Profit, Loss, and Discount
Profit and Loss: Basic Definitions and Terms (Cost Price, Selling Price, Marked Price)
The concepts of profit and loss are fundamental in commerce and everyday transactions involving buying and selling. They describe the financial outcome of a sale based on the difference between the price at which something was acquired and the price at which it was sold.
Key Terms:
Understanding the terminology is essential for working with profit and loss problems:
1. Cost Price (CP)
The Cost Price (CP) is the total expenditure incurred to acquire an article and make it ready for sale. This includes the initial purchase price of the article as well as any additional expenses that are necessary before the sale can occur. These additional expenses are often referred to as overhead expenses. Examples of overhead expenses include costs for transportation, labour charges, repair costs, packaging, installation, etc.
$\boldsymbol{\text{Total Cost Price (CP) = Purchase Price + Overhead Expenses}}$
When overhead expenses are not mentioned, the purchase price is considered as the Cost Price.
2. Selling Price (SP)
The Selling Price (SP) is the price at which an article is sold by the seller to the buyer. It is the revenue generated from the sale of the article.
3. Marked Price (MP) or List Price
The Marked Price (MP), also known as the List Price or the Tag Price, is the price that is indicated or printed on the article or its packaging. This is the price that the seller initially quotes for the article. Discounts offered by the seller are usually calculated on the Marked Price.
Profit and Loss Defined:
The outcome of a sale, in terms of profit or loss, is determined by comparing the Selling Price (SP) and the Cost Price (CP) of the article.
- Profit (or Gain): Profit occurs when the Selling Price (SP) of an article is greater than its Cost Price (CP). The amount of profit is the difference between the Selling Price and the Cost Price.
- Loss: Loss occurs when the Selling Price (SP) of an article is less than its Cost Price (CP). The amount of loss is the difference between the Cost Price and the Selling Price.
- Neither Profit nor Loss (Break-even): If the Selling Price (SP) of an article is exactly equal to its Cost Price (CP), there is no financial gain or loss from the transaction. The seller simply recovers the total cost incurred.
$\boldsymbol{\text{Profit = Selling Price (SP) - Cost Price (CP)}}$
... (i)
(This is applicable when $\text{SP} > \text{CP}$)
$\boldsymbol{\text{Loss = Cost Price (CP) - Selling Price (SP)}}$
... (ii)
(This is applicable when $\text{CP} > \text{SP}$)
$\text{If SP = CP, there is neither Profit nor Loss.}$
Example 1. A vendor bought a table for $\textsf{₹ } 2500$. He spent $\textsf{₹ } 300$ on its transportation and repair. He then sold the table for $\textsf{₹ } 3100$. Find his profit or loss amount.
Answer:
Initial purchase price of the table $= \textsf{₹ } 2500$.
Overhead expenses (transportation and repair) $= \textsf{₹ } 300$.
Total Cost Price (CP) = Purchase Price + Overhead Expenses
CP $= \textsf{₹ } 2500 + \textsf{₹ } 300 = \textsf{₹ } 2800$
Selling Price (SP) $= \textsf{₹ } 3100$.
Now, compare the Selling Price (SP) and the Cost Price (CP):
SP $(\textsf{₹ } 3100)$ is greater than CP $(\textsf{₹ } 2800)$.
Since $\text{SP} > \text{CP}$, there is a Profit.
Calculate the amount of Profit using formula (i):
Profit $= \text{SP} - \text{CP}$
Profit $= \textsf{₹ } 3100 - \textsf{₹ } 2800 = \boldsymbol{\textsf{₹ } 300}$
The vendor made a profit of $\boldsymbol{\textsf{₹ } 300}$.
Competitive Exam Notes:
These basic definitions are the foundation of all profit and loss problems.
- CP vs. SP: The core comparison is always between CP and SP to determine profit or loss.
- Overhead Expenses: Remember to add overhead expenses to the purchase price to get the actual CP.
- MP and Discount: The Marked Price comes into play when discounts are discussed. Discount is typically calculated on MP.
- Formulas: Memorize the basic formulas: Profit = SP - CP (if SP > CP) and Loss = CP - SP (if CP > SP).
Calculating Profit and Loss Amount
The absolute monetary value of the profit or loss from a transaction is simply the difference between the Selling Price (SP) and the Cost Price (CP). The determination of whether it's a profit or loss depends on which price is greater.
The formulas for calculating the profit or loss amount are:
- If $\text{SP} > \text{CP}$, there is a Profit.
$\boldsymbol{\text{Profit} = \text{SP} - \text{CP}}$
... (iii)
- If $\text{CP} > \text{SP}$, there is a Loss.
$\boldsymbol{\text{Loss} = \text{CP} - \text{SP}}$
... (iv)
- If $\text{SP} = \text{CP}$, there is no profit or loss (Break-even). The profit/loss amount is 0.
Example 1. A shopkeeper bought a chair for $\textsf{₹ } 1500$ and sold it for $\textsf{₹ } 1350$. Find the amount of profit or loss.
