Profit, Loss & Discount

Profit and loss is a percentage application that appears every year in CDS. Keep the cost price (CP) as the base for profit/loss, and the marked price (MP) as the base for discount.

Core formulae

• Profit = SP - CP; Loss = CP - SP.
• Profit% = (Profit/CP) x 100; Loss% = (Loss/CP) x 100 (always on CP).
• SP = CP x (1 + Profit%/100); SP = CP x (1 - Loss%/100).
• Discount = MP - SP; Discount% = (Discount/MP) x 100 (always on MP).

Useful results

• Successive discounts of a% and b% = a + b - (ab/100) percent.
• If an article is sold at two prices giving the same numeric profit% and loss%, there is a net LOSS of (common%)^2/100.
• Dishonest dealer using a false weight: gain% = (error / (true value - error)) x 100.

Exam Tricks & Tips

• 🎯 Profit/Loss percentages are ALWAYS on CP; discount is ALWAYS on MP — fix the base first.
• 🎯 Two successive discounts a% and b% = a + b - ab/100 (note the minus, unlike successive rises).
• 🎯 Equal profit% and loss% on the same item = a net loss of (x^2/100) percent, never break-even.
• 🎯 For a false-weight dealer, use gain% = error/(true - error) x 100.
• 🎯 When SP is fixed, a higher CP means less profit — set up SP = CP(1 +/- p/100) and solve for the unknown.
• ❌ Common mistake: calculating profit% on the selling price instead of the cost price.

Expected exam pattern

CP/SP/profit%, marked-price-and-discount, and successive-discount questions; 6-10 marks, all percentage-based.

Quick recap

Base profit/loss on CP and discount on MP, use SP = CP(1 +/- p/100), apply a + b - ab/100 for successive discounts, and remember the equal-%/net-loss and false-weight results.

Cover the answer, recall, then check. 11 cards on profit, loss and discount.

Q1. On which value is profit% calculated?
A1. Always on the cost price (CP).

Q2. On which value is discount% calculated?
A2. Always on the marked price (MP).

Q3. SP in terms of CP and profit%?
A3. SP = CP x (1 + Profit%/100).

Q4. Two successive discounts a% and b% equal what single discount?
A4. a + b - (ab/100) percent.

Q5. CP = 400, SP = 500. Profit%?
A5. (100/400) x 100 = 25%.

Q6. An item sold at equal profit% and loss% (say 10% each) gives what net result?
A6. A net loss of (10^2)/100 = 1%.

Q7. Formula for a dishonest dealer's gain using a false weight?
A7. gain% = error / (true value - error) x 100.

Q8. MP = 800, discount 25%. SP?
A8. 800 x 0.75 = 600.

Q9. If SP = 540 at a 10% loss, find CP.
A9. CP = 540 / 0.90 = 600.

Q10. Successive discounts of 20% and 10% equal?
A10. 20 + 10 - (200/100) = 28%.

Q11. Profit = SP - CP; what is loss?
A11. Loss = CP - SP.

Profit, Loss and Discount is one of the most concrete and frequently tested CDS arithmetic topics. Every question is built from a few relationships between cost price, selling price, marked price, and percentages.

Core idea / what this tests

This tests calculation of profit or loss (always as a percentage of cost price), and the interplay of marked price, discount, and selling price. Master the base of each percentage and these questions become mechanical.

Deep explanation

Beginner — the core formulas

Let CP = cost price, SP = selling price.

• Profit = SP − CP (when SP > CP); Loss = CP − SP (when CP > SP).
• Profit% = (Profit ÷ CP) × 100; Loss% = (Loss ÷ CP) × 100. The base is always CP.
• SP = CP × (1 + Profit%/100) for profit, or × (1 − Loss%/100) for loss.

Intermediate — marked price and discount

• Marked Price (MP) is the listed price before discount; Discount is a percentage of MP, not of CP.
• SP = MP × (1 − Discount%/100). Shopkeepers mark up the CP, then give a discount on the MP — so the two percentages have different bases, a common trap.
• Discount% = (Discount ÷ MP) × 100.

Advanced — combined and special cases

• Successive discounts of a% then b% combine like successive percentage changes: net discount = a + b − ab/100.
• False weight: if a dishonest trader uses a weight of w grams instead of 1000 g while selling at cost, Gain% = [(1000 − w)/w] × 100.
• Two articles sold at the same SP, one at +x% and one at −x%: there is always a net loss of (x/10)² % (x²/100 %). E.g., ±10% gives a 1% loss regardless of the prices.

Worked example

A shopkeeper marks an article 40% above cost and then gives a 25% discount. Find the profit percentage.

Step 1 — assume CP = 100 (convenient base). Markup 40% → MP = 100 × 1.40 = 140.
Step 2 — apply 25% discount on MP: SP = 140 × (1 − 0.25) = 140 × 0.75 = 105.
Step 3 — profit = SP − CP = 105 − 100 = 5, so Profit% = (5 ÷ 100) × 100 = 5%.
Answer: 5% profit. Note the markup was on CP (100) but the discount was on MP (140) — using the right base for each is the whole game.

CDS relevance

Profit-loss-discount questions are near-guaranteed in CDS and are fast once you fix the base of each percentage. The "assume CP = 100" technique turns most problems into simple multiplication, saving time under the exam clock.

Exam tricks & shortcuts

• Assume CP = 100 to make percentages concrete.
• Profit/Loss% base = CP; Discount% base = MP — keep them separate.
• Successive discounts: a + b − ab/100.
• Same SP with +x% and −x% ⇒ net loss = (x²/100)%, always.
• Markup then discount: SP = CP × (1 + markup) × (1 − discount).

Key formulas

• Profit = SP − CP; Loss = CP − SP; Profit% = (Profit/CP)×100, Loss% = (Loss/CP)×100.
• SP = CP(1 + profit%/100) = CP(1 − loss%/100).
• Marked price and discount: SP = MP(1 − d/100).
• Two articles sold at same SP, one at +x% one at −x%: net loss = x²/100 %.
• Dishonest dealer using false weight: gain% = (error/(true − error))×100.
• CP from SP: CP = SP × 100/(100 ± profit/loss%).

Usage: always compute profit/loss percentage on cost price, not selling price.

Worked Example

Problem: A shopkeeper marks an article 40% above its cost price and then allows a discount of 10% on the marked price. Find his profit percentage.

Solution:
Work with an assumed cost price of 100 for simplicity.

Step 1 — Marked price (MP) = 40% above cost:
MP = 100 + 40% of 100 = 140.

Step 2 — Selling price (SP) after a 10% discount on the marked price:
Discount = 10% of 140 = 14.
SP = 140 − 14 = 126.

Step 3 — Profit = SP − CP = 126 − 100 = 26.
Profit percentage = (Profit / CP) × 100 = (26 / 100) × 100 = 26%.

Answer: The profit percentage is 26%.

Solution note: Discount is always taken on the marked price, never on the cost price — a common mistake.