Question

By selling oranges at the rate of Rs 72 per dozen, Suhel loses 10% of his investment. What would be the percentage of her gain or loss, if she sold them at Rs 600 per hundred?

Answer 1

Hint: Firstly, we have to find the selling price (SP) of 1 orange using the unitary method. Then, we will assume the cost price (CP) of one orange will be x. We will find the value of x by substituting the values in the formula for percentage of loss which is given by $\%\text{Loss}=\dfrac{\left( CP-SP \right)}{CP}\times 100$ . Then, we will find the CP of 100 oranges by multiplying the value of x by 100. We will then check whether the SP of 100 oranges is less than or greater than the CP of 100 oranges. If the SP is less than CP, then loss has been incurred. If the SP is greater than CP, then profit has occurred. Finally, we have to find the percentage of loss or percentage of gain accordingly.

Complete step by step answer:
We are given that the selling price (SP) of a dozen oranges, that is, 12 oranges is Rs 72.
$\Rightarrow \text{SP of 12 oranges}=\text{Rs}.72$
Therefore, by unitary method, we can find the SP of 1 orange. We have to divide 72 by 12.
$\begin{align}
  & \Rightarrow \text{SP of 1 orange}=\dfrac{\text{Rs}.72}{12} \\
 & \Rightarrow \text{SP of 1 orange}=\text{Rs}\text{.6} \\
\end{align}$
We are given the percentage of loss as 10%.
$\Rightarrow \text{loss}%=10\%$
Let the cost price (CP) of one orange be x. We know that percentage of loss is given by
$\%\text{Loss}=\dfrac{\left( CP-SP \right)}{CP}\times 100...\left( i \right)$
Let us substitute the values in the above formula.
$\Rightarrow 10=\dfrac{\left( x-6 \right)}{x}\times 100$
We have to find the value of x. Let us take the denominator of the RHS to the LHS.
$\Rightarrow 10x=\left( x-6 \right)100$
We have to apply distributive property on the RHS.
$\Rightarrow 10x=100x-600$
Let us collect like terms on one side and unlike terms on the other.
$\begin{align}
  & \Rightarrow 600=100x-10x \\
 & \Rightarrow 600=90x \\
\end{align}$
We have to move the coefficient of x to the RHS.
$\begin{align}
  & \Rightarrow \dfrac{600}{90}=x \\
 & \Rightarrow x=\dfrac{20}{3} \\
\end{align}$
Therefore, the CP of 1 orange is $\dfrac{20}{3}$ . Let us find the CP of 100 oranges by multiplying $\dfrac{20}{3}$ by 100.
$\begin{align}
  & \Rightarrow \text{CP of 100 oranges}=\dfrac{20}{3}\times 100 \\
 & \Rightarrow \text{CP of 100 oranges}=\dfrac{2000}{3} \\
\end{align}$
We are given that SP of 100 oranges is Rs 600. We can see that the SP of 100 oranges is less than CP of 100 oranges. Therefore, Suhel got a loss. Let us find the percentage of loss. By substituting the values in the formula (i).
$\begin{align}
  & \Rightarrow \%\text{Loss}=\dfrac{\left( \dfrac{2000}{3}-600 \right)}{\dfrac{2000}{3}}\times 100 \\
 & \Rightarrow \%\text{Loss}=\dfrac{\dfrac{2000-1800}{3}}{\dfrac{2000}{3}}\times 100 \\
 & \Rightarrow \%\text{Loss}=\dfrac{\dfrac{200}{3}}{\dfrac{2000}{3}}\times 100 \\
 & \Rightarrow \%\text{Loss}=\dfrac{200}{3}\times \dfrac{3}{2000}\times 100 \\
\end{align}$
Let us cancel the common factors and terms.
$\begin{align}
  & \Rightarrow \%\text{Loss}=\dfrac{\require{cancel}\cancel{200}}{\require{cancel}\cancel{3}}\times \dfrac{\require{cancel}\cancel{3}}{{{\require{cancel}\cancel{2000}}_{1\require{cancel}\cancel{0}}}}\times 10\require{cancel}\cancel{0} \\
 & \Rightarrow \%\text{Loss}=10\% \\
\end{align}$
Hence, the percentage of loss is 10%.

Note: Students must be thorough with the formulas of percentage of loss and percentage of gain. The formula for percentage of gain or profit is given by $\dfrac{\left( SP-CP \right)}{CP}\times 100$ . Students must compare the SP and CP to conclude whether to find the percentage of loss or gain.