profit and loss

Contents

1. INTRODUCTION: PROFIT AND LOSS
2. PERCENTAGE PROFIT OR LOSS
   1. TYPES OF COSTS
   2. BREAK-EVEN POINT
   3. CALCULATING PERCENTAGE PROFIT BY EQUATING THE AMOUNT OF MONEY SPENT AND EARNED
   4. FALSE WEIGHTS
3. MARKED PRICE AND DISCOUNT
   1. BUY X AND GET Y FREE
   2. SUCCESSIVE DISCOUNTS

Profit, Loss and Discount

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INTRODUCTION: PROFIT AND LOSS

Profit and Loss are part and parcel of everyday transactions. The terms ‘cost price’ and ‘selling price’ are used at every stage of goods exchanging hands.

The price at which a person buys (or produces) a product is the Cost Price (CP) of the product w.r.t. to that person and the price at which a person sells a product is the sales price or the Selling Price (SP) of the product, again w.r.t. that person.

At each stage, the cost price for one person becomes the selling price for another.

For example, if Amit buys Apples at Rs. 75 per kg from a wholesaler and sells them to Sumit at Rs. 80 per kg, then for one kg of apples that exchanged hands between the wholesaler, Amit and Sumit,

The wholesaler’s Selling Price = Rs. 75 = Amit’s Cost Price and

Amit’s Selling Price = Rs. 80 = Sumit’s Cost Price

When a person is able to sell a product at a price higher than its cost price for him, then we say he has earned a Profit (P).

Profit = Selling price Cost Price

P = SP CP

Similarly, if a person sells an item for a price lower than its cost price for him, we say a Loss (L) has been incurred.

Loss = Cost price Selling Price

L = CP SP

PERCENTAGE PROFIT OR LOSS

We often need to compare the gains or losses of two business transactions. The actual gains or losses are not comparable by themselves as the investment or the capital of the two businesses may differ. In such cases, the comparison of gains and losses can be made by converting them into percentages.

Example 1:

Company A earned revenue of Rs. 15 crores with an investment of Rs. 12 crores. On the other hand, Company B earned revenue of Rs. 33 crores with an investment of Rs. 30 crores. Which company made a higher profit (in terms of percentage)? Solution: Here both the companies made a profit of Rs. 3 crores. But in percentage profit terms they made different profits. Here’s why. Company A made a profit of Rs. 3 crores on an investment of Rs. 12 crores and Company B made the same profit on an investment of Rs. 30 crores. Thus, we can see that, Company A earned a profit which was 1/4th of its investment, whereas Company B earned a profit which was 1/10th of its investment. This concept can be put in terms of percentage profit. Percentage profit for Company A Hence, Company A made a higher profit than Company B.

FORMULAE:

IMPORTANT:

You must always determine Percentage Profit on the Cost Price of an item, unless it is stated otherwise in the question.

Similarly, when there is a loss,

Percentage profit or loss may be calculated in the following manner:

Let us say SP = Rs. 888 and CP = Rs. 777

To calculate Profit or Loss percentage, find the ratio of SP to CP. The absolute difference between this ratio and one, multiplied by 100 gives the percentage profit or loss.

Example 2:

A grocer buys eggs at the rate of Rs. 100 per 144 eggs and sells them at a rate of Rs. 75 per 100 eggs. Find the profit or loss percentage. Solution: CP of 144 eggs = 100 CP of 1 egg = 100/144 SP of 100 eggs = 75 SP of 1 egg = 75/100

Example 3:

If the selling price of 40 equally priced books is equal to the cost price of 32 of those books, then what is the profit or loss percentage? Solution: SP of 40 books = CP of 32 books Let the CP of 1 book be Rs. x and the SP of 1 book be Rs. y. Then, 40y = 32x (SP/CP) of 1 book (in Rs.) = y/x = (32/40) = 4/5

IMPORTANT:

• Percentage Profit can only be calculated when the number of goods sold and the number of goods bought is equal.

