Summary: Simplification Approximation

Table of Contents
1. Introduction
2. Basic Rules of Simplification
3. Modulus of a Real Number
4. Conversion of Decimal Numbers (Approximation)
5. Approximation of Square Roots
6. Fast Tricks for Multiplication and Squaring Near Bases
7. Problem-Solving Notes
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Introduction

• The Quantitative Aptitude section covers topics like Simplification and Approximation, Number Series, Permutation & Combination, Quadratic Equations, Data Interpretation, Data Analysis, Percentage & Average, Ratio & Proportion, Partnerships, Profit & Loss, Simple & Compound Interest and other miscellaneous problems.

Basic Rules of Simplification

• Follow the BODMAS order: Brackets → Order (powers, roots) → Division → Multiplication → Addition → Subtraction.
• Evaluate brackets first; use bracket order: (), {}, [].
• Evaluate exponents and roots next.
• Do division and multiplication left to right (same priority).
• Do addition and subtraction left to right (same priority).

Modulus of a Real Number

• The Modulus (absolute value) |x| is never negative.
• Definition: |x| = x if x ≥ 0; |x| = -x if x < 0.

Conversion of Decimal Numbers (Approximation)

• Round decimals to the nearest convenient value before computing.
• Round percentage values to nearby whole numbers when appropriate, then simplify using the rounded values.

Approximation of Square Roots

• Factor the number under the root.
• Move each pair of identical factors outside the root once.
• If a factor appears an odd number of times, move pairs outside and leave one inside.
• Multiply numbers outside the root together; multiply remaining factors inside the root together.
• For square root of a fraction, simplify numerator and denominator separately.
• Check: (outside number)^2 × (inside number) must equal the original number under the root.

Fast Tricks for Multiplication and Squaring Near Bases

• Multiplying by 99, 999, etc.: subtract 1 from the multiplicand for the left part; take complement to the base (100, 1000, ...) for the right part; join the two parts. Use leading zeros on the right part to match base digits when needed.
• Squaring a number near a base (100, 1000, ...): let d = difference from base. Left part = number ± d (subtract if below base, add if above). Right part = d^2 padded to the number of zeros in the base. Join left and right parts.

Problem-Solving Notes

• Convert word problems into simple equations before solving.
• Use rounding and the above shortcuts to speed up approximate calculations while keeping required accuracy.