Episode 8 — Aptitude and Reasoning / 8.2 — Profit and Loss

8.2.a Concepts and Formulas -- Profit and Loss

1. Fundamental Definitions

1.1 Cost Price (CP)

The Cost Price is the price at which an article is purchased or manufactured. It may include additional expenses like transportation, taxes, or repairs. When overhead expenses are included, the total is called the effective cost price or net cost price.

Effective CP = Purchase Price + Overhead Expenses

Example:

A shopkeeper buys a TV for Rs. 15,000 and pays Rs. 500 for transportation.

Effective CP = 15,000 + 500 = Rs. 15,500

1.2 Selling Price (SP)

The Selling Price is the price at which an article is sold to the buyer.

If SP > CP  -->  Profit (Gain)
If SP < CP  -->  Loss
If SP = CP  -->  No Profit, No Loss

1.3 Marked Price (MP) / List Price

The Marked Price is the price displayed on the product label or tag. The seller may offer a discount on the marked price. The actual transaction happens at the selling price.

Relationship:  CP  --(Markup)-->  MP  --(Discount)-->  SP

2. Profit and Loss -- Core Formulas

2.1 Profit (Gain)

Profit = SP - CP                    (when SP > CP)

Profit % = (Profit / CP) x 100
         = ((SP - CP) / CP) x 100

Example:

An article is bought for Rs. 400 and sold for Rs. 480.

Profit    = 480 - 400 = Rs. 80
Profit %  = (80 / 400) x 100 = 20%

2.2 Loss

Loss = CP - SP                      (when CP > SP)

Loss % = (Loss / CP) x 100
       = ((CP - SP) / CP) x 100

Example:

An article is bought for Rs. 500 and sold for Rs. 450.

Loss    = 500 - 450 = Rs. 50
Loss %  = (50 / 500) x 100 = 10%

2.3 Finding SP When Profit% or Loss% is Given

When there is a Profit:
    SP = CP x (100 + Profit%) / 100

When there is a Loss:
    SP = CP x (100 - Loss%) / 100

Example 1 (Profit):

CP = Rs. 600, Profit% = 25%

SP = 600 x (100 + 25) / 100
   = 600 x 125 / 100
   = Rs. 750

Example 2 (Loss):

CP = Rs. 800, Loss% = 15%

SP = 800 x (100 - 15) / 100
   = 800 x 85 / 100
   = Rs. 680

2.4 Finding CP When SP and Profit% or Loss% are Given

When there is a Profit:
    CP = SP x 100 / (100 + Profit%)

When there is a Loss:
    CP = SP x 100 / (100 - Loss%)

Example 1 (Profit):

SP = Rs. 1,050, Profit% = 5%

CP = 1050 x 100 / (100 + 5)
   = 1050 x 100 / 105
   = Rs. 1,000

Example 2 (Loss):

SP = Rs. 720, Loss% = 10%

CP = 720 x 100 / (100 - 10)
   = 720 x 100 / 90
   = Rs. 800

3. Marked Price, Markup, and Discount

3.1 Markup

The markup is the amount added to the cost price to arrive at the marked price.

Markup = MP - CP

Markup % = (Markup / CP) x 100
         = ((MP - CP) / CP) x 100

MP = CP x (100 + Markup%) / 100

Example:

CP = Rs. 500. The shopkeeper marks it 40% above CP.

MP = 500 x (100 + 40) / 100
   = 500 x 140 / 100
   = Rs. 700

3.2 Discount

The discount is the reduction offered on the marked price.

Discount = MP - SP

Discount % = (Discount / MP) x 100
           = ((MP - SP) / MP) x 100

SP = MP x (100 - Discount%) / 100

Example:

MP = Rs. 700, Discount = 10%

SP = 700 x (100 - 10) / 100
   = 700 x 90 / 100
   = Rs. 630

3.3 The Complete Chain: CP --> MP --> SP

This is the most important relationship to master. A shopkeeper:

  1. Buys at CP
  2. Marks up by some percentage to get MP
  3. Offers discount on MP to get SP
  4. The difference between SP and CP determines Profit or Loss
CP  --[+Markup%]--> MP  --[-Discount%]--> SP  --[compare with CP]--> Profit or Loss

Full Worked Example:

A shopkeeper buys an article for Rs. 500.
He marks it up by 60% and then offers a 20% discount.
Find his profit percentage.