Answer:
Cost Price (CP) of the chair $= \textsf{₹ } 1500$.
Selling Price (SP) of the chair $= \textsf{₹ } 1350$.
Compare SP and CP:
$\textsf{₹ } 1350 < \textsf{₹ } 1500$ (SP is less than CP).
Since $\text{SP} < \text{CP}$, there is a Loss in this transaction.
Calculate the amount of Loss using formula (iv):
Loss $= \text{CP} - \text{SP}$
Loss $= \textsf{₹ } 1500 - \textsf{₹ } 1350 = \boldsymbol{\textsf{₹ } 150}$
The shopkeeper incurred a loss of $\boldsymbol{\textsf{₹ } 150}$.
Example 2. If an article is bought for $\textsf{₹ } 600$ and sold for $\textsf{₹ } 600$, what is the profit or loss amount?
Answer:
Cost Price (CP) $= \textsf{₹ } 600$.
Selling Price (SP) $= \textsf{₹ } 600$.
Compare SP and CP:
$\textsf{₹ } 600 = \textsf{₹ } 600$ (SP is equal to CP).
Since $\text{SP} = \text{CP}$, there is neither profit nor loss.
Profit or Loss amount $= \text{SP} - \text{CP} = \textsf{₹ } 600 - \textsf{₹ } 600 = \boldsymbol{\textsf{₹ } 0}$
There is $\boldsymbol{\textsf{₹ } 0}$ profit or loss.
Competitive Exam Notes:
Calculating the absolute profit or loss amount is straightforward once CP and SP are known. The key is correctly identifying CP (including overheads) and SP.
- Always Compare SP and CP: This is the first step to determine if it's a profit or a loss scenario.
- Profit Formula: SP - CP (Use when SP > CP).
- Loss Formula: CP - SP (Use when CP > SP).
- Break-even: SP = CP, Profit/Loss = 0.
Calculating Profit Percentage and Loss Percentage
While knowing the profit or loss amount is important, the profit or loss percentage provides a standardized measure of profitability relative to the investment made (the Cost Price). This allows for meaningful comparisons between different transactions, regardless of the absolute values involved.
By convention, unless explicitly stated otherwise, Profit Percentage and Loss Percentage are always calculated on the Cost Price (CP).
Profit Percentage (% Profit or % Gain)
If a transaction results in a profit ($\text{SP} > \text{CP}$), the profit percentage is calculated by expressing the profit amount as a percentage of the Cost Price.
$\boldsymbol{\text{Profit Percentage } = \left(\frac{\text{Profit}}{\text{Cost Price (CP)}} \times 100\right)\%}$
... (v)
Using the formula Profit = SP - CP, we can also write:
$\boldsymbol{\text{Profit Percentage } = \left(\frac{\text{SP - CP}}{\text{CP}} \times 100\right)\%}$
Loss Percentage (% Loss)
If a transaction results in a loss ($\text{CP} > \text{SP}$), the loss percentage is calculated by expressing the loss amount as a percentage of the Cost Price.
$\boldsymbol{\text{Loss Percentage } = \left(\frac{\text{Loss}}{\text{Cost Price (CP)}} \times 100\right)\%}$
... (vi)
Using the formula Loss = CP - SP, we can also write:
$\boldsymbol{\text{Loss Percentage } = \left(\frac{\text{CP - SP}}{\text{CP}} \times 100\right)\%}$
In both cases, the denominator is the Cost Price (CP).
Example 1. A retailer buys a watch for $\textsf{₹ } 800$ and sells it for $\textsf{₹ } 1000$. Find his profit percentage.
Answer:
Cost Price (CP) $= \textsf{₹ } 800$.
Selling Price (SP) $= \textsf{₹ } 1000$.
Since $\text{SP} (\textsf{₹ } 1000) > \text{CP} (\textsf{₹ } 800)$, there is a Profit.
Calculate the amount of Profit:
Profit $= \text{SP} - \text{CP} = \textsf{₹ } 1000 - \textsf{₹ } 800 = \textsf{₹ } 200$
Now, calculate the Profit Percentage using formula (v):
$\text{Profit Percentage } = \left(\frac{\text{Profit}}{\text{CP}} \times 100\right)\%$
$\text{Profit Percentage } = \left(\frac{\textsf{₹ } 200}{\textsf{₹ } 800} \times 100\right)\%$
Simplify the fraction and calculate:
$\frac{200}{800} = \frac{2}{8} = \frac{1}{4}$
$\text{Profit Percentage } = \left(\frac{1}{4} \times 100\right)\% = \boldsymbol{25\%}$
The retailer's profit percentage is $\boldsymbol{25\%}$.
Example 2. A shopkeeper sold a product for $\textsf{₹ } 450$, which he had bought for $\textsf{₹ } 500$. Find his loss percentage.
Answer:
Cost Price (CP) $= \textsf{₹ } 500$.
Selling Price (SP) $= \textsf{₹ } 450$.
Since $\text{CP} (\textsf{₹ } 500) > \text{SP} (\textsf{₹ } 450)$, there is a Loss.
