Overview: Profit ,Loss & Discount | CSAT Preparation - UPSC PDF Download

Importance in CSAT

This chapter plays a vital role in Basic Numeracy for CSAT exams. From the analysis of Previous Years’ Papers, it is clear that not only direct questions are asked from this chapter, but its basics are necessary in Data Interpretation too.

In the year 2023 and 2022, 1 question and in the years 2020-2018, 1-2 questions were asked from this chapter.

Introduction

• The profit and Loss formula is 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 these prices, we can calculate the profit gained or the loss incurred for a particular product.

Profit, Loss, and Discount

The amount paid for a product or commodity to purchase it is called a cost price. Also, denoted as CP.
This cost price is further classified into two different categories:

• Fixed Cost: The fixed cost is constant, it doesn’t vary under any circumstances
• Variable Cost: It could vary depending as per the number of units

2. Selling price (SP)

• The amount for which the product is sold is called Selling Price. It is usually denoted as SP. Also, sometimes called a sale price.

3. Profit and Loss

Profit: When an article is sold for more than the cost of the article, then we say there is a Profit or Gain.

• Profit or Gain = S.P - C.P 

Loss: When an article is sold for less than the cost of the article,  then we say there is a Loss.

• Loss = C.P - S.P

The formula for the profit and loss percentage is:

• Profit percentage = (Profit / Cost Price) x 100
• Loss percentage = (Loss / Cost price) x 100

4. Marked Price Formula (MP)

This is basically labeled by shopkeepers to offer a discount to the customers in such a way that,

• Discount = Marked Price – Selling Price
• Discount Percentage = (Discount/Marked price) x 100

Profit and Loss Common Formulas

You have learned until now how to calculate profit as well as loss and also the percentage of them. Here is a list of the most commonly used formulas for all related questions: 

Example: By selling a watch for ₹495, a shopkeeper incurs a loss of 10%. Then, find the cost price of the watch for the shopkeeper.

(a) Rs 545
(b) Rs 550
(c) Rs 555
(d) Rs 565
Solution : (b)
Explanation:

Elementary Formuale on Profit and Loss

By selling an article for ₹a, some loss is incurred, and by selling it for ₹b, there is a gain equal to the loss amount, then the cost price is

Example: By selling an article for ₹50, some loss is incurred, and by selling it for ₹100, then there is a gain equal to the loss amount. Find the cost price of the article.

(a) 100
(b) 50
(c) 75
(d) None of the above

Ans : (c)
Explanation:

Here, a= 50 and b=100

FORMULA 2

If the selling price of a articles is equal to the cost price of b articles, then gain % or loss % is given by

 (if b>a, it is gain and a>b, it is loss)

Example: The SP of 24 articles is equal to the CP of 20 articles. Then, find the loss percentage.

(a) 16%
(b)
(c)
(d) None of these

Ans: (d)

Explanation:

Here, b= 24 and a = 20

FORMULA 3

If the cost price of a article is b and selling price of b article is a, then net gain % or loss % is

(if a>b, it is gain, if b>a, it is loss)

Example:  The CP of 10 articles is ₹5 and selling price of 5 articles is ₹10. Then, find the net gain percentage.

(a) 300%
(b) 200%
(c) 1000%
(d) None of these

Ans: (a)
Explanation:  

Gain percentage=

Here , a= 10 and b = 5

 FORMULA 4

When an article is sold it gains a%, if it is sold for ₹X more, then gain is b%. The cost price of an article is given by

Example: When a bag is sold, it gains 6%. If it is sold for ₹10 more, then the gain is 8%. Then, find the cost price of the bag.

(a) ₹200
(b) ₹300
(c) ₹400
(d) ₹500

Ans : (d)
Explanation:
Cost Price of Bag =

Here, X= 10, a= 6 and b= 8

FORMULA 5

When an article is sold it loses a% and if it is sold for ₹X more, then gain is b%. The cost price of an article is given by

 

Example: When a set of bottles is sold at a loss of 3% and if it is sold for ₹120 more, then gain is 5%. Then, find the cost price of the set of bottles.

