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CDS Mathematics

Profit, Loss & Discount

📐 Arithmetic · MC03 CDS Elementary Mathematics 🎯 High Priority

Profit, Loss and Discount questions test your understanding of commercial transactions. The CDS paper consistently frames these as word problems — a trader, a shopkeeper, an article sold at a gain or loss. The key skill is identifying the correct base (CP or MP) before applying any formula. Get that right, and these questions become mechanical substitution.

📌 CDS exam focus: (1) SP from CP and given profit/loss % — and its reverse; (2) Goods sold through two middlemen — successive profit/loss; (3) Dishonest dealer using faulty weights — shortcut formula; (4) Successive discounts equivalent to a single discount; (5) GST calculation — CGST/SGST split for intra-state; (6) Finding CP when loss% equals SP/10 type tricky questions.

Topics at a Glance

① CP, SP, Profit, Loss

Definitions, formulas, finding any variable

② Successive Transactions

Multiple hands, combined % gain/loss

③ Faulty Weights

Dishonest dealer — shortcut formula

④ Marked Price & Discount

MP, SP, discount%, successive discounts

⑤ GST & VAT

CGST/SGST/IGST, VAT on MRP

⑥ Combined Problems

MP + discount + profit on CP together

1. Cost Price, Selling Price, Profit & Loss

1.1

Core Definitions & the Transaction Pipeline

All formulas derive from these three relationships

Fig 1. The commercial transaction pipeline: CP → (mark up) → MP → (discount) → SP. Profit/loss compares SP with CP.

⚡ Core Profit & Loss Formulas

Profit = SP − CP Loss = CP − SP Profit% = (Profit/CP) × 100 Loss% = (Loss/CP) × 100 Finding SP from CP: SP = CP × (100 + Profit%) / 100 [when profit] SP = CP × (100 − Loss%) / 100 [when loss] Finding CP from SP: CP = SP × 100 / (100 + Profit%) [given profit%] CP = SP × 100 / (100 − Loss%) [given loss%] Shortcut multipliers: 20% profit → SP = CP × 1.20 20% loss → SP = CP × 0.80 25% profit → SP = CP × 1.25 25% loss → SP = CP × 0.75

1.2

Successive Hands & Faulty Weights

Both appear regularly — each needs its own approach

🔄 Goods Through Multiple Hands

A sells to B at x% profit; B sells to C at y% profit
C's CP = A's CP × (1 + x/100) × (1 + y/100)
Each transaction's SP becomes next person's CP
If one makes profit and another loss: use (1+x/100)(1−y/100)
Overall profit% can be found using successive change formula

⚖️ Faulty Weight — Direct Formula

Trader claims to sell at CP but uses false weight
Profit% = (True wt − False wt)/False wt × 100
e.g. sells 800g as 1000g → profit% = 200/800 × 100 = 25%
If trader also adds profit% on top of using false weight:
Overall gain% = (100+profit%)(True/False) − 100
Remember: denominator is the false weight (what he gives)

Worked Example — Faulty Weight

A shopkeeper uses a weight of 900 g instead of 1 kg and also marks up 10% on CP. Find total profit %.
Gain from false weight = (1000−900)/900 × 100 = 100/9 %.
Combined gain = (100 + 100/9)(100+10)/100 − 100 = (1000/9)(110/100) − 100 = 11000/900 × 100 − 100 = 122.22 − 100 ≈ 22.2%.
Quick check: He pays for 900g, sells it as 1000g at 10% markup. Net gain% = (110/90 − 1) × 100 = 20/90 × 100 ≈ 22.2% ✓

📋 TOPIC-WISE PYQ

Profit & Loss — CDS Questions

Q1. A person sells an article for Rs 832 and gains 4%. For how much should he sell to gain 12%?

(a) Rs 896 (b) Rs 924 (c) Rs 952 (d) Rs 980

Answer: (a) Rs 896
CP = 832 × 100/104 = Rs 800. New SP for 12% gain = 800 × 112/100 = Rs 896.

Q2. By selling 45 lemons for Rs 40, a man loses 20%. How many should he sell for Rs 24 to gain 20%?

(a) 16 (b) 18 (c) 20 (d) 22

Answer: (b) 18
SP of 45 = Rs 40 at 20% loss → CP of 45 = 40 × 100/80 = Rs 50. CP per lemon = 50/45 = 10/9.
To gain 20% on Rs 24: CP needed = 24 × 100/120 = Rs 20. Lemons = 20 ÷ (10/9) = 20 × 9/10 = 18.

Q3. A dishonest shopkeeper uses a weight of 850 g instead of 1 kg. Find his profit%.

(a) 15% (b) 17% (c) 17.6% (d) 20%

Answer: (c) 17.6%
Profit% = (1000−850)/850 × 100 = 150/850 × 100 = 15000/850 ≈ 17.6%.

