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Commercial Mathematics - Profit and Loss: Cost Price, Selling Price, Profit/Loss Percentage

Grade 7ICSE

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

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Cost Price (CP): The total amount paid to purchase an item, including any additional overhead expenses like transportation, labor, or repairs. Visually, think of CP as the 'Starting Value' on a timeline of a transaction.

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Selling Price (SP): The price at which an item is finally sold to a customer. If you visualize a vertical bar chart, SP represents the height of the 'Inflow' bar compared to the 'Outflow' bar of the CP.

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Profit (Gain): When the Selling Price is greater than the Cost Price (SP>CPSP > CP), a profit is made. On a number line, Profit is the positive distance or 'gap' extending from the CP point forward to the SP point.

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Loss: When the Cost Price is greater than the Selling Price (CP>SPCP > SP), a loss is incurred. Visually, this represents a 'drop' or a downward step from the initial investment (CP) to the final return (SP).

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Overhead Charges: These are extra expenses like freight, insurance, or renovation costs incurred after buying an item but before selling it. These must be added to the original purchase price to calculate the 'Total Cost Price'. Total CP=Initial Purchase Price+Overhead ExpensesCP = \text{Initial Purchase Price} + \text{Overhead Expenses}.

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Profit and Loss Percentage: This is the calculation of profit or loss relative to every 100100 units of the Cost Price. It allows for a fair comparison between different transactions regardless of their scale. Importantly, the percentage is always calculated using the Cost Price as the base or denominator.

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Break-even Point: A situation where SP=CPSP = CP, resulting in no profit and no loss. Visually, the bars for CP and SP would be of equal height, indicating a neutral financial outcome.

📐Formulae

Profit = SP - CP

Loss = CP - SP

Profit%=(ProfitCP×100)%Profit \% = (\frac{Profit}{CP} \times 100) \%

Loss%=(LossCP×100)%Loss \% = (\frac{Loss}{CP} \times 100) \%

SP=(100+Profit%)100×CPSP = \frac{(100 + Profit \%)}{100} \times CP

SP=(100−Loss%)100×CPSP = \frac{(100 - Loss \%)}{100} \times CP

CP=100(100+Profit%)×SPCP = \frac{100}{(100 + Profit \%)} \times SP

CP=100(100−Loss%)×SPCP = \frac{100}{(100 - Loss \%)} \times SP

💡Examples

Problem 1:

A shopkeeper bought a cycle for Rs 1200 and spent Rs 300 on its repairs. He then sold it for Rs 1800. Find his profit or loss percentage.

Solution:

  1. Calculate Total Cost Price: Total CP=1200+300=Rs1500CP = 1200 + 300 = Rs 1500
  2. Identify Selling Price: SP=Rs1800SP = Rs 1800
  3. Since SP>CPSP > CP, it is a Profit.
  4. Calculate Profit: Profit=SP−CP=1800−1500=Rs300Profit = SP - CP = 1800 - 1500 = Rs 300
  5. Calculate Profit Percentage: Profit%=(ProfitCP×100)%=(3001500×100)%=20%Profit \% = (\frac{Profit}{CP} \times 100) \% = (\frac{300}{1500} \times 100) \% = 20 \%

Explanation:

We first include repair costs into the Cost Price to find the total investment. Since the selling price is higher than the total investment, we calculate the profit amount and then find its percentage relative to the total cost price.

Problem 2:

By selling a mobile phone for Rs 4500, a dealer loses 10%. For what price should he sell it to gain 10%?

Solution:

Step 1: Find the Cost Price. SP=Rs4500SP = Rs 4500, Loss%=10%Loss \% = 10 \% CP=100100−Loss%×SPCP = \frac{100}{100 - Loss \%} \times SP CP=10090×4500=Rs5000CP = \frac{100}{90} \times 4500 = Rs 5000 Step 2: Find the new Selling Price for 10% Gain. New Gain%=10%Gain \% = 10 \% New SP=100+Gain%100×CPSP = \frac{100 + Gain \%}{100} \times CP New SP=110100×5000=Rs5500SP = \frac{110}{100} \times 5000 = Rs 5500

Explanation:

In this two-step problem, we first use the given selling price and loss percentage to find the original cost of the phone. Once the cost price is known, we calculate the new selling price required to achieve the desired profit margin.