Question

A fruit vendor buys 300 apples at Rs 50 per dozen. The vendor sold 200 of them at Rs 50 per 10 apples. At what rate per apple should the vendor sell the remaining apples to make an overall profit of $12\%$?

Answer 1

Hint: First of all find the cost price for 300 apples by using the information of buying the apples at Rs 50 per dozen so 12 apples cost Rs 50 so by unitary method we will calculate how much 1 apple costs then multiply by 300 to get the cost price of 300 apples. Similarly find the selling price of 200 apples by using the selling rate of Rs 50 per 10 apples. Now, we have left with 100 apples so let us suppose that the rate of 100 apples is Rs x per apple so the selling price for 100 apples is the multiplication of x with 100. Now, add the selling price for 200 and 100 apples and then using the formula of profit which is $\text{Profit}=\dfrac{\text{S}\text{.P}\text{.}-\text{C}\text{.P}\text{.}}{\text{C}\text{.P}\text{.}}\times 100$ where S.P. stands for selling price and C.P. stands for the cost price substituting these values in the formula of profit we will get the value of x.

Complete step-by-step answer:
The rate of buying 300 apples is equal to Rs 50 per dozen. Using this information we are going to find the cost price of 300 apples.
As 12 apples cost Rs 50 so by unitary method the cost of 1 apple is the division of 50 by 12.
Cost of 1 apple $=\dfrac{50}{12}$
Cost of 300 apples is calculated by multiplying the cost of 1 apple by 300.
Cost of 300 apples $=\dfrac{50}{12}\times 300$
Solving the above equation we get,
Cost of 300 apples $=\text{Rs}1250$
Hence, the cost price of 300 apples is equal to Rs 1250.
Now, the vendor sold 200 apples at the price of Rs 50 per 10 apples so by unitary method we can find the cost of 1 apple by dividing 50 by 10.
Selling price of 1 apple $=\dfrac{50}{10}=\text{Rs}5$
So, selling price of 200 apples are found by multiplying Rs 5 by 200 we get,
Selling price of 200 apples $=200\times 5=\text{Rs}1000$
Let us assume that the selling price per apple for the remaining 100 apples is Rs x. The selling price for 100 apples is calculated by the multiplication of x by 100.
Selling price for 100 apples $=100x$
Selling price for 300 apples is equal to the addition of selling price for 200 apples and 100 apples.
Selling price for 300 apples $=\text{Rs}\left( 1000+100x \right)$
We have given the profit for 300 apples as $12\%$. We know the formula for profit as:
$\text{Profit}=\dfrac{\text{S}\text{.P}\text{.}-\text{C}\text{.P}\text{.}}{\text{C}\text{.P}\text{.}}\times 100$
In the above formula, S.P. is the selling price and C.P. is the cost price so substituting these values for 300 apples that we have calculated above and profit as $12\%$ in the above equation we get,
$\begin{align}
  & 12=\dfrac{1000+100x-1250}{1250}\times 100 \\
 & \Rightarrow 12=\dfrac{100x-250}{1250}\times 100 \\
\end{align}$
Taking 10 as common from the numerator of the right hand side of the above equation we get,
$12=\dfrac{10\left( 10x-25 \right)}{1250}\times 100$
In the above equation, numerator and denominator has one zero that will be cancelled out and we are left with:
$12=\dfrac{1\left( 10x-25 \right)}{125}\times 100$
On cross multiplying the above equation we get,
$12\left( 125 \right)=\left( 10x-25 \right)\left( 100 \right)$
As you can see that left hand and right hand side of the above equation is divisible by 25 so dividing 25 on both the sides we get,
$\begin{align}
  & \dfrac{12\left( 125 \right)}{25}=\dfrac{\left( 10x-25 \right)\left( 100 \right)}{25} \\
 & \Rightarrow 12\left( 5 \right)=\left( 10x-25 \right)\left( 4 \right) \\
\end{align}$
Dividing 4 on both the sides of the equation we get,
$\begin{align}
  & \dfrac{12\left( 5 \right)}{4}=\dfrac{\left( 10x-25 \right)4}{4} \\
 & \Rightarrow 3\left( 5 \right)=\left( 10x-25 \right) \\
 & \Rightarrow 15=10x-25 \\
\end{align}$
Adding 25 on both the sides we get,
$\begin{align}
  & 15+25=10x \\
 & \Rightarrow 40=10x \\
\end{align}$
Dividing 10 on both the sides we get,
$4=x$
As x is the selling price per apple for the remaining 100 apples so the value of x that we have calculated above is Rs 4.
Hence, the selling price per apple is Rs 4 for 100 apples such that the vendor has made an overall profit of $12\%$.

Note: To solve the above problem, you need to know what dozen means. The point of mistake that could happen is that in the hastiness of solving the problem you consider the selling price of each apple for 200 apples by considering the rate as Rs 50 per dozen apples but actually it is given as Rs 50 per 10 apples. This merge happens because the buying rate of 300 apples is Rs 50 per dozen. So, be careful while solving the above problem. It’s better to write the information of the question first then proceed then such mistakes would not happen.