expanding this we get,

\[\begin{align}

  & \Rightarrow 100y+50-50y=80 \\

 & \Rightarrow 50+50y=80 \\

\end{align}\]

Now, on rearranging and further simplifying we get,

\[\begin{align}

  & \Rightarrow 50y=30 \\

 & \therefore y=\dfrac{3}{5} \\

\end{align}\]

Now, by subtracting this value of y from 1 we get the amount of vegetable oil.

\[\begin{align}

  & \Rightarrow 1-y \\

 & \Rightarrow 1-\dfrac{3}{5} \\

 & \Rightarrow \dfrac{2}{5} \\

\end{align}\]

Thus, the ratio of pure ghee and vegetable oil is:

\[\begin{align}

  & \Rightarrow \dfrac{3}{5}:\dfrac{2}{5} \\

 & \Rightarrow 3:2 \\

\end{align}\]

Hence, the correct option is (b).

Note: It is important to note that from the selling price of the mixture given we need to find the cost of the mixture which contains both ghee and vegetable oil. So, as we know the cost of them separately per kg now we can get the amount of them present in the mixture.
While solving we should not neglect any of the terms and while rearranging we need to be careful about the sign because it changes the complete and so the result also. It is also to be noted that both the amount of ghee and vegetable oil sum up to 1 kg because the cost we got is per kg.