Calculate the amount of Loss:
Loss $= \text{CP} - \text{SP} = \textsf{₹ } 500 - \textsf{₹ } 450 = \textsf{₹ } 50$
Now, calculate the Loss Percentage using formula (vi):
$\text{Loss Percentage } = \left(\frac{\text{Loss}}{\text{CP}} \times 100\right)\%$
$\text{Loss Percentage } = \left(\frac{\textsf{₹ } 50}{\textsf{₹ } 500} \times 100\right)\%$
Simplify the fraction and calculate:
$\frac{50}{500} = \frac{5}{50} = \frac{1}{10}$
$\text{Loss Percentage } = \left(\frac{1}{10} \times 100\right)\% = \boldsymbol{10\%}$
The shopkeeper's loss percentage is $\boldsymbol{10\%}$.
Competitive Exam Notes:
Percentage calculations for profit and loss are central to many problems. The key is always using the Cost Price as the base.
- Base is CP: % Profit and % Loss are almost always calculated on CP. If a question specifies they are on SP, adjust accordingly, but assume CP unless stated.
- Formulas:
- % Profit $= \frac{\text{Profit}}{\text{CP}} \times 100$
- % Loss $= \frac{\text{Loss}}{\text{CP}} \times 100$
- Steps: First, determine if it's profit or loss by comparing SP and CP. Then, calculate the profit/loss amount. Finally, use the appropriate percentage formula with CP in the denominator.
- Relationship between SP, CP, and % Profit/Loss:
- If there is a Profit of $P\%$, then $\text{SP} = \text{CP} \times \left(1 + \frac{P}{100}\right)$.
- If there is a Loss of $L\%$, then $\text{SP} = \text{CP} \times \left(1 - \frac{L}{100}\right)$.
- These inverse formulas are useful for finding SP given CP and % change, or finding CP given SP and % change.
Discount: Concept and Calculations
In retail and commerce, Discount refers to a reduction in the price of an article. It is typically offered on the price marked on the article, which is known as the Marked Price (MP) or List Price. Discounts are used as a strategy to attract customers, clear old stock, or increase sales volume.
The key point about discount is that it is always calculated on the Marked Price.
If a discount of $D\%$ is offered on an article with Marked Price (MP), the amount of discount is calculated as $D\%$ of MP.
$\boldsymbol{\text{Discount Amount} = \text{Discount \% of Marked Price (MP)}}$
... (i)
Using the definition of percentage:
$\boldsymbol{\text{Discount Amount} = \frac{\text{Discount \%}}{100} \times \text{MP}}$
... (ii)
Once the discount amount is known, the Selling Price (SP) of the article after the discount is calculated by subtracting the discount amount from the Marked Price.
$\boldsymbol{\text{Selling Price (SP) = Marked Price (MP) - Discount Amount}}$
... (iii)
Calculating Discount Percentage
If the Marked Price (MP) and the Selling Price (SP) of an article are known, the discount amount can be easily found using formula (iii) rearranged: Discount Amount = MP - SP.
The discount percentage is then calculated by expressing this discount amount as a percentage of the Marked Price (MP).
$\boldsymbol{\text{Discount Percentage } = \left(\frac{\text{Discount Amount}}{\text{Marked Price (MP)}} \times 100\right)\%}$
... (iv)
Substituting Discount Amount = MP - SP into formula (iv):
$\boldsymbol{\text{Discount Percentage } = \left(\frac{\text{MP - SP}}{\text{MP}} \times 100\right)\%}$
... (v)
Example 1. The marked price of a dress is $\textsf{₹ } 3000$. A discount of 15% is offered on it. Find the discount amount and the selling price of the dress.
Answer:
Marked Price (MP) $= \textsf{₹ } 3000$.
Discount Percentage $= 15\%$.
Calculate the Discount Amount:
Discount Amount $= 15\%$ of MP
Discount Amount $= \frac{15}{100} \times \textsf{₹ } 3000$
Simplify and multiply:
Discount Amount $= \frac{15}{\cancel{100}^{\normalsize 1}} \times \cancel{3000}^{\normalsize 30}$
Discount Amount $= 15 \times 30 = 450$
Discount Amount $= \textsf{₹ } 450$.
Calculate the Selling Price (SP):
Selling Price (SP) = Marked Price (MP) - Discount Amount
SP $= \textsf{₹ } 3000 - \textsf{₹ } 450 = \boldsymbol{\textsf{₹ } 2550}$
The selling price of the dress is $\textsf{₹ } 2550$.
Example 2. A bicycle is marked at $\textsf{₹ } 7500$. It is sold for $\textsf{₹ } 6000$ after giving a discount. Find the rate of discount (discount percentage).
Answer:
Marked Price (MP) $= \textsf{₹ } 7500$.
Selling Price (SP) $= \textsf{₹ } 6000$.