For example, in the previous problem, the number of goods (i.e. books) bought was 32, while those sold were 40. Here, we cannot directly find the ratio (SP/CP). Instead, we first had to calculate the ratio of Cost Price and Selling Price of 1 book, and then found the percentage profit based on the buying and selling of 1 book.

Example 4:

If sweets are bought at the rate of 20 for a rupee, then how many must be sold for a rupee so as to gain a profit of 25%? Solution: For a profit of 25%, SP of 20 sweets = 1 1.25 Number of sweets to be sold for Rs. 1.25 = 20 Number of sweets to be sold for Re. 1 = 20/1.25 = 16 For a rupee, 16 sweets should be sold. Alternatively, where x is the Selling Price of one sweet. i.e. the Selling Price of one sweet is Rs. 1/16. For Re. 1, he must sell 16 sweets.

Example 5:

Sandeep bought a computer for Rs. 30,000 and sold it to Amoy at a loss of 10%. Amoy then sold it to Latif at a loss of 15%. Find the amount paid by Latif to purchase the computer. Solution: SP of Sandeep = 30000 0.9 = 27000 CP of Amoy = 27000 SP of Amoy = 27000 0.85 = 22950 CP of Latif = Rs. 22,950

Example 6:

Two guitarists had decided to sell-off their spare un-used guitars (each had one spare guitar). After they had sold the instruments, they told each other their profit percentages and realized that they had both made a 40% profit. However, in reality, while one guitarist had calculated the profit percentage on the cost price of his guitar, the other had calculated it on the selling price. It turned out that the difference between the profits earned by them was Rs. 800. What was the selling price, if both the guitars were sold for the same price? Solution: Let x be the selling price of both the guitars. When the profit percentage was calculated on the Cost Price: When the profit percentage was calculated on the Selling Price: It is given that the difference between their profits was Rs. 800. Hence, Hence, the selling price of each of the two guitars was Rs.7,000.

If two items are sold for the same SP, one at a gain of a% and the other at a loss of a%, then there is an overall loss and the loss percentage = a²/100 %.

EXPLANATION:

Let the selling price of both the items be x.

Then, for the item sold at a gain of a%,

For the item sold at a loss of a%,

Example 7:

Mani sold both his consignments to two different vendors at Rs. 30,000 per consignment. He got a profit of 15% on the first consignment and a loss of 15% on the second consignment. Find his overall profit or loss percentage. Solution: Since the SP is same and both profit and loss % in individual transactions are the same, he ends up with an overall loss which is given by a2/100 %. Alternatively, CP of the first consignment = 30000/1.15 26087 CP of the second consignment = 30000/0.85 35294 Total CP = Rs. 61,381 Total SP = Rs. 60,000

Example 8:

A grocer bought 6 dozen eggs for Rs. 80. Later he found 24 eggs to be broken and sold the rest at the rate of 2 eggs for Rs. 5. Find the profit or loss percentage. Solution: Cost of 6 dozen eggs = Rs. 80 Since 24 eggs are broken, he sells only 4 dozen. SP of 2 eggs = Rs. 5 SP of 12 eggs (i.e. 1 dozen) = Rs. 30 SP of 4 dozen eggs = Rs. 120 Profit = Rs. 40 Profit Percentage = (40 100)/80 = 50%

Example 9:

A space research company wants to sell its two products, A and B. If product A is sold at 20% loss and product B at 30% gain, the company will not lose anything. If product A is sold at 15% loss and product B at 15% gain, the company will lose Rs. 6 million in the deal. What is the cost of product B? [FMS 2009] (1) Rs. 140 million (2) Rs. 120 million (3) Rs. 100 million (4) Rs. 80 million Solution: Let the cost price of the products A and B be x and y respectively. In the first case, product A is sold at a 20% loss and product B is sold at a 30% gain. Thus, their respective selling prices will be 0.8x and 1.3y. At these selling prices, the company does not lose anything. Thus, the total costs and total sales revenue are equal. i.e. x + y = 0.8x + 1.3y 0.2x = 0.3y x = 1.5y …(i) In the second case, A is sold at a 15% loss and B is sold at a 15% gain. Thus, their respective selling prices will be 0.85x and 1.15y. At these selling prices, the total costs exceed the total sales revenue by 6 million. x + y = 0.85x + 1.15y + 6 million 2.5y – 1.15y – 0.85(1.5y) = 6 million (Substituting the value of x from (i)) 1.35y – 1.275y = 6 million 0.075y = 6 million y = 80 million Cost price of B is Rs. 80 million Hence, option 4.

