Profit and loss is one of the basic concepts in Mathematics that will be helpful throughout life. Profit and loss are two terms that are used to determine if a deal is profitable or not. We use these terms very frequently in our daily lives. The simplest approach to determine profit and loss is by assessing the difference between the selling price and the cost price. If the difference is positive then there is a profit whereas if the difference is negative then there is a loss. If the selling price is less than the cost price, then a difference between the cost and selling prices is called loss. The price at which a particular article is bought is called its cost price. The price at which a particular article is sold is called its selling price. In this article, we will discuss the profit and loss concept with formulas and solved examples.

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Profit and Loss: Definition

Cost Price \(\left({CP} \right)\): The amount paid to buy a product or the price at which a product is made is known as cost price.

Selling Price \(\left({SP} \right)\): The price at which a product is sold is the selling price.

Profit: If the selling price is greater than the cost price, then the difference between the selling price and cost price is called profit.

If \(SP > CP,\) i.e., in the case of profit, \({\text{Profit}}\, = SP – CP.\)
Example: A man bought an article at \(₹30\) and sold it at \(₹35.\)
Given, \(CP = ₹30\) and \(SP = ₹35\) here \(SP > CP\)
Therefore, it is profit. That is \({\rm{profit}} = 35 – 30 = ₹5\)
Hence, profit \(= ₹5\)

Loss: If the selling price is less than the cost price, then the difference between the cost price and the selling price is called loss.

If \(CP > SP,\) i.e., in case of Loss, \({\text{Loss}} = CP – SP.\)
Example: A man bought an article at \(₹30\) and sold it at \(₹25.\)
Given, \(CP = ₹30\) and \(SP = ₹25\) here \(CP > SP\)
Therefore, it is loss. That is \({\text{Loss = 30 – 25 = ₹5}}\)
Hence, loss \(= ₹5\)

Profit and Loss: Formula

Some of the commonly used formulas for profit and loss are provided below:

Profit

If a product’s selling price is greater than its cost price, there is a profit in the business. The basic formula used for computing the profit is:

\({\text{Profit}} = {\text{Selling}}\,{\text{price}}\, – {\text{Cost}}\,{\text{price}}.\)
\({\text{Overall}}\,{\text{again}} = {\text{Combined}}\,{\text{SP}}\, – {\text{Combined}}\,{\text{CP}}\)

Loss

If the selling price is lesser than the cost price, there is a loss in the business. The basic formula used for determining the loss is:

\({\text{Loss}} = {\text{Cost}}\,{\text{price}}\, – {\text{Selling}}\,{\text{price}}\)
\({\text{Overall}}\,{\text{loss}} = {\text{Combined}}\,{\text{CP}} \,- {\text{Combined}}\,{\text{SP}}\)

Profit and Loss: Percentage Formula

The profit or loss percentage is computed by using the following formulas, which indicate that the profit or loss in a deal is always calculated on its cost price:

\({\text{Profit}}\% = \frac{{{\text{ Profit }}}}{{{\text{ Cost Price }}}} \times 100\) and \({\text{Loss}}\% = \frac{{{\text{ Loss }}}}{{{\text{ Cost Price }}}} \times 100\)

Note: Profit \(\% \) or Loss \(\% \) are calculated on the cost price.

Use of Profit and Loss Calculation

Profit and Loss calculations are used in mathematics to determine the price of a commodity in the market and understand how profitable a business is. Every product has a Cost Price And A Selling Price. Based on the values of those prices, we can calculate the profit earned or the loss incurred for a specific product.

Importance of Profit and Loss

Profit and loss statements are important because many companies must complete them by law or association membership. A profit and loss declaration also helps the company’s leadership team (including its executive committee) understand the business’s net income, which could be helpful in decision-making procedures.

For example, a company owner may decide to restore the building or business expansion based on the profit margins of a company.

Profit and Loss: Calculation

Profit and loss formulas are used to compute profit or loss that has been achieved by selling a specific article. The formulas mostly find application in business and financial dealings. Profit and loss as a percentage is usually a measure to illustrate how much profit or loss a trader incurred from any deal.

1. When the selling price of any article sold is greater than the cost price (the price at which the article was initially bought), gain or profit is made.
2. When the selling price of any article sold is lesser than the cost price (the price at which the article was initially bought), the loss is made.

