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Population Problems

ITBP Constable · Quantitative Aptitude

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Core Concept

Read this first — the foundation of the topic

→CORE CONCEPT

Population problems follow the compound growth formula. If a population increases or decreases by a certain percentage each year, you apply that percentage repeatedly, not just once. This is different from simple interest — it's like compound interest

💡KEY RULES

Population grows or shrinks by a fixed percentage each year 2. The percentage applies to the NEW population each year, not the original 3. Use the compound formula, not simple addition/subtraction 4. Decrease and increase work the same way mathematically

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Formula Block

Memorise — at least one formula appears in every paper

Final Population = Initial Population × (1 + r/100)^n

Where:

- r = rate of increase (use negative r for decrease)

- n = number of years

- If r = 5% increase, use (1 + 5/100) = 1.05

- If r = 10% decrease, use (1 - 10/100) = 0.90

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Exam Patterns

What examiners ask — read before attempting PYQs

Find final population after n years

Find initial population (work backwards)

Find rate of growth

Find time period

Mixed increase and decrease over different years

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Worked Example

Solve this step-by-step before moving on

1

Step 1

Population after Year 1 = 50,000 × (1 + 10/100) = 50,000 × 1.10 = 55,000

2

Step 2

Population after Year 2 = 55,000 × (1 + 20/100) = 55,000 × 1.20 = 66,000 Alternative Direct Method: = 50,000 × 1.10 × 1.20 = 50,000 × 1.32 = 66,000 COMMON MISTAKE: Students add percentages directly: 10% + 20% = 30%, then calculate 50,000 × 1.30 = 65,000. This is WRONG because the 20% applies to the increased population, not the original. Always multiply the factors for each year.

🔑 Key Points

Population problems use compound growth formula: Final = Initial × (1 + r/100)^n
Percentage always applies to the CURRENT population, not the original amount
For decrease, use (1 - r/100) in the formula instead of (1 + r/100)
Multiple years with different rates: multiply all factors together for direct calculation
Never add percentages directly; always use multiplication of decimal factors
If asked for initial population, rearrange formula: Initial = Final ÷ (1 + r/100)^n

📌 Exam Facts

Population formula: Final = Initial × (1 + r/100)^n where r is annual rate and n is years
For 10% increase, multiply by 1.10; for 10% decrease, multiply by 0.90
If population increases by p% one year and q% next year, combined factor = (1 + p/100) × (1 + q/100)
Compound population growth applies the percentage to the NEW amount each year, not original
For population decrease problems, the formula remains the same but r is treated as negative
Quick check: 50,000 population growing at 10% annually for 2 years = 50,000 × 1.21 = 60,500

🚀 60-Second Revision

Formula: Final Population = Initial × (1 + r/100)^n — this is compound, not simple
Trap: Never add percentages from different years. Multiply the growth factors instead
Decrease: Use negative r or write (1 - r/100) — both methods give same answer
Multi-year: For different rates each year, write as Initial × (1.10) × (1.20) × (0.95) etc.
Reverse: If given final population, divide backwards: Initial = Final ÷ [(1 + r/100)^n]
Quick mental check: 10% increase twice ≈ 21% total (not 20%), because second 10% acts on larger base

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