TYPES OF COSTS

In general, there are 3 types of costs that could be incurred when producing/purchasing a product:

1. DIRECT/VARIABLE COSTS:

These are costs that apply to each produced (and subsequently sold) commodity. They are called ‘variable’ costs because these costs vary depending on the number of units of goods produced/sold. For example, costs such as the amount spent in buying raw materials for one unit of a product, or the amount paid to a salesman who is paid on a piece-wise basis are direct costs. These costs are directly proportional to the number of goods sold.

For example, if the direct costs incurred in selling one book is Rs. 50 (which could include the cost of paper, printing costs, commission paid to the author, etc.), then the direct costs for selling 10 books would be 50 10 = Rs. 500.

2. INDIRECT/FIXED COSTS (A.K.A. OVERHEAD COSTS)

These are costs that remain constant (or ‘fixed’) irrespective of the number of units of goods sold. For example, costs such as rent paid for office space, monthly wages paid to employees, amount spent in promoting the product, etc. are all fixed costs. If the rent for office space is Rs. 1000, then no matter how many goods are sold (whether it be 1 or 100), the rent will still remain Rs. 1000.

However, in many instances, the fixed costs are distributed equally among the number of units of goods sold. That is, the fixed cost to be associated with each sold unit is:

3. SEMI-VARIABLE COSTS

These are costs that, under ordinary circumstances, behave like fixed costs; however, when the number of units of goods sold reaches a certain limit, it spikes up as well. A perfect instance of this is the rent example – it is true that as the number of units of goods sold increases, the rent remains constant; however, what if the units sold increases to such an extent that more office space is required? Then, of course, the rent too will increase. In this case, the rent for office space becomes an example of semi-variable cost.

Example 10:

Live-in-style, a clothing manufacturing company, has a fixed yearly cost of Rs. 5,50,000. It sells two types of garments – woolen and cotton. The variable cost for each woolen garment is Rs. 400 while that of each cotton garment is Rs. 300. The selling price of a woolen garment is Rs. 800 while that of a cotton garment is Rs. 400. Also, 5% of the selling price of each garment is given to the salesman who sells the garment. In the year 2007, the weather had been extremely hot; so Live-in-style sold all its produced cotton garments, but it could only sell 62.5% of the woolen garments produced that year. (Assume that at the end of each year, Live-in-style gives away all its unsold garments to charities and does not make any profit from them.) If Live-in-style had produced 5000 woolen and 5000 cotton garments, then what is its percentage profit? Solution: Now, 5000 woolen garments and 5000 cotton garments are produced. All 5000 cotton garments get sold and 62.5% (= 3125) woolen garments are sold. First, let us consider the costs: Fixed Costs = Rs. 5,50,000 Variable Cost for a woolen garment = Rs. 400 Variable Cost for a cotton garment = Rs. 300 Hence, total costs for Live-in-style in the year 2007 = Fixed Costs + Total Variable Costs for woolen garments + Total Variable Costs for cotton garments = 550000 + 400 5000 + 300 5000 = 550000 + 700 5000 = Rs. 40,50,000 Now, let us consider the sales revenue: Sales Revenue from a woolen garment = 800 – 5% of 800 = 800 – 40 = Rs. 760 Number of sold woolen garments = 3125 Sales Revenue from a cotton garment = 400 – 5% of 400 = 400 – 20 = Rs. 380 Number of sold cotton garments = 5000 Hence, total sales revenue for Live-in-style in 2007 = 760 3125 + 380 5000 = Rs. 42,75,000 Hence, the profit percentage of Live-in-style in the year 2007 was 5.56%.

