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In this section we will study more on Profit and Loss topic.  Let’s begin with the basics and then we will try to understand each question type with example.

Whenever we purchase or sold any object, either profit or loss happens. This topic helps us to solve such problems.

Let’s quickly understand the terms used in this topic.

1.            Cost Price (CP): The cost at which the object is purchased.

2.            Selling Price (SP): The Cost at which the object is sold.

3.            Profit or Gain: If the selling price is more than cost price then the seller earns a profit.

4.            Loss: If the selling price is less than cost price then the seller suffers a loss.

Basic Formulae:

The formulae that make the very basic of Profit and loss are:

1.            Profit or gain = SP – CP

2.            Loss = CP – SP

3.            Gain % = (Gain/CP) × 100

4.            Loss % = (Loss/CP) × 100

Let’s solve some examples from RRB NTPC Quant questions to see how to apply these formulae.

E.g. 1: If Reena sold 12 mobile phones for Rs. 188,160 which costs Rs. 14,056 per phone, what was the total profit and profit % made by her?

Here, SP = Rs. 188,160

CP = 14,056 × 12 = Rs. 168,672

We know that,

Profit = SP – CP

⇒ Profit = 188,160 – 168,672

= Rs. 19,488

∴ Profit % = (Profit/CP) × 100

= (19,488/168,672) × 100

= 11.55% (Ans).

E.g. 2: Q purchased a table for Rs. 13,500 and sold it for Rs. 11,000. What is his loss percentage?

Here, CP = Rs. 13,500 and SP = Rs. 11,000

Loss = CP – SP

= 13,500 – 11,000

= Rs. 2,500

Loss % = (Loss/CP) × 100

= (2500/13,500) × 100

= 18.59% (Ans).

Note: Profit and loss are always calculated on CP.

Formulae derived from Percentage Change concept:

If there is a percentage change in a quantity, then:

Final quantity = Initial quantity × (100 ± Percentage increase/decrease)/100

We know the Initial amount is the Cost price on which the profit % or loss % is applied to get the selling price. Therefore,

SP = CP × (100 ± Profit %/Loss %)/100

OR

SP= CP × MF (Multiplying Factor)

We have discussed Multiplying Factor under percentage topic

Let’s solve an example

E.g.: If a person sold an article for Rs. 1,300 and made a profit of 30%, find the cost price of the article.

Here, SP = Rs. 1300

Profit % = 30%

By formula:

SP = CP × (100 + Profit %)/100

⇒ 1300 = CP × (100 + 30)/100

⇒ 1300 = CP × 130/100

⇒ 1300 × 100/130 = CP

⇒ CP = 10 × 100 = Rs. 1000 (Ans).

By MF:

For 30% increase, MF = (100 + 30)/100 = 13/10

SP= CP × MF

⇒ 1300 = CP × 13/10

⇒ 1300 × 10/13 = CP

⇒ CP = 100 × 10 = Rs. 1000 (Ans).

If C.P. of m articles = S.P. Of n articles, then- Profit or Loss % = {(m – n)/n} × 100%.

Let’s understand this with an example from RRB NTPC Quant questions.

E.g.: If the cost price of 5 cars is equal to selling price of 4 cars. Find the percentage of profit or loss?

Let CP of 1 car be 1 rupee.

⇒ CP of 5 cars = 5 rupees = SP of 4 cars

⇒ SP of 1 car = Rs. 5/4 = 1.25

Now we have CP = 1 and SP = Rs. 1.25

⇒ Profit = Re 0.25

⇒ Profit % = (0.25/ 1) × 100 = 25%

By formula:

Profit Percentage = {(m – n)/n} × 100%

= {(5 – 4)/4} × 100%

= 25% (Ans).

If ‘a’ part of an article is sold at x % profit/loss, ‘b’ part at y % profit/loss and ‘c’ part at z % profit/loss of Rs. R, then-   C.P. of entire article = (R × 100)/(ax + by +cz).

Let’s solve an example from the NTPC Railway Exam previous year questions.

