Percentages • Topic 1 of 3

Percentage Calculations and Profit & Loss

What is Percentage?

Percentage means "per hundred" (from Latin "per centum"). A percentage is a fraction with denominator 100. The symbol % represents "out of 100".

Key Conversions:

Convert FromToMethodExample
FractionPercentageMultiply by 100%\(\frac{3}{4} = 0.75 \times 100\% = 75\%\)
PercentageFractionDivide by 100%\(40\% = \frac{40}{100} = \frac{2}{5}\)
DecimalPercentageMultiply by 100%\(0.65 \times 100\% = 65\%\)
PercentageDecimalDivide by 100\(25\% = 25 \div 100 = 0.25\)

Percentage of a Quantity:

To find \(x\%\) of a quantity \(Q\), calculate: \(\frac{x}{100} \times Q\)

Example: 20% of 250 = \(\frac{20}{100} \times 250 = 0.2 \times 250 = 50\)

What is Profit and Loss?

In business, items are bought at Cost Price (CP) and sold at Selling Price (SP).

  • Profit occurs when SP > CP
  • Loss occurs when SP < CP

Important Formulas:

ConceptFormula
ProfitSP - CP
LossCP - SP
Profit Percentage\(\frac{\text{Profit}}{CP} \times 100\%\)
Loss Percentage\(\frac{\text{Loss}}{CP} \times 100\%\)
SP when Profit% given\(CP \times (1 + \frac{\text{Profit%}}{100})\)
SP when Loss% given\(CP \times (1 - \frac{\text{Loss%}}{100})\)
CP when Profit% given\(\frac{SP \times 100}{100 + \text{Profit%}}\)
CP when Loss% given\(\frac{SP \times 100}{100 - \text{Loss%}}\)

Real-life Example: If you buy a pen for ₹10 (CP) and sell it for ₹15 (SP), your profit is ₹5, and profit percentage = \(\frac{5}{10} \times 100\% = 50\%\).

Percentage, Profit and Loss100%75%75 out of 100Profit & Loss FormulasProfit = SP − CP (if SP > CP)Loss = CP − SP (if CP > SP)Profit % = (Profit/CP) × 100Loss % = (Loss/CP) × 100SP = CP × (1 + P/100)SP = CP × (1 − L/100)Example: CP = ₹800, SP = ₹920 → Profit = ₹120 → Profit% = (120/800)×100 = 15%Conversion: x% = x/100, e.g. 35% = 35/100 = 0.35 = 7/20
Solved Examples
1
Worked Example

Convert \(\frac{3}{8}\) to a percentage. Also, find 35% of 240.

Solution
  • Step 1: Fraction to percentage: Multiply by 100%
  • Step 2: \(\frac{3}{8} \times 100\% = \frac{300}{8}\% = 37.5\%\)
  • Step 3: 35% of 240 = \(\frac{35}{100} \times 240\)
  • Step 4: \(= 0.35 \times 240 = 84\)

Answer: $\frac{3}{8} = 37.5\%$; 35% of 240 = 84

2
Worked Example

A shopkeeper buys a watch for ₹800 and sells it for ₹960. Find the profit percentage.

Solution
  • Step 1: CP = ₹800, SP = ₹960
  • Step 2: Profit = SP - CP = 960 - 800 = ₹160
  • Step 3: Profit% = \(\frac{\text{Profit}}{CP} \times 100\% = \frac{160}{800} \times 100\%\)
  • Step 4: \(= 0.2 \times 100\% = 20\%\)

Answer: Profit percentage = 20%

3
Worked Example

A dealer sold a bicycle at a loss of 12%. If the selling price was ₹2640, find the cost price.

Solution
  • Step 1: Loss% = 12%, SP = ₹2640
  • Step 2: Loss% means SP = CP × (1 - \(\frac{12}{100}\)) = CP × 0.88
  • Step 3: \(2640 = CP \times 0.88\)
  • Step 4: \(CP = \frac{2640}{0.88} = \frac{2640 \times 100}{88} = \frac{264000}{88}\)
  • Step 5: \(CP = 3000\)

Answer: Cost price = ₹3000

Key Points

  • Percentage means "out of 100" (per hundred)
  • To convert fraction to %: multiply by 100%
  • Profit = SP - CP (when SP > CP)
  • Loss = CP - SP (when CP > SP)
  • Profit% = (Profit/CP) × 100%
  • Loss% = (Loss/CP) × 100%
  • Always calculate percentages relative to Cost Price
Quick Check — 4 MCQs
Tap an option to check your answer0 / 4
Q1."Percent" means:
Explanation: Per hundred.
Q2.$50\%$ as a fraction is:
Explanation: $\tfrac{50}{100}=\tfrac12$.
Q3.Profit $=$
Explanation: Selling $-$ Cost.
Q4.Profit$\%$ is calculated on the:
Explanation: Cost price.
Practice more MCQs → 📝 Assignment · 35 marks