Skip to main content
Mathematical Skills 3% exam weight

Percentages and Profit-Loss

Part of the MAT study roadmap. Mathematical Skills topic mathem-004 of Mathematical Skills.

→ See in Roadmap Build 1-Month Plan

Percentages and Profit-Loss

🟢 Lite — Quick Review (1h–1d)

Rapid summary for last-minute revision before your exam.

Percentages and Profit-Loss — Quick Facts

Percentage Basics:

  • To convert fraction to percentage: multiply by 100
  • To convert percentage to fraction: divide by 100
  • $75% = \frac{75}{100} = \frac{3}{4}$

Common Fraction-Decimal-Percentage Equivalents (MUST memorise for MAT):

FractionDecimalPercentage
1/20.550%
1/30.333…33.33%
1/40.2525%
1/50.220%
1/60.1666…16.67%
1/70.142857…14.29%
1/80.12512.5%
1/90.111…11.11%
1/100.110%
1/110.0909…9.09%
1/120.0833…8.33%
1/150.0666…6.67%

Profit and Loss Fundamentals:

  • Cost Price (CP): The price at which an article is purchased
  • Selling Price (SP): The price at which an article is sold
  • Profit = SP − CP (when SP > CP)
  • Loss = CP − SP (when CP > SP)
  • Profit % = $\frac{\text{Profit}}{\text{CP}} \times 100$
  • Loss % = $\frac{\text{Loss}}{\text{CP}} \times 100$

⚡ MAT Exam Tip: Always identify whether the base for percentage calculation is Cost Price or Selling Price — this is the most common mistake in profit-loss questions.

Quick Formulas:

  • SP = CP × (1 + Profit%/100)
  • SP = CP × (1 − Loss%/100)
  • CP = SP / (1 + Profit%/100)
  • CP = SP / (1 − Loss%/100)

🟡 Standard — Regular Study (2d–2mo)

For students who want genuine understanding.

Percentages and Profit-Loss — Deep Dive

Percentage Change and Comparison

When a quantity changes from $A$ to $B$:

  • Increase = $\frac{B-A}{A} \times 100%$
  • Decrease = $\frac{A-B}{A} \times 100%$

Successive Percentage Changes: When a quantity changes by $x%$ then by $y%$, the final value = original × (1 + x/100) × (1 + y/100)

⚡ MAT Shortcut: For successive increases of $a%$, $b%$, $c%$, total = $(1 + a/100)(1 + b/100)(1 + c/100) - 1$, expressed as percentage.

Example: A number increases by 20%, then decreases by 10%. Final = $100 \times 1.20 \times 0.90 = 108$ Net increase = 8% (NOT 10%!)

Discount and Marked Price

  • Marked Price (MP): The price printed on the article (before discount)
  • Discount = MP − SP
  • Discount % = $\frac{\text{Discount}}{\text{MP}} \times 100$

Relationship: SP = MP × (1 − Discount%/100)

Important: Some sellers first give discount, then add GST/VAT on the discounted price — this is different from giving a single combined discount.

Break-Even and False Weights

False weight problems (classic MAT topic!):

  • If a trader uses a false weight of $x$ kg instead of 1 kg while selling, they gain: $\frac{(1-x)}{x} \times 100%$
  • If weight is $a$ kg when it should be $b$ kg: Gain % = $\frac{b-a}{a} \times 100$

Example: A shopkeeper uses 900g instead of 1kg. His gain = $\frac{1000-900}{900} \times 100 = \frac{100}{900} \times 100 = 11.11%$

⚡ MAT PYQ Pattern: False weight problems appear almost every year. Formula: gain = $\frac{\text{actual weight} - \text{fake weight}}{\text{fake weight}} \times 100$.

Population and Interest Problems

Population growth/decline:

  • After $n$ years at $r%$ per annum: $P_n = P_0 \times (1 \pm r/100)^n$

Simple Interest: $SI = \frac{P \times R \times T}{100}$ Compound Interest: $A = P \times (1 + R/100)^T$

Difference between CI and SI for 2 years = $P \times (R/100)^2$

Mixture and Alligation

Alligation method (MUST for MAT):

  • Two ingredients with costs $c_1$ and $c_2$ mixed to get mean cost $\bar{c}$
  • Ratio of quantities = $|c_2 - \bar{c}| : |\bar{c} - c_1|$

⚡ MAT Shortcut: Alligation = $\frac{\text{cheap quantity}}{\text{expensive quantity}} = \frac{d - m}{m - c}$ where $d$ = expensive, $c$ = cheap, $m$ = mean price.

🔴 Extended — Deep Study (3mo+)

Comprehensive coverage for students on a longer study timeline.

Percentages and Profit-Loss — Complete Theory

Equation-Based Percentage Problems

Many MAT questions require setting up equations:

Example: A shopkeeper sells an article at 20% profit. If he had bought it at 10% less and sold it at Rs 5 less, he would have made 25% profit. Find the cost price.

Let CP = $x$. Original SP = $1.2x$ New CP = $0.9x$ New SP = $1.2x - 5$

Profit = $\frac{1.2x - 5 - 0.9x}{0.9x} \times 100 = 25%$ $\frac{0.3x - 5}{0.9x} = 0.25$ $0.3x - 5 = 0.225x$ $0.075x = 5$ $x = \frac{5}{0.075} = 66.67$

Markup vs Margin

  • Markup = profit as % of CP
  • Margin = profit as % of SP

If markup = 20%, then margin = $\frac{20}{120} \times 100 = 16.67%$

This distinction is tested in MAT’s business mathematics section.

Compound vs Simple Interest Comparison

For 2 years: CI − SI = $P \times (R/100)^2$

For 3 years: CI − SI = $P \times \left(\frac{R}{100}\right)^2 \times \left(3 + \frac{R}{100}\right)$

⚡ MAT PYQ: If the difference between CI and SI on a sum at 10% per annum for 2 years is Rs 25, find the sum. Solution: $P \times (10/100)^2 = 25$ → $P \times 0.01 = 25$ → $P = 2500$

Depreciation

Value after $n$ years at $r%$ depreciation: $V_n = P \times (1 - r/100)^n$

Business Mathematics in MAT

MAT often tests:

  • Partnership profit sharing (based on capital and time)
  • Commission calculations
  • Tax-inclusive pricing
  • Successive discounts vs single equivalent discount

⚡ Single Equivalent Discount: Two discounts $d_1%$ and $d_2%$: Equivalent = $1 - (1-d_1/100)(1-d_2/100)$ For 20% and 10% off: $1 - 0.80 \times 0.90 = 1 - 0.72 = 28%$ (NOT 30%!)

Content adapted based on your selected roadmap duration. Switch tiers using the selector above.