E.g.: ‘A’ bought 100 eggs in Rs. 600. He sold 25 eggs at 10% profit, another 25 eggs at 25% loss and the left 50 eggs at 20% profit. Find the overall profit/ loss % made by A.

CP of one egg = 600/100 = Rs. 6

Profit1 = 1/10 of (25 × 6) = 1/10 of 150 = Rs. 15

Loss = ¼ of (25 × 6) = Rs. 37.5

Profit2 = 1/5 of (50 × 6) = Rs. 60

Overall Profit= Profit1 – Loss + Profit2

= 15 – 37.5 + 60

= 75 – 37.5

= Rs. 37.5

Overall Profit % = (37.5/600) × 100

= 37.5/6 % = 6.25% (Ans).

By formula,

CP = (R × 100)/(ax + by +cz)            [Here, a= 25/100 = ¼. Similarly, b= ¼ and c= ½]

⇒ 600 = (R × 100)/(10/4 – 25/4 + 20/2)

⇒ 600 × (10 – 25 + 40)/4 = R × 100

⇒ (600 × 25)/400 = R

⇒ R = 25 × 3/2

⇒ R = 75/2

⇒ R = Rs. 37.5

Hence,

Overall Profit % = (37.5/ 600) × 100 %

= 6.25% (Ans).

Miscellaneous

If we have an SP after x% profit/loss, then the SP for y% loss/profit is-

SP2 = (SP1/MF1) × MF2

Let’s see some examples from RRB NTPC to understand this type in a better way.

E.g. 1: A man sells a table for Rs. 4200 at 25% loss. At what price must he sell to get a profit of 25%?

MF for 25% loss, MF1 = ¾

MF for 25% profit, MF2 = 5/4

∵ SP1 = CP × MF1

⇒ 4200 = CP × 3/4

⇒ CP = 4200 × 4/3

⇒ CP = 1400 × 4

⇒ CP = Rs. 5600

Now,

∴ Required selling price, SP2 = CP × MF2

= 5600 × 5/4

= 1400 × 5

= Rs. 7000 (Ans).

By formula,

Required selling price, SP2 = (SP1/ MF1) × MF2

= (4200/¾) × 5/4            [Reciprocal Rule of fractions]

= (4200 × 4 × 5)/(3 × 4)

= 1400 × 5

= Rs. 7000 (Ans).

E.g. 2: Mr. Ashwani sold a bus for Rs. 510 with a loss of 15%. At what price should the bus be sold to get a profit of 15 %?

MF for 15% loss, MF1 = 17/20

MF for 15% profit, MF2 = 23/20

By formula,

Required selling price, SP2 = (SP1/ MF1) × MF2

= (510/17/20) × 23/20

= (510 × 20 × 23)/(17 × 20)

= 510 × 23/17

= 30 × 23

= Rs. 690 (Ans).

Problems on dozen:

In problems in which we are given the CP or SP in dozen we will try to get the CP or SP for one article/dozen.

E.g. 1: If the price of 2.5 dozen apples is Rs. 300 and the selling price of each apple is Rs. 15. Find the profit%?

Step 1: Find out the CP for 1 article.

CP of 2.5 dozen = Rs. 300

CP of 1 dozen = Rs. 300/2.5

CP of 1 apple = 300/(2.5 × 12)

= 300/30

= Rs. 10

Step 2: Apply the relevant formula

CP of 1 apple = Rs. 10

SP of 1 apple = Rs. 15

Profit% = {(SP – CP)/CP} × 100

= {(15 – 10)/10} × 100

= (5/10) × 100

= 50% (Ans).

E.g. 2: A man buys 144 oranges for Rs. 360 and sells them at a gain of 10%. At what rate per dozen does he sell them?

In this case we need to find the SP of 1 dozen.

CP of 12 dozen oranges = Rs. 360      (Given)

Multiplying factor for 10% gain = 11/10

SP = CP × MF

= 360 × 11/10

= 36 × 11 = Rs. 396

No. of oranges = 144

⇒ No. of dozen of oranges = 144/12 = 12

∴ SP of 1 dozen oranges = Total SP/No. of Dozen

= 396/12

= Rs. 33 (Ans).