Question

A shopkeeper buys a kg of rice at Rs x per kg and another b kg of rice at Rs y per kg. If he sells the mixture at z per kg. Find expression for his profit percent.

Answer 1

Hint:
Cost Price (C.P.): Price at which an article is purchased.
Selling Price (S.P.): Price at which an article is sold by a shopkeeper.
If the selling price is more than cost price then it is known as profit.
$Profit = S.P.({\text{Selling price}}) - C.P.({\text{Cost price}})$
If the selling price is less than cost price then it is known as loss.
$Loss = C.P.({\text{ Cost price}}) - S.P.({\text{Selling price}})$
Profit percentage is calculated on cost price.
Profit% = $ \dfrac{{S.P. - C.P.}}{{C.P.}} \times 100$
Loss percentage is calculated on cost price.
Loss% = $\dfrac{{C.P. - S.P.}}{{C.P.}} \times 100$
For evaluation of total cost price add cost of both types of rice. For selling price, multiply the total amount of mixture i.e. (a +b) kg with its rate. It is the sum of weights of given rice.

Complete step by step solution:
According to the question,
Cost price of a kg of rice = Rs. x
Cost price of b kg of rice = Rs. y
Selling price of the mixture = Rs. z
Quantity of the mixture = (a + b) kg
Now, Total cost price = Cost of a kg rice + cost of another b kg of rice
$
= a \times x + b \times y \\
= ax + by \\
$
Total selling price $ = \left( {{\text{Quantity of mixture}}} \right) \times {\text{Rate}}$
$
= (a + b) \times z \\
= az + bz \\
$
Profit = Selling Price – Cost price
$
= az + bz - ax - by \\
= (a + b)z - (ax + by) \\
$
\[\% {\text{Profit = }}\left[ {\dfrac{{{\text{Profit}}}}{{{\text{Cost Price}}}} \times 100} \right]\% \]
$\% {\text{Profit}} = \dfrac{{\left( {a + b} \right)z - \left( {ax + by} \right)}}{{ax + by}} \times 100\% $

Note:
A shopkeeper sells two similar items A and B. If A is sold at a gain of x% and B is sold at a loss of x%, then the shopkeeper always incurs a loss.