Calculating Cost Price from Loss

The initial situation involves selling $50$ mangoes for $₹30$ with a $10\%$ loss.

• Selling Price ($SP$) = $₹30$ for $50$ mangoes.
• Loss percentage = $10\%$.
• This means the Selling Price ($SP$) is $100\% - 10\% = 90\%$ of the Cost Price ($CP$).

We can write this as an equation:

$SP = 0.90 \times CP$

$₹30 = 0.90 \times CP$

Now, solve for $CP$:

$CP = \frac{₹30}{0.90} = \frac{300}{9} = \frac{100}{3}$

So, the Cost Price ($CP$) for $50$ mangoes is $₹\frac{100}{3}$.

Determining Selling Price for Target Gain

The goal is to achieve a $20\%$ gain.

• Target Gain percentage = $20\%$.
• The target Selling Price ($SP_{target}$) should be $100\% + 20\% = 120\%$ of the Cost Price ($CP$).

Calculate the target $SP$ for $50$ mangoes:

$SP_{target} = 1.20 \times CP$

$SP_{target} = 1.20 \times ₹\frac{100}{3} = \frac{12}{10} \times \frac{100}{3} = \frac{4}{10} \times 100 = ₹40$

Therefore, to gain $20\%$, $50$ mangoes must be sold for $₹40$.

Calculating Mangoes for Target Price

We need to find out how many mangoes should be sold for $₹28$ to achieve the $20\%$ gain.

We know:

• $50$ mangoes correspond to a selling price of $₹40$ for a $20\%$ gain.

Let $x$ be the number of mangoes to be sold for $₹28$. We can use a proportion:

$\frac{\text{Number of Mangoes}}{\text{Selling Price}} = \text{constant}$

$\frac{50}{₹40} = \frac{x}{₹28}$

Solve for $x$:

$x = \frac{50 \times 28}{40} = \frac{5 \times 28}{4} = 5 \times 7 = 35$

Thus, $35$ mangoes should be sold for $₹28$ to gain $20\%$.