Unit 2: Percentage and Profit and Loss

I. Orientation — The Percentage Framework

A percentage expresses a quantity as a share of 100. The word “percent” means “per hundred,” so (x\%) represents (x/100). Percentage methods allow quantities with different totals to be compared on a common scale and form the mathematical basis of profit, loss, discount, population change, election results, marks, and measurement error.

• Base quantity: The quantity relative to which a percentage is calculated; it represents 100%.
• Compared quantity: The part, change, profit, loss, or error expressed relative to the base.
• Percentage multiplier: An increase of (r\%) multiplies a quantity by (1+r/100), while a decrease multiplies it by (1-r/100).
• Choice of denominator: The denominator depends on context: cost price for profit or loss, marked price for discount, actual value for percentage error, and total votes for vote share.
• Units: A percentage has no physical unit, although the quantities used to calculate it must have compatible units.
• Direction of change: Percentage increase and percentage decrease are not interchangeable because they generally use different base values.

II. Fundamental Percentage Methods — Calculation, Comparison, and Conversion

Percentage methods translate parts, totals, ratios, fractions, and changes into a common per-hundred form.

A. Calculation of percentage

Percentage calculation determines what share one quantity forms of another, or finds a specified percentage of a quantity.

• Basic relationship:

Percentage = (Part / Whole) × 100
Part       = (Percentage / 100) × Whole
Whole      = (Part × 100) / Percentage

• Part is the selected amount.
• Whole is the corresponding total or base.
• Percentage is the part per hundred.

• Finding a percentage of a number: To find (p\%) of (N), multiply (N) by (p/100).
• Worked example: If 45 of 60 students attend a lecture, the attendance percentage is:

Attendance percentage = (45 / 60) × 100 = 75%

• Decimal method: Convert the percentage to a decimal before multiplication; for example, (18\%=0.18).

B. Percentage comparison

Percentage comparison measures a difference relative to a specified reference quantity.

• Comparison formula:

Percentage difference relative to B = [(A - B) / B] × 100

• A is the quantity being compared.
• B is the reference or original quantity.

• “More than” statement: If (A) is (p\%) more than (B), then (A=B(1+p/100)).
• “Less than” statement: If (A) is (p\%) less than (B), then (A=B(1-p/100)).
• Reverse comparison: If (A) is 25% more than (B), then (A=1.25B), so (B) is (0.25B/1.25B=20\%) less than (A).
• Key distinction: Equal absolute differences can produce unequal percentages when their reference quantities differ.

C. Percentage to fraction conversion

A percentage is converted to a fraction by placing it over 100 and reducing it to lowest terms.

• Conversion rule:

p% = p / 100

• p is the numerical percentage.

• Common conversions:
  • (50\%=1/2), (25\%=1/4), (75\%=3/4).
  • (20\%=1/5), (12.5\%=1/8), (33\frac{1}{3}\%=1/3).
  • (66\frac{2}{3}\%=2/3), (6.25\%=1/16).
• Worked example: (37.5\%=37.5/100=375/1000=3/8).
• Practical value: Fraction equivalents often permit faster mental calculation; (12.5\%) of 240 becomes (1/8) of 240, or 30.

D. Successive percent change

Successive percentage changes must be applied multiplicatively because each change acts on the newly obtained value.

• General formula:

Final value = Initial value × (1 + a/100) × (1 + b/100)
Net change% = a + b + (ab/100)

• a and b are signed percentage changes: positive for increases and negative for decreases.

• Two increases: Increases of (a\%) and (b\%) give a net increase of (a+b+ab/100\%).
• Increase and decrease: An increase of (a\%) followed by a decrease of (b\%) gives (a-b-ab/100\%).
• Equal opposite rates: An (r\%) rise followed by an (r\%) fall causes a net decrease of (r^2/100\%).
• Worked example: A value increased by 20% and then reduced by 10% becomes (1.20\times0.90=1.08) times its original value, a net increase of 8%.

III. Percentage Applications — Results, Elections, Population, and Error

Applied percentage problems require the correct total, original value, or actual value to be identified before calculation.

A. Problems based on marks

Marks problems use total marks as the base for scores and pass marks, unless another reference is explicitly stated.

• Score percentage:

Score% = (Marks obtained / Maximum marks) × 100

• Pass mark: If the pass percentage is (p\%) and maximum marks are (M), the pass mark is (pM/100).
• Failure or excess margin: Compare the candidate’s score with the pass mark in marks, not automatically as a percentage.
• Worked example: In an examination worth 800 marks, a student obtaining 584 scores ((584/800)\times100=73\%).
• Multiple subjects: Add marks obtained and maximum marks separately before calculating the overall percentage; averaging subject percentages is valid only when all subjects carry equal maximum marks.

B. Problems based on elections

Election problems express votes secured, rejected, or not cast as percentages of the relevant voter total.

• Successive bases: Registered voters, votes cast, and valid votes may be different quantities.
• Valid votes:

Valid votes = Votes cast - Invalid votes
Candidate's share% = (Candidate's votes / Valid votes) × 100

• Two-candidate election: If one candidate receives (p\%) of valid votes, the other receives ((100-p)\%), provided there are exactly two candidates.
• Worked example: Of 20,000 registered voters, 80% vote and 5% of votes cast are invalid. Valid votes are (20,000\times0.80\times0.95=15,200).
• Victory margin: Subtract the candidates’ vote counts; do not confuse the vote margin with the winner’s percentage share.

C. Problems based on population

Population change is usually compounded because each period’s rate applies to the population at the beginning of that period.

• Growth model:

Pₙ = P₀(1 + r/100)ⁿ

• P₀ is the initial population.
• Pₙ is the population after n periods.
• r is the percentage growth per period.
• n is the number of periods.

