Important Formulas: Quantitative Reasoning
Arithmetic and quantitative reasoning problems frequently rely on a compact set of standard formulas and clear definitions. The following document collects those formulas, explains their meaning, gives short examples where useful, and offers practical tips for applying them in logical-reasoning and data-interpretation tasks. Retain this page as a quick reference while practising problem solving.
Basic Arithmetic Operations
- Addition: a + b = c
- Subtraction: a - b = c
- Multiplication: a × b = c
- Division: a ÷ b = c
Averages, Weighted Average and Related Concepts
Averages (Arithmetic Mean)
Definition: The arithmetic mean of n numbers x1, x2, ..., xn is the sum of the numbers divided by n.
Formula: (x1 + x2 + ... + xn) ÷ n
Example with worked solution:
Find the average of 4, 7 and 11.
Sum of the numbers = 4 + 7 + 11 = 22
Number of observations = 3
Average = 22 ÷ 3 = 22/3 = 7.333...
Weighted Average
Formula: (w1·x1 + w2·x2 + ... + wn·xn) ÷ (w1 + w2 + ... + wn)
Use weighted average when different observations contribute unequally to the overall mean (for example, averages of batches of different sizes).
Percentages and Conversions
- Percentage (part of whole): (part ÷ whole) × 100
- Percent increase / decrease: ((New - Old) ÷ Old) × 100
- Conversions:
- Percentage to decimal = Percentage ÷ 100
- Decimal to percentage = Decimal × 100
- Percentage to fraction = Percentage ÷ 100, then simplify
- Fraction to percentage = (Numerator ÷ Denominator) × 100
Ratio and Proportion
Ratio: a : b expresses the relative sizes of two quantities.
Proportion: a ÷ b = c ÷ d or a : b = c : d
When solving proportion problems, cross-multiply: a·d = b·c.
Speed, Distance and Time
- Distance: Speed × Time
- Speed: Distance ÷ Time
- Time: Distance ÷ Speed
When speeds are given for different segments, use total distance ÷ total time to compute overall average speed. For harmonic averages of equal distances, use the harmonic mean formula for speeds.
Simple and Compound Interest (Growth)
Simple Interest
Formula: I = P × R × T ÷ 100
Here I is the interest, P the principal, R the annual rate (percentage), and T the time in years.
Example with worked solution:
Find simple interest on ₹5,000 at 6% per annum for 2 years.
I = 5000 × 6 × 2 ÷ 100
I = 5000 × 12 ÷ 100
I = 600
Compound Interest
Formula (amount): A = P(1 + r/n)(n·t)
Here A is the amount after time t years, P the principal, r the annual decimal rate, and n the number of compounding periods per year. For annual compounding (n = 1) use A = P(1 + r)t.
Notes: For interest rates given in percentages, convert r% to decimal by dividing by 100.
Example with worked solution:
Find the amount for P = ₹10,000 at 5% compounded annually for 2 years.
Convert rate to decimal: r = 5 ÷ 100 = 0.05
A = 10000 × (1 + 0.05)2
A = 10000 × 1.1025 = 11025
Profit, Loss and Discounts
- Profit: Selling Price - Cost Price
- Loss: Cost Price - Selling Price
- Profit %: (Profit ÷ Cost Price) × 100
- Loss %: (Loss ÷ Cost Price) × 100
- Marked price, discount and net price relations: Net Price = Marked Price - Discount
Fractions: Operations and Conversions
- Addition: a/b + c/d = (ad + bc) ÷ bd
- Subtraction: a/b - c/d = (ad - bc) ÷ bd
- Multiplication: (a/b) × (c/d) = (a·c) ÷ (b·d)
- Division: (a/b) ÷ (c/d) = (a·d) ÷ (b·c)
- Decimal to fraction: Write decimal as numerator and denominator as power of 10, then simplify
- Fraction to decimal: Divide numerator by denominator
- Fraction to percentage: (Fraction) × 100
- Percentage to fraction / decimal: Percentage ÷ 100
Number Theory: LCM, GCD, Primes, Factors
- Least Common Multiple (LCM): Smallest positive integer divisible by each of the numbers
- Greatest Common Divisor (GCD) / HCF: Largest positive integer that divides each of the numbers exactly
- Prime numbers: Integers greater than 1 with exactly two positive divisors: 1 and itself
- Factors / Divisors: Numbers that multiply to give the original number
Use prime factorisation to compute LCM and GCD quickly: LCM uses the highest powers of primes; GCD uses the lowest powers common to all numbers.
Powers and Roots
- Square: a² = a × a
- Cube: a³ = a × a × a
- Square root: √a is a number b such that b² = a
- Cube root: ∛a is a number b such that b³ = a
Permutations, Combinations and Probability
Permutations (ordered selections): nP r = n! ÷ (n - r)!
Combinations (unordered selections): nC r = n! ÷ [r! (n - r)!]
Basic probability: For equally likely outcomes, Probability(Event) = (Number of favourable outcomes) ÷ (Total number of outcomes)
Permutations and combinations are widely used in counting arguments for logical-reasoning questions.
Data Interpretation: Measures of Central Tendency and Dispersion
- Mean (arithmetic mean): Sum of observations ÷ Number of observations
- Median: Middle value when observations are arranged in order; for even n, median is average of the two middle values
- Mode: Most frequently occurring value(s) in the dataset
- Range: Maximum - Minimum
- Variance (population): σ² = (1/N) Σ(xi - μ)²
- Standard deviation (population): σ = √σ²
For grouped data, use class mid-points to approximate mean and variance. Quick estimations and table reading are frequently required in data-interpretation questions.
Practical Tips and Strategies
- Unit checks: Always confirm that units of distance, time, rate and percentages are consistent before applying formulas.
- Simplify percentages: Convert percentages to fractions or decimals when convenient (for repeated multiplications use decimal form).
- Compound interest tricks: For small rates and short periods, use linear approximations (A ≈ P(1 + rt)) only if error tolerance allows.
- Use factorisation: Prime factorisation helps compute LCM/GCD and simplifies fraction arithmetic quickly.
- Estimate before exact computation: For multiple-choice questions, narrow options by approximation first.
- When to use permutations vs combinations: Ask whether order matters. If yes, use permutations; if not, use combinations.
- Speed problems with multiple segments: Work with total distance and total time rather than averaging speeds directly unless distances are equal.
- Keep formulas memorised but understand their derivations: Understanding reduces errors under exam time pressure.
Short Worked Example: Percentage Change vs Absolute Change
A price rises from ₹200 to ₹260. Compute the percentage increase.
Absolute increase = 260 - 200
Absolute increase = 60
Percentage increase = (60 ÷ 200) × 100
Percentage increase = 0.3 × 100 = 30%
Short Worked Example: Combination and Probability
From a set of 5 distinct objects, two are chosen at random. What is the probability that a particular object A is chosen?
Total number of ways to choose 2 from 5 = 5C2 = 10
Number of choices that include A: choose the other object from the remaining 4, so 4 ways
Probability = 4 ÷ 10 = 2 ÷ 5 = 0.4
Use this page as a checklist when solving quantitative reasoning and data-interpretation problems. Memorise the core formulas, practise their application in varied contexts and always check units and reasonableness of answers for faster, more accurate problem solving.
FAQs on Important Formulas: Quantitative Reasoning
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