Step 1: Find MP
    MP = 500 x (100 + 60) / 100 = 500 x 160 / 100 = Rs. 800

Step 2: Find SP
    SP = 800 x (100 - 20) / 100 = 800 x 80 / 100 = Rs. 640

Step 3: Find Profit
    Profit = SP - CP = 640 - 500 = Rs. 140

Step 4: Find Profit %
    Profit % = (140 / 500) x 100 = 28%

Alternatively, using the combined formula:
    Net effect = Markup% - Discount% - (Markup% x Discount%) / 100
               = 60 - 20 - (60 x 20) / 100
               = 60 - 20 - 12
               = 28% profit

4. Successive Discounts

When two or more discounts are applied one after another (not added together), they are called successive discounts.

4.1 Two Successive Discounts

If discounts are d1% and d2%, applied successively:

SP = MP x (100 - d1) / 100 x (100 - d2) / 100

Equivalent Single Discount % = d1 + d2 - (d1 x d2) / 100

Example:

MP = Rs. 1,000. Successive discounts of 20% and 10%.

Method 1: Step-by-step
    After 1st discount: 1000 x 80/100 = Rs. 800
    After 2nd discount: 800 x 90/100  = Rs. 720

Method 2: Equivalent single discount
    Equivalent discount = 20 + 10 - (20 x 10)/100
                        = 30 - 2 = 28%
    SP = 1000 x (100 - 28)/100 = Rs. 720

Note: Two successive discounts of 20% and 10% are NOT equal to a
      single discount of 30%. The equivalent discount is 28%.

4.2 Three Successive Discounts

For three successive discounts d1%, d2%, d3%:

SP = MP x (100 - d1)/100 x (100 - d2)/100 x (100 - d3)/100

To find equivalent single discount:
    Step 1: Combine d1 and d2 --> d_combined = d1 + d2 - (d1 x d2)/100
    Step 2: Combine d_combined and d3 similarly

Example:

MP = Rs. 2,000. Successive discounts of 10%, 20%, and 10%.

After 1st discount: 2000 x 90/100  = Rs. 1,800
After 2nd discount: 1800 x 80/100  = Rs. 1,440
After 3rd discount: 1440 x 90/100  = Rs. 1,296

Equivalent single discount = (2000 - 1296)/2000 x 100 = 35.2%

5. Profit/Loss on the Same Article Sold Twice

5.1 Selling at Profit and Loss -- Finding True CP

If an article is sold at X% profit and Y% loss, and the difference in selling prices is given:

Let CP = C

SP1 = C x (100 + X) / 100       (profit case)
SP2 = C x (100 - Y) / 100       (loss case)

Difference = SP1 - SP2

C = Difference x 100 / (X + Y)

Example:

If an article is sold at 20% profit, the SP is Rs. 160 more than when
it is sold at 20% loss. Find the CP.

SP1 - SP2 = CP x (120/100) - CP x (80/100)
160       = CP x (120 - 80)/100
160       = CP x 40/100
CP        = 160 x 100/40 = Rs. 400

6. Two Articles Sold at Same SP -- One at Profit, One at Loss

This is a classic exam problem.

6.1 When Profit% = Loss% = x%

If two articles are sold at the same SP, one at x% profit and
the other at x% loss:

There is ALWAYS a LOSS in this case.

Loss % = x^2 / 100   (i.e., x squared divided by 100)

Example:

Two articles are each sold for Rs. 600.
One is sold at 20% profit, the other at 20% loss.
Find the overall profit or loss %.

Loss % = (20)^2 / 100 = 400/100 = 4% loss

Verification:
    Article 1: SP = 600, Profit = 20%
        CP1 = 600 x 100/120 = Rs. 500

    Article 2: SP = 600, Loss = 20%
        CP2 = 600 x 100/80  = Rs. 750

    Total CP = 500 + 750 = Rs. 1,250
    Total SP = 600 + 600 = Rs. 1,200
    Loss     = 1250 - 1200 = Rs. 50
    Loss %   = (50/1250) x 100 = 4%

7. Dishonest Dealer Problems

A dishonest dealer cheats by using faulty weights (giving less than the stated quantity) or by other deceptive practices.

7.1 Using False Weights

Profit % = ((True Weight - False Weight) / False Weight) x 100

Equivalently:
Profit % = (Error / (True Weight - Error)) x 100

Where Error = True Weight - False Weight

Example:

A shopkeeper claims to sell at cost price but uses a weight of 900g
instead of 1 kg (1000g).

Error = 1000 - 900 = 100g

Profit % = (100 / 900) x 100 = 11.11%

Explanation: He charges for 1000g but gives only 900g.
His cost is for 900g, but he receives payment for 1000g.