(a) ₹500
(b) ₹1500
(c) ₹1000
(d) ₹750

Ans: (b)

Explanation: 

Here, X= 120, a=3 and b=5

Profit and Loss Tricks

Now let us look at some additional tricks or formulas that can be used to solve maths problems based on gain and loss: 

• SP = {(100 + P%)/100} x CP
• SP = {(100 – L%)/100} x CP
• CP = {100/(100 + P%)} x SP
• CP = {100/(100 – L%)} x SP
• SP = MP - Discount
• For false weight, profit percentage will be P% = (True weight – false weight/ false weight) x 100.
• When there are two successful profits say m% and n%, then the net percentage profit equals to (m+n+mn)/100
• When the profit is m% and loss is n%, then the net % profit or loss will be: (m-n-mn)/100
• If a product is sold at m% profit and then again sold at n% profit then the actual cost price of the product will be: CP = [100 x 100 x P/(100+m)(100+n)].
  In case of loss, CP = [100 x 100 x P/(100-m)(100-n)]
• If P% and L% are equal then, P = L and %loss = P2/100

Discount

The reduction allowed on the marked price of an article is called a discount. Discount is always calculated on the marked price.

Selling price = Marked price - Discount

Discount = Marked price - Selling price

If the discount allowed is r%, then

Marked Price

The price on the label is called the marked price or the list price. For example, if a pen is purchased for ₹10 and then the selling price is marked as ₹15, then ₹15 is the marked price or list price.

Successive Discount

Suppose the marked price of an article is P, and a discount of r₁% is given on it. Then, on the reduced price, a further discount of r₂% is given. It is said that the successive discounts of r₁% and r₂% are given on the article.

Therefore, the selling price of the article after r₂% and r₁% discount is:

Example: A retail store offered a discount of 25% on every article purchased. Later they announced an additional discount of 35% on every article purchased. Find the total discount percentage availed by the customers.

(a) 40%
(b) 60%
(c) 51.25%
(d) 55%

Ans: (c)
Explanation: 

1st discount offered, r₁₌ 25%

1st discount offered, r₂₌ 35%

 Use formula for successive discount= 

Using False Weight in Place of Actual Weight

By using a false weight of x grams for a kilogram and selling it at cost price, the shopkeeper’s gain percentage can be calculated as follows:

Example: A dishonest dealer professes to sell his goods at cost price but he uses a weight of 930g for a kg weight. Then, find his gain per cent.

Solution: We know that, the gain percent when false weight isused 

Some Solved Problems

To understand the concept let's first open a shop say a local retail stationery shop.

Example 1: Now suppose a student shows up and wants to buy 5 gel pens worth ₹ 5 each. The cost price of one pen is ₹ 4. So, what is the profit earned by the shopkeeper in net terms and its profit percentage?

This ₹ 5 is the selling price, the price at which the commodity is sold to its buyer. And ₹ 4 is the price at which the shopkeeper has bought these pens from his supplier and this price is known as Cost Price.
Thus, this difference between the price at which the shopkeeper buys his pen and at which it is sold is known as profit/ loss earned. Now if Selling Price > Cost Price then he will earn profit and if Selling Price < Cost Price, then he will earn loss.
Therefore, Profit = Selling Price – Cost Price = ₹ 5 – ₹4 = ₹1 per pen
Now,
i.e.

And consequently, Loss = Cost Price – Selling Price
i.e.

Example 2: Suppose another customer comes to the shop and bought 2 registers worth ₹ 50 each and a pencil box from him. And this time the shopkeeper has earned 40% profit on the registers. He earned a profit of ₹10 on pencil box and the profit% on pencil box is 20%. Then what is the cost price and profit on register and selling price and cost price of pencil box?

In this case, we are given S.P. = ₹ 50 for each register and profit% =  40%. Let C.P. be x.
⇒ Profit % = {(50 – x)/x} * 100
⇒ 40 = {(50 – x)/x} * 100
⇒ 4x = 500 – 10x
⇒ 14x = 500
⇒ x = 35.71
⇒ Profit = 50 – 35.71 = 14.29

What if the profit of 40% is on selling price instead?
⇒ Profit = profit% * S.P = 0.4* 50 = ₹20
C.P. = S.P. – Profit = ₹50 – ₹20 = ₹30

Let's now move on to calculate S.P. and C.P. of pencil box. Let C.P. of pencil box be y
∵ Profit % = (profit/ Cost price) *100
⇒ 20 = (10/y) *100
⇒ 2y = 100
⇒ y = ₹ 50
Therefore, S.P. = ₹50 + ₹10 = ₹60

Example 3: Since not many customers showed up on the first day of the shop. Therefore, to popularize the shop the shopkeeper puts up discount of 20% on all the products. The first customer shows up and bought a packet of pencils and 3 erasers and still making up the profit of 30% on both the items. Then what is the actual cost price of both the items when the pencil is marked as ₹30 and the eraser ₹ 5 each?