Calculate the Discount Amount:
Discount Amount = MP - SP
Discount Amount $= \textsf{₹ } 7500 - \textsf{₹ } 6000 = \textsf{₹ } 1500$
Calculate the Discount Percentage:
Using the discount percentage formula (iv):
$\text{Discount Percentage } = \left(\frac{\text{Discount Amount}}{\text{MP}} \times 100\right)\%$
$\text{Discount Percentage } = \left(\frac{\textsf{₹ } 1500}{\textsf{₹ } 7500} \times 100\right)\%$
Simplify the fraction and calculate:
$\frac{1500}{7500} = \frac{15}{75} = \frac{\cancel{15}^1}{\cancel{75}_5} = \frac{1}{5}$
$\text{Discount Percentage } = \left(\frac{1}{5} \times 100\right)\% = \boldsymbol{20\%}$
The discount percentage is $\boldsymbol{20\%}$.
Competitive Exam Notes:
Discount is always applied to the Marked Price (MP). This is a critical point to remember.
- Base for Discount: Always calculate discount amount and discount percentage on the MP.
- Key Formulas:
- Discount Amount = MP - SP
- SP = MP - Discount Amount
- % Discount $= \frac{\text{Discount Amount}}{\text{MP}} \times 100$
- Relationship SP and MP with % Discount:
- If there is a discount of $D\%$, then $\text{SP} = \text{MP} \times \left(1 - \frac{D}{100}\right) = \text{MP} \times \left(\frac{100 - D}{100}\right)$.
- Conversely, $\text{MP} = \text{SP} \times \left(\frac{100}{100 - D}\right)$.
- These formulas allow for direct calculation of SP from MP (and vice versa) if the discount percentage is known.
Relationship between CP, SP, MP, Profit/Loss, and Discount
In a typical business transaction involving a marked price and a discount, the Cost Price (CP), Selling Price (SP), and Marked Price (MP) are interconnected through the concepts of profit/loss and discount. Understanding these relationships is crucial for solving problems that involve multiple stages of pricing.
Summary of Key Relationships:
Let's summarize the fundamental connections we have discussed:
- Profit/Loss relates CP and SP:
- Profit = SP - CP (if SP > CP)
- Loss = CP - SP (if CP > SP)
- Discount relates MP and SP:
- Discount Amount = MP - SP (since MP is usually > SP when a discount is given)
- Percentage calculations:
- % Profit = $\left(\frac{\text{Profit}}{\text{CP}} \times 100\right)\%$
- % Loss = $\left(\frac{\text{Loss}}{\text{CP}} \times 100\right)\%$
- % Discount = $\left(\frac{\text{Discount Amount}}{\text{MP}} \times 100\right)\%$
Derived Formulas Relating CP, SP, MP, and Percentages:
We can derive direct relationships between CP, SP, and MP when percentage profit/loss and percentage discount are involved. These formulas are very useful for solving problems efficiently.
Formulas for SP based on CP and % Profit/Loss:
If there is a profit of $P\%$ on the CP, it means the Selling Price is the Cost Price plus $P\%$ of the Cost Price.
$\text{SP} = \text{CP} + P\% \text{ of CP} = \text{CP} + \left(\frac{P}{100} \times \text{CP}\right)$
Factor out CP:
$\text{SP} = \text{CP} \left(1 + \frac{P}{100}\right) = \text{CP} \left(\frac{100+P}{100}\right)$
$\boldsymbol{\text{SP} = \text{CP} \times \left(\frac{100 + \text{ % Profit}}{100}\right)}$
... (vi)
If there is a loss of $L\%$ on the CP, it means the Selling Price is the Cost Price minus $L\%$ of the Cost Price.
$\text{SP} = \text{CP} - L\% \text{ of CP} = \text{CP} - \left(\frac{L}{100} \times \text{CP}\right)$
Factor out CP:
$\text{SP} = \text{CP} \left(1 - \frac{L}{100}\right) = \text{CP} \left(\frac{100-L}{100}\right)$
$\boldsymbol{\text{SP} = \text{CP} \times \left(\frac{100 - \text{ % Loss}}{100}\right)}$
... (vii)
Formulas for CP based on SP and % Profit/Loss:
We can rearrange formulas (vi) and (vii) to find CP when SP and the percentage profit or loss are known.
From (vi), if there is a profit of $P\%$:
$\boldsymbol{\text{CP} = \text{SP} \times \left(\frac{100}{100 + \text{ % Profit}}\right)}$
... (viii)
From (vii), if there is a loss of $L\%$:
$\boldsymbol{\text{CP} = \text{SP} \times \left(\frac{100}{100 - \text{ % Loss}}\right)}$
... (ix)
Formula for SP based on MP and % Discount:
If there is a discount of $D\%$ on the MP, the Selling Price is the Marked Price minus $D\%$ of the Marked Price.
