📊 Estimation & Bounds Playbook

Master the art of quick approximation to solve problems efficiently and verify answers. Learn to use orders of magnitude, inequalities, and growth comparisons strategically.

🎯 When to Estimate

Always Estimate:

• Large numbers: When exact calculation is tedious
• Verification: To check if your answer is reasonable
• Time pressure: When you need a quick answer
• Answer choices: To eliminate obviously wrong options

Consider Estimation:

• Complex expressions: When exact evaluation is hard
• Word problems: To check if answer makes sense
• Geometry problems: To verify scale and proportions
• Any problem: When you feel uncertain about your answer

📏 Orders of Magnitude

Powers of 10

• $10^0 = 1$: Basic unit
• $10^1 = 10$: Ten
• $10^2 = 100$: Hundred
• $10^3 = 1,000$: Thousand
• $10^6 = 1,000,000$: Million
• $10^9 = 1,000,000,000$: Billion

Quick Magnitude Checks

• Is answer too big?: Compare to reasonable bounds
• Is answer too small?: Check against minimum values
• Does scale make sense?: Use physical intuition
• Are units appropriate?: Verify dimensional analysis

Magnitude Example

Problem: Estimate $2^{20}$

Estimation:

• $2^{10} = 1,024 \approx 1,000 = 10^3$
• $2^{20} = (2^{10})^2 \approx (10^3)^2 = 10^6 = 1,000,000$
• Answer: Approximately 1 million

🔢 Common Approximations

Square Roots

• $\sqrt{2} \approx 1.414$: Common in geometry
• $\sqrt{3} \approx 1.732$: Common in triangles
• $\sqrt{5} \approx 2.236$: Golden ratio related
• $\sqrt{10} \approx 3.162$: Decimal system

Powers

• $2^{10} \approx 1,000$: Computer science
• $3^4 = 81$: Common power
• $5^3 = 125$: Common cube
• $7^2 = 49$: Common square

Logarithms

• $\log_{10} 2 \approx 0.301$: Common logarithm
• $\log_{10} 3 \approx 0.477$: Common logarithm
• $\log_{10} 5 \approx 0.699$: Common logarithm
• $\log_{10} 7 \approx 0.845$: Common logarithm

📈 Growth Rate Comparisons

Exponential vs Polynomial

• Exponential grows faster: $2^n$ vs $n^2$ for large $n$
• Polynomial grows faster: $n^3$ vs $2^n$ for small $n$
• Crossover point: Where growth rates change
• Asymptotic behavior: What happens as $n \to \infty$

Growth Rate Example

Problem: Which is larger: $2^{100}$ or $100^2$?

Comparison:

• $2^{100} = (2^{10})^{10} \approx (10^3)^{10} = 10^{30}$
• $100^2 = 10,000 = 10^4$
• Since $10^{30} > 10^4$, we have $2^{100} > 100^2$

🎯 Bounding Techniques

Upper Bounds

• Find maximum possible value: What’s the largest it could be?
• Use known inequalities: Apply standard bounds
• Test extreme cases: What happens at limits?
• Apply constraints: Use problem restrictions

Lower Bounds

• Find minimum possible value: What’s the smallest it could be?
• Use known inequalities: Apply standard bounds
• Test extreme cases: What happens at limits?
• Apply constraints: Use problem restrictions

Bounding Example

Problem: Find bounds for $f(x) = x^2 - 4x + 3$ on $[0, 4]$

Bounding:

• Upper bound: $f(0) = 3$ and $f(4) = 3$, maximum at $x = 2$: $f(2) = -1$
• Lower bound: Minimum value is $-1$ at $x = 2$
• Range: $[-1, 3]$

⚡ Quick Estimation Strategies

Rounding

• Round to nearest power of 10: $1,234 \approx 1,000$
• Round to nearest integer: $3.7 \approx 4$
• Round to nearest fraction: $0.333 \approx \frac{1}{3}$
• Round to nearest decimal: $2.718 \approx 2.7$