Some important formulas related to profit and loss are:

1. Profit percentage \(\left({P\% } \right) = \left({\frac{{{\text{Profit}}}}{{{\text{Cost Price}}}}} \right) \times 100\)
2. Loss percentage \(\left({L\% } \right) = \left({\frac{{{\text{Loss}}}}{{{\text{Cost Price}}}}} \right) \times 100\)
3. Selling price \( = \left\{{\left({\frac{{100 + P\% }}{{100}}} \right)} \right\} \times CP\) (when \(SP > CP\))
4. Selling price \( = \left\{{\left({\frac{{100 – L\% }}{{100}}} \right)} \right\} \times CP\) (when \(SP < CP\))
5. Cost price \( = \left\{{\left({\frac{{100}}{{100 + P\% }}} \right)} \right\} \times SP\) (when \(SP > CP\))
6. Cost price \( = \left\{{\left({\frac{{100}}{{100 – L\% }}} \right)} \right\} \times SP\) (when \(SP < CP\))

Note:

1. Convert fraction to per cent and per cent to fraction wherever required.
2. Profit or loss percentage is conveyed as a fraction with \(CP\) in the denominator.

Solved Examples – Profit and Loss

These solved examples will allow the students to understand problem sums and attend them independently.

Q.1. A man bought two washing machine sets at \(₹15000\) each. He sold one at a profit of \(15\% \) and the other at a loss of \(15\% .\) Find whether he made an overall profit or loss.
Ans: Combined cost price \( = 15000 + 15000 = ₹30000\)
Profit on one washing machine \(15\% \) of \(15000 = \frac{{15}}{{100}} \times 15000 = ₹2250\)
\(SP\) of one washing machine \( = CP + {\text{Profit}} = ₹\left({15000 + 2250} \right) = ₹17250\)
Loss on the other washing machine \(15\% .\) of \(15000 = \frac{{15}}{{100}} \times 15000 = ₹2250\)
\(SP\) of the other washing machine \(= CP – {\text{Loss}} = ₹\left({15000 – 2250} \right) =₹ 12250\)
Combined \(SP = ₹\left({17250 + 12250} \right) = ₹30000\)
So, Combined \(CP = \) Combined \(SP\)
Hence, no overall profit or loss.

Q.2 A shopkeeper bought eggs at \(₹42,\) per dozen and sells \(5\) eggs for \(₹20.\) Find his gain or loss per cent.
Ans: \(CP\) of \(12\) eggs \(= ₹42,\) therefore \(CP\) of \(1\) egg \(=₹ \frac{{42}}{{12}} =₹ \frac{7}{2} = ₹3.50\)
Therefore \(SP\) of \(5\) eggs \(= ₹20\) and \(SP\) of \(1\) egg \(₹\frac{{20}}{5} = ₹4\)
Profit on the sale of \(1\) egg \( =₹ \left({4 – 3.50} \right) =₹ 0.50\)
Therefore, profit percentage \( = \frac{{{\text{ Profit }}}}{{{\text{CP}}}} \times 100\% = \frac{{0.50}}{{3.50}} \times 100\% = \frac{{100}}{7}\% = 14\frac{2}{7}\% \)
Hence, his profit percentage is \(14\frac{2}{7}\% .\)

Q.3. By selling a TV for \(₹18858,\) a shopkeeper incurs a loss of \(₹3592.\) Find his loss per cent.
Ans: \(SP\) of TV \( =₹ 18858\) and loss \( =₹ 3592\)
Therefore, \(CP = SP + {\text{Loss}} = ₹\left({18858 + 3592} \right) =₹ 22450\)
Thus, \(\operatorname{loss} \% = \frac{{{\text{loss}} \times 100}}{{{\text{CP}}}}\)
\( = \frac{{3592 \times 100}}{{22450}} = 16\% \)
Hence, his loss per cent is \(16\% .\)