$\text{SP} = \text{MP} - D\% \text{ of MP} = \text{MP} - \left(\frac{D}{100} \times \text{MP}\right)$
Factor out MP:
$\text{SP} = \text{MP} \left(1 - \frac{D}{100}\right) = \text{MP} \left(\frac{100-D}{100}\right)$
$\boldsymbol{\text{SP} = \text{MP} \times \left(\frac{100 - \text{ % Discount}}{100}\right)}$
... (x)
Formula for MP based on SP and % Discount:
We can rearrange formula (x) to find MP when SP and the percentage discount are known.
$\boldsymbol{\text{MP} = \text{SP} \times \left(\frac{100}{100 - \text{ % Discount}}\right)}$
... (xi)
Direct Relationship between CP, MP, % Profit, and % Discount:
In many problems, you are given the percentage profit (or loss) made after a certain percentage discount has been offered on the marked price. We can establish a direct relationship between CP and MP in such scenarios.
We have two different ways to express the Selling Price (SP):
$\text{SP} = \text{CP} \times \left(\frac{100 + \text{ % Profit}}{100}\right)$
(From formula vi)
$\text{SP} = \text{MP} \times \left(\frac{100 - \text{ % Discount}}{100}\right)$
(From formula x)
Since both expressions are equal to the same SP, we can equate them:
$\text{CP} \times \left(\frac{100 + \text{ % Profit}}{100}\right) = \text{MP} \times \left(\frac{100 - \text{ % Discount}}{100}\right)$
Multiply both sides by 100:
$\text{CP} \times (100 + \text{ % Profit}) = \text{MP} \times (100 - \text{ % Discount})$
Rearrange the terms to find the ratio of MP to CP:
$\boldsymbol{\frac{\text{MP}}{\text{CP}} = \frac{100 + \text{ % Profit}}{100 - \text{ % Discount}}}$
... (xii)
This is a powerful formula that directly links Marked Price and Cost Price based on the percentage profit and percentage discount.
If there is a loss of $L\%$ instead of a profit of $P\%$, the relationship becomes:
$\boldsymbol{\frac{\text{MP}}{\text{CP}} = \frac{100 - \text{ % Loss}}{100 - \text{ % Discount}}}$
... (xiii)
(Replace $100 + \text{% Profit}$ with $100 - \text{% Loss}$ in formula xii).
Example 1. A shopkeeper allows a discount of 10% on the marked price and still makes a profit of 20%. If the cost price of the article is $\textsf{₹ } 450$, find the marked price.
Answer:
Given: Discount % = 10%, Profit % = 20%, CP = $\textsf{₹ } 450$.
We need to find the Marked Price (MP).
Method 1: Step-by-step Calculation
First, find the Selling Price (SP) using the CP and Profit %.
Using formula (vi): $\text{SP} = \text{CP} \times \left(\frac{100 + \text{ % Profit}}{100}\right)$
SP $= \textsf{₹ } 450 \times \left(\frac{100 + 20}{100}\right) = \textsf{₹ } 450 \times \frac{120}{100}$
SP $= \textsf{₹ } 450 \times \frac{6}{5} = \textsf{₹ } \cancel{450}^{\normalsize 90} \times \frac{6}{\cancel{5}^{\normalsize 1}} = \textsf{₹ } 90 \times 6 = \textsf{₹ } 540$
The Selling Price is $\textsf{₹ } 540$.
Now, use the SP and Discount % to find the Marked Price (MP).
Using formula (xi): $\text{MP} = \text{SP} \times \left(\frac{100}{100 - \text{ % Discount}}\right)$
MP $= \textsf{₹ } 540 \times \left(\frac{100}{100 - 10}\right) = \textsf{₹ } 540 \times \frac{100}{90}$
Simplify and multiply:
MP $= \textsf{₹ } 540 \times \frac{10}{9} = \textsf{₹ } \cancel{540}^{\normalsize 60} \times \frac{10}{\cancel{9}^{\normalsize 1}} = \textsf{₹ } 60 \times 10 = \textsf{₹ } 600$
Marked Price $= \textsf{₹ } 600$.
Method 2: Using the Direct Relationship Formula (xii)
Given: Profit % = 20%, Discount % = 10%, CP = $\textsf{₹ } 450$. Find MP.
Using the formula $\frac{\text{MP}}{\text{CP}} = \frac{100 + \text{ % Profit}}{100 - \text{ % Discount}}$:
$\frac{\text{MP}}{\textsf{₹ } 450} = \frac{100 + 20}{100 - 10}$
$\frac{\text{MP}}{450} = \frac{120}{90}$
Simplify the fraction on the right side:
$\frac{120}{90} = \frac{12}{9} = \frac{4}{3}$
$\frac{\text{MP}}{450} = \frac{4}{3}$
Solve for MP:
$\text{MP} = 450 \times \frac{4}{3}$
MP $= \cancel{450}^{\normalsize 150} \times \frac{4}{\cancel{3}^{\normalsize 1}} = 150 \times 4 = 600$
Marked Price $= \textsf{₹ } 600$.
Both methods confirm the marked price is $\boldsymbol{\textsf{₹ } 600}$. The direct formula method (Method 2) is generally quicker for this type of problem.