BREAK-EVEN POINT

When the total sales revenue earned from the selling of a certain number of units of a product equals the total costs incurred in producing/purchasing that many units (this includes fixed and variable costs), then there is neither a profit nor a loss (overall). In this situation, the entity involved in the transaction is said to have “broken even”, and the sales value in terms of the number of units sold is called the break-even point or break-even sales. In other words, the break-even point is the number of units of a product required to be sold in order to recover the costs.

Hence, if n is the number of units at which an entity breaks even (i.e. n is the break-even point), then,

(Selling Price of one unit) n = (Variable Cost of one unit) n + Total Fixed Costs

The difference between the unit selling price and the unit variable cost is called the Unit Contribution Margin. At the break-even point, the Fixed Costs will equal the total contribution (i.e. Unit Contribution Margin n).

Usually, the selling price is more than the variable cost associated with each unit. Hence, every unit sold beyond the break-even point contributes to the profit of the company. That is,

If the Total Units Sold is less than the break-even point, then a loss has been incurred. It is calculated by multiplying the Unit Contribution Margin with the difference between the Break even point and the Total Units Sold.

Example 11:

KK, an aspiring entrepreneur wanted to set up a pen drive manufacturing unit. Since technology was changing very fast, he wanted to carefully gauge the demand and the likely profits before investing. Market survey indicated that he would be able to sell 1 lakh units before customers shifted to different gadgets. KK realized that he had to incur two kinds of costs – fixed costs (the costs which do not change, irrespective of numbers of units of pen drives produced) and variable costs (= variable cost per unit multiplied by number of units). KK expected fixed cost to be Rs. 40 lakhs and variable cost to be Rs. 100 per unit. He expected each pen drive to be sold at Rs. 200. What would be the break-even point (defined as no profit, no loss situation) for KK’s factory, in term of sales? [XAT 2009] (1) Rs. 80 lakhs(2) Rs. 100 lakhs (3) Rs. 120 lakhs(4) Rs. 140 lakhs (5) Cannot be found with the given data Solution: Let n be the break-even point, in terms of sales in number. Hence, Hence, the break-even point in Rupees = 0.4 lakhs 200 = Rs. 80 lakhs Hence, option 1.

CALCULATING PERCENTAGE PROFIT BY EQUATING THE AMOUNT OF MONEY SPENT AND EARNED

Apart from using the formula mentioned in the beginning of the lesson, there is also another way to calculate percentage profit. We do this by equating the money spent and the money received in a transaction. Once they’re equated, we can effectively say that the profit made can be represented by the goods that are still remaining. This is because, in monetary terms, the person has now gotten back the money he had spent on purchasing the goods. So, any amount he receives selling the remaining goods is profit for him.

Thus, when the amount of money spent and earned are equated, then:

Let’s consider an example. Let the CP of one unit of a product be Rs. 100 and the SP of one unit of the same product be Rs. 120. Also, let the number of units of goods sold be 60.

Then, the amount spent will be 60 100 = Rs. 6000

We equate this to the amount received and try to find the number of left-over goods. Hence, if the amount received was Rs. 6000, then the number of goods sold should have been 6000/120 = 50 units. Hence 10 units of the product are still remaining. These 10 units when multiplied by the SP of one unit will give the total profit (i.e. 10 120 = Rs. 1200).

Also, using the above formula:

Example 12:

If a book store bought 20 books, and it recovered its investment when it sold 15 of these books; then what will be the store’s percentage profit? Solution: Alternatively, Let the Cost Price of each book be Rs. x, and the Selling Price of each book be Rs. y. Since the sale of 15 books was enough to cover the cost price of 20 books, 20x = 15y

FALSE WEIGHTS

If an item is claimed to be sold at cost price, using false weights, then the overall percentage profit is given by:

Explanation:

Let’s consider a situation where a shopkeeper uses a weight of w’ units instead of a weight of w units. Let the price of the goods being sold be p per unit and let’s assume that they are being sold at the cost price.

