3  Profit, Loss and Discount

3.1 Introduction

This chapter develops a comprehensive toolkit for profit–loss–discount (PLD) questions.
We will cover:
- Core definitions and percentage measures
- Quick ratio forms and multiplier models
- Markup and discount combinations
- Successive changes and equivalent discount
- Break-even and target profit
- Mixtures of items, free/defective articles
- Taxes, overheads, depreciation
- Solved examples, traps, and practice sets

3.2 Core Definitions

  • Cost Price (CP): Cost of acquiring/producing the article.
  • Selling Price (SP): Price at which it is sold.
  • Marked Price (MP): Printed/list price before discount.
  • Discount (D): Reduction on MP.
  • Profit (Gain): \(SP-CP\) (if positive).
  • Loss: \(CP-SP\) (if positive).

3.2.1 Percentage Measures

  • Profit% (on CP):
    \[ \frac{SP-CP}{CP}\times100 \]

  • Loss% (on CP):
    \[ \frac{CP-SP}{CP}\times100 \]

  • Discount% (on MP):
    \[ \frac{MP-SP}{MP}\times100 \]

Always check the base. Exam traps often hinge on CP vs MP.

3.3 Quick Ratio and Multiplier Forms

  • Profit situation: \(SP=CP(1+p/100)\)
  • Loss situation: \(SP=CP(1-\ell/100)\)
  • Discount situation: \(SP=MP(1-d/100)\)

Reverse:
- \(CP=\tfrac{SP}{1+p/100}\) or \(\tfrac{SP}{1-\ell/100}\)
- \(MP=\tfrac{SP}{1-d/100}\)

Thumb rule: Convert every sentence to a multiplier on the correct base.

3.4 Markup and Discount Together

If markup = \(m\%\) on CP and discount = \(d\%\) on MP:
\[ SP=CP(1+m/100)(1-d/100) \]

Effective profit%:
\[ \big[(1+m/100)(1-d/100)-1\big]\times100 \]

Example: Markup 25%, discount 20%.
SP = \(CP\times1.25\times0.8=CP\). → No profit, no loss.

3.5 Successive Percentage Changes

If two changes \(x\%\) and \(y\%\):
\[ \text{Net \%} = x+y+\frac{xy}{100} \]

Example: Two successive discounts of 20% and 15%.
Net discount = \(20+15-3=32\%\).

3.6 Equivalent Discount

  • Two discounts \(a\%\) and \(b\%\):
    \[ d_{eq}=a+b-\frac{ab}{100} \]
  • Three discounts \(a\%,b\%,c\%\):
    \[ d_{eq}=1-(1-a/100)(1-b/100)(1-c/100) \]

3.7 Linking Markup, Discount, Profit

If markup = \(m\%\) and discount = \(d\%\):
\[ (1+m/100)(1-d/100)=1+p/100 \]
Solve for required profit \(p\) or markup \(m\).

3.8 Break-even and Margin

  • Break-even: SP=CP.
  • Profit margin on SP:
    \[ \frac{SP-CP}{SP}\times100 \]
  • To achieve target profit \(p\%\), set \(SP=CP(1+p/100)\).

3.9 Mixtures and Weighted CP

For items with costs \(c_1,c_2,\dots,c_n\):

Overall profit%:
\[ \frac{S-\sum c_i}{\sum c_i}\times100 \]

If sold at same SP per item: use average CP.

3.10 Free and Defective Articles

  • Buy \(x\), get \(y\) free ⇒ effective discount:
    \[ \frac{y}{x+y}\times100 \]

  • If \(k\%\) defective: profit% falls since fewer items are saleable.

Example: 100 pens at cost 10 each. 10% defective. Desired profit=20%.
CP=1000. Revenue target=1200. Good pens=90. SP per pen=1200/90=13.33.

3.11 Taxes, Overheads, Depreciation

  • With GST \(t\%\): Billed price = \(SP(1+t/100)\).
  • Overheads: add before computing profit.
  • Depreciation: Value after \(n\) years at \(d\%\):
    \[ V=V_0(1-d/100)^n \]

3.12 Solved Examples

  1. Article sold at 690 after 23% discount. MP?
    \(690/0.77=896.10\).

  2. Markup needed for 25% profit after 20% discount?
    \((1+m)(0.8)=1.25\) → \(m=56.25\%\).

  3. Successive discounts 30% and 10% on 500.
    SP=500×0.7×0.9=315. Net discount=37%.

  4. Buy 4 get 1 free. Effective discount=20%.

  5. 200 bulbs @30 each, 10% defective. Profit 25% overall.
    CP=6000. Revenue=7500. Good bulbs=180. SP=7500/180=41.67.

  6. Item sold at 10% loss. If SP increased by 63, profit=5%. Find CP.
    Let CP=x. 0.9x+63=1.05x → x=420.

  7. Profit is 25% of SP. Find profit% on CP.
    SP=100, profit=25, CP=75. Profit%=25/75×100=33.33%.

  8. Article listed at 2000, discount 10%, GST 18%.
    SP=1800. Bill=1800×1.18=2124.

3.13 Common Traps

  • Wrong base (CP vs SP vs MP).
  • Adding discounts linearly.
  • Freebies change quantity base, not tag price.
  • Margin vs profit confusion.
  • Always check with CP=100 for quick sanity.

3.14 Practice Set – Level 1

  1. CP=640, profit=12.5%. Find SP.
  2. SP=750, loss=20%. Find CP.
  3. MP=1200, discount 25%. Find SP and discount.
  4. Successive discounts 10% and 15% on 2000. Find SP and net discount.
  5. Markup 40%, discount 30%. Find effective profit%.

3.15 Practice Set – Level 2

  1. To earn 30% profit after 20% discount, find markup%.
  2. Trader sells at 15% profit but gives 1 in 10 free. Effective profit%?
  3. Tea mix: 2 kg at 200, 3 kg at 300. Mixture sold at 280/kg. Profit%?
  4. Machine depreciates 10% annually. Value after 3 yrs is 14580. Find initial.
  5. Goods marked 60% above CP. To earn 20% profit, what discount?

3.16 Practice Set – Level 3

  1. Two successive equal discounts \(a\%\) result in 51% of MP. Find \(a\).
  2. Dealer sells two items at same SP, one 25% gain, other 20% loss. Net result?
  3. Dishonest dealer: 900 g weight, 10% discount, CP=100/kg, MP=150. Effective profit%?
  4. Watch sold at 10% loss. If bought 10% less and sold 27 more, profit=20%. Find CP.
  5. Article with MP=P. Discounts 10%, 20%, 25% lead to SP=972. Find P.

3.17 Answer Key (Outline)

Level 1: 1) 720; 2) 937.5; 3) SP=900, D=300; 4) SP=1530, net=23.5%; 5) -2%.
Level 2: 6) 62.5%; 7) ≈6.11%; 8) 4%; 9) 20000; 10) 25%.
Level 3: 11) 30%; 12) 2% loss; 13) 23.33% profit; 14) 750; 15) 1800.

3.18 Summary

  • Translate words → multipliers.
  • Successive % changes: use formula or chained multipliers.
  • Free/defective items: adjust quantity base.
  • Differentiate margin on SP vs profit on CP.
  • Most PLD problems reduce to 2–3 clean multiplier steps.