In this blog we will continue with Profit and Loss topic. We will see Discount and Marked Price based concepts and problems and also some problems based on Faulty Weight. 

Discount and Marked Price are always heard of when we borrow or sell something. These words are a part of our daily life.

So, let’s begin with the quick definitions of Discount and Marked Price and basic formulae based on them. 

• Discount and Marked Price:

The amount of rebate given on the price (Marked Price) of an article is called discount.

Marked Price (MP), as its name suggests, is the price marked on the articles. It’s basically the initial intended selling price without any discount and is determined by the sellers.

The discount that we get from the side of the sellers is on the marked price only.

Let’s quickly see some basic formulae related to discount and marked price.

1. Discount = MP – SP
2. Discount% = {(MP – SP)/MP} × 100
3. SP = MP × (1 – Discount%/100) 

Let’s take an example from RRB NTPC Exam 2018 to see the application of these formulae.

• E.g.: An article was sold for Rs. 3,600 at a discount of 10%. Find the selling price if the discount was 15%.

By formula 3,

SP = MP × (1 – Discount%/100)

⇒ 3600 = MP × (1 – 10/100)

⇒ 3600 = MP × 90/100

⇒ 3600 = MP × 9/10

⇒ 3600 × 10/9 = MP

⇒ MP = 400 × 10

⇒ MP= Rs. 4000

Now when discount is 15%,

SP = MP × (1 – Discount%/100)

      = 4000 × (1 – 15/100)

      = 4000 × 85/100

      = 4000 × 17/20

      = 200 × 17

      = Rs. 3400 (Ans).

• Successive Discounts:

Single discount equivalent to two successive discounts a% and b% can be calculated by-

ab principle — {a + b – (ab/100)} % 

• E.g.1: Which single discount will be equal to two successive discounts of 12% and 5%?

Let the Marked Price be Rs. 100.

Then after 12% discount, SP = 88

Further 5% discount is again applied on SP.

∴ Required SP = 88 × 95/100 = 88 × 19/20 = 4.4 × 19 = 83.6.

Hence, equivalent discount = 100 – 83.6 = 16.4% (Ans).

By formula,

Equivalent discount = {a + b – (ab/100)} %

                            = {12 + 5 – (12 × 5/100)} %

                            = {12 + 5 – (60/100)} %

                            = (17- 0.6) %

                            = 16.4% (Ans). 

• E.g. 2: Ram Naresh got two successive discounts, a bag with a list price of Rs. 400 is available at Rs. 160. If the second discount is 20%, find the first discount.

Discount % = {(MP – SP)/MP} × 100 %

                    = {(400 – 160)/400} × 100 %

                    = (240/400) × 100 %

                    = 240/4 %

                    = 60%

By ab principle,

Equivalent Discount = a + b – ab/100

⇒ 60 = a + 20 – 20a/100

⇒ 60 – 20 = a – a/5

⇒ 40 = 4a/5

⇒ a = 40 × 5/4

⇒ a = 50

∴ First Discount = 50% (Ans). 

• If there is a discount of d% on marked price and there is a profit of p%, then the percentage by which marked price is above the cost price is–

{(d + p)/(100 – d)} × 100 %

We will solve an example to understand the type of questions asked on this formula.

• E.g.: After allowing a discount of 20% on Marked price Kishore makes a profit of 12%. What percentage is the Marked price above the Cost price?

Let the CP be Rs. 100

Then, SP = Rs. 112                       [12% profit on CP]

Let the marked price be Rs. x.

After getting a 20% discount on Marked price the selling price was 112.

For 20% discount, MF = 4/5

Therefore,

Marked Price × MF of Discount = Selling Price

⇒ x × 4/5 = 112

⇒ x = 112 × 5/4

⇒ x = 28 × 5

⇒ x = 140.

∴ MP = Rs. 140

Hence, Required percentage = (MP – CP)/ CP} × 100 %

                                              = (140 – 100)/ 100} × 100 %

                                              = 40% (Ans). 

By formula,

Required percentage = {(d + p)/(100 – d)} × 100 %

                                  = {(20 + 12)/(20 – 12)} × 100 %

                                  = (32/8) × 100 %

                                  = 40% (Ans). 

• If Marked price is a% more than the Cost price and a discount of d% is allowed, then–

Profit %/ Loss % = [{a × (100 – d)}/100] – d

We will solve some examples based on the above formula. 

• E.g.: Aparna changes the marked price of an item to 50% above its CP. What % of discount allowed in approximately to gain 10%?

Let the CP be Rs. 100.

Therefore, MP = Rs. 150

Gain = 10%

⇒SP = Rs. 110

Hence,

Discount % = {(MP – SP)/MP} × 100 %

                    = {(150 – 110)/150} × 100 %

                    = (40/150) × 100 %

                    = 40 × 2/3 %

                    = 80/3 %

                    = 26.66% or 27% approx (Ans).

By formula,

Putting all the known values in it

Profit % = [{a × (100 – d)}/100] – d

⇒ 10 = [{50 × (100 – d)}/100] – d

⇒ 10 = {(100 – d)}/2} – d

⇒ 10 = 50 – d/2 – d

⇒ d + d/2 = 50 – 10

⇒ 3d/2 = 40

⇒ d = 40 × 2/3 %

⇒ d = 80/3 %

⇒ d = 26.66% or 27% approx (Ans) 

False Weight Problems:

• If a dishonest shopkeeper sells his items at CP but he uses false weight, then-

Profit % = {(True weight – False weight)/False weight} × 100 

• E.g. 1: A dishonest shopkeeper sells his good at cost price but he uses weight of 900 g for every 1 kg weight. Find his gain percent.

Here, the profit that shopkeeper gets is 100 g on every 900 g he sells.

Therefore, Profit% = (100/900) × 100 = 1/9 × 100 = 11.11% (Ans).

By formula,

Profit% = {(True weight – False weight)/False weight} × 100

              = {(1000 – 900)/900} × 100

              = (100/900) × 100

              = 1/9 × 100

              = 11.11% (Ans). 

• E.g. 2: Deepak sells all his goods available in his shops at the cost price, but he cheats his customer and gives 20% less goods he should give. Find his % profit.

Suppose, true quantity of goods = 100 units

Then, the false quantity = 80 units

By formula,

Profit % = {(True weight – False weight)/False weight} × 100

               = {(100 – 80)/80} × 100

               = (20/80) × 100

               = ¼ × 100

               = 25% (Ans).