Q.4. A fruit vendor bought \(5\) lemons for \(₹5\) and sold at the rate of \(4\) lemons for \(₹5.\) Calculate:
a) Profit in selling \(60\) lemons
b) Profit \(\% \)
Ans: a) \(CP\) of \(5\) lemons \(= ₹5\)
\(CP\) of \(1\) lemon \( = \frac{5}{5} = ₹1\)
\(CP\) of \(60\) lemons \(= 1 \times 60 = ₹60\)
\(SP\) of \(4\) lemons \(= ₹5\)
\(SP\) of \(1\) lemon \( = \frac{5}{4}\)
\(SP\) of \(60\) lemons \(= \frac{5}{4} \times 60 = ₹75\)
Profit \(= ₹75\, -\, ₹60 = ₹15\)
Hence, the profit is \(₹15.\)
b) Profit \(\% = \frac{{{\text{ Profit }}}}{{{\text{CP}}}} \times 100\)
\( = \frac{{15}}{{60}} \times 100 = 25\% \)
Hence, the profit percentage is \(25\% .\)

Q.5. A shopkeeper sold a shirt for \(₹720\) and made a profit of \(20\% .\) Find:
a) The cost price of a shirt
b) Profit made by her in selling \(10\) shirts
Ans: a) Let the cost price of a shirt be \(₹100\)
Profit was \(20\% .\)
So, profit on \(1\) shirt \(₹20\) (because the \(CP\) is taken as \(₹100\))
Hence, the \(SP\) of \(1\) shirt \( = ₹100 +₹ 20 = ₹120\)
Let the actual cost price \( = ₹x\)
Thus, \(100:120::x:720\)
\( \Rightarrow 120 \times x = 100 \times 720\)
\( \Rightarrow x = \frac{{100 \times 720}}{{120}} = ₹600\)
b) Profit on \(1\) shirt \( =₹ 720 \,-\, ₹600 = ₹120\)
So, the profit for \(10\) shirts \(120 \times 10 = ₹1200\)

Summary

In this article, we have learned the basics of profit and loss, profit and loss formula, profit and loss percentage formula, use of profit and loss calculation, the importance of profit and loss, and how to calculate profit and loss. Solved examples on profit and Loss and FAQs help in quickly revising the concept.

The learning outcome of this article is how profit and loss formulas are used to calculate the money made/ profit or loss that has been made by selling a particular product.

Frequently Asked Questions – Profit and Loss

Frequently asked questions related to profit and loss are listed as follows:

Q.1. What is the profit formula in math?
Ans: If an article’s selling price is greater than the cost price, then the difference between the selling price and cost price is called profit.
If \(SP > CP,\) i.e., in the case of profit, \({\text{Profit=}}SP – CP.\)

Q.2. What is the formula of \(SP\) and \(CP\)?
Ans: Cost Price \(\left({CP} \right)\): The amount paid to buy a product or the price at which a product is made is known as cost price.
\({\text{Cost price = Selling price – Profit}}\) (with profit)
\({\text{Cost price = Selling price + Loss}}\)
Selling Price \(\left({SP} \right)\): The price at which a product is sold is known as the selling price.
\({\text{Selling price = Cost price + Profit}}\)
\({\text{Selling price = Cost price – Loss}}\)

Q.3. What is the formula for calculating profit and loss?
Ans: Profit:
If a product’s selling price is greater than its cost price, there is a profit in the business. The basic formula used for computing the profit is:
\({\text{Profit = Selling Price – Cost Price}}{\text{.}}\)
\({\text{Overall gain = Combined SP – Combined CP}}\)
Loss:
If a product’s selling price is lesser than the cost price, there is a loss in the business. The basic formula used for determining the loss is:
\({\text{Loss = Cost Price – Selling Price}}\)
\({\text{Overall loss = Combined CP – Combined SP}}\)

Q.4. How do you calculate profit and loss percentage?
Ans: Profit and loss percentage can be calculated using the following formulas:
1. Profit percentage \(\left({P\% } \right) = \left({\frac{{{\text{Profit}}}}{{{\text{Cost}}\,{\text{Price}}}}} \right) \times 100\)
2. Loss percentage \(\left({L\% } \right) = \left({\frac{{{\text{Loss}}}}{{{\text{Cost}}\,{\text{Price}}}}} \right) \times 100\)

Q.5. Explain the importance of profit and loss for students.
Ans: Profit and loss formulas help students calculate the profit or loss that has been made by selling a particular product.
In general, profit and loss statements are important because many companies must complete them by law or organization participation. A profit and loss declaration also helps a company’s management team compute its net income, which is useful in business decision making.