Competitive Exam Notes:
Problems involving CP, SP, MP, profit/loss, and discount require a clear understanding of how these terms are linked. Use the derived formulas to save time.
- CP is the Base for Profit/Loss: Profit/Loss is always calculated on CP. $\text{SP} = \text{CP} \times \frac{100 \pm \text{%Change}}{100}$.
- MP is the Base for Discount: Discount is always calculated on MP. $\text{SP} = \text{MP} \times \frac{100 - \text{%Discount}}{100}$.
- SP is the Link: SP connects CP (via profit/loss) and MP (via discount).
- Direct Formula: The formula $\frac{\text{MP}}{\text{CP}} = \frac{100 \pm \text{ % Profit/Loss}}{100 - \text{ % Discount}}$ is extremely useful for quickly relating MP and CP when percentages are given. Use $(100 + \text{ % Profit})$ for profit and $(100 - \text{ % Loss})$ for loss in the numerator.
- Read Carefully: Pay attention to whether the percentage given is profit/loss (on CP) or discount (on MP).
Advanced Problems in Profit, Loss, and Discount
Building upon the basic definitions and relationships between Cost Price (CP), Selling Price (SP), Marked Price (MP), profit, loss, and discount, this section explores more complex problems that require combining these concepts and applying the derived formulas and principles effectively. These problems often involve multiple steps or indirect relationships between the terms.
Example 1. A shopkeeper allows a discount of 15% on the marked price of an article and makes a profit of 19%. If the cost price of the article is $\textsf{₹ } 600$, find its marked price.
Answer:
Given: Cost Price (CP) $= \textsf{₹ } 600$, Profit Percentage $= 19\%$, Discount Percentage $= 15\%$.
We need to find the Marked Price (MP).
Method 1: Using Selling Price (SP) as an intermediate step.
First, calculate the Selling Price (SP) using the Cost Price (CP) and the Profit Percentage.
If there is a profit of 19%, the Selling Price is 19% more than the Cost Price. Using the formula $\text{SP} = \text{CP} \times \left(\frac{100 + \text{ % Profit}}{100}\right)$:
SP $= \textsf{₹ } 600 \times \left(\frac{100 + 19}{100}\right)$
SP $= \textsf{₹ } 600 \times \frac{119}{100}$
Calculate the Selling Price:
SP $= \textsf{₹ } \cancel{600}^{\normalsize 6} \times \frac{119}{\cancel{100}^{\normalsize 1}}$
SP $= 6 \times 119 = 714$
SP $= \textsf{₹ } 714$
The Selling Price of the article is $\textsf{₹ } 714$.
Now, use the Selling Price (SP) and the Discount Percentage to find the Marked Price (MP).
We know that the SP is obtained after a 15% discount on the MP. This means the SP is $(100 - 15)\% = 85\%$ of the MP. Using the formula $\text{SP} = \text{MP} \times \left(\frac{100 - \text{ % Discount}}{100}\right)$:
$\textsf{₹ } 714 = \text{MP} \times \left(\frac{100 - 15}{100}\right)$
$\textsf{₹ } 714 = \text{MP} \times \frac{85}{100}$
To find MP, multiply both sides by the reciprocal of $\frac{85}{100}$ (which is $\frac{100}{85}$):
$\boldsymbol{\text{MP} = \textsf{₹ } 714 \times \frac{100}{85}}$
... (xiv)
Simplify the fraction $\frac{100}{85}$ by dividing numerator and denominator by 5: $\frac{20}{17}$.
MP $= \textsf{₹ } 714 \times \frac{20}{17}$
Now, simplify $\frac{714}{17}$ (note that $17 \times 4 = 68$, $71-68=3$, bring down 4, $34$, $17 \times 2 = 34$. So $17 \times 42 = 714$).
MP $= \textsf{₹ } \cancel{714}^{\normalsize 42} \times \frac{20}{\cancel{17}^{\normalsize 1}}$
MP $= 42 \times 20 = 840$
Marked Price $= \textsf{₹ } 840$.
Method 2: Using the direct relationship formula.
We have the formula relating MP and CP directly when profit percentage and discount percentage are given: $\frac{\text{MP}}{\text{CP}} = \frac{100 + \text{ % Profit}}{100 - \text{ % Discount}}$.
Given: CP = $\textsf{₹ } 600$, % Profit = 19, % Discount = 15.
$\frac{\text{MP}}{\textsf{₹ } 600} = \frac{100 + 19}{100 - 15}$
$\frac{\text{MP}}{600} = \frac{119}{85}$
Simplify the fraction $\frac{119}{85}$ by dividing numerator and denominator by their GCD, 17 ($119 = 7 \times 17$, $85 = 5 \times 17$).
$\frac{119}{85} = \frac{\cancel{119}^{\normalsize 7}}{\cancel{85}^{\normalsize 5}} = \frac{7}{5}$
$\frac{\text{MP}}{600} = \frac{7}{5}$
Solve for MP:
$\boldsymbol{\text{MP} = 600 \times \frac{7}{5}}$
MP $= \cancel{600}^{\normalsize 120} \times \frac{7}{\cancel{5}^{\normalsize 1}} = 120 \times 7 = 840$
Marked Price $= \textsf{₹ } 840$.
