Question

A tradesman finds that by selling a bicycle for $ Rs.75 $ , which he had bought for $ Rs.x $ , he gained $ x\% $ . Find the value of $ x $ .
 $ \left( a \right){\text{ 20}} $
 $ \left( b \right){\text{ 35}} $
 $ \left( c \right){\text{ 50}} $
 $ \left( d \right){\text{ 75}} $

Answer 1

Hint:
This question is based on profit and profit percentage. In this, we will first find the equation for profit and then from that, we will find the profit percentage by using the formula. And then solving the equation we will get the cost price.

Formula used:
Profit can be calculated by using
 $ Profit\left( {Rs.} \right) = \dfrac{{Profit\% }}{{100}} \times C.P $
And also $ Profit = S.P - C.P $
Here,
 $ S.P $ , will be the selling price
 $ C.P $ , will be the cost price

Complete step by step solution:
First of all we will see the values given to us
 $ S.P = Rs.75 $ , $ C.P = Rs.x $ and $ Profit = x\% $
So now by using the profit formula,
We can say profit will be equal to $ Profit\left( {Rs.} \right) = Rs.\left( {75 - x} \right) $ , let’s name it equation $ 1 $
And also the $ Profit\left( {Rs.} \right) = \dfrac{x}{{100}} \times x $ , let’s name it equation $ 2 $
Now from both the equations, we get
 $ \Rightarrow 75 - x = \dfrac{{{x^2}}}{{100}} $
And on solving the above equation, we get
 $ \Rightarrow {x^2} + 100x - 7500 = 0 $
Now on solving the equation furthermore for the value of $ x $ , we get
 $ \Rightarrow {x^2} + 150x - 50x - 7500 = 0 $
Now on taking the common, we get
 $ \Rightarrow x\left( {x + 150} \right) - 50\left( {x + 150} \right) = 0 $
So we get
 $ \Rightarrow \left( {x - 50} \right)\left( {x + 150} \right) = 0 $
And on solving, we get
 $ \Rightarrow x = 50{\text{ and x = - 150}} $
Since the cost price can never be negative so
 $ \Rightarrow x = 50 $
Hence, $ 50 $ will be the value for the $ x $ .

Therefore, the option $ \left( c \right) $ will be correct.

Note:
The quadratic equation with which we had encountered while solving this problem can also be solved by using the discriminant. The formula for it- let assume there is the quadratic equation as $ a{x^2} + bx + c $ then the value of $ x $ can is calculated by using the formula $ x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}} $ .
And also the discriminant can be positive or negative or also it can be zero. By using this formula we can solve any complex formula by the use of this. The only thing we need to keep in mind is while solving we have to be cautious for the math part as an error can make the whole solution wrong.