7.2 Dishonest Dealer Who Also Sells Above/Below CP

When a dealer uses false weights AND sells at x% profit (or loss):

Effective Profit % = ((True Weight / False Weight) x (100 + x)/100 - 1) x 100

For selling at cost price, x = 0.
For selling at a loss of x%, replace +x with -x.

Example:

A dealer claims to sell at 10% loss but uses a weight of 800g
instead of 1000g. Find actual profit/loss %.

Effective multiplier = (1000/800) x (100 - 10)/100
                     = 1.25 x 0.9
                     = 1.125

Effective Profit % = (1.125 - 1) x 100 = 12.5% profit

Despite claiming a loss, the dealer actually makes a 12.5% profit
due to the false weights.

8. Buy X Get Y Free

When a seller offers "Buy X, Get Y Free":

The customer pays for X items but receives (X + Y) items.

Effective discount on each item = (Y / (X + Y)) x 100

For the seller:
    CP is for (X + Y) items
    Revenue is for X items only
    Loss % per item = (Y / (X + Y)) x 100   (if sold at listed CP)

Example:

A shop offers "Buy 3 Get 1 Free." What is the effective discount?

X = 3, Y = 1

Effective Discount = (1 / (3 + 1)) x 100 = (1/4) x 100 = 25%

The customer effectively gets a 25% discount.

Extended Example:

A shopkeeper marks his goods 40% above CP and offers "Buy 4 Get 1 Free."
Find his profit or loss %.

Step 1: Let CP of each item = Rs. 100
    MP of each item = Rs. 140

Step 2: Customer buys 4, gets 1 free = 5 items
    Total CP to shopkeeper = 5 x 100 = Rs. 500
    Revenue received       = 4 x 140 = Rs. 560

Step 3: Profit = 560 - 500 = Rs. 60
    Profit % = (60/500) x 100 = 12%

9. Profit/Loss in Transactions Involving Multiple Goods

9.1 Buying and Selling at Different Rates

If a person buys 'a' items for Rs. P and sells 'b' items for Rs. Q:

    CP per item = P / a
    SP per item = Q / b

    Profit/Loss % = ((SP per item - CP per item) / CP per item) x 100

Example:

A man buys 12 oranges for Rs. 80 and sells 10 oranges for Rs. 80.

CP per orange = 80/12 = Rs. 6.67
SP per orange = 80/10 = Rs. 8.00

Profit per orange = 8.00 - 6.67 = Rs. 1.33
Profit %          = (1.33 / 6.67) x 100 = 20%

Shortcut: Profit % = ((12 - 10) / 10) x 100 = 20%
(When amounts are same, Profit% = ((buy qty - sell qty) / sell qty) x 100)

9.2 General Shortcut for Same Amount Transactions

If Rs. X buys 'a' items and Rs. X sells 'b' items:

    If a > b: Profit % = ((a - b) / b) x 100
    If a < b: Loss %   = ((b - a) / b) x 100

Mnemonic: "Buy more, sell less = Profit"
          "Buy less, sell more = Loss"

10. Partnership and Profit Sharing

When two or more people invest in a business, profit is shared in the ratio of their capital x time investments.

If A invests Rs. C1 for T1 months and B invests Rs. C2 for T2 months:

    A's share : B's share = C1 x T1 : C2 x T2

Example:

A invests Rs. 40,000 for 12 months.
B invests Rs. 60,000 for 8 months.
Total profit = Rs. 50,000. Find each person's share.

A's investment = 40,000 x 12 = 4,80,000
B's investment = 60,000 x 8  = 4,80,000

Ratio = 4,80,000 : 4,80,000 = 1 : 1

A's share = 50,000 x (1/2) = Rs. 25,000
B's share = 50,000 x (1/2) = Rs. 25,000

11. Break-Even Point

The break-even point is when total revenue equals total cost (no profit, no loss).

Break-Even Quantity = Fixed Costs / (SP per unit - Variable Cost per unit)

Example:

Fixed cost = Rs. 10,000
Variable cost per unit = Rs. 50
SP per unit = Rs. 75

Break-Even Quantity = 10,000 / (75 - 50) = 10,000 / 25 = 400 units

The seller must sell 400 units to cover all costs.

12. Summary of Key Relationships

                     +Markup%                  -Discount%
             CP  ──────────────>  MP  ──────────────>  SP
             ^                                          |
             |                                          |
             +──────────── Profit or Loss ──────────────+
                    (always measured against CP)


Key Rules:
1. Profit/Loss % --> always on CP
2. Discount %    --> always on MP
3. Markup %      --> always on CP
4. SP > CP       --> Profit
5. SP < CP       --> Loss
6. SP = CP       --> No Profit No Loss