Both methods yield the same result. The direct formula method is often faster for competitive exams.
Example 2. By selling an article for $\textsf{₹ } 1700$, a shopkeeper makes a profit of 25%. Find the cost price of the article.
Answer:
Given: Selling Price (SP) $= \textsf{₹ } 1700$, Profit Percentage $= 25\%$.
We need to find the Cost Price (CP).
Method 1: Using the formula.
If there is a profit of 25% on CP, then SP is (100 + 25)% = 125% of CP.
Using the formula $\text{CP} = \text{SP} \times \left(\frac{100}{100 + \text{ % Profit}}\right)$:
CP $= \textsf{₹ } 1700 \times \left(\frac{100}{100 + 25}\right)$
CP $= \textsf{₹ } 1700 \times \frac{100}{125}$
Simplify the fraction $\frac{100}{125}$ by dividing numerator and denominator by 25: $\frac{4}{5}$.
CP $= \textsf{₹ } 1700 \times \frac{4}{5}$
... (xv)
Calculate the Cost Price:
CP $= \textsf{₹ } \cancel{1700}^{\normalsize 340} \times \frac{4}{\cancel{5}^{\normalsize 1}}$
CP $= 340 \times 4 = 1360$
Cost Price $= \textsf{₹ } 1360$.
Method 2: Using percentage concept directly.
A profit of 25% on CP means the Selling Price is 125% of the Cost Price.
125% of CP = SP
$\frac{125}{100} \times \text{CP} = \textsf{₹ } 1700$
Simplify the fraction $\frac{125}{100} = \frac{5}{4}$.
$\frac{5}{4} \times \text{CP} = \textsf{₹ } 1700$
Solve for CP by multiplying both sides by $\frac{4}{5}$:
$\boldsymbol{\text{CP} = \textsf{₹ } 1700 \times \frac{4}{5}}$
This leads to the same calculation as Method 1, resulting in CP = $\textsf{₹ } 1360$.
The cost price of the article is $\boldsymbol{\textsf{₹ } 1360}$.
Example 3. A shopkeeper marks his goods 40% above the cost price. If he sells them at a discount of 20%, find his profit percentage.
Answer:
Let the Cost Price (CP) be $\textsf{₹ } 100$ for ease of calculation based on percentages.
The goods are marked 40% above CP. This percentage increase is applied to CP to find MP.
Marked Price (MP) $= \text{CP} + 40\% \text{ of CP} = \text{CP} \times \left(1 + \frac{40}{100}\right) = \text{CP} \times \frac{140}{100}$
If CP = $\textsf{₹ } 100$, then MP $= \textsf{₹ } 100 \times \frac{140}{100} = \textsf{₹ } 140$.
A discount of 20% is allowed on the Marked Price (MP = $\textsf{₹ } 140$).
Discount Amount $= 20\% \text{ of MP} = \frac{20}{100} \times \textsf{₹ } 140 = \frac{1}{5} \times 140 = \textsf{₹ } 28$
The Selling Price (SP) is obtained by subtracting the discount amount from the MP.
Selling Price (SP) $= \text{MP} - \text{Discount Amount} = \textsf{₹ } 140 - \textsf{₹ } 28 = \textsf{₹ } 112$
Now, compare the Selling Price (SP) and the original Cost Price (CP) to find the profit or loss.
CP $= \textsf{₹ } 100$, SP $= \textsf{₹ } 112$.
Since $\text{SP} (\textsf{₹ } 112) > \text{CP} (\textsf{₹ } 100)$, there is a Profit.
Profit Amount $= \text{SP} - \text{CP} = \textsf{₹ } 112 - \textsf{₹ } 100 = \textsf{₹ } 12$.
Profit Percentage is calculated on CP:
$\text{Profit Percentage } = \left(\frac{\text{Profit Amount}}{\text{CP}} \times 100\right)\%$
$\text{ % Profit} = \left(\frac{\textsf{₹ } 12}{\textsf{₹ } 100} \times 100\right)\% = \boldsymbol{12 \%}$
The retailer's profit percentage is $\boldsymbol{12\%}$.
Method 2: Using the direct relationship formula.
Let the profit percentage be $P\%$.
The goods are marked 40% above the cost price. This implies that the Marked Price is $(100+40)\% = 140\%$ of the Cost Price. So, $\frac{\text{MP}}{\text{CP}} = \frac{140}{100}$.
The discount percentage is 20%.
Using the formula $\frac{\text{MP}}{\text{CP}} = \frac{100 + \text{ % Profit}}{100 - \text{ % Discount}}$:
$\frac{140}{100} = \frac{100 + P}{100 - 20}$
$\frac{140}{100} = \frac{100 + P}{80}$
Simplify the fraction on the left side: $\frac{140}{100} = \frac{14}{10} = \frac{7}{5}$.
