2  Percentages

2.1 Introduction

A percentage expresses a number as a fraction of 100.
It is the language of growth rates, discounts, profits, marks, and data interpretation.

This chapter builds a complete toolkit: - Conversions between fractions, decimals, and percentages
- Percentage change, successive changes, undoing changes
- Reverse percentages (finding original values)
- Base-change traps
- Percentage points vs percent
- Ratio–percentage links
- Applications: profit–loss, discounts, population growth, mixtures
- Fast mental models and exam-style practice problems

2.2 Core Definitions and Conversions

2.2.1 Definition

\[ \text{Percentage} = \frac{\text{Part}}{\text{Whole}} \times 100\% \]

2.2.2 Conversion Rules

  • Fraction → % : multiply by 100
  • % → Fraction : divide by 100
  • Decimal → % : multiply by 100
  • % → Decimal : divide by 100

Mini-table (to memorize):
- \(\tfrac12=50\%\), \(\tfrac13\approx33\tfrac{1}{3}\%\), \(\tfrac23\approx66\tfrac{2}{3}\%\)
- \(\tfrac14=25\%\), \(\tfrac34=75\%\)
- \(\tfrac15=20\%\), \(\tfrac25=40\%\), \(\tfrac35=60\%\), \(\tfrac45=80\%\)
- \(\tfrac18=12.5\%\), \(\tfrac{3}{8}=37.5\%\), \(\tfrac{5}{8}=62.5\%\), \(\tfrac{7}{8}=87.5\%\)
- \(\tfrac19\approx11.11\%\), \(\tfrac{1}{11}\approx9.09\%\), \(\tfrac{1}{12}\approx8.33\%\), \(\tfrac{1}{16}=6.25\%\), \(\tfrac{1}{20}=5\%\)

2.3 Percentage Change

2.3.1 Single Change

If a value moves from \(A\) to \(B\): \[ \%\ \text{change} = \frac{B-A}{A}\times 100\% \] - Increase if \(B>A\); decrease if \(B<A\).

Example: \(50 \to 60\) ⇒ \(\frac{10}{50}\times100=20\%\) increase.

2.3.2 Successive Changes

Two successive changes \(x\%\) and \(y\%\) combine to: \[ \text{Net \%} = x + y + \frac{xy}{100} \] For more than two:
\[ \big(1+\tfrac{x}{100}\big)\big(1+\tfrac{y}{100}\big)\big(1+\tfrac{z}{100}\big)\cdots -1 \]

Example: +20% then −10% ⇒ \(20+(-10)+\frac{20\cdot(-10)}{100}=8\%\) net increase.

2.3.3 Undoing a Change

  • If decreased by \(d\%\), required increase to restore = \(\frac{d}{100-d}\times100\%\).
  • If increased by \(i\%\), required decrease = \(\frac{i}{100+i}\times100\%\).

Example: −20% then restore? \(\frac{20}{80}\times100=25\%\).

2.4 Reverse Percentages

When final value \(F\) after an increase/decrease is known:

  • Increase \(x\%\): \(F=O(1+\tfrac{x}{100}) \;\Rightarrow\; O=\frac{F}{1+x/100}\)
  • Decrease \(x\%\): \(F=O(1-\tfrac{x}{100}) \;\Rightarrow\; O=\frac{F}{1-x/100}\)

Examples
- After 10% discount, price=540 ⇒ \(O=540/0.9=600\).
- Population after +5% becomes 21000 ⇒ \(O=21000/1.05=20000\).

2.5 Base-Change Traps

Percent is base-dependent.
“A is 25% of B” means \(A=0.25B\). But “B is what % of A?” is \(\frac{B}{A}\times100\), a different number.

Example: A=20% of B ⇒ \(A=0.2B\). Then \(B/A=5\) ⇒ B is 500% of A.

2.6 Percentage Points vs Percent

Change from 8% to 10%:
- +2 percentage points
- \(\frac{2}{8}\times100=25\%\) relative increase

Always use the correct phrase in DI/word problems.

2.7 Part–Whole–Complement

If \(x\%\) is one part, the complement is \((100-x)\%\).

Example: 35% failed ⇒ 65% passed.

2.9 Multipliers (Fast Model)

  • Increase by \(p\%\) ⇒ multiply by \((1+p/100)\)
  • Decrease by \(p\%\) ⇒ multiply by \((1-p/100)\)

Example: “Price +25%, then −20%” ⇒ \(1.25\times0.8=1.0\) → back to original.

