Understanding the Profit Percentage Problem

This problem involves calculating the profit percentage based on a relationship between the selling price (SP) of one quantity of items and the cost price (CP) of another quantity. We are given that the selling price of 84 items is equivalent to the cost price of 105 items.

Setting up the Equation

Let's denote the cost price of one item as $C$ and the selling price of one item as $S$.

According to the question:

Selling Price of 84 items = Cost Price of 105 items

Mathematically, this can be represented as:

$$ 84 \times S = 105 \times C $$

Calculating the Ratio of SP to CP

To find the profit percentage, we first need to determine the relationship between the selling price ($S$) and the cost price ($C$) of a single item. We can rearrange the equation from the previous step:

$$ \frac{S}{C} = \frac{105}{84} $$

Now, let's simplify this fraction. Both 105 and 84 are divisible by 21:

• $105 \div 21 = 5$
• $84 \div 21 = 4$

So, the simplified ratio is:

$$ \frac{S}{C} = \frac{5}{4} $$

This means that for every item, the selling price ($S$) is $\frac{5}{4}$ times its cost price ($C$).

Calculating the Profit

Profit is defined as Selling Price minus Cost Price ($SP - CP$). In terms of a single item:

Profit per item = $S - C$

Substitute the relationship $S = \frac{5}{4}C$ into the profit formula:

Profit per item = $ \frac{5}{4}C - C $

To subtract, find a common denominator:

Profit per item = $ \frac{5}{4}C - \frac{4}{4}C $

Profit per item = $ \frac{5-4}{4}C = \frac{1}{4}C $

So, the profit made on each item is equal to $\frac{1}{4}$ of its cost price.

Calculating the Profit Percentage

The formula for profit percentage is:

$$ \text{Profit Percentage} = \left( \frac{\text{Profit}}{\text{Cost Price}} \right) \times 100\% $$

Using the profit per item we just calculated:

$$ \text{Profit Percentage} = \left( \frac{\frac{1}{4}C}{C} \right) \times 100\% $$

The $C$ in the numerator and denominator cancels out:

$$ \text{Profit Percentage} = \left( \frac{1}{4} \right) \times 100\% $$

$$ \text{Profit Percentage} = 0.25 \times 100\% $$

$$ \text{Profit Percentage} = 25\% $$

Conclusion

The percentage of profit gained in the transaction is 25%. This calculation demonstrates how to find profit percentage when given the relationship between the selling price quantity and the cost price quantity.