Question

Eesha bought two varieties of rice, costing $50\text{Rs/kg }\!\!\,And\,\!\!\text{ 60Rs/kg}$ and mixed them in the ratio.
Then she sold the mixture at $70\text{Rs}$ making a profit of 20 percent, what was the ratio of the mixture?
A.$2:7$
B.$3:8$
C.$1:5$
D.$1:10$

Answer 1

Hint:In this type of question there is an inverse relation between two quantities. In this question the relation is between price & quantity of rice.
Formula used:
\[\dfrac{\text{Quantity of Dearer}}{\text{Quantity of Cheaper}}=\dfrac{\text{Mean price}-\text{Price of cheaper}}{\text{Price of dearer}-\text{Mean price}}\]
$\text{Selling Price}=\text{Cost Price}\left( 1+\dfrac{\text{P}}{100} \right)$
S.P=Selling price
C.P=Cost Price
P=Profit

Complete step-by-step answer:
 It is given that the price of Dearer quantity rice is \[\text{60 Rs/kg}\] and the price of cheaper variety also given that the selling price of mixture is \[\text{70 Rs/kg}\] and she makes the profit of \[20%\]
Price of Dearer quantity$=60\text{ Rs/}kg$
Price of Cheaper quantity$=50\text{ Rs/}kg$
Selling price of mixture$=70\text{ Rs/}kg$
Profit$=20%$
We have to find the cost price of mixture (i.e. Mean price of the mixture) because cost price of the two varieties is given. As we can compare the same things therefore we have to change either every quantity in cost price and selling price.
Now
S.P=C.P$\left( 1+\text{P/100} \right)$
$\Rightarrow $ 70=C.P$\left( 1+20/100 \right)$
$\Rightarrow 70=\text{C}\text{.P(1+1/5)}$
$\Rightarrow 70=6/5$ C.P
$\Rightarrow 5\times 70/3=$ C.P
$\Rightarrow 175/3=$ C.P
\[\begin{align}
  & \dfrac{\text{Quantity of Dearer}}{\text{Quantity of Cheaper}}=\dfrac{\text{Mean price}-\text{Price of cheaper}}{\text{Price of dearer}-\text{Mean price}} \\
 & \Rightarrow \dfrac{\text{Quantity of Dearer}}{\text{Quantity of Cheaper}}=\dfrac{\dfrac{175}{3}-50}{60-\dfrac{175}{3}} \\
 & \Rightarrow \dfrac{\text{Quantity of Dearer}}{\text{Quantity of Cheaper}}\text{=}\dfrac{\dfrac{175-150}{3}}{\dfrac{180-175}{3}} \\
 & \Rightarrow \dfrac{\text{Quantity of Dearer}}{\text{Quantity of Cheaper}}=\dfrac{\dfrac{25}{3}}{\dfrac{5}{3}} \\
 & \Rightarrow \dfrac{\text{Quantity of Dearer}}{\text{Quantity of Cheaper}}=\dfrac{25}{3}\times \dfrac{3}{5} \\
 & \Rightarrow \dfrac{\text{Quantity of Dearer}}{\text{Quantity of Cheaper}}=\dfrac{5}{1} \\
\end{align}\]
So option (c) is the correct option

Additional information:
 This type of question is the inverse relation between two quantities like $km/hr$ that means relation between distance and time it is not speed in allegation & mixture.

Note: We need to find the cost price of mixture as in allegation and mixture we can solve the question only if all the three quantities are in the same form. Because the cost price of two varieties of rice is given so we have to find the cost price of the mixture in order to solve the question. In allegation the quantity and price is inversely proportional if price is given then we can find the quantity and vice versa.