2.10 Applications

2.10.1 Discounts and Markups

  • After discount \(d\%\): \(SP=MP(1-d/100)\)
  • Markup \(m\%\) then discount \(d\%\):
    \[ SP=CP(1+m/100)(1-d/100) \]

2.10.2 Profit/Loss (on CP)

\[ \text{Profit\%}=\frac{SP-CP}{CP}\times100 \]

2.10.3 Population/Compound Growth

Population after \(n\) years at growth \(r\%\):
\[ P=P_0(1+r/100)^n \]

2.10.4 Mixtures

Mix \(x\) liters of \(p\%\) solution with \(y\) liters of \(q\%\):
Final% = \(\frac{xp+yq}{x+y}\).

2.11 Solved Examples

  1. Convert 0.375 to % ⇒ 37.5%.
  2. Successive change: +30%, then −20% ⇒ net +4%.
  3. Restore original: decrease 25%, need +33.33%.
  4. Reverse %: Final SP=18000 after 10% discount ⇒ MP=20000.
  5. Part–whole: 60% passed English, 70% Maths, 20% failed both ⇒ 80% passed at least one.
  6. Base-change: A=25% more than B. Then B is 20% less than A.
  7. Salary puzzle: +10% then +10% ⇒ net +21%.
  8. Population: 20000 grows 3% annually for 5 years ⇒ ≈23185.
  9. Mixture: 4L of 25% + 6L of 40% acid ⇒ 34%.
  10. Marks: 360/600=60%. Needed 65% ⇒ short by 30 marks.

2.12 Quick Patterns and Mental Math

  • Percent of a percent: \(a\%\text{ of }b\%=\tfrac{ab}{100}\%\).
  • \(x\%\) of \(y = y\%\) of \(x\).
  • 10% = move decimal left.
  • 5% = half of 10%.
  • 1% = divide by 100.
  • 12.5% = \(\tfrac{1}{8}\), 33.33% = \(\tfrac{1}{3}\).

2.13 Common Traps

  • Always identify the base (CP, MP, SP, total, previous year).
  • Do not add % across different bases.
  • Distinguish percentage points vs percent change.
  • For reverse % problems, divide by the multiplier, not subtract.

2.14 Practice Sets

2.14.1 Set A – Fundamentals

  1. Convert (a) \(\tfrac{3}{5}\) to %, (b) 0.64 to %, (c) 22.5% to fraction.
  2. Find (a) 15% of 480, (b) 125% of 56, (c) 2.5% of 8000.
  3. Value falls from 840 to 714. Find % decrease.
  4. After 12.5% decrease, price=2100. Find original.
  5. Increase 40% then 10%. Net % change?

2.14.2 Set B – Word Problems

  1. In a class, 40% are girls. If girls=36, find total.
  2. Shop announces 20%+10% successive discounts. Equivalent discount?
  3. Book costs 25% more than pen. Pen is what % less than book?
  4. Exam out of 120 marks. Student scores 75%. How many more for 85%?
  5. Population increases 12% then decreases 5%. Net % change?

2.14.3 Set C – Reverse and Base Change

  1. Item costs 5400 after 10% discount and 18% GST. Find MP before discount and tax.
  2. A’s salary 20% more than B’s. By what % is B less than A?
  3. Two successive decreases of 20% and \(x\%\) give 36% net decrease. Find \(x\).
  4. Score improved from 64% to 72%. Relative % increase? % points increase?
  5. 800 surveyed; 62.5% prefer X. How many prefer Y?

2.15 Answer Key (Concise)

Set A
1. (a) 60% (b) 64% (c) 9/40
2. (a) 72 (b) 70 (c) 200
3. 15%
4. 2400
5. 54% increase

Set B
6. 90 students
7. 28%
8. 20% less
9. 12 marks more
10. 6.4% increase

Set C
11. MP ≈ 5085
12. 16.67%
13. \(x=20\%\)
14. 12.5% relative; 8 percentage points
15. 300

2.16 Summary

  • Treat percentages as multipliers.
  • Successive changes: \(x+y+\tfrac{xy}{100}\) or use chained multipliers.
  • Reverse % requires division by the multiplier.
  • Base-change traps appear frequently in exams.
  • Memorize key fraction–percent pairs for speed.