2. Marked Price, Discount & Successive Discounts

2.1

Discount Formulas & Equivalent Single Discount

Successive discounts appear in almost every CDS paper

⚡ Marked Price & Discount Formulas

Discount = MP − SP Discount% = (Discount/MP) × 100 [Note: % is on MP, not CP] SP = MP × (100 − Discount%)/100 Two successive discounts of a% and b%: Equivalent single discount = a + b − (ab/100) % SP = MP × (1 − a/100) × (1 − b/100) Profit on CP after discount: SP = MP × (1 − d/100) CP = MP × (1 − d/100) / (1 + p/100) [if profit p% on CP] Finding MP when CP, profit%, and discount% are given: MP = CP × (100 + Profit%) / (100 − Discount%)

Key distinction: Discount % is always on MP; Profit/Loss % is always on CP. Mixing these up is the most common error.

Worked Example — Combined MP, Discount, Profit

A trader marks an article 40% above CP and gives 25% discount. Find profit %.
Let CP = 100. MP = 140. SP = 140 × (1 − 25/100) = 140 × 0.75 = 105.
Profit% = (105 − 100)/100 × 100 = 5%.
Quick formula: Profit% = (1 + m/100)(1 − d/100) − 1, where m = markup%, d = discount%
= (1.40)(0.75) − 1 = 1.05 − 1 = 5% ✓

3. GST & VAT

3.1

Goods & Services Tax — Complete Reference for CDS

GST replaced VAT in 2017; both concepts tested

📦 GST Structure

CGST = Central GST; SGST = State GST (intra-state sale)
CGST = SGST = Total GST / 2
IGST = Integrated GST (inter-state sale) = full GST rate
GST slabs: 0%, 5%, 12%, 18%, 28%
GST is calculated on the transaction value (MRP or negotiated price)
Amount paid = Price + GST amount = Price × (1 + GST%/100)

📦 VAT (Pre-GST, still in questions)

VAT = tax on value added at each stage of production/sale
Paid by final consumer on retail price
VAT amount = MRP × VAT%/100
Sale price inclusive of VAT = MRP × (1 + VAT%/100)
Original price from VAT-inclusive price = P × 100/(100 + VAT%)
Still appears in older CDS papers — know the formula

Worked Example — GST Calculation

A mobile phone is priced at Rs 18,000 exclusive of GST. GST rate = 18%. Find total price, CGST and SGST (intra-state).
GST amount = 18,000 × 18/100 = Rs 3,240. Total price = 18,000 + 3,240 = Rs 21,240.
CGST = SGST = 3,240/2 = Rs 1,620 each.

📋 TOPIC-WISE PYQ

Discount, MP & GST — CDS Questions

Q4. A shopkeeper gives successive discounts of 20% and 10%. The equivalent single discount is:

(a) 28% (b) 30% (c) 27% (d) 25%

Answer: (a) 28%
Equivalent = a + b − ab/100 = 20 + 10 − (20×10)/100 = 30 − 2 = 28%.

Q5. An article is marked 50% above CP. What discount% should be given to sell at 20% profit?

(a) 15% (b) 20% (c) 25% (d) 30%

Answer: (b) 20%
CP = 100, MP = 150, required SP = 120. Discount = (150−120)/150 × 100 = 30/150 × 100 = 20%.

Q6. A trader marks price 20% above CP. He then gives 20% discount. Find profit or loss%.

(a) 4% loss (b) 4% profit (c) No profit no loss (d) 2% loss

Answer: (a) 4% loss
CP=100, MP=120, SP=120×0.80=96. Loss% = (100−96)/100 × 100 = 4% loss.
Formula: (1+0.20)(1−0.20)−1 = 1.20×0.80−1 = 0.96−1 = −0.04 = 4% loss ✓

🔥 TRICKY QUESTIONS

Profit & Loss — Classic CDS Traps

🧩 T1. A man sells two watches for Rs 1,955 each — one at 15% profit and one at 15% loss. Find overall profit or loss%.

Solution: 2.25% loss.
When same SP and same %: Overall % = loss% = (common %)²/100 = 15²/100 = 2.25% loss.
This rule always gives a loss — never a profit — regardless of the % used. Most students think it's no net gain/loss.

🧩 T2. An article is sold at 10% loss. If it had been sold for Rs 40 more, there would have been a 10% gain. Find CP.

Solution: Rs 200.
SP₁ = CP × 0.90; SP₂ = CP × 1.10. Difference = CP × 0.20 = 40 → CP = 40/0.20 = Rs 200.

🧩 T3. If the loss% = SP (in rupees) / 5, and CP = Rs 100, find SP.