$\frac{7}{5} = \frac{100 + P}{80}$
Solve for $100 + P$. Multiply both sides by 80:
$100 + P = \frac{7}{5} \times 80$
$100 + P = 7 \times \frac{80}{5} = 7 \times 16 = 112$
Solve for $P$:
$\boldsymbol{P = 112 - 100 = 12}$
The profit percentage is $\boldsymbol{12\%}$.
This method is generally quicker as it directly leads to the profit percentage.
Example 4. A dishonest shopkeeper claims to sell goods at cost price but uses a false weight of 950 gm instead of 1 kg. Find his profit percentage.
Answer:
The shopkeeper states that he sells goods at CP, meaning the price charged to the customer is equal to the Cost Price of the quantity he *claims* to sell (1 kg).
However, he gives a different quantity (950 gm) for that price.
Let the Cost Price of 1 gram of goods be $\textsf{₹ } c$.
Cost Price (CP) of 1 kg (1000 gm) $= 1000 \times \textsf{₹ } c$.
The shopkeeper charges the customer the price of 1000 gm. So, the Selling Price (SP) for the quantity he sells is equal to the CP of 1000 gm.
SP $= 1000 \times \textsf{₹ } c$.
The shopkeeper actually sells 950 gm. The Cost Price for the quantity he actually sells is the cost of 950 gm.
Effective CP $= 950 \times \textsf{₹ } c$.
Compare SP and Effective CP:
SP $(1000c)$ > Effective CP $(950c)$. There is a Profit.
Profit Amount $= \text{SP} - \text{Effective CP} = (1000c - 950c) = 50c$.
Profit Percentage is calculated on the Cost Price of the quantity *actually sold* (Effective CP).
$\text{Profit Percentage } = \left(\frac{\text{Profit Amount}}{\text{Effective CP}} \times 100\right)\%$
$\text{ % Profit} = \left(\frac{50c}{950c} \times 100\right)\%$
Cancel out $c$ (assuming $c \neq 0$) and simplify the fraction $\frac{50}{950} = \frac{5}{95} = \frac{1}{19}$.
$\text{ % Profit} = \left(\frac{1}{19} \times 100\right)\%$
$\text{ % Profit} = \frac{100}{19}\%$
Convert the improper fraction to a mixed number:
$\frac{100}{19} = 5 \frac{5}{19}$
% Profit $= 5\frac{5}{19}\%$.
The shopkeeper's profit percentage is $\boldsymbol{5\frac{5}{19}\%}$.
Alternative Method: Using the formula for dishonest weighing.
When a shopkeeper sells goods at CP but uses a false weight, the profit comes from the quantity difference. The formula for percentage profit is:
$\boldsymbol{\text{ % Profit} = \left(\frac{\text{Error}}{\text{False Weight}} \times 100\right)\%}$
... (xvi)
True Weight (what should be given) $= 1 \text{ kg} = 1000$ gm.
False Weight (what is actually given) $= 950$ gm.
Error in weight $= \text{True Weight} - \text{False Weight} = 1000 \text{ gm} - 950 \text{ gm} = 50$ gm.
Using formula (xvi):
$\text{ % Profit} = \left(\frac{50}{950} \times 100\right)\%$
$= \left(\frac{\cancel{50}^{\normalsize 1}}{\cancel{950}^{\normalsize 19}} \times 100\right)\%$
$= \frac{100}{19}\% = 5\frac{5}{19}\%$
The profit percentage is $\boldsymbol{5\frac{5}{19}\%}$. This formula is very efficient for this specific type of problem.
Competitive Exam Notes:
Advanced problems often combine profit/loss calculation with discount or involve scenarios like dishonest practices. Practice applying the derived formulas and understanding the underlying concepts.
- Interconnecting CP, SP, MP: Most problems involve moving between these three prices using percentage changes. SP is the bridge between CP (+ Profit/%Profit or - Loss/%Loss) and MP (- Discount/%Discount).
- Direct Formula ($\frac{\text{MP}}{\text{CP}}$): $\frac{\text{MP}}{\text{CP}} = \frac{100 \pm \text{ % Profit/Loss}}{100 - \text{ % Discount}}$ is invaluable for directly finding MP given CP (or vice versa) when profit/loss and discount percentages are known. Use +P for profit, -L for loss in the numerator.
- Dishonest Practices: Problems involving false weights or measures are common. The key is to identify the *effective* Cost Price (CP of the quantity actually delivered) and the Selling Price (price charged for the quantity claimed). Profit % is always on the *effective* CP. The formula $\left(\frac{\text{Error}}{\text{False Value}} \times 100\right)\%$ is useful for percentage profit/loss in such cases (where error is difference between true and false values, and false value is what is actually given/measured).
- Successive Discounts/Markups: Treat consecutive percentage changes using multipliers or the net change formula $(a+b+ab/100)$ for percentages. Remember that discounts are successive decreases applied to the decreasing base price.
- Reading Comprehension: Pay extreme attention to the wording of the problem - which percentage is based on which price (CP, SP, or MP)?