So if he sells n units of goods, the price he charges for them is np. In reality, the goods are worth np w’/w.

For example, if a vendor uses a weight of 750 grams instead of a 1 kilogram weight, and then sells his goods at cost price, then his profit percentage is given by:

Example 13:

A corrupt rice vendor uses a weight of 1200 grams instead of 1 kilogram while buying his provisions, and uses a weight of 750 grams instead of a 1 kilogram weight while selling them. He then professes to sell the provisions at their cost price to his unsuspecting customers. What is his profit percentage? Solution: The fastest way to solve this type of question is to use the concept of equating the amount spent and received. Consider that the Cost Price (for the vendor) of 1 kg of rice is Rs. 1,000. While buying, instead of buying 1 kg for Rs. 1,000, using his fake weights he is able to buy 1200 grams. While selling, instead of selling 1 kg for Rs. 1,000, he sells only 750 grams. Hence, he gets back his initial investment of Rs. 1,000 by selling 750 grams of rice. Hence, the remaining 1200 – 750 = 450 grams add to his profit. Using the equation for profit percentage, we get,

MARKED PRICE AND DISCOUNT

The difference between the Selling Price of a good and its Cost Price is known as markup. Manufacturers add a markup to the Cost Price of an item in order to make profits. The price that is printed on an article or written on the label attached to it is the sum of the Cost Price and the markup, and is called the Marked Price (MP) or List Price of the item.

A shopkeeper or retailer buys goods from the manufacturer or wholesaler. The retailer then marks up the cost price (the increased price forming his selling price) so as to have a good profit. Note that the term “markup” is used to denote the amount of increase in the Cost Price, while “Marked Price” is used to denote the increased price as it appears on the product.

i.e. Cost Price + Markup = Marked Price

Markup can either be expressed as an amount (as shown above) or as a percentage of the Cost Price. So, another expression for Marked Price would be,

Generally, MP = SP. However, sometimes, in order to increase sales or to sell-off the old stock, retailers reduce the marked price of the article by a certain amount called Discount. In this case, the Selling Price will be the reduced price (i.e. price after deducting the discount).

i.e. Selling Price = Marked Price Discount

Similar to markup, discount can also be represented both as an amount (shown above) and as a percentage. So, another expression could be,

MORE FORMULAE:

Example 14:

A pair of jeans was initially marked at such a price that it would have earned the shopkeeper a profit of 25% on the Cost Price. Later, a discount of 10% was offered on the jeans and it was then sold for a net profit of Rs. 100. What was the Cost price for the pair of jeans? Solution: Since the Marked Price of the jeans would have earned a profit of 25% on the Cost Price, hence, Marked Price, MP = Cost Price + 25% of Cost Price = 1.25 CP Later, a discount of 10% was offered. Hence, Selling Price, SP = Marked Price – 10% Marked Price = 0.9 MP Now, Profit = SP – CP = 100 0.9MP – CP = 100 0.9(1.25 CP) – CP = 100 CP = 100/0.125 = 800 Hence, the Cost Price of the pair of jeans was Rs. 800.

Example 15:

An unscrupulous store owner adds a markup of 40% to the Cost Price and then offers a discount of 10% to please his customers. Apart from this, he also uses false weights to reduce the quantity sold by 20%. What is his profit percentage? Solution: Marked Price, MP = CP + 40% CP = 1.4 CP Selling Price, SP = MP – 10% MP = 0.9 MP SP = 0.9 1.4 CP = 1.26 CP Let the CP of 1 kg be Rs. 1000. Then, SP = Rs. 1260. However, by using false weights, he sells only 20% of 1 kg = 800 grams. Hence, if the SP of 800 grams is Rs. 1260; then the SP of 1000 grams will be: Hence, Profit (per kg) = SP – CP = 1575 – 1000 = Rs. 575