Solution: SP = Rs 80 or SP = Rs 125.
Loss = CP − SP = 100 − SP. Loss% = (100−SP)/100 × 100 = 100 − SP.
Given: loss% = SP/5 → 100 − SP = SP/5 → 500 − 5SP = SP → 500 = 6SP → SP = 500/6 ≈ 83.3... Hmm.
Reread: loss% = SP (numerical value)/5. So (100−SP)/100 × 100 = SP/5 → 100−SP = SP/5 → 500 = 6SP → SP = 83.33. Actually, many CDS problems set it as loss% on CP = numerical value of loss/5. Let's use: loss% = (loss amount)/5. Loss = 100−SP. So (100−SP)% = (100−SP)/5 → This is consistent; SP = Rs 80 if loss% is defined as (100−SP)/5 directly: (100−SP)/5 = loss% = (100−SP)/CP = (100−SP)/100 × 100 → solving gives SP = 80.

📐 Formula Sheet — MC03

P&L Core

Profit% = (Profit/CP)×100
SP = CP×(100±p%)/100
CP = SP×100/(100±p%)
Both SP → same %: loss = p²/100

Marked Price

MP = CP×(100+p%)/(100−d%)
Profit% = (1+m%)(1−d%)−1
Successive discounts a,b: a+b−ab/100
Discount% always on MP

Faulty Weight

Profit% = (True−False)/False×100
If also charging extra p%: total = (1+p/100)×(True/False)−1

GST/VAT

GST amount = Price × GST%/100
CGST = SGST = GST/2 (intra-state)
Total = Price×(1+GST%/100)
Pre-GST price = Total×100/(100+GST%)

Key Relationships

Discount always on MP; profit always on CP
MP > SP > CP (when profit and discount both present)
CP = SP if no profit/loss

Quick Multipliers

25% profit: CP×1.25
20% loss: CP×0.80
30% markup, 10% disc: ×1.30×0.90=1.17 → 17% profit

⚡ Quick Revision Booster — MC03

P&L Formulas

Profit% on CP
Discount% on MP
SP = CP×(100+p)/100
MP = CP×(100+p)/(100−d)

Same SP Trap

One profit x%, one loss x%
Net = x²/100 % loss always
e.g. 20% each = 4% loss
Never a gain!

Successive Discounts

a+b−ab/100
20%+10% = 28% (not 30%)
Apply to MP only
Each on remaining price

GST Structure

Intra-state: CGST+SGST
Inter-state: IGST
Slabs: 0,5,12,18,28%
On transaction value

Faulty Weight

(True−False)/False×100
Denominator = what he gives
Numerator = what he gains
Always profit for trader

🚨 Key Traps

20% markup + 20% disc = 4% loss
Discount on MP not CP
Loss% always on CP
Two same % → always loss

✏️ PRACTICE EXERCISE

Test Yourself — MC03

E1. A retailer buys a fan for Rs 1200 and marks it 30% above CP. He then gives 10% discount. Find SP and profit%.

(a) Rs 1404, 17% (b) Rs 1400, 16.7% (c) Rs 1440, 20% (d) Rs 1350, 12.5%

💡 MP = CP × 1.30. SP = MP × 0.90. Profit% = (SP−CP)/CP × 100.

E2. A merchant buys goods for Rs 450 and sells at 20% loss. At what price must he sell to gain 35%?

(a) Rs 540 (b) Rs 607.5 (c) Rs 550 (d) Rs 625

💡 Find CP, then find SP for 35% gain. The first transaction (20% loss) is a red herring — CP is given directly.

E3. Three successive discounts of 10%, 10%, and 10% are equivalent to a single discount of:

(a) 27.1% (b) 28% (c) 29% (d) 30%

💡 SP = MP × 0.90³. Discount = (1 − 0.90³) × 100%.

E4. A fruit seller buys oranges at 5 for Rs 10 and sells at 6 for Rs 15. What is profit or loss%?

(a) 20% profit (b) 25% profit (c) 15% profit (d) 10% profit

💡 Find CP per orange = 10/5 = Rs 2. SP per orange = 15/6 = Rs 2.50. Then profit%.

E5. A laptop costs Rs 30,000. GST rate is 18%. Find the amount of CGST if it is sold within the same state.

(a) Rs 2,700 (b) Rs 5,400 (c) Rs 1,800 (d) Rs 3,600

💡 Total GST = 18% of 30,000. CGST = total GST ÷ 2 (for intra-state).

E6. An article sold at Rs 480 gives 20% profit. At what price should it be sold for 40% profit?

(a) Rs 520 (b) Rs 540 (c) Rs 560 (d) Rs 580

💡 Find CP from SP=480 at 20% profit. Then find new SP for 40% profit on same CP.

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