Finding the Fraction to Multiply Selling Price (S.P.) to Get Cost Price (C.P.) with 20% Loss

This question asks us to determine a fraction. When we multiply the selling price (S.P.) by this fraction, we should get the cost price (C.P.). This situation occurs when there is a loss of 20%.

Understanding Loss Percentage in Commercial Mathematics

In topics like profit and loss, the loss percentage is always calculated with respect to the cost price (C.P.). A loss occurs when the selling price (S.P.) is less than the cost price (C.P.).

The formula for loss percentage is:

$\text{Loss}\% = \left( \frac{\text{Loss}}{\text{C.P.}} \right) \times 100$

Also, we know that:

$\text{Loss} = \text{C.P.} - \text{S.P.}$

Relating Cost Price, Selling Price, and Loss Percentage

Given that the loss is 20%, we can write the relationship between C.P., S.P., and loss. The loss is 20% of the C.P.

$\text{Loss} = 20\% \text{ of C.P.}$

$\text{Loss} = \frac{20}{100} \times \text{C.P.}$

$\text{Loss} = 0.20 \times \text{C.P.}$

Now, substitute this into the formula $\text{Loss} = \text{C.P.} - \text{S.P.}:

$0.20 \times \text{C.P.} = \text{C.P.} - \text{S.P.}$

We want to find the fraction that multiplies S.P. to get C.P. This means we need to rearrange the equation to express C.P. in terms of S.P.

Let's move the S.P. term to the left side and the $0.20 \times \text{C.P.}$ term to the right side:

$\text{S.P.} = \text{C.P.} - (0.20 \times \text{C.P.})$

Factor out C.P. from the right side:

$\text{S.P.} = \text{C.P.} \times (1 - 0.20)$

$\text{S.P.} = \text{C.P.} \times (0.80)$

Convert the decimal $0.80$ into a fraction:

$0.80 = \frac{80}{100} = \frac{8}{10} = \frac{4}{5}$

So, the relationship is:

$\text{S.P.} = \text{C.P.} \times \frac{4}{5}$

Finding the Required Fraction

The question asks for the fraction by which S.P. must be multiplied to get C.P. Let this unknown fraction be $f$.

We need to find $f$ such that:

$\text{S.P.} \times f = \text{C.P.}$

From the previous step, we have $\text{S.P.} = \text{C.P.} \times \frac{4}{5}$. We can rearrange this to express C.P. in terms of S.P.:

Divide both sides by $\frac{4}{5}$:

$\text{C.P.} = \text{S.P.} \div \frac{4}{5}$

Dividing by a fraction is the same as multiplying by its reciprocal:

$\text{C.P.} = \text{S.P.} \times \frac{5}{4}$

Comparing this with the required equation $\text{S.P.} \times f = \text{C.P.}$, we can see that the fraction $f$ is $\frac{5}{4}$.

Thus, the selling price (S.P.) must be multiplied by $\frac{5}{4}$ to get the cost price (C.P.) when there is a 20% loss.

Summary of Relationship

When there is a 20% loss:

• S.P. is 80% of C.P. ($S.P. = \frac{4}{5} C.P.$)
• C.P. is S.P. divided by 80%, which is S.P. multiplied by $\frac{1}{80\%} = \frac{1}{4/5} = \frac{5}{4}$. ($C.P. = \frac{5}{4} S.P.$)

Verification

Let's take an example. Suppose C.P. = ₹100.

Loss = 20% of ₹100 = ₹20.

S.P. = C.P. - Loss = ₹100 - ₹20 = ₹80.

Now, let's multiply the S.P. (₹80) by the fraction $\frac{5}{4}$:

₹$80 \times \frac{5}{4} = ₹\frac{80 \times 5}{4} = ₹\frac{400}{4} = ₹100$

This value ₹100 is the original C.P., which confirms our fraction is correct.

Conclusion

To get the cost price (C.P.) from the selling price (S.P.) when there is a 20% loss, the selling price must be multiplied by the fraction $\frac{5}{4}$.

Revision Table: Cost Price, Selling Price, and Loss

Additional Information on Profit and Loss Calculations

Profit and loss calculations are fundamental in commercial mathematics. Understanding the relationship between C.P., S.P., profit, and loss is crucial.

• When there is a profit, S.P. is greater than C.P. Profit = S.P. - C.P. Profit% = $(\text{Profit} / \text{C.P.}) \times 100$.
• S.P. with Profit: $S.P. = C.P. \times (1 + \frac{\text{Profit}\%}{100})$.
• C.P. with Profit: $C.P. = \frac{S.P.}{1 + \frac{\text{Profit}\%}{100}}$.
• S.P. with Loss: $S.P. = C.P. \times (1 - \frac{\text{Loss}\%}{100})$.
• C.P. with Loss: $C.P. = \frac{S.P.}{1 - \frac{\text{Loss}\%}{100}}$.

These formulas help in quickly converting between C.P. and S.P. given the profit or loss percentage. In our case, with a 20% loss, $1 - \frac{20}{100} = 1 - 0.20 = 0.80 = \frac{4}{5}$. So, $S.P. = C.P. \times \frac{4}{5}$ and $C.P. = \frac{S.P.}{4/5} = S.P. \times \frac{5}{4}$.