Understanding the Profit Problem

The problem asks us to find the gain percent when selling a certain amount of cloth. We are told that by selling 33 metres of cloth, a gain equal to the selling price of 11 metres is made. Let's break this down to find the cost price and then calculate the gain percent.

Defining Key Terms

In profit and loss problems, we often use these terms:

• Cost Price (CP): The price at which an article is bought.
• Selling Price (SP): The price at which an article is sold.
• Gain or Profit: The difference between the Selling Price and the Cost Price when SP > CP. $\text{Gain} = \text{SP} - \text{CP}$.
• Gain Percent: The gain expressed as a percentage of the Cost Price. $\text{Gain Percent} = \left( \frac{\text{Gain}}{\text{CP}} \right) \times 100\%$.

Setting up the Problem

Let's assume the Selling Price of 1 metre of cloth is $\text{Rs. } s$.

Based on this assumption, we can find the Selling Price of the total quantity sold:

• Selling Price of 33 metres ($\text{SP}_{33}$) = $33 \times s = 33s$.

The problem states that the gain made is equal to the Selling Price of 11 metres.

• Gain = Selling Price of 11 metres ($\text{SP}_{11}$) = $11 \times s = 11s$.

Calculating the Cost Price

We know that Gain = SP - CP. We can rearrange this formula to find the Cost Price:

CP = SP - Gain

In this case, we are dealing with the Cost Price of 33 metres ($\text{CP}_{33}$) and the Selling Price of 33 metres ($\text{SP}_{33}$). The gain corresponds to the sale of these 33 metres.

So, the Cost Price of 33 metres is:

$\text{CP}_{33} = \text{SP}_{33} - \text{Gain}$

Substitute the values we found:

$\text{CP}_{33} = 33s - 11s$

$\text{CP}_{33} = 22s$

So, the Cost Price of 33 metres of cloth is equal to the Selling Price of 22 metres of cloth (which is $22s$).

Calculating the Gain Percent

Now that we have the Gain and the Cost Price, we can calculate the Gain Percent using the formula:

$\text{Gain Percent} = \left( \frac{\text{Gain}}{\text{CP}} \right) \times 100\%$

Substitute the values for Gain and $\text{CP}_{33}$:

$\text{Gain Percent} = \left( \frac{11s}{22s} \right) \times 100\%$

We can cancel out the variable $s$ from the numerator and the denominator, as long as $s \ne 0$ (which it must be for a real-world price):

$\text{Gain Percent} = \left( \frac{11}{22} \right) \times 100\%$

Simplify the fraction $\frac{11}{22}$:

$\frac{11}{22} = \frac{1}{2}$

So, the Gain Percent is:

$\text{Gain Percent} = \frac{1}{2} \times 100\%$

$\text{Gain Percent} = 50\%$

Thus, the gain percent is 50%.

Revision Table: Profit and Loss Concepts

Additional Information: Alternative Approach

Sometimes, problems like these can be conceptualized by thinking in terms of quantities instead of price per unit.

Let CP of 1 metre be $\text{CP}_m$ and SP of 1 metre be $\text{SP}_m$.

Selling 33 metres means the total SP is $33 \times \text{SP}_m$.

The Cost Price of 33 metres is $33 \times \text{CP}_m$.

The gain is the SP of 11 metres, which is $11 \times \text{SP}_m$.

Using the formula Gain = SP - CP for the total transaction:

$11 \times \text{SP}_m = (33 \times \text{SP}_m) - (33 \times \text{CP}_m)$

Rearrange the terms to group SP and CP:

$33 \times \text{CP}_m = (33 \times \text{SP}_m) - (11 \times \text{SP}_m)$

$33 \times \text{CP}_m = 22 \times \text{SP}_m$

Now, we can find the ratio of $\text{SP}_m$ to $\text{CP}_m$:

$\frac{\text{SP}_m}{\text{CP}_m} = \frac{33}{22} = \frac{3}{2}$

This means that for every Rs. 2 of cost price, the selling price is Rs. 3. The gain per metre is $\text{SP}_m - \text{CP}_m = \frac{3}{2}\text{CP}_m - \text{CP}_m = \left(\frac{3}{2} - 1\right)\text{CP}_m = \frac{1}{2}\text{CP}_m$.

Gain Percent $= \left( \frac{\text{Gain per metre}}{\text{CP per metre}} \right) \times 100\%$

Gain Percent $= \left( \frac{\frac{1}{2}\text{CP}_m}{\text{CP}_m} \right) \times 100\%$

Gain Percent $= \left( \frac{1}{2} \right) \times 100\% = 50\%$

This confirms the result obtained earlier. This method highlights that the gain of SP of 11 metres on selling 33 metres is equivalent to saying that the cost price of 33 metres is equal to the selling price of $33 - 11 = 22$ metres. If 33 metres are bought at the price of 22 metres' selling price, the gain is the selling price of $33 - 22 = 11$ metres on a cost equivalent to the selling price of 22 metres. The gain percent is thus $\left(\frac{11}{22}\right) \times 100\% = 50\%$.