Percentages

A percentage is a way of expressing a number as a fraction of 100. The word 'percent' comes from the Latin phrase 'per centum', meaning 'out of one hundred'. When we write 45%, we mean exactly 45 out of every 100 parts, or the fraction 45/100 = 0.45. Percentages are one of the most universally used mathematical tools in everyday life, appearing in bank interest rates, exam scores, tax calculations, sale discounts, and statistical data. Understanding percentages requires connecting three related representations: the percentage form (e.g., 30%), the fraction form (e.g., 30/100 = 3/10), and the decimal form (e.g., 0.30). Moving fluently between these three forms is the foundation of all percentage calculations. For instance, a store advertising '20% off' is saying it will reduce the price by 20/100 = 1/5 of the original price — a fact that becomes immediately practical when you want to know how much you actually save. In commercial mathematics, percentages are essential for calculating profit and loss, simple and compound interest, tax, commission, and discounts. Before tackling those applications, you must master the basic percentage calculations: finding a percentage of a quantity, expressing one quantity as a percentage of another, and finding the original value when a percentage of it is known. This article covers these three fundamental skills with clear formulas, worked examples, and visual aids.

Key takeaways
  • Converting Between Forms
  • Finding a Percentage OF a Quantity
  • Expressing One Quantity as a Percentage of Another
  • Finding the Original Value (Reverse Percentage)
  • Percentage Increase and Decrease

Converting Between Forms

To convert a percentage to a decimal, divide by 100 (move the decimal point two places left): 75% = 0.75, 8% = 0.08, 130% = 1.30. To convert a decimal to a percentage, multiply by 100: 0.45 → 45%, 0.07 → 7%. To convert a fraction to a percentage, divide the numerator by the denominator and multiply by 100: 3/8 = 0.375 × 100 = 37.5%.

Finding a Percentage OF a Quantity

The word 'of' in mathematics means multiply. To find P% of a quantity Q, use the formula: Answer = (P/100) × Q. For example, 35% of 240 = (35/100) × 240 = 0.35 × 240 = 84. This is the most common percentage calculation and appears in problems about discounts ('find 15% off 80′),tax(′calculate1380'), tax ('calculate 13% tax on 200'), and interest ('earn 5% on $1000').

Expressing One Quantity as a Percentage of Another

To express quantity A as a percentage of quantity B, use: Percentage = (A/B) × 100. For example, a student scores 54 out of 72 on a test. Their percentage score = (54/72) × 100 = 0.75 × 100 = 75%. Always check that both quantities are in the same units before dividing. If a class has 12 girls out of 30 students total, then girls represent (12/30) × 100 = 40% of the class.

Finding the Original Value (Reverse Percentage)

Sometimes you know the result after a percentage change but need the original. If P% of an unknown original value gives result R, then: Original = R ÷ (P/100) = (R × 100)/P. More commonly, if a value after an increase of P% is R, then the original = R / (1 + P/100). If a value after a decrease of P% is R, then the original = R / (1 − P/100). For example, a shirt costs 68aftera1568 after a 15% discount. Original price = 68 / (1 − 0.15) = 68 / 0.85 = 80.

Percentage Increase and Decrease

A percentage change is calculated as: Percentage Change = ((New Value − Original Value) / Original Value) × 100. If the result is positive, it is a percentage increase; if negative, it is a percentage decrease. For example, a price rises from 50to50 to 63: Percentage change = ((63 − 50)/50) × 100 = (13/50) × 100 = 26% increase. To apply a percentage increase of P% to a value V, multiply by (1 + P/100). To apply a decrease of P%, multiply by (1 − P/100). These multipliers are called scale factors.

The Key Equation

Percentage=AB×100\text{Percentage} = \frac{A}{B} \times 100
  • A — Part value (any unit (must match B)): The quantity you want to express as a percentage — the part or subset.
  • B — Whole value (any unit (must match A)): The reference quantity — the total or whole amount you are comparing against.
  • 100 — Scaling factor (dimensionless): Multiplying by 100 converts the pure decimal ratio into a percentage figure.

Worked Example

Problem. A laptop originally costs $840. A store offers a 12.5% discount during a sale. (a) Calculate the discount amount. (b) Find the sale price. (c) Express the sale price as a percentage of the original price.

Discount=12.5100×840=$105,Sale Price=840×0.875=$735\text{Discount} = \frac{12.5}{100} \times 840 = \$105, \quad \text{Sale Price} = 840 \times 0.875 = \$735

Answer. Discount = 105;Saleprice=105; Sale price = 735; Sale price is 87.5% of the original price.

Figure
Diagram
Percentage Bar Model: 12.5% Discount on $840Original Price: $840 = 100%Sale Price: $735 = 87.5%$10512.5%Converting between forms:Percentage12.5%÷ 100 ↓ ↑ × 100Decimal: 0.125Fraction12.5/100= 1/8simplify ÷ 12.5Scale Factor1 − 0.125= 0.875× original price
Origins · 1 minSimon Stevin and how this was discovered
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Simon Stevin
1548–1620 · Flemish (Belgian)

Simon Stevin was a Flemish mathematician and engineer who made the decimal system practical for everyday use. In his 1585 work 'De Thiende' (The Tenth), he demonstrated that all calculations involving fractions could be performed using decimal fractions, laying the conceptual groundwork for expressing quantities per hundred — percentages — in a streamlined numerical way. Before Stevin's work, merchants used cumbersome fractions; his decimal notation made proportional calculations, including per-hundred reckoning, far more accessible. Stevin wrote 'De Thiende' not for mathematicians but for 'stargazers, surveyors, carpet-makers, wine-gaugers, mint-masters, and all merchants' — proof that he wanted mathematics to be practical for everyone.

  • Ratio and Proportion — Percentages are a special case of ratio (a ratio to 100), so mastering ratios deepens your understanding of how percentages compare quantities.
  • Simple Interest — Simple interest calculations use percentages directly: I = PRT/100, where R is an annual percentage rate.
  • Profit, Loss, and Discount — Profit margins, loss percentages, and shop discounts are direct applications of the percentage formulas introduced here.
  • Compound Interest — Compound interest applies percentage increases repeatedly, using the scale factor (1 + r/100) raised to a power — a direct extension of percentage increase.