A comprehensive guide to fractional thinking, financial reasoning, and percentage-based comparisons
Percentages are ubiquitous in science, finance, statistics, and everyday life. They allow us to express numbers as parts of 100, making them intuitive, comparable, and scalable. From analysing data in a research paper to calculating interest on a bank account, understanding percentages is essential for university-level science students and professionals alike.
This lecture explores:
- What a percentage is and how to convert between forms
- Calculating percentages of quantities
- Expressing one value as a percentage of another
- Percentage increase and decrease
- Reverse percentage problems
- Simple interest calculations
- Compound interest calculations (step-by-step and formula-based)
- Depreciation
- Common mistakes and tips
- Practice exercises
By the end of this lecture, you will confidently handle percentages in scientific, financial, and statistical contexts, interpret data, and solve real-world problems with clarity.
Explore the basics at BBC Bitesize – Percentages.
1. What is a Percentage?
A percentage is a number expressed as a fraction of 100. The symbol % literally means “per hundred.”
1.1 Converting Between Forms
| Representation | Example |
| Fraction | 3/4 |
| Decimal | 0.75 |
| Percentage | 75% |
- 40% = 40/100 = 0.40
- 0.75 = 75%
- 3/4 = 75%
1.2 Calculating a Percentage of a Quantity
To calculate x% of a number, multiply the number by x/100.
Example 1: Calculate 40% of £590.
Answer: £236
Example 2: Find 12% of 250.
Practice percentage calculations at Khan Academy – Percentages.
2. Expressing One Value as a Percentage of Another
Sometimes we need to express A as a percentage of B.
Example 1: Tim scored 26 out of 40 on a test.
Example 2: Compare quantities:
- 50 out of 200 → 50 / 200 × 100 = 25%
- 75 out of 150 → 75 / 150 × 100 = 50%
Applications:
- Exam scores
- Survey results
- Statistical comparisons
3. Percentage Increase and Decrease
3.1 Percentage Increase
To increase a number by x%:
Example 1: Increase 80 by 15%.
Answer: 92
3.2 Percentage Decrease
To decrease a number by x%:
Example 2: Decrease 200 by 25%.
Answer: 150
Explore percentage change at MathsIsFun – Percentages.
4. Reverse Percentage Problems
Sometimes the final value is known, and we need to find the original amount.
Formula:
Example: A jacket costs £45 after 20% off. What was the original price?
Answer: £56.25
Applications:
- Retail pricing
- Tax calculations
- Data interpretation
5. Simple Interest
Simple interest is interest calculated only on the original principal.
- Rate = interest rate per year (%)
- Principal = original amount of money
- Time = number of years
Example: 4% per annum on £5000 for 1 year.
Total amount:
Worked Example: 5% per year on £1200 for 3 years.
Total amount: £1200 + £180 = £1380
6. Compound Interest
Compound interest is calculated on the principal and any previously earned interest.
6.1 Year-by-Year Method
Example: £6000 at 6.5% for 3 years
- Year 1: 6.5% of 6000 = 390 → Balance = 6390
- Year 2: 6.5% of 6390 = 415.35 → Balance = 6805.35
- Year 3: 6.5% of 6805.35 = 442.35 → Balance = 7247.70
Total interest: 7247.70 − 6000 = 1247.70
6.2 Quick Method (Multiplier Formula)
Example: £6000 at 6.5% for 3 years
- Multiplier = 1 + 0.065 = 1.065
- Final balance = 6000 × 1.065³ = 7247.70
Explore compound interest at BBC Bitesize – Compound Interest.
7. Depreciation
Depreciation is a percentage decrease over time, often used for cars, electronics, or equipment.
Formula:
Example: A car worth £15,000 depreciates 15% per year for 2 years.
- Multiplier = 0.85
- Value after 2 years = 15000 × 0.85² = £10,837.50
Note: If the rate changes yearly, calculate sequentially for each year.
8. Common Mistakes to Avoid
- Confusing the percentage of a number with the percentage change
- Forgetting to convert percentages to decimals
- Using simple interest when compound interest is required
- Misapplying reverse percentage logic
Tip: Always clarify whether the problem involves an increase, a decrease, or a proportion.
9. Real-World Applications
9.1 Science
- Growth of bacterial populations (% increase over time)
- Radioactive decay (% decrease per unit time)
9.2 Finance
- Bank accounts (simple vs compound interest)
- Loans and mortgages
- Investment growth and depreciation of assets
9.3 Data Analysis
- Exam score comparisons
- Market share (%)
- Experimental results reporting
9.4 Statistics
- Proportions in surveys and polls
- Probability expressed as percentages
- Change in datasets over time
10. Practice Exercises
- Calculate percentages:
- 25% of £480
- 18% of 320
- 3/8 as a percentage
- Express one value as a percentage of another:
- 45 out of 60
- 150 out of 400
- 27 out of 50
- Percentage increase/decrease:
- Increase 120 by 20%
- Decrease 350 by 15%
- Increase 450 by 8%
- Reverse percentage problems:
- Sale price £36 after a 10% discount → Original price?
- Final population 900 after 20% growth → Original population?
- Simple interest:
- £8000 at 3.5% per annum for 4 years
- £1500 at 6% per annum for 2.5 years
- Compound interest:
- £5000 at 5% for 3 years
- £2500 at 7% for 5 years
- Depreciation:
- Laptop worth £1200, depreciates 10% per year for 3 years
- Vehicle worth £20,000, depreciates 12% first year, 8% second year
11. Closing Thoughts: Thinking in Hundreds
Percentages allow us to think in hundreds, a scale that is intuitive, comparable, and widely understood. They are invaluable in:
- Financial reasoning
- Scientific measurement
- Statistical analysis
- Everyday calculations
Mastering percentage logic gives you a flexible, powerful toolkit for analysing change, growth, decay, interest, and proportional relationships.
By combining percentage calculations with simple and compound interest and understanding reverse percentages, you gain a robust method for interpreting data and making decisions based on clear quantitative reasoning.
Support the Archive
This archive is freely shared as a communal act of care.
If you’d like to support its continuation, consider purchasing a companion PDF set for £1 per lecture, with an associated quiz, via Payhip, with the final price depending on the number of lectures in the set, available only once the full series is complete.