• Population decline: Replace the factor with ((1-r/100)^n).
• Reverse calculation: Divide the present population by the relevant growth factors to find an earlier population.
• Worked example: A population of 50,000 growing by 4% annually becomes (50,000(1.04)^2=54,080) after two years.
• Changing rates: For unequal annual rates, multiply separate factors rather than using the arithmetic average rate.

D. Percentage error

Percentage error measures the absolute error relative to the accepted or actual value.

• Formula:

Absolute error   = |Measured value - Actual value|
Percentage error = (Absolute error / |Actual value|) × 100

• Absolute value: The magnitude is reported as non-negative even when the measurement is below the actual value.
• Overestimation and underestimation: The sign of measured minus actual identifies direction, while percentage error normally reports magnitude.
• Worked example: If a length measured as 49 cm is actually 50 cm, the absolute error is 1 cm and the percentage error is ((1/50)\times100=2\%).
• Reference value: Percentage error uses the actual value, whereas percentage change uses the original value.

IV. Price and Profit Structure — Commercial Foundations

Profit-and-loss calculations track how the acquisition cost of an article relates to its listed and actual sale prices.

A. Concept of cost price, selling price, and marked price

Cost price, selling price, and marked price represent three distinct stages in a commercial transaction.

• Cost price (CP): The amount paid to acquire or produce an article; direct expenses such as transport or repair may be included when stated.
• Marked price (MP): The listed or labelled price before discount.
• Selling price (SP): The amount actually received from the buyer after discount.
• Price relationship:

SP = MP × (1 - d/100)

• d is the discount rate.
• MP is the marked price.
• SP is the selling price.

• Comparison sequence: Discount compares MP with SP, while profit or loss compares CP with SP.
• Worked example: An article marked at ₹1,000 with a 15% discount sells for (1000\times0.85=₹850).

B. Calculation of profit and loss percentage

Profit occurs when selling price exceeds cost price, while loss occurs when cost price exceeds selling price.

• Core formulae:

Profit  = SP - CP
Loss    = CP - SP
Profit% = (Profit / CP) × 100
Loss%   = (Loss / CP) × 100

• Selling-price multipliers: At (p\%) profit, (SP=CP(1+p/100)); at (l\%) loss, (SP=CP(1-l/100)).
• Recovering cost price: At (p\%) profit, (CP=100SP/(100+p)); at (l\%) loss, (CP=100SP/(100-l)).
• Worked example: If CP is ₹750 and SP is ₹900, profit is ₹150 and profit percentage is ((150/750)\times100=20\%).
• Denominator rule: Standard profit and loss percentages are calculated on CP unless the problem explicitly states another basis.

V. Special Commercial Problems — Discounts, Quantities, and Selling Conditions

These problems combine percentage multipliers with altered weights, article counts, or a shared selling price.

A. Problems on discount

Discount is the reduction from marked price, and successive discounts are applied one after another.

• Single discount:

Discount  = MP - SP
Discount% = (Discount / MP) × 100

• Successive discounts: Discounts of (a\%) and (b\%) are equivalent to (a+b-ab/100\%), not (a+b\%).
• Worked example: Successive discounts of 20% and 10% leave (0.80\times0.90=0.72) of MP, giving an equivalent discount of 28%.
• Profit after discount: First calculate SP from MP and discount, then compare SP with CP.
• Required marking: To allow discount (d\%) while earning profit (p\%), use:

MP = CP × (100 + p) / (100 - d)

B. Faulty weight

A trader using less than the stated weight gains because the selling price corresponds to a larger nominal quantity than the quantity delivered.

• Effective comparison: Compare money received for the stated weight with the cost of the actual weight supplied.
• Weight-based formula: If (w) grams are supplied as 1,000 grams at cost price, then:

Gain% = [(1000 - w) / w] × 100

• Worked example: Supplying 900 g as 1 kg while charging at cost price gives a gain of ((100/900)\times100=11.11\%).
• Combined gain: If the trader also raises the price, multiply the price factor and quantity factor; do not simply add their percentages.
• Interpretation: The cost of the actual quantity supplied is the denominator when calculating profit.

C. Number of articles

Article-count problems compare the cost of one number of identical articles with the selling price of another number.

• Unit method: If the CP of (x) articles equals the SP of (y) articles, then:

SP per article / CP per article = x / y
Profit or loss% = [(x - y) / y] × 100

• Direction: If (x>y), there is profit; if (x<y), there is loss.
• Worked example: If the CP of 12 articles equals the SP of 10 articles, then (SP/CP=12/10=1.2), so profit is 20%.
• Equivalent quantity statements: “Sell (n) articles for the cost of (m)” should first be translated into total-value equality.
• Mixed rates: When batches have different costs, total cost must be used instead of assuming one common unit cost.

D. Common selling price

When two articles are sold at the same selling price, their different profit or loss rates imply different cost prices.

• Separate cost prices: For common SP (=S), calculate each CP independently:

CP at p% profit = 100S / (100 + p)
CP at l% loss   = 100S / (100 - l)

• S is the common selling price.
• p is the profit rate.
• l is the loss rate.

• Equal profit and loss rates: If one article is sold at (x\%) profit and another at (x\%) loss for the same SP, the overall result is always a loss.
• Combined result:

Overall loss% = x² / 100

• Worked example: Two articles sold for ₹960 each at 20% profit and 20% loss have cost prices ₹800 and ₹1,200. Total CP is ₹2,000 and total SP is ₹1,920, producing a 4% loss.
• Reason for the loss: The loss is calculated on the higher-cost article, while the equal-rate profit is calculated on the lower-cost article.