What does the underlined marked mean? 

Here marked means Marked Price is the price that is offered to customer before discount basically, discount is just difference between marked price and Selling price i.e. Discount = M.P. – S.P.

In this case, Discount = 0.2 * Price of packet of pencil = 0.2 *30 = ₹ 6
Therefore, S.P = 30 – 6 = ₹ 24
∵ Profit is 30% on C.P., assume C.P. be x
⇒ 0.3 = (24 – x)/x
⇒ 0.3x = 24 – x
⇒ 1.3x = 24
⇒ x = 18.46
Similarly,
Discount = 0.2 * Price of 3 erasers = 0.2* 3*5 = ₹ 3
Hence, S.P. = (3*5) - 3 = ₹ 12
∵ Profit is 30% on C.P., assume C.P. be y
⇒ 0.3 = (12 – y)/y
⇒ 1.3y = 12
⇒ y (C.P. of 3 eraser) = 9.23
The above method of solving questions was a direct straightforward way of finding the solution. There's also an alternative way of doing so. Let's now learn that alternative method but this time instead of earning profit the seller was earning the loss.

Example 4: Consider this situation, that the stationery sold a parker pen at a loss of 20% for ₹ 100 and a pack of colored sketch pens at the loss of 15% on S.P.  What are the cost price and selling price of both the articles?

First, let's find out the C.P. and S.P. parker pen,
∵ Loss is 20% on C.P.
Let C.P. be x and S.P. be 100
⇒ S.P. = 0.8 of C.P.
⇒ 100 = 0.8x
⇒ 100/0.8 = x
⇒ 125 = x
For sketch pens,
∵ Loss is 15% on S.P.
Let S.P. be y and C.P. be 100
⇒ Loss = 0.15y
⇒ C.P. = 1.15 y
⇒ 100 = 1.15y
⇒ 100/1.15 = y
⇒ y = ₹ 87

Example 5: A watch dealer incurs an expense of Rs. 150 for producing every watch. He also incurs an additional expenditure of Rs. 30,000, which is independent of the number of watches produced. If he is able to sell a watch during the season, he sells it for Rs. 250. If he fails to do so, he has to sell each watch for Rs. 100. If he is able to sell only 1,200 out of 1,500 watches he has made in the season, then he has made a profit of:
(a) ₹ 90000
(b) ₹ 75000
(c) ₹ 45000
(d) ₹ 60000

This question is not basic and direct as the problems given above. This one came in CAT 2016 paper and you should expect to get this level of questions in the exam and not the ones explained above earlier. Now let's see how to solve this problem

Here, first find out cost he has incurred to produce the watches.
∵ He made 1500 watches costing ₹ 150 each and an additional ₹ 30000 expense on them.
⇒ Total Cost = (1500*150) + 30000 = ₹ 255000
∵ He's able to sell 1200 watches in the season = ₹250 each
So, the revenue earned by him during the season = ₹250 * 1200 = ₹ 300000
Also, the leftover 300 pieces of clocks would have been sold by the watchmaker in off-season = 100 each.
Revenue earned through these 300 watches = 300*100 = ₹ 30000
Total Revenue = ₹ 300000 + ₹30000 = ₹330000
Profit = Revenue - Cost = 330000 - 255000 = ₹ 75000

Example 6: Instead of a meter scale, a cloth merchant uses a 120cm scale while buying, but uses an 80cm scale while selling the same cloth. If he offers a discount of 20% on cash payment, what is his overall profit percentage?

This question above is a special one with the faulty dealer. Here, the dealer is earning profit by using a false scale.
To solve this problem, first assume that the price of the cloth is ₹ 1/cm
∵ He's using a 120 cm scale.
⇒ C.P. = (100/120) * ₹1 = ₹ 0.8333/cm
∵ This merchant again uses faulty scale to sell the cloth to his customers. He uses a scale that measures 80cm as 100cm i.e. he sells 80cm for ₹100
Now he also gives a discount of 20% on the cloth.
⇒ His mark up price is ₹100/80cm
∵ S.P. = M.P. - M.P. * Discount% = M.P. (1 - Discount%) = (100/80)*(80/100) = ₹1/cm
Therefore, his profit % = (1 – 0.8333)/0.8333 *100 = 20%