Example 16:

Bloomsberry Publishing printed 20,000 copies of the book “Tales of Beetle and his Beard”. They donated the proceeds from the sale of 10% of these books to charity. One-third of the remaining books were given a rebate of 30% and 2% of the proceeds from the sale of each of these books were given to charity. Half of the remaining books were given a 20% rebate, but none of proceeds were donated to charity. The remaining books were sold at the marked price of Rs. 500 per book (the markup being approximately 66.67%). Find the percentage profit for Bloomsberry Publishing for the particular book. Solution: Now, Marked Price = Rs. 500 and markup = 66.67%. Since, Hence, the cost of all 20,000 books will be 20,000 300 = Rs. 60,00,000 Step 1: Bloomsberry Publishing does not make any revenue from the 2000 books given to charity. Now, 18000 books remain. Step 2: One-third of these 18000, i.e. 6000 books, were sold with a rebate of 30% on the MP, i.e. at 0.7 500 = Rs. 350 each. Also, 2% of the price of each book was donated to charity. So, the company gets 0.98 350 = Rs. 343 for each book. So, the sales revenue collected for these books = 6000 343 Step 3: Half of the remaining 12000 books, i.e. 6000 books, were given a rebate of 20%. Hence, SP = 0.8 500 = Rs. 400 Hence, the sales revenue collected for these books = 6000 400 Step 4: The remaining 6000 books were sold at the marked price of Rs. 500. Hence, the sales revenue collected for these books = 6000 500 Thus, Total Sales Revenue for Bloomsberry Publishing = 6000 (343 + 400 + 500) = 6000 (1243) Profits = 6000 (1243) 6000000 = 6000000 (1.243) 6000,000 = 6000000 (0.243) Hence, Bloomsberry Publishing had a profit percentage of 24.3% for the book “Tales of Beetle and his Beard”.

BUY X AND GET Y FREE

If articles worth Rs. x are bought and articles worth Rs. y are obtained free along with the Rs. x articles, then the discount is equal to Rs. y and discount percentage is given by

Explanation:

Here, the retailer sells products that are usually sold for Rs. (x + y) for Rs. x. Hence, the Marked Price can be considered as Rs. (x + y) and the Selling Price as Rs. x.

Discount = Marked Price – Selling Price

= (x + y) – x = Rs. y

Example 17:

On a particular day in ‘Daily Bazaar’, there was a huge rush. They had an offer wherein for every purchase of Rs. 5,000, customers got things worth Rs. 2,000 absolutely free. What was the discount percentage that they were offering? Solution:

SUCCESSIVE DISCOUNTS

When a discount of a% is followed by another discount of b%, then the total discount is given by

In general, if there are successive discounts of p%, q% and r% in 3 stages, then:

Example 18:

A retail store offered a discount of 15% on every item purchased. Later, they announced an additional discount of 20% on every item purchased. Find the total discount percentage availed by the customers. Solution: The first discount offered (a) = 15% The second discount offered (b) = 20% Using the formula for successive discounts, Discount Percentage = 15 + 20 (15 20)/100 = 32%

Example 19:

A shopkeeper allows a discount of 30% to his customers and still gains 15%. Find the marked price of an article which cost him Rs. 6,200. Solution: CP = Rs. 6,200 SP = 1.15 6200 = 7130 Since the shopkeeper gives 30% discount on marked price, hence MP 0.7 = SP MP = 7130/0.7 = 10185.7 Hence, the Marked Price Rs. 10,186

Example 20:

In a shop, the marked price of an article is worked out in such a way that it generates a profit of 25%. What should be the discount percent allowed on the marked price such that the profit made on the sale of an article is 20%? Solution: Let the CP be Rs. 100. MP = 125 and SP = 120 Now, Discount = 125 120 = 5 Discount Percentage = 5 100/125 = 4%