Approximation Techniques

• Linear approximation: $f(x) \approx f(a) + f’(a)(x-a)$
• Taylor series: Use first few terms
• Binomial approximation: $(1+x)^n \approx 1 + nx$ for small $x$
• Exponential approximation: $e^x \approx 1 + x$ for small $x$

Estimation Example

Problem: Estimate $\sqrt{101}$

Estimation:

• Use linear approximation: $f(x) = \sqrt{x}$, $f’(x) = \frac{1}{2\sqrt{x}}$
• At $x = 100$: $f(100) = 10$, $f’(100) = \frac{1}{20} = 0.05$
• $f(101) \approx f(100) + f’(100)(101-100) = 10 + 0.05(1) = 10.05$
• Answer: Approximately 10.05

🧮 Inequality Techniques

Basic Inequalities

• Arithmetic-Geometric Mean: $\frac{a+b}{2} \geq \sqrt{ab}$ for $a, b > 0$
• Cauchy-Schwarz: $(a_1b_1 + a_2b_2)^2 \leq (a_1^2 + a_2^2)(b_1^2 + b_2^2)$
• Triangle inequality: $|a + b| \leq |a| + |b|$
• Jensen’s inequality: For convex functions

Inequality Example

Problem: Find the minimum value of $x^2 + y^2$ given $x + y = 1$

Solution:

• By Cauchy-Schwarz: $(x^2 + y^2)(1^2 + 1^2) \geq (x + y)^2$
• $(x^2 + y^2)(2) \geq 1^2 = 1$
• $x^2 + y^2 \geq \frac{1}{2}$
• Minimum value: $\frac{1}{2}$ (achieved when $x = y = \frac{1}{2}$)

📊 Answer Choice Elimination

Using Estimation

• Eliminate extreme values: Too big or too small
• Check reasonableness: Does answer make sense?
• Use bounds: Narrow down the range
• Test special cases: What happens at limits?

Estimation Example

Problem: Find the value of $\frac{1}{1.01}$

Answer choices: A) 0.99, B) 0.990, C) 0.9901, D) 0.9902, E) 0.9903

Estimation:

• $\frac{1}{1.01} = \frac{1}{1 + 0.01} \approx 1 - 0.01 = 0.99$
• But this is too rough, use better approximation
• $\frac{1}{1+x} \approx 1 - x + x^2$ for small $x$
• $\frac{1}{1.01} \approx 1 - 0.01 + 0.0001 = 0.9901$
• Answer: C) 0.9901

🎯 Problem-Specific Strategies

Algebra Problems

• Test extreme values: $x = 0, 1, -1$
• Use symmetry: Even/odd functions
• Check monotonicity: Increasing/decreasing
• Apply bounds: Use known inequalities

Geometry Problems

• Use similar triangles: Proportional relationships
• Apply area formulas: Known area relationships
• Check angles: Sum to 180° in triangles
• Use coordinate geometry: Distance formulas

Number Theory Problems

• Use modular arithmetic: Remainder patterns
• Apply divisibility rules: 2, 3, 5, 9, 11
• Check prime factors: Prime factorizations
• Use bounds: Upper and lower limits

Counting/Probability Problems

• Use symmetry: Symmetric cases
• Apply bounds: Probability between 0 and 1
• Check extreme cases: What happens at limits?
• Use complementary counting: Count what you don’t want

⚡ Quick Reference

Estimation Checklist:

• What’s the order of magnitude? Is answer too big or too small?
• What are reasonable bounds? Upper and lower limits
• Can I use known approximations? Common values and formulas
• Does the answer make sense? Physical and mathematical intuition
• Can I eliminate choices? Use estimation to narrow down

Common Approximations:

• $\sqrt{2} \approx 1.414$
• $\sqrt{3} \approx 1.732$
• $\pi \approx 3.1416$
• $e \approx 2.718$
• $\log_{10} 2 \approx 0.301$

Bounding Techniques:

• Test extreme values: $x = 0, 1, -1, \infty$
• Use known inequalities: AM-GM, Cauchy-Schwarz
• Apply constraints: Problem restrictions
• Check monotonicity: Increasing/decreasing functions

Prev: Diagramming & Markup | Next: Topic Routing Heuristics | Back to: Strategy Guide